IEEE Transactions on Antennas and Propagation [volume 60 number 2 part II]

Citation preview

FEBRUARY 2012

VOLUME 60

NUMBER 2

IETPAK

(ISSN 0018-926X)

PART II OF TWO PARTS

PAPERS

Antennas and Resonators A Broadband VHF/UHF Double-Whip Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Ding, B.-Z. Wang, G.-D. Ge, and D. Wang Wideband Dielectrically Guided Horn Antenna with Microstrip Line to H-Guide Feed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Wong, A. R. Sebak, and T. A. Denidni CPW-Fed Cavity-Backed Slot Radiator Loaded With an AMC Reflector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Joubert, J. C. Vardaxoglou, W. G. Whittow, and J. W. Odendaal The Use of Simple Thin Partially Reflective Surfaces With Positive Reflection Phase Gradients to Design Wideband, Low-Profile EBG Resonator Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Ge, K. P. Esselle, and T. S. Bird Omnidirectional Linearly and Circularly Polarized Rectangular Dielectric Resonator Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. M. Pan, K. W. Leung, and K. Lu Substrate Integrated Composite Right-/Left-Handed Leaky-Wave Structure for Polarization-Flexible Antenna Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Dong and T. Itoh Design and Characterization of Miniaturized Patch Antennas Loaded With Complementary Split-Ring Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Dong, H. Toyao, and T. Itoh Dual-Band Circularly Polarized Microstrip RFID Reader Antenna Using Metamaterial Branch-Line Coupler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y.-K. Jung and B. Lee Small-Size Shielded Metallic Stacked Fabry–Perot Cavity Antennas With Large Bandwidth for Space Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. A. Muhammad, R. Sauleau, and H. Legay A Simple Technique for the Dispersion Analysis of Fabry-Perot Cavity Leaky-Wave Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Mateo-Segura, M. García-Vigueras, G. Goussetis, A. P. Feresidis, and J. L. Gómez-Tornero Analyzing the Complexity and Reliability of Switch-Frequency-Reconfigurable Antennas Using Graph Models . . . . . . . . . . . . . . . . . . . . . . . . . J. Costantine, Y. Tawk, C. G. Christodoulou, J. C. Lyke, F. De Flaviis, A. Grau Besoli, and S. E. Barbin Free Space Radiation Pattern Reconstruction from Non-Anechoic Measurements Using an Impulse Response of the Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Koh, A. De, T. K. Sarkar, H. Moon, W. Zhao, and M. Salazar-Palma Electric Field Amplification inside a Porous Spherical Cavity Resonator Excited by an External Plane Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. A. Bernhardt and R. F. Fernsler

719 725 735 743 751 760 772 786 792 803 811 821 832

(Contents Continued on p. 717)

(Contents Continued from Front Cover) Arrays A 76 GHz Multi-Layered Phased Array Antenna Using a Non-Metal Contact Metamaterial Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Kirino and K. Ogawa Beam Switching Reflectarray Monolithically Integrated With RF MEMS Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. Bayraktar, O. A. Civi, and T. Akin Design and Implementation of a Closed Cylindrical BFN-Fed Circular Array Antenna for Multiple-Beam Coverage in Azimuth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. J. G. Fonseca Rapidly Convergent Representations for Periodic Green’s Functions of a Linear Array in Layered Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Van Orden and V. Lomakin A Novel Strategy for the Diagnosis of Arbitrary Geometries Large Arrays . . . . . . . . . . . . . . . . . A. Buonanno and M. D’Urso Predicting Sparse Array Performance From Two-Element Interferometer Data . . . . . . . . . . . . . . J. A. Nessel and R. J. Acosta Linear Aperiodic Array Synthesis Using an Improved Genetic Algorithm . . . . . . . . . . L. Cen, Z. L. Yu, W. Ser, and W. Cen Beamformer Design Methods for Radio Astronomical Phased Array Feeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Elmer, B. D. Jeffs, K. F. Warnick, J. R. Fisher, and R. D. Norrod Experimental Results for the Sensitivity of a Low Noise Aperture Array Tile for the SKA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. E. M. Woestenburg, L. Bakker, and M. V. Ivashina Direction Finding With Partly Calibrated Uniform Linear Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Liao and S. C. Chan Numerical and Inverse Techniques Calculation of MoM Interaction Integrals in Highly Conductive Media . . . . . . . . . J. Peeters, I. Bogaert, and D. De Zutter Electromagnetic Scattering From General Bi-Isotropic Objects Using Time-Domain Integral Equations Combined With PMCHWT Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Z.-H. Wu, E. K.-N. Yung, D.-X. Wang, and J. Bao Efficient Surface Integral Equation Using Hierarchical Vector Bases for Complex EM Scattering Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. P. Zha, Y. Q. Hu, and T. Su Accelerated FDTD Analysis of Antennas Loaded by Electric Circuits . . . . . . . . . . . . . . . . . . . . . . . Y. Watanabe and H. Igarashi An Angle-Dependent Impedance Boundary Condition for the Split-Step Parabolic Equation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. R. Sprouse and R. S. Awadallah A Nested Multi-Scaling Inexact-Newton Iterative Approach for Microwave Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Oliveri, L. Lizzi, M. Pastorino, and A. Massa Fast and Shadow Region 3-Dimensional Imaging Algorithm With Range Derivative of Doubly Scattered Signals for UWB Radars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Kidera and T. Kirimoto High-Resolution ISAR Imaging by Exploiting Sparse Apertures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Zhang, Z.-J. Qiao, M.-D. Xing, J.-L. Sheng, R. Guo, and Z. Bao Nondestructive Material Characterization of a Free-Space-Backed Magnetic Material Using a Dual-Waveguide Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. W. Hyde, M. J. Havrilla, A. E. Bogle, and E. J. Rothwell Evaporation Duct Height Estimation and Source Localization From Field Measurements at an Array of Radio Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Zhao Extrapolation of Wideband Electromagnetic Response Using Sparse Representation . . . . . . .. . . . . . . H. Zhao and Y. Zhang Wireless A Wearable Two-Antenna System on a Life Jacket for Cospas-Sarsat Personal Locator Beacons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. A. Serra, P. Nepa, and G. Manara Analysis of Cellular Antennas for Hearing-Aid Compatible Mobile Phones . . . . . . . . . . . . . P. M. T. Ikonen and K. R. Boyle A Mobile Communication Base Station Antenna Using a Genetic Algorithm Based Fabry-Pérot Resonance Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Kim, J. Ju, and J. Choi Development of Novel 3-D Cube Antennas for Compact Wireless Sensor Nodes . . . . . . . . . . I. T. Nassar and T. M. Weller Influence of the Hand on the Specific Absorption Rate in the Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.-H. Li, M. Douglas, E. Ofli, B. Derat, S. Gabriel, N. Chavannes, and N. Kuster Demonstration of a Cognitive Radio Front End Using an Optically Pumped Reconfigurable Antenna System (OPRAS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Tawk, J. Costantine, S. Hemmady, G. Balakrishnan, K. Avery, and C. G. Christodoulou Evaluation of a Statistical Model for the Characterization of Multipath Affecting Mobile Terminal GPS Antennas in Sub-Urban Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Ur Rehman, X. Chen, C. G. Parini, and Z. Ying A Mixed Rays—Modes Approach to the Propagation in Real Road and Railway Tunnels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Fuschini and G. Falciasecca Optimum Wireless Powering of Sensors Embedded in Concrete . . . . . . . . . . . . . . . . . . . . . . . S. Jiang and S. V. Georgakopoulos Portable Real-Time Microwave Camera at 24 GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. T. Ghasr, M. A. Abou-Khousa, S. Kharkovsky, R. Zoughi, and D. Pommerenke Is Orbital Angular Momentum (OAM) Based Radio Communication an Unexploited Area? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. Edfors and A. J. Johansson

840 854 863 870 880 886 895 903 915 922 930 941 952 958 964 971 984 997 1009 1020 1026

1035 1043 1053 1059 1066 1075 1084 1095 1106 1114 1126

(Contents Continued on p. 718)

(Contents Continued from p. 717) COMMUNICATIONS

A Circularly Polarized Ring-Antenna Fed by a Serially Coupled Square Slot-Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T.-N. Chang, J.-M. Lin, and Y. G. Chen A Pseudo-Normal-Mode Helical Antenna for Use With Deeply Implanted Wireless Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. H. Murphy, C. N. McLeod, M. Navaratnarajah, M. Yacoub, and C. Toumazou A Novel Folded UWB Antenna for Wireless Body Area Network . . . . . . . . . . . . . . . . . C.-H. Kang, S.-J. Wu, and J.-H. Tarng Hybrid Mode Wideband Patch Antenna Loaded With a Planar Metamaterial Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Ha, K. Kwon, Y. Lee, and J. Choi Explicit Relation Between Volume and Lower Bound for Q for Small Dipole Topologies . . . . .. . . . G. A. E. Vandenbosch On the Generalization of Taylor and Bayliss n-bar Array Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . S. R. Zinka and J. P. Kim Amplitude-Only Low Sidelobe Synthesis for Large Thinned Circular Array Antennas . . . . . . . .. . . . . . . . W. P. M. N. Keizer Power Synthesis for Reconfigurable Arrays by Phase-Only Control With Simultaneous Dynamic Range Ratio and Near-Field Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Buttazzoni and R. Vescovo Design and Experiment of a Single-Feed Quad-Beam Reflectarray Antenna . . . . P. Nayeri, F. Yang, and A. Z. Elsherbeni Oblique Diffraction of Arbitrarily Polarized Waves by an Array of Coplanar Slots Loaded by Dielectric Semi-Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. L. Tsalamengas and I. O. Vardiambasis Analysis of Radiation Characteristics of Conformal Microstrip Arrays Using Adaptive Integral Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W.-J. Zhao, L.-W. Li, E.-P. Li, and K. Xiao Generalized Multilevel Physical Optics (MLPO) for Comprehensive Analysis of Reflector Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Letrou and A. Boag Fast Dipole Method for Electromagnetic Scattering From Perfect Electric Conducting Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Chen, C. Gu, Z. Niu, and Z. Li An Efficient Hybrid GO-PWS Algorithm to Analyze Conformal Serrated-Edge Reflectors for Millimeter-Wave Compact Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Muñoz-Acevedo and M. Sierra-Castañer Time-Domain Microwave Imaging of Inhomogeneous Debye Dispersive Scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. G. Papadopoulos and I. T. Rekanos

1132 1135 1139 1143 1147 1152 1157 1161 1166 1171 1176 1182 1186 1192 1197

CALL FOR PAPERS

Call for Papers: Special Issue on Antennas and Propagation at Millimeter and Sub-millimeter Waves . . . . . . . . . .. . . . . . . . . .

1203

IEEE ANTENNAS AND PROPAGATION SOCIETY All members of the IEEE are eligible for membership in the Antennas and Propagation Society and will receive on-line access to this TRANSACTIONS through IEEE Xplore upon payment of the annual Society membership fee of $24.00. Print subscriptions to this TRANSACTIONS are available to Society members for an additional fee of $36.00. For information on joining, write to the IEEE at the address below. Member copies of Transactions/Journals are for personal use only. ADMINISTRATIVE COMMITTEE S. R. BEST, President T. S. BIRD, President Elect J. S. TYO, Secretary-Treasurer 2012 2013 2014 2015 M. ANDO* R. D. NEVELS* M. SALAZAR PALMA* Y. M. M. ANTAR D. ERRICOLO D. B. DAVIDSON J. T. BERNHARD* M. EL-SHENAWEE L. JOFRE S. MACI S. K. RAO C. PICHOT M. MARTINEZ-VÁZQUEZ M. W. SHIELDS N. RILEY D. F. SIEVENPIPER

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A Broadband VHF/UHF Double-Whip Antenna Xiao Ding, Bing-Zhong Wang, Member, IEEE, Guang-Ding Ge, and Duo Wang

Abstract—This paper presents a broadband VHF/UHF double-whip antenna with one lossless matching scheme combining two methods, embedded transmission line matching method and lumped-distributed hybrid matching method. By adjusting the length of the embedded transmission line, the combination of double-whip antenna and the transmission line can achieve resonance, thus realize a coarse matching. By adding a lumped-distributed hybrid matching network at the feeding point of the double-whip antenna, we can further improve the matching for the double-whip antenna. Moreover, based on the two-step matching scheme, a double-whip antenna has been designed and fabricated. Measured results show that, the VSWRs of the double-whip antenna, with the electrical lengths of and at the minimum operation frequency respectively, are less than 2 over a 17:1 octave bandwidth, and the horizontal gains of the antenna are between 4.2 dB and 6.8 dB. Thanks to its high gain, broadband and low reflection, the proposed double-whip antenna in this paper is ideal for application in vehicle wireless communication. Index Terms—Broadband matching network, double-whip antenna, VHF/UHF antenna.

I. INTRODUCTION

F

EATURING of its characteristics like small size, simple structure and omni-direction, whip antenna have been widely used in ultra-short wave, shortwave, and VHF/UHF wireless communication. Actually, one distinctive characteristic of whip antenna lies in its very small radiation resistance and very large negative reactance in low frequency. Consequently, it gains a large Q factor and a narrow working bandwidth, and causes most of the energy difficult to radiate and only oscillating around the antenna instead. Under this circumstance, direct feeding to the antenna from feed line would make the receiver or transmitter in system front-end fail or even breakdown because of large reflection. To solve this practical problem of over-reflection, or mismatch indeed, many methods have been discussed by scholars. One proposal uses lumped or distributed load on antenna to improve the current distribution on the surface of the antenna. In [1], a LR loaded wire monopole with a matching network achieves a 20:1 bandwidth, VSWR less than 3.0 and system gain greater than

Manuscript received March 02, 2011; revised July 07, 2011; accepted August 26, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported in part by the High-Tech Research and Development Program of China (No. 2008AA01Z206), in part by the Research Fund for the Doctoral Program of Higher Education of China (No. 20100185110021), and in part by the National Natural Science Foundation of China (No. 61071031), and Project 9140A01020110DZ0211. The authors are with the Institute of Applied Physics, University of Electronic and Science Technology of China, Chengdu 610054, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173141

dBi. Another idea takes embedded broadband matching network consisting of some lumped components to eliminate the imaginary part of the antenna impedance. In [2], a 2-meter broadband whip antenna with electronically switching three different matching networks could operate over the frequency range of 23–60 MHz with VSWR less than 3.5 and system gain greater than dB. Reference [3] simultaneously uses load technology and “on-body” matching network to realize a VHF/UHF whip antenna with VSWR less than 2 and system gain greater than 0 dBi. A third one utilizes fractal technique for the improvement of the impedance characteristics of a variety of VHF/UHF antennas [4]. And in recent years, the emergency of the study on metamaterial has inspired some scholars to design VHF/UHF metamaterials antennas [5]. Moreover, [6] matches a conical antenna with the aid of transmission line. By adding a section of the transmission line to form a resonant structure with the conical antenna, more efficient operation at low frequencies is obtained. However, all above methods have more or less specific deficiencies in implementation. For example, the load will inevitably reduce the antenna radiation efficiency and its structural strength; the broadband matching networks of lumped components would sacrifice system gain for bandwidth matching; and the cost of the metamaterials is usually far too high. In order to get higher gain, wider bandwidth and lower reflection, this paper researches a VHF/UHF double-whip antenna and two lossless matching methods. Two individual whips, which are connected by two sections of transmission lines as a double-whip antenna, work in the upper and lower bands of 30–520 MHz, respectively. By adjusting the length of the embedded transmission lines, the whip and the transmission line could get resonant, and the antenna can approximately meet the engineering requirement for impedance matching. This method provides a coarse impedance match for the double-whip antenna. In order to further match the double-whip antenna, two matching networks are designed respectively. A traditionally lumped matching networks and a new lumped-distributed hybrid matching network are added at the feeding point of the antenna separately. These two matching networks working over different frequency bands can get a higher gain and better impendence matching for the double-whip antenna. The designed double-whip antenna with those two lossless matching methods has a good performance in VHF/UHF frequency bands. II. COARSE IMPEDANCE MATCH OF ANTENNA

THE

DOUBLE-WHIP

A. Double-Whip Antenna The structure of the double-whip antenna is shown in Fig. 1(a). It consists of two whip antennas with heights m and m respectively, and their diameters

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Fig. 2. (a) Element 2. (b) Equivalent circuits of (a).

B. Elements Design In the following theoretic analysis, we take Element 2 as an example, and Fig. 2 depicts its configuration. From Fig. 2(b), the input impedance of Element 2 at the joint port is (2) where is the reflection coefficient at the port of the whip with length . For the purpose of matching, must be real in (2) and the exponent must satisfy the condition , ( .). So the solutions to is given by (3) Because the electrical length of the embedded transmission line can not be negative, and the periodicity of the with , so one solution to is

Fig. 1. (a) Double-whip antenna. (b) Equivalent circuit of (a).

(4) are both 2.5 cm. Two embedded transmission lines with lengths and and diameters and are used to connect these whips and the distances between the embedded lines and the ground plane are and , respectively. The joint point of the two embedded lines is connected to two matching networks through an electrical switch. At the other ports of the matching networks is the feeding point. Fig. 1(b) shows the equivalent circuits of the double-whip antenna in Fig. 1(a). is the input impedance of the whip antenna of height and the whip of is the input impedance at the joint point, and and are the characteristic impedances of the embedded transmission lines of lengths and , which can be calculated by the following formula of parallel-wire conductor line (1)

The double-whip antenna can be viewed as two parallel elements connected at the joint point in Fig. 1(a), where Element 1 consists of a whip antenna with the height and one embedded transmission line with the length ; Element 2 consists of a whip antenna with the height and one embedded transmission line with the length .

Because the input impendence varies with frequency, from (4) we can notice that: for any whip antenna, in order to match it with the aid of the embedded transmission line, the electrical length of the embedded line should be variable as the working frequency changes. However, as a matter of fact, the length of the embedded line must be fixed in engineering design. In other word, for a given length and characteristic impedance of an embedded transmission line, it can only match the whip antenna in a narrow band. In order to give attention to broadband matching, we can take an average value of over the whole matching bandwidth as the length of the embedded transmission line, so that we can obtain an approximate matching for the element. Fig. 3 gives the input impedance of the whip antenna m, whose first resonance frequency is at 250 MHz. Fig. 3 can give more apparent illustration about the impedance variation in broadband range: from the first resonance frequency, as frequency increases, the fluctuation of the real part of the input impedance almost becomes moderate in a certain range of 29 while its imaginary part approaches zero. Accordingly, we set the characteristic impendence of the transmission line to be 29 , and take the result of into (3). Then we could calculate the theoretic solution to the transmission line

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Fig. 5. 1.6 m whip antenna impedance Fig. 3. 0.3 m whip antenna impedance

.

.

Fig. 4. Comparison of reflection coefficients before and after embedding the transmission line for element 2.

electrical length after taking the average value processing over the bandwidth, which is . In the practically engineering design, we determine the physical length at the center frequency MHz and get cm from . Further, if the radius of embedded line is set as cm, we can work out the distance between the connection embedded line and the ground from (1) as cm. Fig. 4 shows the reflection coefficients before and after embedding this embedded transmission line between the whip and a 50 feeding line. As shown in the solid line with circular, in the frequency range of 180–410 MHz, VSWR is less than 3 and in the frequency range of 410–520 MHz, VSWR is less than 3.5. And when the frequency is below 150 MHz, Element 2 would reflect all the feeding current. In order to make the double-whip antenna work below 150 MHz, the design of Element 1 should give main concerns to work over the lower frequency band. Fig. 5 gives the input impedance of the whip antenna m, whose first resonance frequency is at 46.8 MHz. In the frequency range of 30–150 MHz, the real part of the input resistance shakes around ohms from 30 to 55 MHz and from 100 to 140 MHz, but changes acutely between 55 MHz and 100 MHz. In order to get a fixed physical length of embedded line from (3) and considering the whole band’s coarse matching, we

Fig. 6. Comparison of reflection coefficients before and after embedding the transmission line for element 1.

take 42 ohms as the characteristic impedance of the embedded line, i.e., ohms. Then we calculate (3) and take the average value of . In the practically engineering design, we determine the physical length at the center frequency MHz and get m from . Fig. 6 shows the reflection coefficients before and after embedding the transmission line between the whip and a 50 feeding line. From Fig. 6, after embedding the matching transmission line, the VSWR2:1 bandwidth increases and the first resonance frequency decreases to about 30 MHz. C. Double-Whip Design From Fig. 4, we know that Element 2 would reflect all the feeding current as the frequency below 150 MHz and partially reflect feeding current as the frequency above 150 MHz. From Fig. 6 the first resonance frequency of Element 1 has decreased to about 30 MHz. Based on the analysis in the above element design, we parallel connect the two elements in the arrangement as shown in Fig. 1(a), and the actual antenna can be obtained as shown in Fig. 7. In Fig. 7, two embedded transmission lines with black and white coating respectively are shown on the top of the ground and two ivory-white plastic fixed parts at the bottom of the monopole are used to connect the quadrate metal pedestals which are below the ground. Finally, we put the

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Fig. 7. Double-whip antenna. Fig. 9. Absolute vale of a 47 pF monolithic ceramic capacitor impedance as a function of frequency.

below the joint port, as shown in Fig. 1(a) and analyzed in the next section. III. FURTHER IMPROVEMENT ON BROADBAND MATCHING

Fig. 8. Comparison of measure reflection coefficients before and after the matching transmission line.

latter two matching networks into the quadrate metal pedestal. At the joint point of the two embedded transmission line, when lower frequency signal is coming, it will be reflected by Element 2 and flow to Element 1 primarily. On the other hand, Element 1 will reflect higher frequency signal and Element 2 will pass higher frequency signal mainly. Fig. 8 shows the comparison of measured reflection coefficients, in which the dotted line stands for the whip antenna of 1.6 m-height without the embedded transmission line, the triangular-solid-line for the whip antenna of 0.3 m-height without the embedded transmission line and the solid line is for the double-whip antenna with two embedded matching transmission lines. From the comparison, we can see that by simply using lossless transmission lines for coarse matching, more than 50% of the frequencies sampled over the band of 30–520 MHz have VSWRs less than 2, and more than 75% of the frequencies have VSWRs less than 3. At present, the double-whip antenna with two lossless embedded transmission lines can work from 30 MHz to 520 MHz. But as shown in Fig. 8, the matching results cannot yet meet practical project needs, such as , especially at the lower frequencies. In order to further improve the impedance matching performance of the double-whip antenna, two lumped-distributed hybrid matching networks are added

Traditional method of broadband matching often loads with lumped components and treats them independent of frequency. So this method could match the real part of the impedance to 50 and the imaginary part to zero with several LC matching networks. However, this traditional method ignores the frequency variability of lumped components, which does exist in practical design and would cause a lower gain and a narrower matching bandwidth. Reference [7] announces this phenomenon. Example 1–4 in [7, p. 19] shows that the capacitance from a real capacitor strictly obeys the rule in the frequency range from DC to several hundred MHz. However, when frequency increases further, the capacitance would not obey . And for a real inductor in Example 1–5 in [7, p. 25], the situation is the same. A 47 pF monolithic ceramic capacitor which is made in China and used in our design, is measured by a Network Analyzer E5071C to obtain its capacitor impedance. Fig. 9 gives the absolute vale of the impedance of this capacitor as a function of frequency. In Fig. 9, when the frequency increases to about 100 MHz, the test capacitance will not obey . Considering the above characteristics of lumped and distributed electronic components at VHF/UHF band, traditional matching method would not be effective at the whole operation band. So, we design two matching networks in this paper which work at 30–120 MHz and 120–520 MHz respectively. For the 30–120 MHz matching network, the impedance characteristic of the lumped components has not changed yet, shown as in Fig. 9. A traditional lumped matching network is implemented. And in the frequency range of 120–520 MHz, an effective matching method of “lumped-distributed hybrid matching” is implemented. The topology structures of the two matching networks are presented in Fig. 10. Fig. 10(a) is the broadband lumped matching network for the range of 30–120 MHz, which shares the same principle as traditional matching networks. Fig. 10(b) gives the proposed “lumped-distributed matching”

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TABLE I VALUES OF THE ELEMENTS IN 30–120 MHZ MATCHING NETWORK

TABLE II VALUES OF THE ELEMENTS IN 120–520 MHZ MATCHING NETWORKS

Fig. 10. Topology structure of matching network, (a) 30–120 MHz broadband matching network, (b) 120–520 MHz broadband matching network.

PCB Layout for the range of 120–520 MHz. It consists of six parts: and are distributed capacitors; is a third-order ladder impedance converter, which, together with , forms one parallel multi-level impedance matching circuit. This matching circuit, functioning as the multi-level damping network of lumped components, can assuage the fierce fluctuation of antenna impedance in low frequency; is one LC low-frequency filter circuit which can filter out part of the input signal with frequency lower than 120 MHz; is the blocking capacitors, and is the 50 matching line for input and output ports. The “lumped-distributed hybrid matching” approach owns at least following advantages: (1) lossless matching to improve the system efficiency; (2) distributed components are used to achieve a broadband matching because their characteristics would not vary with frequency; (3) simple matching network structure can avoid electromagnetic compatibility problems from complicated lumped matching circuit; and (4) planar microstrip line structure of matching networks is easy for conformal design. Both matching networks are printed on microstrip board with dielectric constant 2.2 and thickness 1 mm. Table I shows the values of the lumped electronic components in 30–120 MHz matching network and Table II shows the dimensions and values of the distributed electronic components in 120–520 MHz “lumped-distributed” matching network. The final measurement results for the antenna structure are presented in Figs. 11–13. The reflection coefficients at the feeding port are depicted in Fig. 11. After further matching with “lumped-distributed hybrid network”, the reflection coefficients over the whole frequency bandwidth are generally less than dB . Finally, we give the test results of horizontal gain of the double-whip antenna shown in Fig. 12, and the measured radiation patterns in E-plane and H-plane at frequencies of 45, 170, 330, 500 MHz shown in Fig. 13.

Fig. 11. Measured reflection coefficients of the double-whip antenna, (a) 30–120 MHz, (b) 120–520 MHz.

IV. CONCLUSION This paper proposes a VHF/UHF double-whip antenna with two lossless matching methods. This antenna system has merits as follows. First, it has a very small size, which is convenient for vehicular concealment. The electrical lengths of the two whips are and at the minimum operation frequency, respectively. Second, it has a very wide operation band that

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[3] X. Ding, B.-Z. Wang, G. Zheng, and X.-M. Li, “Design and realization of a GA-optimized VHF/UHF antenna with ‘on-body’ matching network,” IEEE Antenna Wireless Propag. Lett., vol. 9, pp. 303–307, 2010. [4] J. M. González-Arbesú, S. Blanch, and J. Romeu, “Are space filling curves efficient small antennas,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 147–150, 2003. [5] H. Lizuka and P. S. Hall, “Left-handed dipole antennas and their implementations,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1246–1253, 2007. [6] S. Sheldon and W. P. K. Ronold, “Compact conical antenna for wideband coverage,” IEEE Trans. Antennas Propag., vol. 42, no. 3, pp. 436–439, 1994. [7] L. Reinhold and B. Pavel, RF Circuit Design: Theory and Applications. Englewood Cliffs, NJ: Prentice Hall, 2000.

Fig. 12. Measured horizon gains of the double-whip antenna.

Xiao Ding was born in Sichuan Province, China, May, 1982. He received the B.S. and M.S. degrees in communication engineering and electromagnetic field and microwave engineering, respectively, from Guilin University of Electronic Science and Technology (GUET), China. He is currently working toward the Ph.D. degree at the University of Electronic Science and Technology of China (UESTC), Chengdu, since 2009. His research interests include short-wave, ultrashort-wave wire antennas and its broadband matching technology, millimeter-wave antenna and phased array.

Bing-Zhong Wang (M’06) received the Ph.D. degree in electrical engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1988. He joined the UESTC in 1984 where he is currently a Professor. He has been a Visiting Scholar at the University of Wisconsin-Milwaukee, a Research Fellow at the City University of Hong Kong, and a Visiting Professor in the Electromagnetic Communication Laboratory, Pennsylvania State University, University Park. His current research interests are in the areas of computational electromagnetics, antenna theory and technique, electromagnetic compatibility analysis, and computer-aided design for passive microwave integrated circuits.

Fig. 13. Measured radiation patterns of the double-whip antenna at different frequencies, (a) 45 MHz, (b) 170 MHz, (c) 330 MHz, (d) 500 MHz.

covers VHF and UHF band from 30 to 520 MHz with measured VSWRs less than 2. Third, compared to other ways of lossy matching, our lossless matching methods result in relatively higher horizontal gains, which are between 4.2 to 6.8 dBi in the operation band. This designed antenna can be widely used in vehicular, shipboard and civil mobile communication with high gain, wide band and low reflection. REFERENCES [1] S. D. Rogers, C. M. Butler, and A. Q. Martin, “Design and realization of GA-optimized wire monopole and matching network with 20:1 bandwidth,” IEEE Trans. Antennas Propag., vol. 51, no. 3, pp. 493–502, Mar. 2003. [2] K. Yegin and A. Q. Martin, “Very broadband loaded monopole antennas,” in Proc. IEEE Antennas and Propag. Soc. Int. Symp., Montreal, QC, Canada, Jul. 1997, vol. 1, pp. 232–235.

Guang-Ding Ge is currently working toward the Ph.D. degree in the Institute of Applied Physics, University of Electronic Science and Technology of China, Chengdu, China. His main research interests include microwave circuits, antenna theory and design, time reversal technique, compact and wideband antennas and arrays for wireless communications systems.

Duo Wang was born in 1986 in Chongqing, China. He received the B.S. and M.S. degrees from the University of Electronic Science and Technology of China, in 2008 and 2011, respectively. He is currently pursuing the Ph.D. degree in the Institut National des Sciences Appliquées de Rennes, France. His current research interests include the technique of Electromagnetic Time Reversal and the design of microwave antenna.

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Wideband Dielectrically Guided Horn Antenna with Microstrip Line to H-Guide Feed Michael Wong, Member, IEEE, Abdel Razik Sebak, Fellow, IEEE, and Tayeb A. Denidni, Senior Member, IEEE

Abstract—The design, simulation, and measurement of a complete microstrip line-fed dielectrically guided horn antenna are presented. The proposed antenna achieves similarly high gains as compared to traditional air-filled horn antennas, is simpler than a typical array design, and can easily be fabricated using typical two dimensional substrate machining processes. An H-guide, operating in the fundamental 00 mode, slowly tapers into a “gapped” H-guide, or dielectrically guided horn, where a large air gap separates the center dielectric and metallic plates. A wideband Bézier shaped microstrip to H-guide transition feeding structure is fabricated using a low loss Rogers 5880 substrate and integrated with the proposed antenna. The fabricated prototype operates from 8 to 16 GHz with a peak gain of approximately 16 dBi.

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Index Terms—Aperture antennas, dielectric waveguides, feeds, microstrip transitions, millimeter wave antennas.

I. INTRODUCTION ORN antennas appear in many different forms, such as dielectric-filled horn antennas [1], metamaterial-lined horn antennas [2], corrugated circular horn antennas [3], or even our recently proposed Substrate Integrated Waveguide (SIW) planar slot antenna [4]. Typically, metallic air-filled horn antennas are formed using expensive machining processes to obtain the precise angles required for highly predictable antenna patterns. Gain standard horns are typically made in this fashion using metallic walls on the top, bottom, and sides. In the design proposed in this paper, however, waves are mostly guided by the dielectric near the mouth of the horn, or aperture, thus reducing the dependence upon the horn angles and precise dimensions, and removing the need for metallic sidewalls. As is shown in this paper, such control can allow the design of a high gain wide bandwidth antenna, while reducing diffracted fields by concentrating the fields away from metallic edges. Such reduction in diffracted fields immediately reduces spill-over radiation and consequently lowers the sidelobe levels. In addition, by using this dielectric wave guiding property, the

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Manuscript received November 18, 2010; revised April 05, 2011; accepted July 25, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. M. Wong is with Research in Motion (RIM), Kanata, ON, Canada (e-mail: [email protected]). A. R. Sebak is with Concordia University in Montreal, Quebec and with Prince Sultan Advanced Technological Research Institute (PSATRI), King Saud University, 11451 Riyadh, Saudi Arabia (e-mail: [email protected]). T. A. Denidni is with INRS-EMT, Montreal, QC H5A-1K6, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173123

magnitude and the phase of the electric fields over the aperture can be controlled. Neglecting fringing effects due to metallic edges, this control of the near field pattern allows some control of the far field radiation pattern. To achieve these very desirable properties, a wide bandwidth, single mode, thin H-guide is used as a feed mechanism to excite the horn, which is in turn fed by a microstrip to H-guide transition. We have briefly discussed the H-guide horn antenna concept in a conference paper [5] for a slightly different design. While the H-guide has existed for many years [6], it is only recently that the miniaturization of such dielectric waveguides, for the example, in the form of the Non-Radiative Dielectric (NRD) waveguide [7], or the Dielectric Image Guide (DIG) [8] has become of interest. In addition, the excitation of waves within substrates has recently been proposed in the form of surface wave launchers [9], SIW horns [4], or other methods to form compact, high gain antennas. The fence guide [10] has been used to form horn antennas as well. Other recent research in the area of horn antennas aims to reduce the sidelobe level through the use of periodic structures, or metamaterials along the metallic walls of the horn [2]. These structures reduce the magnitude of the electric fields close to the edges, thus reducing fringing effects at the mouth of the horn and consequently reducing the sidelobe levels. This paper begins with a description of a thin microstrip-fed H-guide design, where the metallic plate separation is close to . Theoretical, simulated, and measured results for an 8 to 18 GHz back-to-back Bézier shaped microstrip to H-guide transition using Rogers 5880 substrate are then presented. We have previously discussed a low frequency (3 to 7 GHz) transition using FR4 in [11]. Finally, theoretical, simulated, and measured results for the H-guide horn antenna and its aperture are then presented. II. H-GUIDE The goal of this section is to describe the parameters for the H-guide to be used in the proposed microstrip line-fed dielectrically guided horn antenna. In this case, the dimensions are chosen to force a single mode, where all other propagating modes are lossy, or evanescent. The design, fabrication, and testing of the thin H-guide are made possible through the use of a low profile microstrip to H-guide transition, which is discussed in the next section. Consider the H-guide [6] shown in Fig. 1. The electric field lines of the dominant mode are shown, where the electric field is directed in the y-direction for z-propagation. Cut-off

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Fig. 1. Side view of the H-guide structure operating in the fundamental mode. The size of the arrows represents the magnitude of the electric field. Propagation is in the z direction.

Fig. 2. Two dimensional view of the fields for the a) TE00 vs the b) TE10 modes.

frequencies can be found for various modes using equations in this section. A comparison with the next higher mode is shown in Fig. 2. The distance “ ” between the plates is made small, so that the electric field parallel to the plates is forced to zero as shown in Fig. 1. The magnetic field for this mode will then exist in the xz plane for propagation in the z direction. The guide wavelength for even TE modes in the H-guide can , the be found by solving the following equations for solution to the [6] and [12]

(1) where variable is the width of the dielectric as shown in Fig. 1, is the transverse wavenumber inside the dielectric, and is the first transverse wavenumber outside the dielectric. is the relative dielectric constant, and is the propagation constant in free space. is then substituted into the characteristic equation inside the dielectric to find , the guide wavelength for the solution as follows:

(2)

Fig. 3. Insertion loss over a 10.0 cm long, 62 mil (1.575 mm) thick substrate for an H-guide that is 10 mm wide for the TE00 and TE10 modes. The cutoff frequency is marked with a dotted line at 13.7 GHz.

For this design, an H-guide made of a 10 mm wide section of dielectric is formed out of 62 mil (1.575 mm thick) Rogers . The guide wavelength, , for even TE modes 5880 within a single H-guide can be found by solving the equations for , the allowed even mode following procedures in [6], [12], using (1), where at 13.7 GHz, found to be equal to 16.4 mm. Since the distance between the top and bottom metallic plates of the substrate is only 1.575 mm , higher order modes in the vertical direction ( , , etc.) are highly evanescent. Consequently, TM modes, where the electric fields are parallel to the metallic plates, are also highly evanescent. The waveguide is therefore predominantly single-mode up to the cutoff frequency of the second mode ( , , odd) at 13.7 GHz as estimated by the expression [12] (3) where is the speed of light in free space. Using Ansoft HFSS numerical simulation software [13], it is possible to plot the attenuation curves for the first two modes, and over a length of 10 cm, as shown in Fig. 3. In the figure, above a frequency of approximately 15 GHz, the is attenuated only slightly, which confirms that the calculation of the cutoff frequency of 13.7 GHz is reasonably accurate. Note, however, that the cutoff frequency is not as well defined as for a rectangular metallic waveguide, where boundaries conditions end abruptly at metallic sidewalls. III. BÉZIER-CURVE SHAPED MICROSTRIP TO H-GUIDE TRANSITION An obvious choice of common transmission lines to excite the single mode in the thin H-guide is the microstrip line, because of the small vertical dimension of 1.575 mm proposed in the previous section. Transitions such as those proposed in [14] and similar transitions are intended for Non-Radiative Dielectric (NRD) structures, cannot excite the intended H-guide mode, require two separate low-loss substrates,

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Fig. 5. Profiled horn antenna shape.

Fig. 4. (a) Top view of the Wedge Radial Waveguide with metallic side walls. Cylindrical waves are the preferred modes. (b) Top view of the Wedge Radial Waveguide with air walls. Cylindrical waves are the preferred modes.

and are inherently narrowband because of their resonant behavior. The solution proposed here is a non-resonant, wideband, low-loss Bézier-curve shaped microstrip to H-guide transition using a low-loss Rogers 5880 substrate, previously discussed in [11] using a low-cost FR4 substrate. Consider the air-filled metallic wedge radial waveguide shown in Fig. 4(a). On the left and right walls, metallic boundaries, or perfect electric conductors (PEC), enforce the tangential electric fields to be zero. The dominant mode in such a structure is a radial mode [12]. A line of equal phase is shown as a dotted line in the figure for propagation in the direction. For the design of a transition to the fundamental H-guide mode as shown in Fig. 1, this is not an efficient choice, since both the microstrip line and the H-guide do not have side metallic walls. The transition from the mouth of this transition to and H-guide would therefore be very abrupt. Now, consider the same waveguide-based transition, with a height of 1.575 mm and filled with air, except with virtual perfect magnetic conductors (PMC) instead of PEC for walls, as shown in Fig. 4(b). The virtual PMC boundaries enforce the magnetic fields to be zero, and hence again, radial waves are the preferred modes of operation. The lines of equal phase will be identical for propagation in the direction, however, no

metallic side edges are needed, thus improving the transition from the microstrip to the H-guide. Only a fraction of the waves that are leaving the mouth of the transition, however, are propdirection towards the H-guide. Most agating in the desired waves propagate at an angle to that axis, which is energy that will be lost at the transition. The profiled transition shape shown in Fig. 5 that is discussed in [11] is used as a solution to encourage the propagation of planar, equal phase waves at the mouth of the transition, as opposed to radial waves. Planar waves, as opposed to radial waves, are less lossy at the mouth of the transition because the rectangular waveguide shape in this region prefers planar waves. The smooth transition from radial waveguide to rectangular waveguide requires a profiled shape, as shown in Fig. 5. To form the required profiled shape, several different formulas have been presented [15], such as sine squared, exponential, hyperbolic, or polynomial curves. In this design, however, a class of cubic spline curves, called the Bézier curve, is used. The Bézier curve is flexible enough to approximate various different curves while maintaining a smooth shape and allows the specification of the slope, or direction, at both sides of the transition. The parametric form of the cubic Bézier curve [16] using a 3rd degree Bernstein polynomial over points P0 through P3 is given by

(4) where the parameter, , varies between 0 and 1 To examine the direction of the curve at its endpoints, consider the derivative of B with respect to

(5)

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Fig. 6. Control points for Bezier curve.

Fig. 7. A microstrip to H-guide transition with tapered dielectric.

We see that at the endpoints, where the parameter t is 0 or 1, the direction of the Bézier curve can be found by taking the difference between the two points, P0 and P1, or P2 and P3, respectively. Consequently, as shown in Fig. 6, by putting P0 and axis, the beginning of the curve becomes P1 in line with the parallel to the axis. The same property is applied to points P2 and P3 as shown in the same figure. To form a smooth transition from a microstrip line to an H-guide, it is proposed that the transition be considered in several stages as shown in Fig. 7. The microstrip line tapers slowly into a wide microstrip line in Section I. To maintain a smooth transition, the slope in this section is enforced to be parallel to the direction of the microstrip line. In Section II, the microstrip line crosses over the dielectric and the profiled “radial” waveguide as described in Fig. 5. Section III is the H-guide. At the beginning of this section, the slope of the transition is enforced to be parallel to the direction of the H-guide to form the rectangular waveguide section of the transition. In addition, Sections I to II are smoothed by slowly tapering the air gap in the dielectric as compared to the abrupt air gap

Fig. 8. (a) Dimension of the Bezier transition top sheet. Bézier curve control points P0 (2.425, 90.0), P1 (2.425, 62.0), P2 (25.0, 62.0), and P3 (25.0, 40.0) are shown, where units are in mm. b) Dimension of the Bezier transition board cutouts. Bézier curve control points P0 (6.6, 90.0), P1 (6.6, 65.25), P2 (30.0, 55.3125), and P3 (35.0, 47.0) are shown, where units are in mm.

originally proposed in [11]. Dimensions for the inner and outer curves are shown in the caption in Fig. 8. A simulated plot of the magnitude of the electric fields are shown in Fig. 9. For this prototype, the actual top sheet is cut from a thin brass sheet and has finite thickness. The brass layer touches the microstrip line. For the middle layer, the board is made out of Rogers 5880 with a microstrip line for testing the connectors before the final assembly. The bottom layer is formed out of a copper-plated FR4 sheet, where the metallic copper layer must touch the middle layer. A photograph of the assembled prototype and measurements of the insertion and return loss are shown in Fig. 10.

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Fig. 9. Magnitude of the electric fields within the back-to-back microstrip to H-guide transition.

Fig. 10. Photograph of the assembled prototype.

Fig. 11. Comparison of simulated vs. measured insertion and return loss.

The prototype shown in Fig. 10 was tested and measured. The insertion and return loss are plotted in Fig. 11. The match is fairly good up to 15 GHz, however, the measured insertion loss tends to fall away as 18 GHz approaches. The discrepancies between measured and simulated results are due to some additional loss due to the coaxial to microstrip connectors and their soldering on the board that was not taken into account in the simulation. The same connector also causes degradation in the return loss from roughly 25 dB to 15 dB. IV. ANTENNA DESIGN Using the Bézier shaped microstrip to thin H-guide transition discussed in the previous section, the antenna design becomes relatively straightforward. One possible design has been discussed in [5], where the usage was proposed over a small bandwidth only. In this design, we show through simulation that this antenna exceeds a return loss of 10 dB from 8 to 18 GHz

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Fig. 12. Gapped H-Guide. The magnitude and direction of the electric field is represented by arrows. The effective dielectric constant in region I is higher than that of region II.

while achieving a gain that varies between 12 and 18 dBi. However, due to manufacturing imperfections the fabricated prototype achieves a maximum of about 16 dBi at only 16 GHz. Some advantages of this design are that the wave exiting the aperture is nearly planar, and is mostly concentrated in the dielectric, away from metallic edges in the vertical direction, and with no metallic side walls required. The concentration away from metallic edges in the vertical direction reduces sidelobes, as does the absence of metallic side walls. Let us first consider the groove guide as proposed in [17]. If an air gap is added between the dielectric and top metallic plate, the effective dielectric constant of the dielectric is simply reduced, so that guided propagation is still possible. We have briefly discussed this property in [5]. Now, consider the case where an air gap is added above and below the dielectric so that the dielectric floats between the two metallic plates and propagation is still possible. The proposed dual air gap configuration is shown in Fig. 12. The width of the dielectric slowly increases to compensate for the loss in effective dielectric constant as the air gap widens. A rough approximation for the reflection coefficient seen at the input can be formulated as shown in Fig. 13. If Section I approaches a propagation constant that is equal to an H-guide , then . Each subsequent section then has an effective propagation constant, , and length . The reflection coefficient, with reference to the start of the taper, is then given as [18]

(6)

The contribution of each discrete reflection in (6) is not a simple task and planned for discussion in future studies. For this paper, a numerical simulation is, therefore, shown in Fig. 14, where the angle is varied, while the taper in the dielectric remains constant. In this parametric study, the radiation boundary is placed at the aperture of the horn, so that the effects of the aperture on

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Fig. 13. Segmentation of an H-guide taper. Each section has an effective propagation constant, , and length l . Fig. 15. Improved dual air gap H-guide aperture horn antenna design using ,b : , , 1.575 mm thick Rogers 5880. a , , , = ,  , .

= 10 mm = 1 575 mm SX = 100 mm PH1 = 50 mm PH2 = 50 mm HW = 32 mm RH = HW 2 = 16 mm = 15 = 18 V. ANTENNA PROTOTYPE SIMULATION AND MEASUREMENT

Fig. 14. Variation in angle of metallic plates. Inner dielectric tapers from 10 mm to 30 mm for each run. The horn angle is measured from the horizontal plane to the plane of one of the metallic plates.

the reflection coefficient are removed. The variations in Fig. 14 are therefore due only to the change in metallic plate angles. We can conclude that for an optimal angle for operation from 6 to 18 GHz, where a return loss of 10 dB or better is required, is equal to 8 degrees in the vertical plane, close to the H-guide to gapped H-guide transition. This corresponds to a dielectric taper from 10 mm to 30 mm, given that Rogers 5880 is used as a substrate, and the H-guide plate separation distance is equal to 1.575 mm. A two-stage design, with a smaller angle to improve the return loss, and a large angle, is used to widen the aperture and thus improve the gain. Finally, to speed up the wave at the aperture and thus adjust the phase, a curved shape with radius HW/2 is cut into the dielectric. The top and side views of the antenna are shown in Fig. 15.

In this section, the antenna prototype fabrication, return loss, and radiation pattern measurements are discussed. The antenna was first simulated using a waveport that directly feeds the H-guide in HFSS as shown in Fig. 16. This type of feeding forces the fundamental mode and therefore allows the full potential of the antenna design to be seen. The simulation was then performed using the microstrip to H-guide transition as discussed in previous sections as shown in Fig. 17. This is a more realistic case for comparison with measured data, since the thin H-guide must be fed in some way. At lower frequencies, all modes higher than the fundamental are evanescent, and therefore, any higher order modes that are introduced by the transition are naturally attenuated through the structure. However, at higher frequencies, other modes may propagate freely, so the that modes that may be introduced by the transition or fabrication imperfections may propagate and affect the radiation pattern. These effects are most notable in the sidelobes. Finally, the assembled prototype is shown in Fig. 18. The return loss plots for the three cases are shown in Fig. 19. When comparing the return loss of two simulated results, it can be seen that the transition is so effective that is does not introduce any significant penalty in the return loss. When comparing the measured results with the simulated ones, it can be concluded that the transition has been reasonably well fabricated since the variations and trends in the return losses are all similar. Radiation patterns in azimuth and elevation for simulated and measured data are shown in Figs. 20through 25 for 9, 12, and 15 GHz. The sidelobes have been degraded once the microstrip to

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Fig. 17. (a) Antenna simulation using microstrip to H-guide transition. (b) Electric field at 15 GHz within antenna simulation.

Fig. 18. Photograph of antenna prototype including microstrip to H-guide transition.

Fig. 16. a) Antenna Simulation using waveport. b) Electric fields within antenna at 15 GHz in the horizontal plane within antenna simulation using a waveport. c) Electric fields at 15 GHz in the vertical plane within antenna simulation using a waveport.

H-guide transition has been introduced, however, the measured data matches the simulated data reasonably well if the transition is included. Some additional degradation in the sidelobes is visible at 15 GHz in the elevation pattern, however, this is mostly due to unexpected additional leakage from the microstrip to H-guide transition. The transition from coaxial connector to microstrip line, which is not included in the simulation, caused many small sidelobes to appear at 15 GHz. We have shown this to be the

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Fig. 19. Return loss ( S11) for the measured prototype compared to the simulations using a waveport, and using the microstrip to H-guide transition.

Fig. 20. Azimuth pattern at 9 GHz.

Fig. 22. Azimuth pattern at 12 GHz.

Fig. 23. Elevation pattern at 12 GHz.

Fig. 21. Elevation pattern at 9 GHz. Fig. 24. Azimuth pattern at 15 GHz.

cause of these ripples by covering this transition with foam and aluminum foil, which significantly reduced these sidelobe levels in subsequent measurements. Because of the crudeness of this approach, these results were not included. In future designs, a shielded transition may be used. Simulated data for 18 GHz is shown in Figs. 26 and 27 using a waveport only. A simulation that includes the transition was not possible at 18 GHz due to memory limitations in the computer being used. The measured data at 18 GHz showed non-ideal

sidelobe levels as compared to simulated results and therefore have not been shown here. The sidelobe levels at 18 GHz using the waveport, however, show promising results. The peak measured gain is approximated by first subtracting the power received from a gain stardard horn antenna from the power received with the H-guide horn antenna, where the transmit antenna is a dual-ridged horn antenna. This delta, as shown in Fig. 28, is then compared to the known gain

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Fig. 25. Elevation pattern at 15 GHz. Fig. 28. The actual measured power received from a gain standard metallic air-filled horn antenna is compared to the power received from the H-guide horn antenna in 0.1 GHz steps as shown above. The power received is at the bore sight (center) without any re-pointing where the transmit antenna is a wideband dual-ridged horn antenna.

TABLE I GAINS

Fig. 26. Simulated azimuth pattern with waveport at 18 GHz. Measured gain is approximated by comparing received power with a gain standard horn antenna and applying the difference to the known gain. Gain is approximated 12 GHz because the received power of the gain standard was not available.

VI. CONCLUSION

Fig. 27. Simulated elevation pattern with waveport at 18 GHz.

standard’s gain. The peak measured gain as shown in Table I exceeds the simulated results partly because the higher order modes created by the transition change the fields at the aperture, causing a narrower main beam. Errors can also partly be due to measurement and calibration errors. The simulated gain with the transition is slightly higher than the gain with the waveport due to the distortion of the beam as seen in Figs. 20to 25. A gain table in Table I outlines these results.

In this paper, a new wideband high gain H-guide horn antenna, and a microstrip to H-guide transition have been presented and discussed. The proposed new antenna has some unique advantages compared to traditional horn antennas, such as lower dependence upon precise angles, similar gains, and simpler designs. This antenna also successfully demonstrates that the thin single H-guide is a useful transmission line. Measured data including return loss and radiation patterns have confirmed that simulated results are reasonably accurate. VII. ACKNOWLEDGEMENT The authors would like to thank all the technicians who assisted in the fabrication and measurement of prototypes discussed in this paper. Namely, J. Landry from Concordia University, and T. Antonescu and M. Thibault from Ecole Polytechnique in Montreal, QC, Canada. A. R. Sebak would like to thank the King Saud University and the National Plan for Sciences and Technology (NPST) for funds through Research Grant 09ELE858-02.

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REFERENCES [1] E. Lier and A. Kishk, “A new class of dielectric-loaded hybrid-mode horn antennas with selective gain: Design and analysis by single mode model and method of moments,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pt. 1, pp. 125–138, Jan. 2005. [2] E. Lier, “Review of soft and hard horn antennas, including metamaterial-based hybrid-mode horns,” IEEE Trans. Antennas Propag. Mag., vol. 52, no. 2, pp. 31–39, Apr. 2010. [3] M. Abbas-Azimi, F. Mazlumi, and F. Behnia, “Design of broadband constant-beamwidth conical corrugated-horn antennas [tAntenna Designer’s Notebook],” IEEE Trans. Antennas Propag. Mag., vol. 51, no. 5, pp. 109–114, Oct. 2009. [4] M. Wong, A. R. Sebak, and T. A. Denidni, “A broadside substrate integrated horn antenna,” in Proc. 2008 IEEE Int. Symp. Antennas and Propagation, San Diego, CA, Jul. 5–12,, 2008, pp. 1–4. [5] M. Wong, A. R. Sebak, and T. A. Denidni, “Gapped radial H-guide aperture antenna,” in Proc. 2009 IEEE Int. Antennas Propagation Symp., June 1–5, 2009, pp. 1–4. [6] F. J. Tischer, “H guide with laminated dielectric slab,” IEEE Trans. Microw. Theory Tech., vol. 18, no. 1, pp. 9–15, Jan. 1970. [7] T. Yoneyama and S. Nishida, “Nonradiative dielectric waveguide for millimeter wave integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. 29, no. 11, pp. 1188–1192, Nov. 1981. [8] A. S. Al-Zoubi, A. A. Kishk, and A. W. Glisson, “A linear rectangular dielectric resonator antenna array fed by dielectric image guide with low cross polarization,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 697–705, Mar. 2010. [9] H. F. Hammad, Y. M. M. Antar, A. P. Freundorfer, and S. F. Mahmoud, “Uni-planar CPW-fed slot launchers for efficient TM0 surface wave excitation,” IEEE Trans. Microw. Theory Techn., vol. 51, no. 4, pp. 1234–1240, Apr. 2003. [10] F. J. Tischer, “Domino-type microwave and millimeter-wave systems,” in Proc. 15th European Microwave Conf., Oct. 1985, pp. 721–725. [11] M. Wong, A. R. Sebak, and T. A. Denidni, “Wideband Bézier curve shaped microstrip to H-guide transition,” Electron. Lett., vol. 45, no. 24, pp. 1250–1252, Nov. 19, 2009. [12] R. E. Collin, Field Theory of Guided Waves, 2nd ed. Piscataway, NJ: IEEE Press, 1991, pp. 712–716. [13] Ansoft HFSS ver. 10.1.2, Ansoft Corporation. Pittsburg, PA, Build, Sep. 28, 2006. [14] L. Han, K. Wu, and R. G. Bosisio, “An integrated transition of microstrip to nonradiative dielectric waveguide for microwave and millimeter-wave circuits,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 7, pt. 1, pp. 1091–1096, Jul. 1996. [15] A. D. Olver and B. Philips, “Profiled dielectric loaded horns,” in Proc. Eighth Int. Conf. Antennas and Propagation, 1993, vol. 2, pp. 788–791. [16] W. C. Song, S. C. Ou, and S. R. Shiau, “Integrated computer graphics learning system in virtual environment: Case study of Bezier, B-spline and NURBS algorithms,” in Proc. 2000 IEEE Int. Conf. Information Visualization, Jul. 19–21, 2000, pp. 33–38. [17] F. J. Tischer, “The groove guide, a low-loss waveguide for millimeter waves,” IEEE Trans. Microw. Theory Tech., vol. 11, no. 5, pp. 291–296, Sep. 1963. [18] M. Wong, A. R. Sebak, and T. A. Denidni, “Analysis, simulation, and measurement of square periodic H-guide structures,” IET Microw., Antenna. Propag., to be published. Michael Wong (S’05–M’10) received the B.Sc. degree in electrical engineering from Queen’s University, Kingston, ON, Canada, in 1997 and the M.Sc. and Ph.D. degrees, both in electrical engineering, from Concordia University, Montreal, QC, Canada, in 2006 and 2010, respectively. From 1998 to 2004, he was a Systems Engineering Associate with Mobile Satellite Ventures (now LightSquared), Ottawa, ON, Canada, working on traffic engineering for mobile satellite telephony with the MSAT satellites. During this time, he also performed consulting work with Telesat, a Canadian satellite operator, to study the effects of interference from solar effects and terrestrial services on fixed satellite services. He is currently with Research in Motion (RIM) working on calibration software for board level testing with the next generation Blackberry smartphones.

Abdel Razik Sebak (F’10) received the B.Sc. degree (with honors) in electrical engineering from Cairo University, Cairo, Egypt, , in 1976 and the B.Sc. degree in applied mathematics from Ein Shams University, Cairo, in 1978. He received the M.Eng. and Ph.D. degrees from the University of Manitoba, Winnipeg, MB, Canada, in 1982 and 1984, respectively, both in electrical engineering. From 1984 to 1986, he was with the Canadian Marconi Company, working on the design of microstrip phased array antennas. From 1987 to 2002, he was a Professor in the Electrical and Computer Engineering Department, University of Manitoba. He is a Professor of Electrical and Computer Engineering, Concordia University, Montreal, QC, Canada. His current research interests include phased array antennas, computational electromagnetics, integrated antennas, electromagnetic theory, interaction of EM waves with new materials and bio-electromagnetics. Dr. Sebak received the 2000 and 1992 University of Manitoba Merit Award for outstanding Teaching and Research, the 1994 Rh Award for Outstanding Contributions to Scholarship and Research, and the 1996 Faculty of Engineering Superior Academic Performance. He is a Fellow of IEEE. Dr. Sebak has served as Chair for the IEEE Canada Awards and Recognition Committee (2002–2004) and the Technical Program Chair of the 2002 IEEE-CCECE and 2006 ANTEM conferences.

Tayeb A. Denidni (M’98–SM’04) received the B.Sc. degree in electronic engineering from the University of Setif, Setif, Algeria, in 1986, and the M. Sc. and Ph.D. degrees in electrical engineering from Laval University, Quebec City, QC, Canada, in 1990 and 1994, respectively. From 1994 to 1996, he was an Assistant Professor with the engineering department, Université du Quebec in Rimouski (UQAR), Quebec, Canada. From 1996 to 2000, he was also an Associate Professor at UQAR, where he founded the Telecommunications laboratory. Since August 2000, he has been with the Personal Communications Staff, Institut National de la Recherche Scientifique (INRS), Université du Quebec, Montreal, QC, Canada. He founded the RF laboratory, INRS-EMT, Montreal, for graduate student research in the design, fabrication, and measurement of antennas. He possesses ten years of experience with antennas and microwave systems and is leading a large research group consisting of two research scientists, five Ph.D. students, and three M.S. students. Over the past ten years, he has graduated numerous graduate students. He has served as the Principal Investigator on numerous research projects on antennas for wireless communications. Currently he is actively involved in a major project in wireless of PROMPT-Quebec (Partnerships for Research on Microelectronics, Photonics and Telecommunications). His current research interests include planar microstrip filters, dielectric resonator antennas, electromagnetic-bandgap (EBG) antennas, antenna arrays, and microwave and RF design for wireless applications. He has authored over 100 papers in refereed journals. He has also authored or coauthored over 150 papers and invited presentations in numerous national and international conferences and symposia. From 2006 to 2007, Dr. Denidni was an associate editor for IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS. From 2008 to 2010, he served as an associate editor for IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He is a member of the Order of Engineers of the Province of Quebec, Canada. He is also a member of URSI (Commission C).

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CPW-Fed Cavity-Backed Slot Radiator Loaded With an AMC Reflector Johan Joubert, Senior Member, IEEE, J. (Yiannis) C. Vardaxoglou, Senior Member, IEEE, William G. Whittow, and Johann W. Odendaal, Senior Member, IEEE

Abstract—A low profile coplanar waveguide (CPW) fed printed slot antenna is presented with uni-directional radiation properties. The slot antenna radiates above a closely spaced artificial magnetic conducting (AMC) reflector consisting of an array of rectangular patches, a substrate and an electric ground plane. The electromagnetic bandgap (EBG) performance of the cavity structure between the upper conducting surface in which the slot is etched, and the ground plane at the bottom of the reflector, is investigated using an equivalent waveguide feed in the place of a half-wavelength section of the slot antenna. From the reflection coefficient of the equivalent waveguide feed one can determine the frequency band where minimum energy will be lost due to unwanted radiation from the cavity sides. The dimensions of the cavity were found to be very important for minimum energy loss. Experimental results for the final antenna design (with a size of ), mounted on a back plate, exhibit a 5% impedance bandwidth, maximum gain in excess of 10 dBi, low cross-polarization, and a front-to-back ratio of approximately 25 dB. This low-profile antenna with relatively high gain could be a good candidate for a 2.4 GHz WLAN application. Index Terms—Electromagnetic bandgap materials, periodic structures, slot antennas.

I. INTRODUCTION

S

LOT radiators are suitable candidates for portable units and unobtrusive base stations of mobile communications systems, because of their compactness, flush-mounting and simple structure [1]. Slot radiators are also attractive when an antenna has to be integrated into a metallic surface eg. as single elements or arrays conformal to an airborne structure for electronic warfare applications [2]. When a slot is printed on one side of a substrate, the element will radiate bidirectionally, and an electric conducting surface has to be added at a distance of a quarter-wavelength below the slot as a reflector to achieve optimum uni-directional radiation. If the reflector distance is reduced to achieve a lower profile, which is highly desirable for conformal antennas in most cases, parallel plate modes will be excited that can cause significant energy leakage. A fair amount of work was published on so-called conductor-backed CPW-fed Manuscript received May 05, 2011; revised July 15, 2011; accepted August 16, 2011. Date of current version February 03, 2012. J. Joubert and J. W. Odendaal are with the Centre for Electromagnetism, University of Pretoria, Department of Electrical, Electronic and Computer Engineering, Pretoria 0002, South Africa (e-mail: [email protected]; [email protected]). J. C. Vardaxoglou and W. G. Whittow are with the Centre for Mobile Communications Research, Department of Electronic and Electrical Engineering, Loughborough University, Loughborough LE11 3TU, U.K. (e-mail: [email protected], [email protected]). Digital Object Identifier 10.1109/TAP.2011.2173152

slot antennas, where single or multiple dielectric layers were inserted between the upper conducting surface containing the slot and the reflector [3]–[6]. Energy leakage between the parallel plates because of unwanted modes remains a problem for conductor-backed slot radiators and special techniques have to be used to overcome this, eg. the use of closed cavities behind the slot [1], the placement of shorting pins around the slot [2], or the use of twin slot configurations for phase cancellation [5], [6]. This paper presents results of an investigation of a CPW-fed slot radiator above a metamaterial-based AMC surface as a reflector. The intention was to investigate a single radiating element with a low profile and compact lateral dimensions, in order to improve on results for a similar structure investigated in [7]. Fig. 1 shows the geometry of the proposed structure. It can also be described as a cavity-backed (with open sides) slot radiator loaded with an AMC reflector. An alternative description would be that the radiation is due to the “structural resonance mode” [8] of the entire slot-fed cavity structure loaded with a high-impedance surface. The most important design considerations are: 1) the reflection phase of the AMC reflector—at the centre frequency it should ideally be 0 , and minimum variation as function of frequency will ensure optimum bandwidth of the antenna—the basic principle being that the reflected wave from the reflector should add sufficiently in phase with the resonant slot field in order to achieve a radiation resistance that can be matched to the CPW feed line; 2) the EBG performance of the physical structure between the upper conducting surface in which the slot is etched, and the ground plane at the bottom of the AMC—the propagation of any modes between the two conducting surfaces (which effectively form the parallel plate cavity structure), and subsequent radiation from the edges of the structure, should be minimized over the bandwidth of operation. Results from a number of studies related to the topic of low profile slot antennas with a metamaterial-based reflector to achieve uni-directional radiation have been published in the open literature by other researchers, all reporting some level of success. One has to distinguish between compact cavity-backed slot antennas and single slot radiators embedded in a laterally large parallel plate environment. The effectiveness of the EBG structure used between the plates should be a direct function of the size of the surface area over which the structure is employed around the radiating slot—the larger the area the more effective the EBG surface. As far back as 1999 Shumpert et al. [9] published results of a conductor-backed folded slot (fed with a coaxial transmission line) with an EBG structure consisting of an array of square dielectric cylinders between

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Fig. 1. Printed CPW-fed slot antenna above an AMC reflector; (a) 3-D view, (b) top view, and (c) front view.

the plates. They achieved uni-directional radiation for a single folded slot with a front-to-back ratio of 15 dB and a frequency bandwidth of 7.5%. The dimensions of the structure were —so the thickness of the antenna was still more than , and several wavelengths of the EBG structure were used to achieve significant suppression of the dominant TEM mode between the parallel plates. Elek et al. [10] published results for a uni-directional ringslot antenna. A primary focus of this work was the design of a suitable EBG surface to minimize radiation leakage between the parallel plates of the structure. They opted to use the popular “mushroom” periodic surface (small rectangular patches with vias connecting the centre of all the patches to a common ground plane) proposed by Sievenpiper et al. in [11]. This work [10] contains interesting details of an experimental setup to determine the EBG stopband. They effectively measure/simulate the transmission coefficient through a section of parallel plate waveguide with and without the EBG surface between the parallel plates, and show conclusively that the inclusion of the

EBG surface dramatically suppresses the propagation of the unwanted TEM-mode between the plates. The EBG surface that is closely spaced (much closer than a quarter-wavelength) to the ground plane in which the ring-slot is cut also acts as a quasi-artificial magnetic reflector (independent analysis of the structure has indicated a reflection phase close to 90 at the centre frequency), which allows for a low profile structure with uni-directional radiation. The final ring-slot antenna design had overall dimensions of , but they used an EBG surface of centrally located behind the slot. The overall structure was still quite large laterally (with the slot asymmetrically spaced relative to the antenna ground plane without any motivation), and achieved a good front-toback ratio of 21 dB, and a gain improvement between 2.5 and 2.9 dB (compared to a ring-slot antenna without a reflector) over a 5% impedance bandwidth. Niyomjan and Huang [12] theoretically (no experimental results were presented) investigated a suspended microstrip fed slot antenna on a high impedance surface (HIS), which is just an alternative description of an artificial magnetic conductor. Similar to [10], they also employed the Sievenpiper [11] “mushroom” periodic surface. Their final design had lateral dimensions of approximately , and they used a relatively large gap of 4 mm ( at 10 GHz) between the slot substrate and the AMC surface, which together with the two substrate thicknesses results in a final antenna that is not significantly lower in profile than what can be achieved with a quarter-wavelength spaced electric conducting surface. An example of a compact cavity-backed (with open sides) slot antenna (offset-fed with a microstrip line) with an etched or so-called “uni-planar compact photonic band-gap” (UC-PBG) reflector was introduced by Park et al. [7]. The UC-PBG reflector, consisting of a lattice of square pads, capacitive gaps and narrow lines connecting each cell was also designed to act as a magnetic reflector at the resonant frequency. The dimensions of the structure were very compact and extremely thin . The measured performance of this antenna was however not very good, with a small experimental frequency bandwidth of 1.33%, a front-to-back ratio in the vicinity of 10 dB, poor cross polarization, and only a 1 dB gain improvement with reference to a slot antenna without a reflector. It was decided to design the newly proposed low profile slot antenna with an AMC reflector for a practical frequency band (2.4–2.484 GHz)—that for WLAN base stations. Section II gives details on the design of the AMC reflector consisting of an array of rectangular patches, a substrate and an electric ground plane. The cavity-backed CPW-fed slot antenna design and the EBG performance of the cavity structure (between the upper conducting surface in which the slot is etched and the ground plane at the bottom of the reflector) are discussed in Section III. The EBG performance is investigated using an equivalent waveguide feed for one half-wavelength section of the slot antenna. From the waveguide reflection coefficient a designer can determine the frequency band where minimum energy will be lost due to unwanted modes radiating from the cavity sides. The dimensions of the cavity were found to be very important

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Fig. 2. 3D and front view of unit cell for determining of the resonant frequency (with 0 reflecting phase) of the AMC surface.

for minimum energy loss. Simulated and experimental results for the whole structure are presented in Section IV. II. AMC REFLECTOR DESIGN Artificial Magnetic Conductor (AMC) is the name that was established for the complex EBG structures (the “mushroom” periodic surface above a conducting ground) initially presented as High Impedance Structures (HIS) in [11] because of the specific property that the reflection of plane waves from such a surface occurs such that the reflected wave is in phase with the incident wave. This property would be exhibited by a magnetic conductor (dual of the perfect electric conductor). Different types of AMC reflectors have been investigated since the publication of [11], eg. in [13] a comparative study was performed on four types of AMC surfaces, including the “mushroom” EBG, uniplanar compact EBG, Peano curve, and Hilbert curve. The study shows that the “mushroom” surface has the best bandwidth of the four types considered. Another popular type of AMC is the Jerusalem Cross-based structure [14], but our own simulations have shown that the bandwidth of this structure (implemented on the same substrate) is also substantially less than that of the “mushroom” AMC. A modification of the “mushroom” AMC was proposed in [15]—the vias were removed between the patches and the ground because vias complicate the fabrication of AMC surfaces—the effect of this was negligible on the AMC performance of the structure for normal incidence, but the EBG typically shifts to higher frequencies. In [15] the AMC and EBG characteristics of this new patch array AMC were investigated and a new technique presented to tailor the spectral position of the AMC operation and the EBG. Because of the bandwidth advantage, and the simplicity of fabrication it was decided to use the patch array AMC proposed in [15] for the antenna in this paper—a unit cell of the proposed metasurface is shown in Fig. 2. The performance of an infinite repetition of this basic cell has been simulated using CST Microwave Studio applying the proper boundary conditions (PEC’s on the planes normal the polarization of the incident field, and PMC’s on the planes parallel to the polarization of the incident field) to the unit cell and under normal incidence. The total structure thus comprises of a metal backed dielectric substrate with an array of metal square patches arranged in a periodical form. Design by repetitive analysis resulted in final dimensions for the structure: mm, mm (which implies mm, referring to Fig. 1) and substrate thickness mm (RO4003, , ). The design aim was to get zero reflecting phase at a resonant frequency of 2.45 GHz.

Fig. 3. The simulated reflecting phase response for a patch AMC and a “mushmm, mm room” AMC. In both cases the dimensions were mm and mm (on substrate RO4003, , ).

The reflection phase of the unit cell is shown in Fig. 3. If one includes a metallic via embedded inside the dielectric substrate connecting the metal square patch to the ground plane the well known “mushroom” type AMC is obtained [10], [11]. This structure was also analyzed and the result is also shown in Fig. 3. The reflection phase response of the patch AMC and the “mushroom” AMC were found to be very similar. The reflection phase of the structure is 0 at 2.45 GHz and between in the frequency range from 2.28 GHz to 2.64 GHz (approximately over a 15% bandwidth). The magnitude of the reflection coefficient is close to . The bandwidth of the structure can be improved if one uses a thicker substrate or a substrate with a lower permittivity, but the overall antenna structure will then be thicker (less low-profile), or the unit element size will have to be increased. III. CAVITY-BACKED SLOT ANTENNA DESIGN The CPW-fed cavity-backed slot antenna loaded with the AMC reflector designed in the previous section (see Fig. 1) was designed to operate in the WLAN frequency band (2.4–2.484 GHz). The radiating element basically consists of two half-wavelength slots radiating in phase and in close proximity to each other—each slot fed by one of the slotlines of the CPW feed line. It is sometimes referred to as a full-wavelength printed slot radiator, and this type of element (without the AMC reflector) has previously been characterized in [16], [17]. The design of the final antenna was performed with the assistance of CST Microwave Studio. The radiating slot element substrate was chosen as Rogers RO4003, with , and mm. The antenna design process involved the determination of suitable values for some inter-dependant parameters, specifically , and , and and (all of them indicated in Fig. 1). A. Gap—The Distance Between the Slot Substrate and the AMC Reflector Simulations indicated that the bandwidth of the antenna bemm. This probably hapgins to reduce dramatically for pens because the AMC reflector performance begins to change drastically as the slot substrate comes closer in proximity to

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Fig. 5. Three different test cases for which the EBG performance were invesmm , (b) tigated; (a) Case#1—5 5 patch array AMC mm mm , and (c) Case#2—5 4 patch array AMC mm mm . Case#3—5 3 patch array AMC

Fig. 4. EBG performance test setup consisting of a dielectric-filled rectangular waveguide (in the position of one of the radiating slots) feeding the parallel plate cavity formed by the finite slot ground plane, antenna substrate and AMC reflector.

the AMC structure (the AMC reflector was designed for an air medium in front of the reflector). A final value of mm was used. The overall height of the final antenna was mm , with the free-space wavelength at 2.45 GHz. B.

and

—The Lateral Cavity Dimensions

and , (the lateral cavity dimensions of the antenna), and also the number of patches used to construct the AMC reflector were chosen to minimize the power radiated from the sides of the cavity structure, which contribute towards high sidelobes and a high backlobe in the radiation pattern. This effective EBG performance was found to be strongly influenced by the lateral cavity dimensions (for a specific value of ). The EBG stop band was initially determined using the method proposed in [10], where the slot is removed from the top conducting surface (and replaced with a PEC), and two ports on either side of the parallel plate cavity are defined, and the transmission coefficient then determined. The two remaining sides of the cavity were designated as perfect magnetic walls. The results showed that the stop band did not coincide with the AMC reflector band, with high transmission levels predicted within the 2.4–2.5 GHz band. Upon further investigation it was soon realized that this method does not necessarily determine the EBG performance of our finite cavity structure very accurately—the excitation of parallel plate modes by a slot in the top conducting surface of the finite cavity structure might be very different to that of the structure with the magnetic walls as described in [10]. A new EBG performance test setup was devised, consisting of a dielectric-filled waveguide feeding the finite parallel plate cavity formed by the slot conducting surface, the antenna substrate and the AMC reflector, as shown in Fig. 4. The assumption is that the waveguide aperture will excite parallel plate modes in the cavity very similar to a slot. The simulated reflection coefficient of the waveguide port will then be indicative of the EBG performance of the structure. The dimensions and location of the waveguide feed were chosen to be the same as one half-wavelength of the radiating slot element, and then suitable compact cavity dimensions were determined through repetitive analysis. The EBG performance of three representative cases (see Fig. 5) are discussed: (a) Case#1—5 5 patch array AMC

mm , (b) Case#2—5 4 patch array AMC mm mm , and (c) Case#3—5 3 patch array AMC mm mm . The basic difference between Case#1 and Case#2 is the cavity dimension in the -direction. Case#2 and Case#3 have exactly the same cavity dimensions, but the patch array sizes are different in the two cases, and in all cases mm was used. The waveguide port was excited with a dominant waveguide mode and the reflection coefficient calculated. From Fig. 6 one can see that the difference in the reflection coefficient is significant for Case#1 and Case#2—illustrating the significance of the cavity dimensions. The difference in reflection coefficient is less significant between Case#2 and Case#3, for different array sizes but the same cavity dimensions. Case#3 has the largest reflection coefficient over the 2.4–2.5 GHz band, which is an indication that the waveguide aperture radiates into a very high impedance for that particular structure, thus limiting significant leakage of power between the cavity side walls. Case#3 was chosen for the final antenna design — mm , mm , and a 5 3 patch array AMC. For a compact cavity-backed slot antenna like this the cavity dimensions are the primary parameters that have to be determined to ensure minimum energy leakage. Because of the small cavity size and the close proximity of the slots to the cavity edges, and hence the small surface area around the slot where an EBG surface can be employed, the mode suppression by the periodic structure of the AMC reflector does not seem to play a significant role. For larger structures like those in [7], [9], [10], the mode suppression of the periodic structure of the AMC will play a more significant role, and the newly devised EBG test setup will also be suitable for future designs of such structures. C.

and

—The Slot Dimensions

and , the slot length and width were determined that will provide an impedance match to a 50 CPW feed line. For the 50 CPW feed line the gap between the slotlines was determined as mm and the slotline width as mm. The slot width primarily determines the radiation resistance, and the slot length primarily determines the resonant frequency. The final antenna design slot dimensions were determined as mm and mm. IV. REFERENCE ANTENNA DESIGN For the purpose of gain and bandwidth comparisons a reference antenna without an AMC reflector was also designed using the same substrate and the same lateral dimensions mm and mm. If the final antenna design slot

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Fig. 8. Photograph of the manufactured final antenna design.

Fig. 6. Simulated magnitude of cases shown in Fig. 5.

for the waveguide port—for the three test

Fig. 9. Simulated and measured reflection coefficient tenna design. Fig. 7. Simulated magnitude of for the waveguide port—for an infinite mm. PEC reflector spaced at

dimensions as determined in the previous section are used and the antenna simulated without the AMC reflector, the radiation resistance of the slot is approximately doubled, and the resonant frequency increases. The final reference slot dimensions can be obtained by decreasing the slot width (to mm), for a radiation resistance 50 , and by increasing the slot length (to mm), for resonance at 2.45 GHz. The reference antenna above an infinite PEC reflector spaced at mm was also analyzed. The antenna was still reasonably well matched, but the E-plane radiation patterns were seriously distorted because of significant energy leakage from the sides of the antenna. To confirm the EBG test procedure (in Fig. 4) was performed for the antenna above the PEC reflector, and the waveguide port reflection coefficient is shown in Fig. 7—the level of dB at the design frequency is a clear indication of significant energy leakage. V. RESULTS OF FINAL DESIGN AND DISCUSSION To validate the simulations the slot antenna with the AMC reflector was manufactured and measured. A photograph of the final antenna is shown in Fig. 8. The different layers were fixed with four plastic screws with 3 mm spacers.

for the final an-

A comparison between the simulated and measured reflection coefficient is shown Fig. 9. The two sets of data correspond well, and the impedance bandwidth ( dB return loss) was found to be close to 5%. It has to be mentioned that the impedance bandwidth of the reference slot antenna without any reflector was more than 20%. The much lower bandwidth achieved with the reflector is due to the limited bandwidth of the AMC surface in terms of reflection phase. The overall bandwidth of the final antenna roughly corresponds to the bandwidth of the AMC surface where the reflection phase is between . The electromagnetic bandgap of the structure is significantly broader (define as 2.2–2.6 GHz, from Fig. 6), so if one can improve the reflection phase bandwidth an antenna with better overall bandwidth will be possible. The E- and H-plane radiation patterns of the final antenna design were measured at 2.45 GHz. The simulated and measured patterns are shown in Figs. 10 and 11. The cross-polarization was found to be very low (more than 20 dB down), and the front-to-back ratio acceptable—in the region of 18 dB. To improve the front-to-back ratio the final antenna design was mounted on a back plate with a size mm mm. Fig. 12 shows a photograph of the final antenna design mounted on the back plate. Simulations have shown that the back plate added to the structure does not affect the EBG properties or the matching of the antenna significantly—the was still lower than dB for the entire WLAN band. The

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Fig. 12. Photograph of the manufactured final antenna design mounted on a back plate.

Fig. 10. Simulated and measured E-plane radiation patterns for the final antenna design.

Fig. 13. Simulated and measured E-plane radiation patterns for the final anback plate. tenna design mounted on a

Fig. 11. Simulated and measured H-plane radiation patterns for the final antenna design.

back plate does however change the radiation direction of the small amount of leaked energy from the cavity sidewalls—with less energy being radiated towards the back and sides of the antenna. The simulated and measured E- and H-plane radiation patterns are shown in Figs. 13 and 14. The front-to-back ratio improved to about 25 dB. The realized gain was simulated and measured for the slot antenna without an AMC reflector, the final antenna design (with an AMC reflector and lateral dimensions of 125 mm 100 mm) and the same design mounted on the back plate, and the results are shown in Fig. 15. Good correlation between the simulated and measured values can be observed. The gain of

the final antenna design is approximately 3 dB higher than the gain of the antenna without a reflector, and the gain improved by another dB for the final design mounted on the larger back plate. The measured gain of the final antenna mounted on the back plate varied between 9.8 and 10.3 dB within the 2.4–2.5 GHz band. Also evident in Fig. 15 is the more constant gain over a wider bandwidth (close to 6 dBi) of the reference antenna which has a much larger bandwidth because of the absence of the effect of the AMC reflector. The simulated radiation and total efficiencies of the final antenna mounted on the back plate are shown in Fig. 16—the radiation efficiency is better than 95% and the total efficiency better than 85% for the entire 2.4–2.5 GHz band. VI. CONCLUSIONS A low profile CPW-fed cavity-backed printed slot antenna is presented with uni-directional radiation properties. The slot

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Fig. 16. Simulated radiation and total efficiencies of the final antenna mounted back plate. on the

Fig. 14. Simulated and measured H-plane radiation patterns for the final anback plate. tenna design mounted on a

of approximately 25 dB. This antenna is simple to manufacture using simple photolithography (no vias or solid sidewalls are necessary), and can be used as a flush-mounted single radiator. This low-profile antenna with relatively high gain could be a good candidate for a 2.4 GHz WLAN application. Future investigations will include the design of arrays of printed slots above an AMC reflector, for which the newly devised EBG test procedure will be a valuable design tool. REFERENCES

Fig. 15. Simulated and measured antenna gain for the reference slot antenna without a reflector, the final antenna design with an AMC reflector, and the final back plate. antenna design with an AMC reflector mounted on a

antenna radiates above a closely spaced artificial magnetic conducting reflector consisting of an array of rectangular patches, a substrate and an electric ground plane. The basic AMC reflector design was based on an infinite structure, but the final dimensions of the metamaterial based reflector cavity was determined on the basis of a newly devised EBG performance test for the finite structure. The EBG performance was evaluated with an equivalent waveguide feed in place of one half-wavelength section of the slot antenna. From the waveguide reflection coefficient one can determine the frequency band where minimum energy will be lost due to unwanted modes propagating between the parallel plates and radiating from the cavity sides. Experimental results for the final antenna design (with a size of ), mounted on back plate, exhibit a 5% impedance bandwidth, maximum gain in excess of 10 dBi, low cross-polarization, and a front-to-back ratio

[1] A. Vallecchi and G. B. Gentili, “Microstrip-fed slot antennas backed by a very thin cavity,” Microw. Opt. Tech. Lett., vol. 49, no. 1, pp. 247–250, Jan. 2007. [2] C. Löcker, T. Vaupel, and T. F. Eibert, “Radiation efficient unidirectional low-profile slot antenna elements for X-band applications,” IEEE Trans. Antennas Propagat., vol. 53, no. 8, pp. 2765–2768, Aug. 2005. [3] M. Qiu, M. Simcoe, and G. V. Eleftheriades, “Radiation efficiency of printed slot antennas backed by a ground reflector,” in Proc. 2000 IEEE AP-S Symp. Dig., pp. 1612–1615. [4] J. P. Jacobs, “Self and mutual admittance of CPW-fed slots on conductor-backed two-layer substrate,” Microw. Opt. Tech. Lett., vol. 49, no. 11, pp. 2798–2802, Nov. 2007. [5] J. P. Jacobs, J. Joubert, and J. W. Odendaal, “Radiation efficiency and impedance bandwidth of conductor-backed CPW-fed broadside twin slot antennas on two-layer dielectric substrate,” IEE Proc-Microw. Antennas Propag., vol. 150, pp. 185–190, 2003. [6] M. Qiu and G. V. Eleftheriades, “Highly efficient unidirectional twin arc-slot antennas on electrically thin substrates,” IEEE Trans. Antennas Propagat., vol. 52, no. 1, pp. 53–58, Jan. 2004. [7] J. Y. Park, C.-C. Chang, Y. Qian, and T. Itoh, “An improved low-profile cavity-backed slot antenna loaded with 2-D UC-PBG reflector,” in Proc. IEEE Int. Symp. Antennas Propagat., Boston, MA, Jul. 2001, pp. 194–197. [8] A. O. Karilainen, J. Vehmas, O. Luukkonen, and S. A. Tretyakov, “High-impedance-surface-based antenna with two orthogonal radiating modes,” IEEE Antennas Wireless Propagat. Lett., vol. 10, pp. 247–250, 2011. [9] J. D. Shumpert, W. J. Chappell, and L. B. Katehi, “Parallel-plate mode reduction in conductor backed slots using electromagnetic bandgap substrates,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2099–2104, Nov. 1999. [10] F. Elek, R. Abhari, and G. V. Eleftheriades, “A unidirectional ring-slot antenna achieved by using an electromagnetic bandgap surface,” IEEE Trans. Antennas Propagat., vol. 53, no. 1, pp. 181–190, Jan. 2005. [11] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999.

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[12] G. Niyomjan and Y. Huang, “A suspended microstrip fed slot antenna on high impedance surface structure,” in Proc. EuCAP, Nice, France, Nov. 2006. [13] J. R. Sohn, K. Y. Kim, and H.-S. Tae, “Comparative study on various artificial magnetic conductors for low-profile antenna,” PIER, vol. 61, pp. 27–37, 2006. [14] M. Hosseini and M. Hakkak, “Characteristics estimation for Jerusalem cross-based artificial magnetic conductors,” IEEE Antennas Wireless Prop. Lett., vol. 7, pp. 58–61, 2008. [15] G. Goussetis, A. Feresidis, and J. C. Vardaxoglou, “Tailoring the AMC and EBG characteristics of periodic metallic arrays printed on grounded dielectric substrate,” IEEE Trans. Antennas Propagat., vol. 54, no. 1, pp. 82–89, Jan. 2006. [16] A. Nešić, “Slotted antenna array excited by a coplanar waveguide,” Electron. Lett., vol. 18, pp. 275–276, 1982. [17] H.-C. Liu, T.-S. Horng, and N. G. Alexopoulos, “Radiation of printed antennas with a coplanar waveguide feed,” IEEE Trans. Antennas Propagat., vol. 43, no. 10, pp. 1143–1148, Oct. 1995. Johan Joubert (M’86–SM’05) received the B.Eng., M.Eng. and Ph.D. degrees in electronic engineering from the University of Pretoria, Pretoria, South Africa, in 1983, 1987, and 1991 respectively. From 1984 to 1988 he was employed as a Research Engineer at the Council for Scientific and Industrial Research, Pretoria. In 1988 he joined the Department of Electrical and Electronic Engineering at the University of Pretoria, where he is currently a Professor of Electromagnetism. From July to December 1995 he was Visiting Scholar with the Department of Electrical and Computer Engineering, California State University, Northridge. From July to December 2001 he was Visiting Scientist at Industrial Research Laboratories in Wellington, New Zealand. From July to December 2006 he was Visiting Scholar at the Institut für Höchstfrequenztechnik und Elektronik, Universität Karlsruhe (TH), Germany, and from July to September 2010 he visited Loughborough University, U.K., for a collaborative research project on metamaterials. His research interests include antenna array design and computational electromagnetism. Prof. Joubert is a registered Professional Engineer in South Africa.

J.(Yiannis) C. Vardaxoglou (M’87) received the B.Sc. degree in mathematical physics in 1982 and the Ph.D. degree in electronics in 1985 at the University of Kent, Kent, U.K. He joined Loughborough University as a Lecturer in 1988 and was promoted to Senior Lecturer in 1992 and Professor or Wireless Communications in 1998. He is currently Dean of the School of Electronic, Electrical and Systems Engineering at Loughborough University. He established the Wireless Communications Research (WiCR) group

at Loughborough University and heads the Centre for Mobile Communications Research. He has pioneered research, design and development of frequency selective surfaces (FSS) for communication systems, Metamaterials and low SAR antennas for mobile telephony and has commercially exploited a number of his innovations. He has served as a consultant to various industries, holds 6 patents and is the Technical Director of Antrum Ltd (a University spinout company). He has attracted research funding from industry and has been awarded 18 EPSRC research grants. He has published over 160 refereed journals and conference proceeding papers and has written a book on FSS. Prof. Vardaxoglou was Chairman of the Executive Committee of the IET’s Antennas and Propagation Professional Network in the UK and chaired the IEEE’s distinguish lecturer program of the Antennas and Propagation Society (APS) for 5 years. He was the General Chair of EuCAP’2007. He has chaired numerous IEE/IET events and has been on the Steering Committee of the European Conference on Antennas and Propagation, EuCAP since 2006. He founded the Loughborough Antennas and Propagation Conference (LAPC), which has been running since 2005.

William G. Whittow received the B.Sc. degree in physics from The University of Sheffield, Sheffield, U.K. in 2000 and the Ph.D. degree from the Electrical and Electronics Engineering Department, University of Sheffield, in 2004. He is currently employed as a Research Associate at Loughborough University, U.K. He has published over 60 peer reviewed journal and conference papers in topics related to electromagnetic materials, wearable antennas, VHF antennas, Specific Absorption Rate, FDTD, bioelectromagnetics, phantoms and Genetic Algorithms. Dr. Whittow has been the Coordinating Chair of the Loughborough Antennas & Propagation Conference (LAPC) since 2007.

Johann W. Odendaal (M’90–SM’00) received the B.Eng., M.Eng., and Ph.D. degrees in electronic engineering from the University of Pretoria, Pretoria, South Africa, in 1988, 1990, and 1993, respectively. From September 1993 to April 1994, he was a Visiting Scientist with the ElectroScience Laboratory at the Ohio State University. From August to December 2002, he was a Visiting Scientist with CSIRO Telecommunications and Industrial Physics in Australia. Since May 1994, he has been with the University of Pretoria, where he is currently a Full Professor. His research interests include electromagnetic scattering and radiation, compact range measurements, and signal processing. He is also Director of the Centre for Electromagnetism at the University of Pretoria. Prof. Odendaal is a member of the Antenna Measurement Techniques Association (AMTA) and is registered as a Professional Engineer in South Africa.

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The Use of Simple Thin Partially Reflective Surfaces With Positive Reflection Phase Gradients to Design Wideband, Low-Profile EBG Resonator Antennas Yuehe Ge, Member, IEEE, Karu P. Esselle, Senior Member, IEEE, and Trevor S. Bird, Fellow, IEEE

Abstract—Partially reflecting surfaces (PRS) with positive reflection phase gradients are investigated for the design of wideband, low-profile electromagnetic band gap (EBG) resonator antennas. Thin single-dielectric-slab PRSs with printed patterns on both sides are proposed to minimize the PRS thickness and to simplify fabrication. Three such surfaces, each with printed dipoles on both sides, have been designed to obtain different positive reflection phase gradients and reflection magnitude levels in the operating frequency bands. These surfaces, and the EBG resonator antennas formed from them, are analyzed theoretically and experimentally to highlight the design compromises involved and to reveal the relationships between the antenna peak gain, gain bandwidth, the reflection profile (i.e., positive phase gradient and magnitude) of the surface and the relative dimensions of dipoles. A small feed antenna, designed to operate in the cavity field environment, provides dB) across the opergood impedance matching ( ating frequency bands of all three EBG resonator antennas. Experimental results confirm the wideband performance of a simple, low-profile EBG resonator antenna. Its PRS thickness is only 1.6 mm, effective bandwidth is 12.6%, measured peak gain is 16.2 dBi at 11.5 GHz and 3 dB gain bandwidth is 15.7%. Index Terms—Broadband, electromagnetic band-gap (EBG), Fabry–Perot, frequency selective surface (FSS), high-gain, metamaterial, partially reflecting surface (PRS), resonator antenna, wideband.

I. INTRODUCTION

M

ETAMATERIAL-BASED structures have sparked considerable interest in recent years. The novel properties exhibited by metamaterials, such as electromagnetic band gap (EBG), artificial magnetic conductor behavior and negative refractive index, have been studied as a means of enhancing the performance of microwave and millimeter-wave components including antennas [1]–[12]. Among them, one practically useful application is the use of an EBG structure as a partially

Manuscript received September 29, 2010; revised May 31, 2011; accepted September 14, 2011. Date of publication October 24, 2011; date of current version February 03, 2012. This work was supported by the Australian Research Council. Y. Ge is with the College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China (e-mail: [email protected]). K. P. Esselle is with the Department of Electronic Engineering, Faculty of Science, Macquarie University, Sydney, NSW 2109, Australia. T. S. Bird is with Macquarie University, Sydney, NSW 2109, Australia and also with the CSIRO ICT Centre, Epping, NSW 1710, Australia. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173113

but highly reflecting surface to construct an EBG resonator antenna. A typical EBG resonator antenna is formed by creating an air-filled resonant cavity between an EBG structure, which behaves as a partially reflecting surface (PRS), and a total reflector. The cavity is fed using a small feeding antenna or an array. An early version of EBG resonator antennas uses appropriately spaced multiple dielectric layers to achieve a high gain [1], [12]. Transmission line theory proved sufficient for such designs [12]. Later, more complicated EBG structures were devised to enhance the gain of small antennas [2]–[10]. The structures studied in the past few years include high-permittivity covers [2], 3-D woodpile structures [3], 2-D dielectric rods [4], 2-D metallic rods [2], 2-D dielectric grids [4], 2-D metallic grids [4], [5], 2-D frequency selective surface (FSS) [6]–[10] and magneto-dielectric structures [11]. Among these EBG-resonator antennas, those based on multiple dielectric layers and FSSs have the advantages of structural simplicity and ease of fabrication and mounting. An inherent disadvantage of EBG resonator antennas, however, is the narrow bandwidth due to their typically narrowband resonant cavity. To overcome this, an active reconfigurable PRS antenna [13] was developed to operate in a wide frequency range, from 5.2 to 5.95 GHz. Using a slot antenna array as the excitation of an EBG resonator antenna [14], an improved bandwidth, up to 13.2%, was achieved while maintaining a high gain. However, the changes employed in these approaches [13], [14] to increase bandwidth make the design and fabrication complicated, losing many of the advantages of simplicity and low cost of EBG resonator antennas over conventional array antennas. According to the analysis in [6], a wideband EBG resonator antenna can be realized if the reflection phase of the PRS increases linearly with frequency. Based on this observation, PRSs with positive phase gradients have been realized using two dielectric slabs with an air-gap spacing to make wideband EBG resonator antennas [15]–[19]. Subsequently single-slab PRSs with a positive phase gradient have been proposed for the same purpose [20], [21]. EBG resonator antennas so designed have the advantages of standard passive EBG resonator antennas whilst providing a wider bandwidth. In the present paper, we investigate simple, thin PRSs that can offer positive reflection phase gradients, and provide the wideband performance required by EBG resonator antennas. The PRSs investigated here is composed of a single dielectric slab with two-dimensional arrays printed on both surfaces. The objective is to make the antenna low-profile with the antenna

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Fig. 1. Geometry of the EBG resonator antenna.

height only slightly greater than the cavity height. Here, dipole arrays are considered as an example. The antennas designed by this approach have the advantages of simplicity, low profile, low cost and ease of fabrication and assembling. By appropriately selecting the parameters of the two dipole arrays according to the proposed design method, the PRS exhibits positive reflection phase gradients within a designated frequency band, and hence a wideband EBG resonator antenna could be realized. A monopole antenna, which provides wideband impedance matching ( dB) over the entire bandwidth, has been designed to excite the cavity. This is a crucial step in designing wideband EBG resonator antennas because typical wideband feed antennas (matched in free space) become mismatched when placed in the cavity. Without a wideband feed antenna that is well-matched within the cavity, one can achieve only wideband directivity but not wideband gain. In the next section, we describe the design strategy of a PRS for wideband low-profile EBG resonator antennas. The design approach and the realization of the proposed PRSs are demonstrated in Section III. The design methodology is summarized in Section IV. In Section V, the measured performance of three EBG resonator antenna prototypes are presented with the reflection characteristics of the three PRSs used for them. A wideband feed antenna—an essential feature—is also described. The results are discussed in Section VI. Finally, Section VII concludes the paper with observations. II. PRS DESIGN STRATEGY FOR WIDEBAND LOW-PROFILE ANTENNAS The proposed EBG resonator antenna configuration is shown in Fig. 1. It consists of a small feed antenna, a thin slab PRS with double-sided printing and a PEC ground plane. The PRS forms a resonant cavity between itself and the PEC ground plane. At all frequencies in the band, the resonance condition [6] is satisfied. The antenna has a high directivity, provided that the reflection coefficient magnitude of the PRS is sufficiently large. Usually the cavity resonates only at one frequency, resulting in a narrowband EBG resonator antenna. To build a wideband EBG resonator antenna that maintains the resonance condition over the operating frequency band, the reflection phase of the PRS should increase with frequency, as described in [20]. The reflection phase of a conventional PRS, on the other hand, decreases with frequency. Our first objective in the present work was to design a PRS with a positive reflection phase gradient.

Fig. 2. Directivity versus reflection magnitude of PRSs.

The gain of the antenna is mainly determined by the crosssection of the PRS, the reflection coefficient of the PRS, the cavity height , and the feed antenna. If the PRS and the feed antenna are fixed, the cavity height determines the operating frequency and the antenna gain. To obtain a high gain or directivity, a highly reflective PRS is required. A PRS, made out of a 0.8 mm thick Rogers RT/Duroid 5880 slab, with square copper patch array printed on its bottom surface, was initially considered to investigate the variation of directivity with the reflection magnitude of the PRS. Changing the side length of the patch changed the PRS reflection magnitude. As illustrated by the results in Fig. 2, the antenna directivity increases with the reflection magnitude of the PRS until the reflection level is about dB. Also, a directivity dBi can normally be achieved as long as the reflection is above about dB. Also for the same investigation we considered a single dielectric slab, and changed the permittivity of the dielectric to change its reflection magnitude. A similar directivity variation trend was obtained, although the directivity was slightly lower than that shown in Fig. 2. In summary, when the reflection magnitude of the PRS is greater than about dB, an acceptable antenna directivity and possibly gain can be expected. III. REALIZATION OF A SIMPLE PRS WITH POSITIVE REFLECTION PHASE GRADIENT It is known that objects exhibit significantly different electromagnetic properties at resonance. For example, metamaterials exhibiting an overall negative permittivity or permeability can be realized using resonant rings [22]. The reflection coefficient of a FSS varies significantly in the frequency band around the FSS resonance frequency [20], [23]. Although the reflection phase of a conventional FSS decreases with frequency at most frequencies, it can be made to increase with frequency over a frequency band that is close to the FSS’s resonance frequency [20]. We take the advantage of this phenomenon to design PRSs with increasing phase over a selected frequency band. All the PRSs we investigated for wideband EBG resonator antennas were composed of a single dielectric slab, with arrays of periodic elements printed on both surfaces. Several element geometries, such as dipoles, slots, patches, rings, etc., have been investigated for this purpose.

GE et al.: USE OF SIMPLE THIN PARTIALLY REFLECTIVE SURFACES WITH POSITIVE REFLECTION PHASE GRADIENTS

From our investigations, it was found that, when PRS resonance frequency is empirically given by

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, the (1)

Fig. 3. (a) Proposed PRS structure; (b) a transparent unit cell; (c) characterization model of the PRS.

Fig. 4. Computed reflection magnitude and phase at Surface 1 for unit cells with two same or different dipoles.

The proposed PRS, shown in Fig. 3(a), has two periodic dipole arrays printed on the two sides of the dielectric slab. A transparent unit cell with a surface size of is shown in Fig. 3(b). The dimensions of two dipoles, printed on the two sides of the unit cell, are and , respectively. The characterization of the entire PRS can be reduced to that of a single unit cell as a result of the periodicity. Periodic boundary conditions can be applied to the four side walls of a unit cell [24], as shown schematically in Fig. 3(c). For simplicity only normal incidence was considered and the incident electric field is polarized along the direction of the dipole. Therefore, the boundaries 1 and 3 were set as perfect electric conductors (PEC) and 2 and 4 were set as perfect magnetic conductors (PMCs). The commercial software package, CST Microwave Studio, was used to characterize the PRS. Two PRSs, with the same unit cell dimensions: mm and mm, are studied. Both PRSs use Rogers RT/Duriod 5880 material that has a dielectric constant of 2.2 and a thickness of 1.6 mm. Two sets of dimensions for the two dipoles are applied to the investigation: mm and mm, mm, respectively. In both cases, mm. Computed reflections for Port 1 are plotted in Fig. 4. The phase is referenced at surface 1 where dipole 1 resides. It can be seen that there is a strong resonance at 11.95 GHz when the dipoles on the two sides are identical (i.e., ) and a less strong resonance at 12.35 GHz when they are different (i.e., ). The reflection phase has a sharp drop at the dipole resonance frequency when [24]. Interestingly, it exhibits an increase with frequency within a frequency band around 12.35 GHz when .

where a is a constant in the range 0.9 to 1, is the dipole length, is the PRS resonance frequency and is the substrate dielectric constant. The constant is determined by several factors, such as the thickness of the PRS, the size of the unit cell, the dipole width, etc. The effect of changing the parameters of the unit cell and the dipoles on the resonance frequency of the PRS has been studied. These parameters include the dimensions of the unit cell and the dipoles. A large number of simulations were carried out using CST Microwave Studio for two identical dipoles. For brevity here we will not show the complete results but only present our conclusions. These are: the resonance frequency of the PRS of the unit cell, the decreases with increasing dimensions of the dipoles and the thickness of the dielectric width layer. These effects cannot be directly obtained from formula or the (1) but the constant in (1) decreases when l, w, thickness of the unit cell increases. We have found that when the lengths of the two dipoles are different, the resonance becomes weaker. As a result, the minimum PRS reflection magnitude increases and the reflection phase exhibits a positive gradient at frequencies close to the PRS resonance frequency, as shown in Fig. 4. The frequency band within which the phase increases with frequency can be estimated from the resonance frequencies of the two dipoles (oband , respectively). tained from (1) by setting can be utiThis local inversion of the phase gradient lized to design PRSs for wideband EBG resonator antennas, as described in Section II. We also found that the closer the two dipole resonance frequencies are, the stronger the resonance and the larger the positive gradient of the phase curve. This characteristic affects the performance of the corresponding EBG resonator antenna, as will be demonstrated in Section IV. In addition, we found that the reflection phase at Surface 2 on which dipole 2 is located (found from Port 2 shown in Fig. 3(c)) still decreases with the frequency. In other words, the reflection phase increases only at the surface on which the dipole with a lower resonance frequency is located. For example, if and , Surface 1 resonates and , or at a lower frequency. Then the reflection phase at Surface 1 increases with frequency but the phase at Surface 2 decreases with frequency as usual. IV. DESIGN METHODOLOGY A methodology to design a simple thin PRS with a positive reflection phase gradient, , and the corresponding antenna, is as follows: 1) Layout two 2-D arrays on both sides of a dielectric slab of suitable material. The unit cells can have the same dimensions. 2) Select the elements for the 2-D arrays. Element geometries can be dipoles, slots, patches, rings, etc. They should have different dimensions. Elements on the lower surface should have a lower resonance frequency. Design the elements

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such that each resonates close to an edge of the operating band. If dipoles are chosen, resonance frequencies can be estimated using formula (1). 3) Analyze the reflection phase and magnitude of the surface and fine tune the elements if required. Ensure that the reflection magnitude is sufficiently large at all operating frequencies. 4) Simulate the antenna with the PRS and check the directivity versus frequency. An elementary Hertzian feed antenna may be used at this stage. 5) Design a feed antenna and test the performance of the complete antenna (gain, input matching, pattern etc.) V. WIDEBAND EBG RESONATOR ANTENNA DESIGN AND RESULTS Three PRSs were designed for potential use in wideband EBG resonator antennas by means of the methodology described in Section IV. It was expected that a weaker composite resonance of the PRS would simultaneously produce a positive phase gradient and a sufficiently large reflection minimum. The results in Fig. 4 for were promising but the minimum reflection was still too small. It was obvious that a strong resonance leads to an undesirable dip in the reflection magnitude, as indicated by the curve in Fig. 4. Indeed the strength of the resonance, positive gradient of the reflection phase and the minimum reflection magnitude are related. In this section, we explore this relationship using examples, with an aim to find the right combination to form a wideband yet low-profile EBG resonator antenna. A. Three PRSs and the Corresponding EBG Resonator Antennas The positive gradient of the increasing phase curve, shown in Fig. 4, will vary with the dimensions of the unit cell and the dipoles. Three PRSs have been designed and made on 1.6 mm Rogers RT/Duroid 5880 to illustrate this variation. Then they are used to construct three EBG resonator antennas to understand the connection between antenna performance and PRS reflection characteristics. The dimensions of the three unit cells and the three dipole-pairs that were designed are as follows: PRS1: mm mm, mm, mm, mm. PRS2: mm mm, mm, mm, mm. PRS3: mm mm, mm, mm, mm. The computed reflection coefficients of the three PRSs, obtained using the cavity model in Fig. 3(c) and CST Microwave Studio, are plotted in Fig. 5. Each of the PRSs exhibits positive reflection phase gradients over different parts of the Ku-band. The three frequency ranges are 11.25–12.46 GHz, 11.3–12.6 GHz, and 11.5–12.7 GHz corresponding to PRS1, PRS2, and PRS3, respectively. It is also noted that PRS1 has the strongest resonance and the largest positive gradient, whereas PRS3 has the weakest resonance and the smallest positive gradient. The reflection magnitudes of the three PRSs at 12 GHz are 4.8 dB, 3.8 dB and 2.8 dB, respectively. The conclusion from this is

Fig. 5. Computed reflection magnitude and phase for the three PRSs.

that the closer the resonance frequencies of the two dipoles, the stronger the resonance, the smaller the minimum reflection magnitude and the larger the phase gradient of the increasing phase. By comparison, the optimal reflection phase (of an ideal wideband antenna that would operate from 11.5 GHz to 12.5 GHz) is also plotted in Fig. 5, obtained from the Fabry–Perot resonance condition [6], [20] and assuming a cavity height of mm. Note that the phase of PRS2 is almost ideal when it has a positive gradient. The three PRSs have been fabricated and used to construct three EBG resonator antennas. In these antenna prototypes, all PRSs have overall dimensions of 110 110 mm (about at 12 GHz). The two dipole arrays printed on the two sides of PRS1 includes 12 13 dipoles and those on PRS2 and PRS3 have 12 18 dipoles. The unit cell dimensions are 9 8 mm for PRS1 and 9 6 mm for PRS2 and PRS3. The dipole dimensions are (8.2 1 mm , 7.2 2 mm ), (8.2 1 mm , 7 2 mm ) and (8.2 1 mm , 6.5 2 mm ) for the three cases. The thickness and the dielectric constant of the Rogers RT/Duroid 5880 slabs are 1.6 mm and 2.2, respectively. Each antenna has a large 295 295 mm ground plane. The cavity heights are 12.8 mm for PRS1 and 13.2 mm for both PRS2 and PRS3. B. Feed Antenna and Input Matching A well-designed feed antenna is crucial to the performance of wideband EBG resonator antennas. We designed a monopole antenna, composed of a copper strip printed on the upper surface of a small Rogers RT/Duroid 5880 substrate and fed by a probe, as shown in Fig. 1, to operate over a wide frequency band inside the resonance cavity. This feed antenna is placed 2.4 mm above the ground and has dimensions of 36 36 0.8 mm (the substrate in Fig. 1 has been deliberately shrunk for better visualization). The dimensions of the copper strip are 10 4 mm . The reflection coefficient of the three EBG resonator antenna prototypes have been measured with a network analyzer and their magnitudes are plotted in Fig. 6. The 10-dB return-loss bandwidths are 11.28–15 GHz for the Antenna 1 with PRS1, 11.45–15 GHz for Antenna 2 with PRS2 and 11.7–15 GHz for Antenna 3 with PRS3. The results beyond 15 GHz are not shown because the gain is expected to be low at such frequencies. The reflection coefficient of the feed antenna without the PRS top (i.e., without cavity) is also plotted in Fig. 6 to illustrate the

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Fig. 6. Measured input reflection coefficient magnitude of the three EBG resonator antenna prototypes and the feeding antenna.

loading effect of the cavity on the feed antenna. While interacting with the strong cavity field, the feed antenna successfully operates over a wide frequency band that encompasses the operating bands of all three PRSs. C. Radiation Measurements The three antenna prototypes were measured in a spherical near-field test range to obtain radiation patterns and the gain. The measured antenna directivities and gains for the three antennas are plotted in Fig. 7(a)–(c), respectively. It is seen from Fig. 7(a) that Antenna 1 with PRS1 offers two possible operating frequency bands over the frequency range of 11–13 GHz. The measured radiation patterns of Antenna 2 are plotted in Fig. 8 at 11.1 GHz, 11.6 GHz, 12 GHz, and 12.4 GHz. It can be seen that consistent radiation patterns are achieved at these four frequencies. However the sidelobe levels may be unsatisfactory for some applications when the frequency exceeds 12.6 GHz, as indicated by the radiation pattern at 12.7 GHz as shown in Fig. 8(d). Hence the effective 3-dB antenna bandwidth can be considered as 11.1–12.6 GHz, i.e., 12.6%. This antenna demonstrates the advantages and the limitations of the design method presented here. The radiation patterns of Antenna 3 have the similar properties to those of Antenna 2, whilst a higher peak gain and a narrower effective bandwidth are obtained, which are 18.4 dB and 6.4%, respectively. VI. DISCUSSION OF RESULTS In this section, we discuss the results given in the previous section. First, although the reflection phase of the PRS1 increases with frequency from 11.25 to 12.46 GHz with the largest positive gradient, its stronger dipole resonance and resulting smallest reflection magnitude minimum of 5 dB around 12 GHz (see Fig. 5) lead to a lower directivity and gain around 12 GHz. At the same time, relatively larger directivities and gains are achieved at frequencies where reflection is stronger, and consequently a dual-band antenna instead of a wideband one is obtained. In general, an antenna produced from such a PRS with a positive reflection gradient and a relatively weak minimum reflection magnitude may operate in two frequency

Fig. 7. Measured gains and directivities of three trial EBG resonator antennas with: (a) PRS1; (b) PRS2; and (c) PRS3.

bands. This character can be exploited to design dual-band EBG resonator antennas operating in close frequency bands. On the other hand, PRS2 has a smaller, close-to-ideal positive reflection phase gradient from 11.3 GHz to 12.6 GHz and a larger reflection magnitude at 12 GHz, compared to PRS1. As expected, Antenna 2 with PRS2 achieves a wide 3-dB gain bandwidth, from about 11.1 GHz to 13 GHz, i.e., a bandwidth of 15.7%. From Fig. 7(b), we can see that the directivity bandwidth obtained is even greater. The gain drops sharply at frequencies below 11.5 GHz. This is because the antenna input is

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Fig. 8. Measured radiation patterns of the antenna with PRS2 at: (a) 11.1 GHz; (b) 11.6 GHz; (c) 12 GHz; (d) 12.4 GHz; 12.7 GHz.

mismatched at such frequencies, as shown in Fig. 6. The peak gain is 16.2 dBi at 11.5 GHz. Although a wide bandwidth is achieved in this case, the side-lobe levels within the bandwidth are not consistent when the operating frequency is increased beyond a limit. The increase in the sidelobe levels with frequency has been observed with many wideband EBG resonator antennas [16]. This is because at such frequencies the phase difference between the incident wave and the first reflective wave is more than 180 . The more the frequency changes, the higher is the sidelobe level. This is consistent with beam splitting occurring in EBG resonator antennas at higher frequencies. PRS3 has the smallest positive phase gradient and the largest reflection magnitude around 12 GHz. The Antenna 3 with PRS3 has a 7.4% 3-dB gain bandwidth, from 11.85 GHz to 12.72 GHz. The measured peak gain is 18.4 dBi at 12.3 GHz. Again the sidelobe levels degrade at frequencies above 12.6 GHz and therefore the effective 3-dB bandwidth is about 6.4%. Compared with Antenna 2, this antenna exhibits a narrower bandwidth but has a higher gain over this bandwidth. Compared with a previously designed EBG resonator antenna [9], which has an intentionally high peak gain of 22.15 dBi and a narrow bandwidth of 2.2%, the present antennas with surfaces PRS2 and PRS3 exhibit a similar gain-bandwidth product. It is a common feature in EBG resonator antennas that greater bandwidth is accompanied by a relatively lower peak gain. The example antennas presented here further demonstrates that one needs to compromise between the gain and the bandwidth when designing an antenna using the present method. The cavity heights of the three antennas are 12.8 mm, 13.2 mm, and 13.2 mm, respectively. A small change in the cavity height results in a minor change of the antenna performance. The cavity heights of 12.4 mm and 13.6 mm were also tested in the measurements. When the heights are smaller than the desired values, higher peak gains and narrower bandwidths are obtained, and vice versa. There is about a 1 to 1.5 dB difference between the measured gains and directivities of these antennas within the matched radiation bandwidth. This is due to antenna loss, primarily in the dielectrics. The Wheeler Cap Method was used to determine the

loss of Antenna 3 at 12.3 GHz, and it was found to be about 1 dB, which confirms the difference. A good feed antenna is crucial to the design. The monopole feed antenna used here is well-matched, has a return loss greater than 10 dB over the entire radiation bandwidths of the three antennas when placed inside the strong cavity field. This helps to reduce the gap between the gain and directivity to the low levels mentioned above, in the operating bands. When designing the two resonant elements on the surfaces of the PRS, identical unit cell sizes can be maintained on both surfaces for ease of modeling and design. In addition to the PRS with two dipole arrays, a dielectric layer with two cross-dipole arrays, two slot arrays [21], two cross-slot arrays, two square patch arrays, etc., can also provide similar reflection magnitude and phase properties. The design principle is similar, that the reflection phase at the lower surface, which has elements whose dimensions lead to the lower resonance frequency, should have a positive, close-to-optimal phase gradient and a sufficiently large reflection magnitude. The reflection phase gradient on the other surface can still be negative. This approach is now being applied to several other antennas to obtain a wide bandwidth. VII. CONCLUSION A simple PRS with printed patterns on both sides of a dielectric slab can provide a positive phase gradient over a band of frequencies, to assist with the design of a low profile, wideband EBG resonator antenna. By appropriately designing the resonant elements of the PRS, the positive phase gradient can be optimized to obtain a useful antenna bandwidth (greater than 10%). In the low-profile antenna prototypes presented here, the total thickness of the PRS is only 1.6 mm, or 0.064 (at midband 12 GHz), and hence the antenna height is almost equal to the cavity height. By comparison, the PRS thickness of earlier wideband antenna designs based on spaced double-slab PRSs are much greater ( .) The minimum value of the reflection coefficient magnitude is another crucial factor in the PRS design because a too low reflection, say below dB, weakens the spreading effect of the PRS, creating a dip in the gain versus frequency curve. This happens when the resonance frequencies of the two elements are too close and hence the overall resonance of the composite PRS is relatively strong. On the other hand, too different elements produce a weak composite resonance and a weak positive gradient; the result then is higher gain but narrower gain bandwidth. When the two elements are appropriately designed to keep the minimum reflection above dB and the positive phase gradient close to the optimal value, good gain can be obtained over a wide bandwidth. One such example presented here has a peak gain of 16.2 dBi and an effective bandwidth of 12.6%. When the elements widely differed in geometry, the peak gain rose to 18.4 dBi but the effective bandwidth reduced to 6.4%. When the geometry was closer, the result was a dual-band antenna. The method described in this paper has been used to control the strength of the PRS composite resonance and hence the reflection phase gradient and the minimum reflection magnitude. By following the proposed method, a positive reflection phase gradient was successfully realized using a single slab with two

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dipole arrays printed on its surfaces. Three such PRSs were designed and subsequently utilized to construct and compare EBG resonator antennas. Measurements of these antennas demonstrated the feasibility of the design principle presented in the paper and the compromises involved in antenna design. They further confirm the advantage offered by the positive phase gradient when attempting to increase the bandwidth of EBG resonator antennas.

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[21] Y. Ge, K. P. Esselle, and T. S. Bird, “Partially reflective surfaces for wide-band EBG resonator antennas,” in Proc. Metamaterials, London, U.K., Aug./Sep. 30, 2009. [22] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite media with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, pp. 4184–4187, 2000. [23] Y. Ge, K. P. Esselle, and T. S. Bird, “Designing a partially reflective surface for dual-band EBG resonator antennas,” in Proc. APS, 2010. [24] Y. Zhang, J. Hagen, M. Younis, C. Fischer, and W. Wiesbeck, “Planar artificial magnetic conductors and patch antennas,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2704–2712, Oct. 2003.

REFERENCES [1] A. R. Weily, K. P. Esselle, B. C. Sanders, and T. S. Bird, “High-gain 1D EBG resonator antenna,” Microw. Opt. Technol. Lett., vol. 47, no. 2, pp. 107–114, Oct. 2005. [2] M. Thevenot, C. Cheype, A. Reineix, and B. Jecko, “Directive photonic-bandgap antennas,” IEEE Trans. Microw. Theory Tech., vol. MTT-47, no. 11, pp. 2115–2122, Nov. 1999. [3] A. R. Weily, L. Horvath, K. P. Esselle, B. C. Sanders, and T. S. Bird, “A planar resonator antenna based on a woodpile EBG material,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 216–223, Jan. 2005. [4] Y. Lee, J. Yeo, R. Mittra, and W. Park, “Application of electromagnetic bandgap (EBG) superstrates with controlable defects for a class of patch antennas with spatial angular filters,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 224–235, Jan. 2005. [5] N. Guerin, S. Enoch, G. Tayeb, P. Sabouroux, P. Vincent, and H. Legay, “A metallic Fabry Perot directive antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 220–224, Jan. 2006. [6] A. P. Feresidis and J. C. Vardaxoglou, “High gain planar antenna using optimized partially reflective surfaces,” in IEE Proc. Microw., Antennas Propag., Dec. 2001, vol. 148, no. 6, pp. 345–350. [7] A. P. Feresidis, G. Goussetis, S. Wang, and J. C. Vardaxoglou, “Artificial magnetic conductor surfaces and their application to low-profile high-gain planar antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 209–215, Jan. 2005. [8] Y. Ge and K. P. Esselle, “Low-profile resonant cavity antenna based on an in-phase metamaterial surface,” Microw. Opt. Technol. Lett., vol. 51, no. 3, pp. 731–733, Mar. 2009. [9] Y. Ge, K. P. Esselle, and Y. Hao, “Design of low-profile high-gain EBG resonator antennas using a genetic algorithm,” IEEE Antennas Wireless Propag. Lett., no. 6, pp. 480–483, 2007. [10] Y. Ge and K. P. Esselle, “A resonant cavity antenna based on an optimised thin superstrate,” Microw. Opt. Technol. Lett., vol. 50, no. 12, pp. 3057–3059, Dec. 2008. [11] M. Diblanc, E. Rodes, E. Arnaud, M. Thevenot, T. Monediere, and B. Jecko, “Circularly polarized metallic EBG antenna,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 10, pp. 1–3, Oct. 2005. [12] D. R. Jackson and N. Alexopoulos, “Gain enhancement methods for printed circuits antennas,” IEEE Trans. Antennas Propag., vol. 33, no. 9, pp. 976–987, Sep. 1985. [13] A. R. Weily, T. S. Bird, and Y. J. Guo, “A reconfigurable high-gain partially reflecting surface antenna,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3382–3390, Nov. 2008. [14] A. Weily, K. P. Esselle, T. S. Bird, and B. C. Sanders, “Dual resonator 1-D EBG antenna with slot array feed for improved radiation bandwidth,” in IET Proc. Microw., Antennas Propag., Feb. 2007, vol. 1, no. 1, pp. 198–203. [15] P. Feresidis and J. C. Vardaxoglou, “A broadband high-gain resonant cavity antenna with single feed,” in Proc. EuCAP, Nice, France, 2006. [16] L. Moustafa and B. Jecko, “Broadband high gain compact resonator antennas using combined FSS,” in IEEE Int. Antennas Propag. Symp. Dig., San Diego, CA, Jul. 5–12, 2008, pp. 1301–1304. [17] L. Moustafa and B. Jecko, “EBG structure with wide defect band for broadband cavity antenna applications,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 693–696, Nov. 2008. [18] C. Mateo-Segura, A. P. Feresidis, and G. Goussetis, “Analysis of broadband highly-directive Fabry-Perot cavity leaky-wave antennas with two periodic layers,” in IEEE Int. Antennas Propag. Symp. Dig., Toronto, ON, Canada, July 11–17, 2010. [19] C. Mateo-Segura, A. P. Feresidis, and G. Goussetis, “Highly directive 2-D leaky wave antennas based on double-layer meta-surfaces,” in Proc. EuCAP2010, Barcelona, Spain, 2010. [20] Y. Ge, K. P. Esselle, and T. S. Bird, “Designing a partially reflective surface with increasing reflection phase for wideband EBG resonator antennas,” in IEEE Int. Antennas Propag. Symp. Dig., North Charleston, SC, Jun. 1–5, 2009.

Yuehe Ge (S’99-M’03) received the Ph.D. degree in electronic engineering from Macquarie University, Sydney, Australia, in 2003. Currently, he is a Professor of the College of Information Science and Engineering, Huaqiao University, Xiamen, China. Previously, he was a Research Fellow in the Department of Electronic Engineering, Macquarie University. Before joining Macquarie University, he was an Antenna Engineer at Nanjing Marine Radar Institute, Nanjing, China. His research interests are in the areas of antenna theory and designs for radar and communication applications, computational electromagnetics and optimization methods, metamaterials, and their applications. He has authored and coauthored over 90 journal and conference publications and two book chapters. Dr. Ge received several prestigious prizes from China State Shipbuilding Corporation and China Ship Research and Development Academy, due to his contributions to China State research projects. He received 2000 IEEE MTT-S Graduate Fellowship Awards and 2002 Max Symons Memorial Prize of IEEE NSW Section, Australia, for the best student paper. He is the cowinner of 2004 Macquarie University Innovation Awards-Invention Disclosure Award. He has served as a technical reviewer for over ten international journals and conferences.

Karu P. Esselle (M’92–SM’96) received the B.Sc. degree in electronic and telecommunication engineering (with first class honors) from the University of Moratuwa, Moratuwa , Sri Lanka, and the M.A.Sc. and Ph.D. degrees in electrical engineering from the University of Ottawa, Ottawa, ON, Canada. As a Professor in electronic engineering, Macquarie University, Sydney, he currently heads the department. He is the Immediate Past Associate Dean—Higher Degree Research and the Founding Director of Postgraduate Research Committee in the Division of Information and Communication Sciences. He held these positions from 2003 to 2008 and was also a member of the Division Executive. He has authored over 300 scientific publications, including six invited book chapters and over 16 invited conference presentations. Since 2002, he was involved with research grants and contracts worth about five million dollars. His Ph.D. students have received scholarships worth over 2 million dollars in the same period and his research team members attracted further grants worth about a million dollars. His research interests include periodic and electromagnetic band gap structures including frequency-selective surfaces, metamaterials, broadband and multi-band antennas, biomedical devices, on-body and through-body wireless communication, millimeter-wave and MMIC devices, antenna and EBG applications in mobile and wireless communication systems, ultra-wideband systems, theoretical methods, and lens and focal-plane-array antennas for radio astronomy. His research activities are posted on the web at http://www.engineering.mq.edu.au/research/groups/celane/. He served in all Macquarie University HDR-related committees at the highest level. He is the Director of Electromagnetic and Antenna Engineering, and the Deputy Director of the Research Center on Microwave and Wireless Applications, which was recently expanded after recognized as a Concentration of Research Excellence. He has been invited to serve as an international expert/research grant assessor by several overseas nationwide research funding bodies from the Netherlands, Canada, Finland, Hong-Kong, and Chile. He has been invited by Vice-Chancellors of other universities to assess applications for promotion to full professor level. He has been invited to assess grant applications submitted to Australia’s most prestigious schemes such as Australian Federation Fellowships and Australian Laureate Fellowships. His industry experience includes full-time employment as Design Expert by the Hewlett Packard

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Laboratory, and several consultancies for local and international companies, including Cisco Systems (USA), Cochlear, Optus Networks, Locata (USA)/QX Corporation, ResMed, FundEd, and Katherine-Werke (Germany) through Peter-Maxwell Solicitors. He was an Assistant Lecturer at the University of Moratuwa, a Canadian Government laboratory Visiting Postdoctoral Fellow at Health Canada, a Visiting Professor of the University of Victoria and a Visiting Scientist of the CSIRO ICT Center. He is an Editor of the International Journal of Antennas and Propagation. Prof. Esselle’s recent awards include the 2009 Vice Chancellor’s Award for Excellence in Higher Degree Research Supervision (the first such award ever offered in Macquarie University) and 2004 (Inaugural) Innovation Award for best invention disclose. The CELANE, which he founded, has provided a stimulating research environment for a strong team of researchers including six postdoctoral fellows. His mentees have been awarded six extremely competitive postdoctoral fellowships. Nine international experts who examined the theses of his recent five Ph.D. graduates ranked them in the top 5% or 10% in the world. He has served in technical program committees or international committees for many international conferences. He cochairs the Technical Program Committee of APMC 2011; he was the Publicity Chair of the APMC 2000. He is the Chair of the IEEE New South Wales (NSW) MTT/AP Joint Chapter, Foundation Editor of MQEC, the past Chair of the Educational Committee of the IEEE NSW, and a member of the IEEE NSW Committee.

Trevor S. Bird (S’71–M’76–SM’85–F’97) received B.App.Sc., M.App.Sc., and Ph.D. degrees from the University of Melbourne, Melbourne, Australia. From 1976 to 1978, he was a Postdoctoral Research Fellow at Queen Mary College, University of London, London, U.K., followed by five years as a Lecturer in the Department of Electrical Engineering at James Cook University of North Queensland. During 1982 and 1983, he was a consultant at Plessey Radar, U.K., and in December 1983 he joined CSIRO in Sydney, Australia. He held several positions with CSIRO, including Chief Scientist, ICT Center. He is currently a

CSIRO Fellow and Principal of Antengenuity, a specialist consulting firm, an Adjunct Professor at Macquarie University and a Guest Professor of Shanghai Jiao Tong University. Dr. Bird is a Fellow of the Australian Academy of Technological and Engineering Sciences, the Institution of Electrical Technology (IET), and an Honorary Fellow of the Institution of Engineers, Australia, and also Queens College, University of Melbourne. He received the John Madsen Medal of the Institution of Engineers, Australia, in 1988, 1992, 1995, and 1996 for the best paper published annually in the Journal of Electrical and Electronic Engineering, Australia. In 2001, he was corecipient of the H. A. Wheeler Applications Prize Paper Award of the IEEE Antennas and Propagation Society. He was awarded a CSIRO Medal in 1990 for the development of an Optus-B satellite spot beam antenna and again in 1998 for the multibeam antenna feed system for the Parkes radio telescope. He received an IEEE Third Millennium Medal in 2000 for outstanding contributions to the IEEE New South Wales Section. He received project awards from the Society of Satellite Professionals International (New York) in 2004, the Engineers Australia in 2001, and the Communications Research Laboratory, Japan, in 2000. In 2003, he was awarded a Centenary Medal for service to Australian society in telecommunications and was also named Professional Engineer of the Year by the Sydney Division of Engineers Australia. Since 2006, his biography has been listed in Who’s Who in Australia. He was a Distinguished Lecturer for the IEEE Antennas and Propagation Society from 1997 to 1999, Chair of the New South Wales joint AP/MTT Chapter from 1995 to 1998, and again in 2003, Chairman of the 2000 Asia Pacific Microwave Conference, Member of the New South Wales Section Committee from 1995 to 2005 and was Vice-chair and Chair of the Section in 1999 to 2000 and 2001 to 2002, respectively, Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION from 2001 to 2004, a member of the Administrative Committee of the IEEE Antennas and Propagation Society from 2003 to 2005, a member of the College of Experts of the Australian Research Council (ARC) from 2006 to 2007 and Editor-in-Chief of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION from 2004 to 2010. He has been a member of the technical committee of numerous conferences including JINA, ICAP, AP2000, IRMMW-THz and the URSI Electromagnetic Theory Symposium. Currently, he is a member of the Editorial Boards of the IET Microwaves Antennas and Propagation and the Journal of Infrared, Millimeter and Terahertz Waves, and also Chair of the IEEE Antennas and Propagation Society’s Publication Committee.

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Omnidirectional Linearly and Circularly Polarized Rectangular Dielectric Resonator Antennas Yong Mei Pan, Member, IEEE, Kwok Wa Leung, Fellow, IEEE, and Kai Lu, Student Member, IEEE

Abstract—The rectangular dielectric resonator antenna (DRA) centrally fed by a probe is investigated. Its operating mode is analogous to the 011 mode of a cylindrical DRA. The DRA radiates like an electric monopole, generating omnidirectional linearly polarized (LP) fields. Based on this LP design, a novel omnidirectional circularly polarized (CP) DRA is studied for the first time. Slots are introduced to the sidewalls of the DRA, exciting a degenerate mode for the generation of CP fields. To demonstrate the idea, an omnidirectional CP DRA was designed for WLAN (2.4–2.48 GHz) applications. The reflection coefficient, axial ratio (AR), radiation pattern, and antenna gain are studied, and reasonable agreement between the measured and simulated results is observed. Index Terms—Circular polarization (CP), dielectric resonator antenna (DRA), omnidirectional antenna, slot, TM mode.

I. INTRODUCTION

I

N THE past two decades, the dielectric resonator antenna (DRA) has received tremendous attention due to a number of attractive features such as its small size, light weight, low loss, wide bandwidth, and ease of excitation [1]–[4]. The shape of the DRA can be hemispherical, cylindrical, or rectangular. Among these shapes, the rectangular one has the largest number of design parameters (width, length, and height), facilitating the antenna design. Okaya and Barash [5] divided the modes of the and rectangular DRA into two categories, namely the modes. The fundamental mode has been investigated by different researchers extensively [6]–[9]. It produces a radiation pattern which is similar to that of a magnetic dipole, with the strongest radiation found in the boresight (broadside) direction. It is known that the fundamental omnidirectional modes of the [10] and hemispherical and cylindrical DRAs are the [11], [12] modes, respectively, with their mode indices referring to field variations along the spherical/cylindrical coordinate variables. Both of the modes are excited by a centrally fed probe and radiate like an electric monopole. However, little or no information of the analogous mode was found for the rectangular DRA. In this paper, a rectangular DRA with a square mode for the first time. It cross section is operated in the Manuscript received November 10, 2010; revised March 22, 2011; accepted July 20, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported by a GRF grant from the Research Grants Council of Hong Kong Special Administrative Region, China (Project 116911). The authors are with the State Key Laboratory of Millimeter Waves and Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173122

was found that its radiation pattern is omnidirectional in the horizontal plane as similar to that of an electric monopole. As compared with the linearly polarized (LP) system, the circularly polarized (CP) system allows a more flexible orientation between the transmitter and receiver. Also, it can suppress the multipath problem due to reflections from the building wall and ground surface [13]. As a result, the CP antenna has been used in modern wireless systems extensively. Various CP DRAs were investigated, such as the cross-slot coupled circular disk DRA [14], strip-loaded hemispherical DRA [15], and aperture-fed rectangular stair-shaped DRA [16]. All of these CP DRAs have broadside radiation patterns. However, the omnidirectional CP radiation pattern is sometimes needed because it helps stabilize the signal transmission [17]. Also, it permits the maximum freedom of choice of antenna location and, thus, can cover a larger service area [17]. These features have attracted certain research efforts on investigating the omnidirectional CP antennas. Examples include the vertical sleeve dipole antenna combined with three pairs of tilted parasitic elements [13], the normal mode helical antenna [18], the slotted ring antennas [19], [20] and the conical patch antennas [21], [22]. However, either a balun [13] or a complex radiating element [19]–[21] is needed for these CP antennas. Designing a simple omnidirectional CP antenna is still a challenging problem today. In general, an LP wave can be changed into an elliptically or circularly polarized wave by using a wave polarizer. It should therefore be possible to obtain an omnidirectional CP antenna by adding a wave polarizer to an omnidirectional LP antenna. However, adding an external polarizer will inevitably increase the size and complexity of the antenna. An interesting CP cylindrical DRA has been proposed in [23], which has slots fabricated on its top for exciting a broadside CP radiation mode. In this paper, similar inclined slots are fabricated on the sidewalls of the omnidirectional LP DRA to obtain a novel compact omnidirectional CP DRA. To demonstrate the idea, an omnidirectional CP DRA operating at around 2.4 GHz was designed for WLAN applications. For each of the LP and CP rectangular DRAs, the reflection coefficient, axial ratio (AR) (CP DRA only), radiation pattern, and antenna gain were simulated using Ansoft HFSS. Measurements were done and reasonable agreement between the measured and simulated results was obtained. It is worth mentioning that since the basic characteristics of the DRA are independent of its shape, in principle the proposed idea can also be applied to the cylindrical and hemispherical DRAs, although fabricating a slot on a curved surface is much more difficult than for a flat surface. The organization of this paper is as follows. A study of the centrally probe-fed omnidirectional LP DRA is given

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=

Fig. 2. Simulated reflection coefficient of the DRA for l 15, 16, and 17 mm: a b ,h : mm, and r : mm, " mm. mm, g

= = 39

= 15 = 33

= 12 7

= 0 63

Fig. 1. Configuration of the probe-fed rectangular DRA. (a) Front view. (b) Bottom view.

in Section II. A design example of the proposed CP DRA for WLAN applications is demonstrated and discussed in Section III. Finally, a conclusion is drawn in Section IV. II. LP RECTANGULAR DRA OPERATING IN

MODE

Fig. 1 shows the configuration of the rectangular DRA with a length of , a width of , and a height of . The DRA is centrally fed by a coaxial probe of length and radius . It is extended from the inner conductor of a SMA connector, which has a square flange acting as the (small) ground plane of the antenna. mm. No adThe side length of the flange is given by ditional ground plane is added so that the maximum radiation can be enabled around the direction. Otherwise, more radiation toward the upper region will result, lifting up the beam from the horizontal plane [3]. The vertical current flowing along the probe will generate a circular -field , with . Therefore, the DRA is excited in its TM mode. For symmetry, the two side lengths of the rectangular DRA are set equal (i.e., ). A centrally probe-fed rectangular DRA was designed at 2.4 GHz using Ansoft HFSS. The DRA has a dielectric and dimensions of mm, constant of mm. Fig. 2 shows the simulated reflection coefficient 15, as a function of frequency for different probe lengths of 16, and 17 mm. With reference to the figure, the DRA resonates (minimum ) at around 2.43 GHz for all of the probe lengths, showing that the resonance is caused by the dielectric resonator, not by the coupling probe. Fig. 3 shows the simulated resonant - and -fields inside the DRA. As can be observed from the figure, the -field is circular around the -axis whereas the -field is vertical and strongest along the -axis. The field pattern is very similar to that of the well-known

Fig. 3. Simulated resonant H -field and E -filed inside the LP DRA. (a) H -field. (b) E -filed.

mode of the cylindrical DRA [24], where the first, second, and third indices refer to the azimuthal, radial, and axial directions, respectively. Fig. 4 compares the return loss of our rectangular DRA with that of a cylindrical DRA with the same cross-sectional area and dielectric constant. Both of them are axially fed by the same probe. With reference to the figure, the resonance frequencies of the two DRAs are close to each other. Their radiation patterns were simulated and were found to be almost the same. As a result, our mode is named as the mode. Since our rectangular DRA has a square quasicross section, it can be regarded as a perturbed cylindrical DRA. Other dimensions and dielectric constants were used for the DRA and similar phenomena were observed. Furthermore, no other resonant modes could be found below this mode. Thus, it can be concluded that the excited mode is the fundamental

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Fig. 4. Simulated reflection coefficients of the rectangular DRA and cylindrical DRA with same cross-sectional area and dielectric constant.

Fig. 6. Measured and simulated radiation patterns of the rectangular DRA. The parameters are the same as in Fig. 5.

Fig. 5. Measured and simulated reflection coefficients of the rectangular DRA mm. Other parameters are the same as in Fig. 2. with l

Fig. 7. Measured and simulated antenna gains of the rectangular DRA. The parameters are the same as in Fig. 5.

TM mode of the antenna. It should be emphasized that since is strongest at as shown in Fig. 3(b), the the probe should be placed at the center of the DRA to obtain the strongest excitation of the mode. A prototype of the rectangular DRA was fabricated and tested. The measured and simulated reflection coefficients of the prototype are shown in Fig. 5 and reasonable agreement between them is observed. The discrepancy is caused by experimental tolerances and imperfections including the inevitable airgap between the probe and the hole. With reference to the figure, the measured and simulated 10-dB impedance banddB are given by 10.1% (2.34–2.59 GHz) widths and 10.4% (2.28–2.53 GHz), respectively. The measured resonance frequency is 2.46 GHz, which agrees very well with the simulated (2.43 GHz) value. Fig. 6 shows the measured and simulated field patterns of the TM mode. It can be seen from the radiation patterns that the DRA is a good omnidirectional plane, the co-polarized field is stronger than antenna. In the dB in the the cross-polarized counterpart by direction, whereas the former is also stronger than the latter by plane. The -plane field pattern more than 20 dB in the was also simulated and measured. It was found that the results are similar to that of the plane, which is expected because of the symmetry of the structure. Fig. 7 shows the measured and simulated antenna gains of the omnidirectional DRA. With reference to the figure, the antenna gain varies between 0.49 dBi

and 2.04 dBi across the impedance passband (2.34–2.59 GHz). The measured maximum antenna gain is 2.04 dBi, which is close to that of the half-wave dipole (2.15 dBi).

= 15

III. SLOTTED OMNIDIRECTIONAL CP DRA It is shown in [23] that loading an LP DRA with strategically oriented slots can excite a degenerate mode and, thus, generate CP fields. This idea is used to design our omnidirectional DRA. Similar inclined slots are introduced to the LP rectangular DRA to obtain a novel compact omnidirectional CP DRA. Since the radiation field excited by the probe is predominantly vertically polarized, the slots should be fabricated on a sidewall of the DRA, as shown in Fig. 8. To obtain an omnidirectional antenna, the slots are also fabricated on the three other sidewalls. It was found that it is sufficient to fabricate only a single slot on each sidewall to obtain a good omnidirectional CP antenna. Fig. 9 shows the proposed design for the first time. To demonstrate the idea, an omnidirectional CP DRA for the 2.4-GHz WLAN system was designed and fabricated. Fig. 10 shows two photographs of the prototype. The CP design is based on the LP DRA studied in Section II, and the parameters were tuned to optimize the CP performance, with mm, mm, and mm. Other design parameters are mm and a the same as before. A slot with a width of mm was fabricated on each sidewall of the depth of DRA. The inclination angle of the slots can be determined

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Fig. 8. Dielectric block with inclined slots on its sidewall.

Fig. 11. Measured and simulated reflection coefficients of the omnidirectional CP DRA:, a : mm, h : mm, w : mm, d : mm, l : mm, r : mm, and g : mm.

= 12 4

= 39 4 = 0 63

= 33 4 = 94 = 12 7

= 14 4

Fig. 12. Measured and simulated ARs of the omnidirectional CP DRA in the x direction. The parameters are the same as in Fig. 11.

+

Fig. 9. Configuration of the proposed omnidirectional CP DRA. (a) Perspective view. (b) Front view.

Fig. 10. Prototype of the proposed omnidirectional CP DRA. (a) Photograph showing the top face and sidewalls. (b) Photograph showing the bottom face of the DRA. The launcher is inserted into the hole drilled at the center of the bottom face.

from the DRA and slot dimensions as , which was found to be 48 in our prototype. It should be mentioned that the optimal angle that maximizes the bandwidth is dependent on the dielectric constant of the DRA. Again, the

flange of the SMA connector is used as the (small) ground plane and no external ground plane is added. This CP DRA will generate left-hand CP (LHCP) fields, and right-hand CP (RHCP) fields can be obtained by aligning the slot with the other diagonal of the sidewall. Fig. 11 shows the measured and simulated reflection coefficients of the proposed CP DRA. With reference to the figure, reasonable agreement between the measured and simulated results is observed. The measured and simulated 10-dB impedance bandwidths are 24.4% (2.30–2.94 GHz) and 20.3% (2.34–2.87 GHz), respectively. Fig. 12 shows the measured and simulated ARs of the CP DRA in the direction . Almost the same results were obtained . From the figure, it can for other values of with be found that the measured 3-dB AR bandwidth is given by 7.3% (2.39–2.57 GHz), which agrees reasonably well with the simulated value of 8.2% (2.34–2.54 GHz). The bandwidth is more than enough for the 2.4-GHz WLAN band. It is noted that the entire measured AR passband falls within the impedance passband and, thus, the entire AR passband is usable. This result is very desirable. Fig. 13 shows the measured and simulated ARs as a function of the horizontal angle in the plane. With reference to the figure, the measured result is not as flat as the simulated one due to experimental tolerances and

PAN et al.: OMNIDIRECTIONAL LINEARLY AND CP RECTANGULAR DRAs

Fig. 13. Measured and simulated ARs as a function of the horizontal angle  in the x y plane. The parameters are the same as in Fig. 11 Measured (2.46 GHz) ——— Simulated (2.44 GHz).

0

111111111

755

Fig. 15. Measured and simulated radiation patterns of the omnidirectional CP DRA. The parameters are the same as in Fig. 11.

Fig. 14. Measured and simulated antenna gains of the omnidirectional CP DRA. The parameters are the same as in Fig. 11.

imperfections. Nevertheless, both the measured and simulated results are below 1.5 dB across the entire range of , showing that it is a good omnidirectional CP antenna. Fig. 14 shows the measured and simulated antenna gains, which are similar to those of the LP case. With reference to the figure, good agreement between the measured and simulated results is observed. The measured antenna gain varies between 0.91 dBic and 1.60 dBic across the AR passband (2.39–2.57 GHz). Fig. 15 shows the radiation patterns of the xz and xy planes and very good omnidirectional performance can be observed. Similar results were obtained for the yz-plane pattern as expected and are therefore not included here for brevity. It can be seen from the figure that the LHCP fields are stronger than the crosspolarized (RHCP) fields by about 20 dB, only except for a small region around the axis. The radiation patterns were studied at other frequencies and found to be very stable across the entire passband. It should be mentioned that since the SMA flange (ground plane) is very small, it could be suspicious that there may be currents excited on the outer conductor of the feed structure, resulting in unwanted radiation. In order to check whether it is the case, three cables of different lengths ( 16 mm, 24 mm,

Fig. 16. Measured ARs and gains of the omnidirectional CP DRA with different cable lengths: lc 16 mm, 24 mm, and 33 mm. (a) AR. (b) Gain.

=

and 33 mm) were used for the proposed antenna. The measured ARs, antenna gains for these three cases are shown in Fig. 16. With reference to the figure, similar results are found for all of the three cases. The radiation patterns of the three cases were also examined. It was observed that they are very similar to one another, but the results are not included here for brevity. It is shown from the study that it should be needless to minimize the outer current or add any choke structure to the proposed CP antenna.

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Fig. 17. Simulated reflection coefficient and AR of the omnidirectional CP DRA as a function of frequency for different DRA widths of a 37.4, 39.4, and 41.4 mm. Other parameters are the same as in Fig. 11. (a) Reflection coefficient. (b) AR.

Fig. 18. Simulated reflection coefficient and AR of the omnidirectional CP DRA as a function of frequency for different slot depths of d 10.4, 12.4, and 14.4 mm. Other parameters are the same as in Fig. 11. (a) Reflection coefficient. (b) AR.

A parametric study of the proposed DRA was carried out using HFSS. The effect of the DRA size is discussed first. Fig. 17 shows the reflection coefficient [Fig. 17(a)] and AR [Fig. 17(b)] as a function of frequency for different DRA widths of 37.4, 39.4, and 41.4 mm. With reference to the figure, both the impedance and AR passbands shift downward with an increase of , which is expected because a larger DRA should have a lower resonance frequency. The effect of the height of the DRA was also studied. It was found the changes of the reflection coefficient and varies, showing that can AR are relatively mild as be used as a fine-tuning parameter. Next, the effect of the slot is investigated. Fig. 18 shows the reflection coefficient [Fig. 18(a)] and AR [Fig. 18(b)] for different slot depths of 10.4, 12.4, and 14.4 mm. With reference to the figure, both the impedance and AR passbands shift upward as increases because of a decrease in the effective dielectric constant. When increases from 10.4 mm to 14.4 mm, the AR bandwidth substantially increases from 3.7% to 8.2%, with the optimum AR value desirably decreasing from 1.74 dB to 0.33 dB. It was found that the value of depends on the dielectric constant of the DRA; the larger the dielectric constant, the smaller the value of is needed. Fig. 19 shows on the reflection coefficient and the effect of slot width

AR. As can be observed from the figure, the effect of is similar to that of . Fig. 20(a) and b) shows the reflection coefficient and AR as a function of frequency for different probe lengths, respectively. From the figure, it can be observed that while the reflection coefficient changes significantly as increases from 11.4 mm to 13.4 mm, the AR remains almost unchanged. It suggests optimizing the AR first by changing the DRA and slot sizes, then tune to obtain good match without the need to worry about the AR. This is a very favorable result that can greatly facilitate the design of the CP DRA. The effect of the flange (ground plane) size of the SMA connector is studied. Fig. 21(a) and (b) shows the reflection coefficient and AR, respectively, as a function of frequency 12.7, 22.7, 32.7, and 42.7 mm. for different side lengths of With reference to the figure, as increases from 12.7 mm to 32.7 mm, the impedance and AR bandwidths decrease from 20.3% to 10.6% and from 8.2% to 4.2%, respectively. It can be observed that the AR deteriorates with an increase of . When exceeds a certain value, say 42.7 mm, the entire AR curve is even above the 3-dB level. It is because the boundary condition requires that the tangential -field component be zero on the surface of a (perfect) conductor and only the perpendicular -field component remains. It results in a poor

=

=

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Fig. 19. Simulated reflection coefficient and AR of the omnidirectional CP DRA as a function of frequency for different slot widths of w 8.4, 9.4, and 10.4 mm. Other parameters are the same as in Fig. 11. (a) Reflection coefficient. (b) AR.

Fig. 20. Simulated reflection coefficient and AR of the omnidirectional CP DRA as a function of frequency for different probe lengths of l 11.4, 12.4, and 13.4 mm. Other parameters are the same as in Fig. 11. (a) Reflection coefficient. (b) AR.

AR since two orthogonal components are required to generate CP fields. Finally, the effect of the dielectric constant of the DRA is 10, investigated. Three omnidirectional CP DRAs with 15, 25 were designed for the WLAN system. In each case, the DRA size and probe length were tuned to optimize the input impedance and AR. Fig. 22(a) and (b) shows the reflection coefficients and ARs of the DRAs, respectively. It can be observed from the figure that the 10-dB impedance bandwidth decreases from 22.1% to 11.0% as the dielectric constant increases from 10 to 25. The result is reasonable because using a larger dielectric constant should give a higher Q-factor and, thus, a narrower bandwidth. However, as can be seen from Fig. 22(b), using a low dielectric constant gives a poor AR. It is due to the fact that the difference between the velocities of the two orthogonal components will decrease with a decrease of , making it difficult to obtain the required phase difference of 90 . As a compromise, a medium dielectric constant in the range of 12–20 is suggested for the proposed CP design. Fig. 22(b) shows that a good AR with a reasonable bandwidth can be obtained by using a medium dielectric constant of . Table I shows the optimized antenna dimensions and bandwidths. With reference to the table, a larger DRA is needed for a smaller dielectric constant, which is to be expected.

In addition, the slot size decreases as the dielectric constant increases, which has been discussed before.

=

=

IV. CONCLUSION The centrally probe-fed omnidirectional LP rectangular DRA has been studied and its fundamental omnidirectional mode has mode. Based on been identified as the quasi cylindrical this LP antenna, a novel omnidirectional CP rectangular DRA has been proposed and investigated. Inclined slots have been fabricated on the sidewalls of the LP DR, giving a very compact omnidirectional CP antenna. Due to the perturbation of the slot, the omnidirectional LP field excited by the probe can be resolved into two orthogonal components with different velocities. By tuning the slot size, the two orthogonal field components can be made equal in magnitude but different in phase by 90 . As a result, an omnidirectional CP wave can be obtained. Both the LP and CP DRAs were simulated with HFSS. To verify the simulations, the two DRAs were fabricated and tested. Reasonable agreement between the measured and simulated results has been observed for each case. It has been found that the measured 10-dB impedance bandwidth of the LP DRA is 10.1%. For the CP DRA, the measured impedance bandwidth is 24.4%, but the antenna bandwidth is limited by the measured AR bandwidth of 7.3%.

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TABLE I COMPARISON OF DIMENSIONS AND BANDWIDTHS OF THE DRAS USING DIFFERENT DIELECTRIC CONSTANTS

Fig. 21. Simulated reflection coefficient and AR of the omnidirectional CP DRA as a function of frequency for different side lengths of the SMA flange. Other parameters are the same as in Fig. 11. (a) Reflection coefficient. (b) AR.

A parametric study of the CP DRA has been done and the effects of various design parameters were examined. It has been found that the DRA and slot sizes affect both the input impedance and AR considerably, whereas the probe length primarily alters the input impedance only. It has also been found that the AR bandwidth is narrow when the dielectric constant of the DRA is high. However, the dielectric constant cannot be too low or an unsatisfactory AR would result. Therefore, it is suggested using a medium dielectric constant between 12 and 20. In addition, it has been shown that the CP performance can be destroyed if a large ground plane is used. Since the small flange of a SMA connector is used as the ground plane, the CP design is very compact. The design guideline of the CP DRA is given as follows. First, design the dimension of the DRA for a chosen dielectric constant to obtain the required resonance (operating) frequency. It is followed by designing the slot to give a good AR. It may

Fig. 22. Simulated reflection coefficient and AR of the omnidirectional CP 10, DRA as a function of frequency for different dielectric constants of  15, and 25. The DRA dimensions and probe lengths are given in Table I. (a) Reflection coefficient. (b) AR.

=

need to retune the dimension of the DRA to keep the resonance frequency unchanged. Finally, the probe length is adjusted to match the impedance. Since the AR is virtually not affected by the probe length, the proposed CP DRA can be designed very easily and straightforwardly. ACKNOWLEDGMENT The authors would like to thank the reviewers for their valuable comments. REFERENCES [1] S. A. Long, M. W. McAllister, and L. C. Shen, “The resonant cylindrical dielectric cavity antenna,” IEEE Trans. Antennas Propag., vol. AP-31, no. 3, pp. 406–412, May 1983. [2] Dielectric Resonator Antennas, K. M. Luk and K. W. Leung, Eds. Baldock, U. K: Research Studies Press, 2003. [3] A. Petosa, Dielectric Resonator Antenna Handbook. Norwood, MA: Artech House, 2007.

PAN et al.: OMNIDIRECTIONAL LINEARLY AND CP RECTANGULAR DRAs

[4] R. K. Mongia and P. Bhartia, “Dielectric resonator antennas—a review and general design relations for resonant frequency and bandwidth,” J. Microw. Millimeter-Wave Eng., vol. 4, pp. 230–247, 1994. [5] A. K. Okaya and L. F. Barash, “The dielectric microwave resonator,” Proc. IRE, vol. 50, pp. 2081–2092, 1962. [6] M. S. Al-Salameh, Y. M. M. Antar, and G. Seguin, “Coplanar-waveguide-fed slot-coupled rectangular dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 50, no. 10, pp. 1415–1419, Oct. 2002. [7] R. K. Mongia and A. Ittipiboon, “Theoretical and experimental investigations on rectangular dielectric resonator antennas,” IEEE Trans. Antennas Propag., vol. 45, no. 9, pp. 1348–1355, Sep. 1997. [8] R. M. Baghaee, M. H. Neshati, and J. R. Mohassel, “Rigorous analysis of rectangular dielectric resonator antenna with a finite ground plane,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2801–2809, Oct. 2008. [9] X. L. Liang and T. A. Denidni, “Wideband rectangular dielectric resonator antenna with a concave ground plane,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 367–370, 2009. [10] K. W. Leung, K. W. Ng, K. M. Luk, and E. K. N. Yung, “Simple formula for analysing the centre-fed hemispherical dielectric resonator antenna,” Electron. Lett., vol. 33, no. 6, pp. 440–441, 1997. [11] R. K. Mongia, A. Ittipiboon, P. Bhartia, and M. Cuhaci, “Electricmonopole antenna using a dielectric ring resonator,” Electron. Lett., vol. 29, no. 17, pp. 1530–1531, 1993. [12] S. M. Shum and K. M. Luk, “Stacked annular ring dielectric resonator antenna excited by axi-symmetric coaxial probe,” IEEE Trans. Antennas Propag., vol. 43, no. 8, pp. 889–892, Aug. 1995. [13] K. Sakaguchi, T. Hamaki, and N. Hasebe, “A circularly polarized omnidirectional antenna,” IEICE Trans. Commun., vol. E79-B, no. 11, pp. 1704–1710, 1996. [14] C. Y. Huang, J. Y. Wu, and K. L. Wong, “Cross-slot-coupled microstrip antenna and dielectric resonator antenna for circular polarization,” IEEE Trans. Antennas Propag., vol. 47, no. 4, pp. 605–609, 1999. [15] K. W. Leung and H. K. Ng, “Theory and experiment of circularly polarized dielectric resonator antenna with a parasitic patch,” IEEE Trans. Antennas Propag., vol. 51, no. 3, pp. 405–412, Mar. 2003. [16] R. Chair, S. L. S. Yang, A. A. Kishk, K. F. Lee, and K. M. Luk, “Aperture fed wideband circularly polarized rectangular stair shaped dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1350–1352, Apr., 2006. [17] G. H. Brown and O. M. Woodward, “Circularly-polarized omni- directional antenna,” RCA Rev., vol. 8, pp. 259–269, 1947. [18] H. A. Wheeler, “A helical antenna for circular polarization,” Proc. IRE, pp. 1484–1488, 1947. [19] V. Galindo and K. Green, “A near-isotropic circularly polarized antenna for space vehicles,” IEEE Trans. Antennas Propag., vol. AP-13, no. 6, pp. 872–877, Nov. 1965. [20] J. L. Masa-Campos, J. M. Fernandez, M. Sierra-Perez, and J. L. Fernandez-Jambrina, “Omnidirectional circularly polarized slot antenna fed by a cylindrical waveguide in millimeter band,” Microw. Opt. Technol. Lett., vol. 49, no. 3, pp. 638–642, 2007. [21] K. L. Lau and K. M. Luk, “A wideband circularly polarized conicalbeam patch antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1591–1594, May 2006. [22] J. S. Row and M. C. Chan, “Reconfigurable circularly-polarized patch antenna with conical beam,” IEEE Trans. Antennas Propag., vol. 58, no. 8, pp. 2753–2757, Aug. 2010. [23] L. C. Y. Chu, D. Guha, and Y. M. M. Antar, “Comb-shaped circularly polarized dielectric resonator antenna,” Electron. Lett., vol. 42, no. 14, pp. 785–787, 2006. [24] D. Kajfez and P. Guillon, Dielectric Resonators. Norwood, MA: Artech House, 1986.

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Yong-Mei Pan (M’11) was born in Huangshan, Anhui Province, China, in 1982. She received the B.Sc. and Ph.D. degrees in electrical engineering from the University of Science and Technology of China (USTC), Hefei, in 2004 and 2009, respectively. She is currently a Research Fellow at City University of Hong Kong. Her research interests include dielectric resonator antennas, leaky wave antennas, and metamaterials.

Kwok Wa Leung (S’90–M’93–SM’02–F’11) was born in Hong Kong in 1967. He received the B.Sc. degree in electronics and the Ph.D. degree in electronic engineering from the Chinese University of Hong Kong in 1990 and 1993, respectively. From 1990 to 1993, he was a Graduate Assistant with the Department of Electronic Engineering, the Chinese University of Hong Kong. In 1994, he joined the Department of Electronic Engineering at City University of Hong Kong as an Assistant Professor. Currently, he is a Professor and an Assistant Head of the Department. From January to June, 2006, he was a Visiting Professor in the Department of Electrical Engineering, The Pennsylvania State University, State College, PA. His research interests include RFID tag antennas, dielectric resonator antennas, microstrip antennas, wire antennas, guided wave theory, computational electromagnetics, and mobile communications. Prof. Leung was the Chairman of the IEEE AP/MTT Hong Kong Joint Chapter for the years of 2006 and 2007. He was the Chairman of the Technical Program Committee, 2008 Asia-Pacific Microwave Conference, Hong Kong, the Co-Chair of the Technical Program Committee, 2006 IEEE TENCON, Hong Kong, and the Finance Chair of PIERS 1997, Hong Kong. He was an Editor for HKIE Transactions and a Guest Editor of IET Microwaves, Antennas and Propagation. Currently, he serves as an Associate Editor for IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and received Transactions Commendation Certificates twice in 2009 and 2010 for his exceptional performance. He is also an Associate Editor for IEEE ANTENNAS WIRELESS PROPAGATION LETTERS. He received the International Union of Radio Science (USRI) Young Scientists Awards in 1993 and 1995, awarded in Kyoto, Japan and St. Petersburg, Russia, respectively. He received Departmental Outstanding Teacher Awards twice in 2005 and 2010. He is a Fellow of HKIE.

Kai Lu (S’11) was born in Laoting, Hebei Province, China. He received the B.Eng. and M.Eng. degrees in electronic engineering from the Harbin Institute of Technology (HIT), Harbin, China, in 2006 and 2008, respectively. He is currently working toward the Ph.D. degree at City University of Hong Kong. From August 2008 to July 2009, he was an Antenna Engineer with Beijing Skyway Technologies Co., Ltd., Beijing, China. His research interests include dielectric resonator antennas, Fabry–Perot resonator antennas, Cassegrain antennas, microstrip antennas, and millimeter wave imaging technology.

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Substrate Integrated Composite Right-/Left-Handed Leaky-Wave Structure for Polarization-Flexible Antenna Application Yuandan Dong, Student Member, IEEE, and Tatsuo Itoh, Life Fellow, IEEE

Abstract—An effective development of a composite right–/lefthanded (CRLH) leaky-wave (LW) structure for polarization- flexible antenna applications is presented. The proposed leaky transmission line (TL) is a planar passive circuit built using the substrate integrated waveguide technology. It consists of two symmetrical waveguide lines loaded with series interdigital capacitors which radiate orthogonal 45 linearly polarized waves. Its dispersion, Bloch impedance and radiation characteristics are extracted by applying a comprehensive analysis on the unit cell. Its backfire-to-endfire beam-steering capability through frequency scanning due to the CRLH nature is demonstrated and discussed. It is able to generate arbitrary different polarization states by changing the way of excitation, including linear polarization (LP) and circular polarization (CP). This leaky TL is fabricated by the standard printed-circuit board process. Two broadband couplers are also designed and fabricated for the specified excitation purpose. Six different polarization states, including four LP cases and two CP ones, are experimentally verified. The propagation and radiation parameters, including the S-parameters, radiation patterns, gain, and axial ratio (for CP states) are presented for these modes. Measured results are consistent with the simulation. The proposed LW structure shows some desirable merits, such as the simplicity in design, low-cost fabrication, and beam-steering and polarization-flexible capabilities, providing a high degree of flexibility for the real application. Index Terms—Composite right/left handed (CRLH), leaky- wave antenna (LWA), polarization-flexible antenna, substrate integrated waveguide (SIW).

I. INTRODUCTION

C

OMPOSITE right-/left-handed (CRLH) transmission-line (TL) metamaterials are understood as artificially engineered and structured media that exhibit some unique electromagnetic (EM) properties. They have received significant attention and have enabled numerous applications over the past decade [1]–[4], especially for the radiated-wave devices. Various CRLH leaky-wave (LW) antennas have already been studied and developed based on different techniques [5]–[12]. They all possess a full-space beam-steering capability by varying the frequency. Manuscript received November 23, 2010; revised June 16, 2011; accepted August 05, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported by Honeywell through the UC Discovery Project. The authors are with the Electrical Engineering Department, University of California at Los Angeles, Los Angeles, CA 90095 USA (e-mail: yddong@ee. ucla.edu; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173124

Polarization is an important parameter for antennas. Multipolarized antennas are able to change their polarization state dynamically depending on the requirement. They can be used to mitigate the multipath fading effect encountered in the wireless communication systems and increase the channel capacity. Their polarization can be tailored for specific applications. Many antennas with this function have already been studied in various literature, such as the polarization-agile antennas [13]–[16], reconfigurable antennas [17]–[19], dual-polarized antennas [20]–[22], and polarization diversity antennas [23], [24]. Substrate integrated waveguide (SIW) has been a very popular candidate to realize low-loss, low-cost, and low-profile planar waveguide components and antennas [25]–[30]. In [31], a post-wall, or SIW array for dual polarization has been proposed. A millimeter-wave LW antenna with quad polarization has also been successfully implemented in [32] using the half-mode SIW. In this paper, a CRLH LW structure based on the SIW scheme is developed for the polarization-agile antenna application. The proposed antenna can be mounted on board or other different vehicles providing flexible radiation directions and polarization states. The CRLH feature is achieved by periodically loading the series interdigital capacitors on the waveguide surface. The polarization-flexible functionality is obtained by symmetrically aligning two leaky TLs with orthogonal linear polarizations excited by different inputs. It is noted that previous research [33] was done, explaining the realization of the circular polarization. Some conventional SIW circularly polarized antennas have also been studied in [34]–[36]. Here, substantial work has been added into this research. The proposed structure possesses an ability to generate arbitrary polarization states. Six specific cases among them, including four linear types and two circular types, are experimentally verified. Its main beam can be steered continuously from backward to forward while maintaining a pure polarization state. This paper is organized as follows. The configuration of the proposed LW structure and its polarization-flexible working scheme are illustrated in Section II. The design procedure is demonstrated in Section III, including the designs of the single unit cell, two-element unit cell, leaky TL, and the excitations. Sections IV–VII present their applications to CRLH SIW LW antennas with some different polarization states. Specifically, they are two orthogonal 45 linear polarizations, horizontal (X-directed) polarization, vertical (Y-directed) polarization, and two circular polarizations (left handed and right handed). Section VIII draws the conclusion.

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II. PROPOSED STRUCTURE AND WORKING PRINCIPLE A. Geometrical Layout The geometric configuration of the proposed LW structure is shown in Fig. 1, where the layout of the unit-cell elements [Fig. 1(a) and (b)] and the prototype of the entire LW structure with its orientations in the coordinate system [Fig. 1(c)] are displayed. As shown, the unit cell is surrounded by vias on the two sides which are connected to a solid metallic ground. The interdigital slots etched on the waveguide surface are 45 inclined compared to the propagation direction (X-directed). Two symmetrical leaky TLs are side by side arranged and separated with a small distance to improve the isolation as depicted in Fig. 1(b) and (c). Each of them carries 14 interdigital slots which are periodically etched on the broad wall. They can generate two orthogonal linearly polarized waves. The slot acts like a series capacitor, which, along with the waveguide inherent shunt inductor provided by the vias, creates the necessary condition to support the CRLH operation. A piece of 50- microstrip line along with a taper line for impedance matching is placed at the end of each waveguide to facilitate the outside connection. This LW structure is fabricated on a substrate of Rogers 5880 with a thickness of 50 mils and a relative permittivity of 2.2. Generally, a thick and low dielectric constant substrate can be used to reduce the loss. The metallic via holes are chosen to have a diameter of 0.8 mm and a center-to-center pitch around 1.5 mm. B. Polarization-Flexible Capability The polarization of an EM wave is defined as the orientation of the electric-field vector. The polarization-agile operation scheme for the proposed structure can be explained using Fig. 1(b) and (c) and Fig. 2(a). The two leaky lines radiate two orthogonally polarized waves. The total electric field is the vector addition of the two waves. When only Port 1 (left line) is excited, a guided wave will be transmitted along the left line which produces the linearly polarized wave in the direction. It should be noted that the orthogonal wave will also be generated but in a very weak manner, which is called the cross-polar component. When Port 4 (right line) is fed alone, only the linearly polarized wave along the direction will be produced. When they are illuminated by two equal and in-phase signals simultaneously, the X-polarized (horizontal direction) wave will be produced. Similarly, Y-polarized (vertical direction) waves can be obtained with two inputs of the same magnitude and 180 out of phase. They form a pair of orthogonal linearly polarized modes. When the two lines are equally excited with phase difference, a circularly polarized mode can be generated. Depending on their phase relation (phase delay or advance), left-handed circular polarization (LHCP) or right-handed circular polarization (RHCP) can be implemented. As shown in Fig. 2(a), the feeding control circuit is required in order to implement the desired polarization. And it is noted that arbitrary polarization, including linear, circular, and elliptical types, can be achieved depending on the phase and magnitude relation of the two input excitations. To give a better explanation, Table I summarizes the operation principle of six specific polarization states. They can be generalized into three orthogonal pairs: 1) linearly polarized waves; 2) X- and

Fig. 1. Configurations of the proposed structures. (a) Single CRLH-SIW radiating element. (b) Two-element unit cell of the whole structure. (c) Overall LW antenna prototype.

Fig. 2. (a) Operation principle of the polarization-flexible antenna. (b) Circuit model of the CRLH-SIW element shown in Fig. 1(a).

Y-directed linearly polarized waves; and 3) RHCP and LHCP radiating waves. III. DESIGN PROCEDURES The design process to physically implement this radiating structure with a polarization-flexible function is outlined in this section. Design procedures for the unit cell, traveling-wave lines, and the feeding circuits, including a 90 half-mode

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TABLE I SUMMARY OF SIX SPECIFIC POLARIZATION STATES UNDER DIFFERENT INPUT EXCITATIONS

SIW directional coupler and a two-section rat-race hybrid, are discussed in sequence. All of the full-wave simulations are carried out using Ansoft’s High Frequency Structure Simulator (HFSS) software package. A. Single Unit-Cell Analysis The proposed TL is basically a CRLH structure working in the fast-wave region with a small periodicity compared to the free-space wavelength. The design can be started from analyzing the unit cell. Fig. 2(b) shows the equivalent circuit of the CRLH-SIW unit cell as presented in Fig. 1(a). The surface and the ground can be modeled as a two-wire TL with distributed series inductance and distributed shunt capacitance. The vias provide the shunt inductance. The interdigital capacitor has been introduced into the model as to obtain CRLH behavior. The left-handed (LH) contribution comes from the series capacitor and the shunt inductance . To obtain a continuous beam-scanning performance, the balanced condition is usually required. Note that the series interdigital slot, which is rotated by 45 , also plays the role of a radiating element. Here, a radiation resistor can be introduced in parallel to the series capacitor [37]. Increasing the width and length of the slot could make the radiation more efficient. The dispersion diagram for the proposed unit cell is then investigated carefully based on the HFSS simulation. Two approaches are commonly adopted to extract the dispersion curve. One is based on the -parameters from the fast driven-mode simulation [38]. The other one rests on the eigenmode simulation by applying the periodic boundary condition [39]. For comparison, Fig. 3(a) plots the dispersion curves for the CRLH-SIW unit cell using both of the two methods. The unit-cell dimensions are listed in the caption. Reasonable agreement is obtained. It should be pointed out that the eigenmode simulation shows that actually a very small bandgap (from 8.085 to 8.2 GHz) exists between the LH and right-handed (RH) regions. Also, it is noted that the eigenmode simulation approach is believed to be more accurate although time-consuming. Rich information can be obtained from this figure. The LH and RH regions are separated by the transition frequency (or the bandgap). The air line plotted in the figure gives rise to two distinct regions: the radiating region (fast

wave) below it and the guiding region (slow wave) above the line. Fig. 3(b) presents the simulated Bloch impedance of the unit cell extracted from the -parameters. For a symmetric CRLH unit cell shown in Fig. 2(b) without considering the radiation resistor, the Bloch impedance takes the form [1]

(1) where

(2) and correspond to a zero and a pole of It is seen that . The balanced condition satisfies when . Otherwise, the zero and pole always exist on the Bloch impedance regardless of how is close to . This is consistent with what we observed in Fig. 3(b) where a zero is closely followed by a pole. Note that it is difficult to eliminate this rapid change near the transition frequency. The Bloch impedance value gives some useful information for the final impedance matching. Fig. 3(c) shows a loss analysis for the unit cell, which is calculated using the equation shown in the inset of Fig. 3(c). The normalized leakage constant is also included in the figure. Due to a waveguide propagation mode and a relatively thick (50 mils) and low-permittivity substrate, the dielectric and conductor losses are very small and almost negligible as indicated in the figure. Good radiation efficiency can be envisioned. B. Investigation on Two-Element Unit Cell When symmetrically aligning two leaky TLs to form an antenna with specified polarization, the distance between them is an important factor which ultimately determines the isolation, cross-polarization level, and the grating lobe performance. To this end here, we did some analysis on the radiation characteristics in the plane to obtain an optimal value of the distance between the two leaky TLs. And this information can be

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Fig. 3. (a) Dispersion diagram calculated from the driven-mode and eigenmode simulations for the CRLH-SIW unit cell shown in Fig. 1(a). (b) Bloch impedance obtained using the driven-mode simulation. (c) Calculated different losses and the normalized leakage constant for the unit cell. The parameters of 0.545 mm, 0.4 mm, 12.4 mm, 9.1 mm, the unit cell are 3.1 mm. (The interdigital capacitor has nine fingers.)

Fig. 4. Simulated results for the two-element unit cell with different . plane radiation patterns for (a) Isolation between different ports. (b) The the in-phase excitation case (left) and 180 out-of-phase excitation case (right) plane. The other at the transition frequency. (c) AR observed in the unit-cell dimensions are the same as those shown in the Fig. 3 caption.

obtained at an early stage by investigating the two-element unit cell as shown in Fig. 1(b). Here, the structure is replotted in the inset of Fig. 4(a) for convenience. The radiation boundary condition is applied on a big enough air box containing this two-element unit cell. By changing the separation , varied isolation between the ports and different radiation characteristics can be observed. As shown in Fig. 4(a), when is increased, the isolation between Port 1 and Port 3 is also enhanced. Also, the isolation at upper frequencies is larger than that at lower frequencies because of the smaller wavelength at upper frequencies. The isolation between port 1 and port 4 is also shown for the 10.2-mm case. It is interesting to note that due to the backward coupling in the LH region, the isolation between port

1 and port 4 is weaker at lower frequencies and higher at the upper frequencies compared to that between port 1 and port 3. Fig. 4(b) shows the simulated radiation patterns in the plane at the transition frequency for the in-phase (left figure) and out-of-phase (right figure) excitation cases. It reveals that when is larger, the undesired cross-polarization level is increased and the grating lobe occurs. The mutual coupling as indicated by Fig. 4(a) is not significant due to its traveling-wave nature for which the field is not resonating strongly. However it can still slightly deteriorate the axial ratio (AR) for the circular polarization when this separation is small. Fig. 4(c) shows the simulated AR by exciting the two-element unit cell with 90 phase difference. A larger enables better AR at the

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Fig. 5. Detailed input-matching structure for the leaky TL.

main beam direction. The AR at different frequencies is also plotted for the 10.2-mm case. Circular polarization scanning with the main beam is also observed. Bear in mind that the radiation characteristics for this two-element unit cell are different from those of the whole leaky-wave antenna. The isolation could be decreased when the number of unit cells is increased due to more slot couplings. Therefore, the AR is also different for the whole leaky-wave antenna. Nevertheless, it is still able to provide some useful information which can be used to predict the final performance and guide the design. The aforementioned results show that the choice of the separation should be a compromise between the isolation and crosspolarization. From the simulation, we find that there is an optimal range for , which could match the needs of the isolation and cross-polarization. However, as shown in Fig. 4(b), the desired polarization (co-polarization) only works in the region of 20 to 20 in the plane for both of the two cases. Outside this region, the cross-polar component becomes dominant. This is an important feature of this proposed antenna. C. Design of the LW Line The entire line is a fast-wave radiating structure which can be built by simply cascading the unit cell described in Fig. 3 and viewed as a uniform LW structure essentially. However, connecting with the outside circuits matching network is necessary since the impedance of the unit cell is not simply 50 . From the Bloch impedance shown in Fig. 3(b), the average real part considering the entire fast-wave region is around 40 . The imaginary part appears as capacitive in the LH region and zero in the RH region. For the real part, we use a simple taper line as shown in Fig. 5, converting the impedance from 40 to 50 . For the imaginary part, we can tune the waveguide length between the taper line and the first unit cell ( as indicated by Fig. 5) to match the circuit. At the low frequency (LH region below the waveguide cutoff), the vias provide a shunt inductance which can be used to compensate the initial capacitive value. However, at the frequency above the cutoff (RH region), the vias form an electric wall which supports the propagation of the mode. Thus, by slightly extending the waveguide length before connecting the taper line, the matching in the LH region can be improved and in the RH region, it almost remains the same. D. Coupler Designs for Excitations In order to realize the input excitations listed in Table I, two broadband couplers are designed and fabricated covering the frequency range of interest. The first one is a two-section 3-dB rat-race hybrid providing inphase and 180 out-of-phase outputs

Fig. 6. Measured and simulated performance for the two-section rat-race coupler. (a) Measured -parameters for the out-of-phase case. (b) Measured -parameters for the in-phase case. (c) Measured and simulated phase performance. The structure is shown in the inset. Port 1 is excited for 180 out-of-phase operation. Port 4 is the input port for the in-phase case.

[40]. The second one is a 90 3-dB directional coupler designed on the half-mode SIW scheme [41]. Their detailed design procedures are shown in [40] and [41]. The two-section rat-race hybrid is fabricated on the Rogers 5880 substrate with a thickness of 20 mils. A photo is shown in the inset of Fig. 6(b). It consists of three vertical and four horizontal lines whose impedances are optimized to have good wideband matching from 7 to 11 GHz. Specifically, they are 52.9 74.1 30.8 56.7 66.1 , and 50 . Fig. 6 shows the experimental results for the coupler, including magnitude and phase responses for the out-of-phase and in-phase cases. Over the interested frequency band, small reflection (below 10 dB),

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Fig. 8. Photograph of the fabricated LW antennas. Parameters are 5.2 mm, 5.8 mm, 3.9 mm, and 10.2 mm.

4 mm,

Fig. 9. Measured -parameters of the fabricated antennas shown in Fig. 11. The gray line shows the simulated results for comparison.

Fig. 7. Measured and simulated performances for the 3-dB HMSIW directional coupler. (a) Measured -parameters. (b) Measured and simulated phase performance. The structure is shown in the inset. Port 1 is the input port and Port 4 is isolated.

good amplitude imbalance (less than 0.4 dB), small phase variation (less than 6 ), and large isolation (better than 24 dB) are achieved. The fabricated prototype of the half-mode SIW directional coupler is shown in the inset of Fig. 7(b). It is employed to realize the feedings with 90 phase difference and equal power division. The coupling area is an aperture on the via wall. This coupler is implemented on the Rogers 5880 substrate with a thickness of 50 mils. The measured results are shown in Fig. 7. Good reflection and isolation (below 13.5 dB), balanced outputs, and expected phase difference are achieved covering a frequency band from 7 to 10 GHz. IV.

-POLARIZED LW ANTENNA

Based on the procedures shown before, a single travelingwave antenna with 14 unit cells depicted in Fig. 3 is designed and optimized. Fig. 8 shows a photograph of the LW antennas with the parameters shown in the caption. Two identical LW lines are symmetrically aligned along the X-direction. We fabricated and measured these antennas in our laboratory. Fig. 9 shows the measured -parameters for each of the two antennas. Basically, they are in agreement with the simulation (gray dash line). The LH and RH regions are separated by the transition frequency of 8.2 GHz. As observed, the entire TL is not perfectly balanced which is in agreement with eigenmode simulation on the unit cell. The small difference is due to this being a finitely

Fig. 10. Measured radiation patterns in the different frequencies.

plane for co-polarization at

long leaky line which cannot guarantee a periodic boundary condition for the unit cell. The matching in the RH region is better than the LH region which is consistent with the Bloch impedance analysis. Fig. 10 shows the normalized radiation patterns of the second antenna measured in a far-field chamber. It is important to bear in mind that its co-polar direction is 45 rotated with respect to plane (scanning plane or co-polarization plane). In the the measurement, the standard linearly polarized horn antenna as the transmitter is rotated by 45 to match the co-polar direction of the LW antenna. Its full-space beam-steering performance by frequency scanning is verified experimentally. The beam width is larger at the lower frequencies due to the larger leakage constant and decrease of the antenna equivalent aperture size. To check the directivity and efficiency, we also measured this antenna in a near-field chamber in our High-Frequency Center. Fig. 11 compares the normalized broadside patterns at 8.2 GHz

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Fig. 11. Comparison of the simulated and measured radiation patterns in the plane at 8.2 GHz. Measured patterns include the results obtained from the far-field chamber and the near-field chamber.

Fig. 12. Simulated and measured directivity, gain, and the measured efficiency for the second linearly polarized antenna.

obtained from the simulation, the far-field measurement and the near-field measurement. The cross-polarization is also plotted. A reasonable agreement is observed. The cross-polar level is 15 dB in the measurement and 23.5 dB in the simulation. Fig. 12 shows the simulated and measured directivity, realized gain, and measured efficiency. The measured efficiency is low at 8.2 GHz and 10 GHz because of the large reflection and termination loss. When it radiates completely, this antenna should be able to provide an average efficiency around 80% as that obtained at 7.8 and 8.7 GHz. By improving the impedance matching and increasing the number of unit cells, better efficiency can be achieved. V. X-POLARIZED LW ANTENNA A. Simulation The X-polarized wave is obtained by equally feeding the two leaky lines with in-phase excitations. In the simulation setup, two identical signals are directly applied at port 1 and port 4 [Fig. 1(c)], respectively. To avoid handling large structures using HFSS, the coupler here is not included in the simulation. Fig. 13 shows the simulated gain pattern in the plane (scanning plane), including both co-polarization and cross- polarization. Low cross-polar level and beam scanning are observed.

Fig. 13. Simulated patterns in terms of realized gain for the X-polarized anplane in the (a) LH region, (b) broadside, and (c) RH region. tenna in the

plane is 10 dB in the The realized maximum gain in the LH region and 12 dB in the RH region approximately. B. Experimental Results To measure the antenna performance under the X-polarized condition, we cascaded the fabricated rat-race hybrid and the two-element LW antenna as the prototype shown in the inset of Fig. 14. The -parameters are first measured using an Agilent 8510C network analyzer. The whole structure is fed at port 4, and Fig. 14 presents the measured -parameters. As predicted, low reflection coefficient and good isolation are achieved. Fig. 18 shows the normalized radiation patterns at five different frequencies obtained from the near-field measurement. The main polarization is in the plane and the cross-polarization level is higher than that observed in the simulation which is due

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Fig. 14. Measured -parameters of the X-polarized LW antenna.

Fig. 16. Simulated patterns in terms of the realized gain for the Y-polarized plane in (a) the LH region, (b) broadside, and (c) RH antenna in the region.

to the fact that the coupler performance is not ideal. Also, the two fabricated radiating TLs are not identical. The measured gains at 7.5, 7.8, 8.2, 8.7, and 10 GHz are 9.72, 9.48, 7.76, 11.1, and 11.75 dBi, respectively. Overall, the experimental results are consistent with the simulation. VI. Y-POLARIZED LW ANTENNA A. Simulation

Fig. 15. Measured and normalized radiation patterns for the X-polarized anplane in (a) the LH region, (b) broadside, and (c) RH region. tenna in the

Fig. 16 shows the simulated gain patterns which are polarized in the Y-direction. Two signals with 180 out of phase are directly applied at port 1 and port 4 [Fig. 1(c)] without including the coupler. The -plane coincides with the plane (scanning plane), and full-space beam scanning is also observed.

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Fig. 17. Measured -parameters of the Y-polarized LW antenna.

Fig. 19. Measured and simulated radiation patterns in the plane at 8.2 GHz for (a) the X-polarized LW antenna and (b) the Y-polarized LW antenna.

B. Experimental Results Fig. 17 shows the measured -parameters. In this case, the entire structure, as indicated in the inset of Fig. 17, is fed at port 1, and other ports are terminated with the 50- load. Its radiation patterns are measured in the near-field chamber, and Fig. 18 shows the normalized results. The increase of the cross-polarization level is also found in the measurement after introducing the rat-race hybrid. The measured gains at 7.5, 7.8, 8.2, 8.7, and 10 GHz are 8.21, 10.8, 10.01, 11.78, and 11.13 dBi, respectively. It is interesting to find that at the broadside (8.2 GHz), the gain for the Y-polarized wave is higher than that observed in the X-polarized case. To find the reason, we checked the simulated and measured patterns in the plane at 8.2 GHz for both of the two cases. They are plotted in Fig. 19, together with the 3-D patterns, in the inset. It is seen from the 3-D radiation patterns that the beam in the plane for the X-polarized case is much wider than that in the Y-polarized case. The reason is that the cross-polar component is more significant for the X-polarized case. This result is in good agreement with the analysis on the two-element unit cell shown in Fig. 4(b). VII. CIRCULARLY POLARIZED LW ANTENNA Fig. 18. Measured and normalized radiation patterns for the Y-polarized anplane in (a) the LH region, (b) broadside, and (c) RH region. tenna in the

The circular polarization is achieved by exciting two orthogonally polarized radiating lines with 90 phase difference. For

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Fig. 20. Measured -parameters of the entire circularly polarized LW antenna.

Fig. 22. Measured AR of the circularly polarized antenna in the plane in (a) the LH region, (b) broadside at 8.2 GHz for different , and (c) RH region.

Fig. 21. Measured radiation patterns of the circularly polarized antenna in the plane in (a) the LH region, (b) broadside at 8.2 GHz, and (c) RH region.

simplicity here, only some measured results are provided. The detailed analysis and performance can be found in [33]. In the

measurement, the half-mode SIW 90 directional coupler is connected with the leaky lines to provide the required excitation. The whole structure, as displayed in the inset of Fig. 20, is fed at port 1, resulting in an RHCP. It is worth noting that LHCP can also be obtained by feeding the whole structure at port 4. The measured -parameters are shown in Fig. 20. It is seen that the total reflection is below 11 dB in the entire region. The isolation experiences a peak around 8.2 GHz and deteriorates below 7.2 GHz. This is reasonable since the reflected waves from the two leaky lines arrive at port 1 with 180 phase difference; thus, they cancel each other. However, they are in-phase when arriving at port 4; thus, it is the superposition of the two waves due to the 90 directional coupler. Therefore, the behaves similar to the reflection of the single radiating line.

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Fig. 22 shows the normalized radiation patterns measured in the near-field chamber at five different frequencies. Both the co-polarization (RHCP) and cross-polarization (LHCP) are provided. Beam scanning is verified. The measured directivity is 10.45 dBi at 7.5 GHz, 10.71 dBi at 7.8 GHz, 11.52 dBi at 8.2 GHz, 13.22 dBi at 8.7 GHz, and 14.556 dBi at 10 GHz. The directivity in the RH region is higher than that in the LH region due to the lower leakage constant shown by Fig. 3(c) and the decrease of the wavelength, which lead to a more effective and larger aperture. The measured AR at the aforementioned frequencies is plotted in Fig. 22. We also plotted the simulated AR for different separations between the two TLs in Fig. 22(b) in order to give a comparison with the AR obtained from the two-element unit-cell simulation shown in Fig. 4(c). Overall, the simulated AR for the entire LW antenna deteriorates a little compared to the unit-cell simulation because of the increased slot coupling. This adopted distance is an optimal value for the entire structure. It is seen that at the main beam direction, the obtained AR is always below 3 dB. Discrepancy is observed between the simulated and measured AR which is predictable since the 3-dB coupler is not perfect and bandlimited. Also, the fabrication error could result in two different leaky lines which would affect the AR. In general, the antenna performance is satisfactorily characterized by high directivity, low cross-polar level, and good circular polarization. VIII. CONCLUSION We have implemented a frequency-scanning CRLH leaky TL structure used for polarization-agile traveling-wave antenna applications. It is a low-loss passive component which can be realized on a planar substrate by a low-cost printed-circuit board (PCB) process. Depending on the input excitation, it can support arbitrary polarization states. We experimentally verified six specified cases, including two pairs of orthogonal linearly polarized modes and one pair of orthogonal circularly polarized modes. The measured and simulated results show that full-space beam steering through frequency change is achieved including the broadside direction. Good efficiency is also observed. This structure can be a potential candidate for wireless communication applications requiring polarization diversity. REFERENCES [1] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. Hoboken, NJ: Wiley/IEEE, 2005. [2] G. V. Eleftheriades and K. G. Balmain, Negative Refraction Metamaterials: Fundamental Principles and Applications. Hoboken, NJ: Wiley/IEEE, 2005. [3] N. Engheta and R. W. Ziolkowski, Electromagnetic Metamaterials: Physics and Engineering Explorations. Hoboken, NJ: Wiley/IEEE, 2006. [4] T. J. Cui, R. Liu, and D. R. Smith, Metamaterials: Theory, Design and Applications. New York: Springer, 2010. [5] T. Kokkinos, C. D. Sarris, and G. V. Eleftheriades, “Periodic FDTD analysis of leaky-wave structures and applications to the analysis of negative-refractive-index leaky-wave antennas,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1619–1630, Jun. 2006. [6] R. Goto, H. Hiroyuki, and M. Tsuji, “Composite right/left-handed transmission lines based on conductor-backed coplanar strips for antenna application,” in Proc. 36th Eur. Microw. Conf., U.K., Sep. 2006, pp. 1040–1043.

[7] Y. Weitsch and T. Eibert, “A left-handed/right-handed leaky-wave antenna derived from slotted rectangular hollow waveguide,” in Proc. Eur. Microw. Conf., Munich, Germany, Oct. 2007, pp. 917–920. [8] T. Ueda, N. Michishita, M. Akiyama, and T. Itoh, “Dielectric-resonator-based composite right/left-handed transmission lines and their application to leaky wave antenna,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 10, pp. 2259–2268, Oct. 2008. [9] T. Ikeda, K. Sakakibara, T. Matsui, N. Kikuma, and H. Hirayama, “Beam-scanning performance of leaky-wave slot-array antenna on variable stub-loaded left-handed waveguide,” IEEE Trans. Antennas Propag., vol. 56, no. 12, pp. 3611–3618, Dec. 2008. [10] S. Paulotto, P. Baccarelli, F. Frezza, and D. Jackson, “Full-wave modal dispersion analysis and broadside optimization for a class of microstrip CRLH leaky-wave antennas,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 2826–2837, Dec. 2008. [11] T. Kodera and C. Caloz, “Uniform ferrite-loaded open waveguide structure with CRLH response and its application to a novel backfire-to-endfire leaky-wave antenna,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 4, pp. 784–795, Apr. 2009. [12] Y. Dong and T. Itoh, “Composite right/left-handed substrate integrated waveguide and half mode substrate integrated waveguide leaky-wave structures,” IEEE Trans. Antennas Propag., vol. 59, no. 3, pp. 767–775, Mar. 2011. [13] D. H. Schaubert, F. G. Farrar, A. Sindoris, and S. Hayes, “Microstrip antenna with frequency agility and polarization diversity,” IEEE Trans. Antennas Propag., vol. AP-29, no. 1, pp. 118–123, Jan. 1981. [14] P. Haskins, P. S. Hall, and J. S. Dahele, “Polarization-agile active patch antenna,” Electron. Lett., vol. 30, no. 2, pp. 98–99, Jan. 1994. [15] S. Gao, A. Sambell, and S. S. Zhong, “Polarization-agile antennas,” IEEE Antennas Propag. Mag., vol. 48, no. 6, pp. 28–37, Jun. 2006. [16] F. Ferrero, C. Luxey, R. Staraj, G. Jacquemod, M. Yedlin, and V. Fusco, “A novel quad-polarization agile patch antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1562–1566, May 2009. [17] M. Fries, M. Grani, and R. Vahldieck, “A reconfigurable slot antenna with switchable polarization,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 11, pp. 490–493, Nov. 2003. [18] Y. Sung, T. Jang, and Y. Kim, “A reconfigurable microstrip antenna for switchable polarization,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 11, pp. 534–536, Nov. 2004. [19] B. Kim, B. Pan, S. Nikolaou, Y. Kim, J. Papapolymerou, and M. M. Tentzeris, “A novel single-feed circular microstrip antenna with reconfigurable polarization capability,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 630–638, Mar. 2008. [20] K. L. Wong, H. C. Tung, and T. W. Chiou, “Broadband dual-polarized aperture-coupled patch antennas with modified H-Shaped coupling slots,” IEEE Trans. Antennas Propag., vol. 50, no. 2, pp. 188–191, Feb. 2002. [21] Y. X. Guo, K. W. Khoo, and L. C. Ong, “Wideband dual-polarized patch antenna with broadband baluns,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 78–83, Jan. 2007. [22] K. S. Ryu and A. A. Kishk, “Wideband dual-polarized microstrip patch excited by hook shaped probes,” IEEE Trans. Antennas Propag., vol. 56, no. 12, pp. 3645–3649, Dec. 2008. [23] Y. F. Wu, C. H. Wu, D. Y. Lai, and F. C. Chen, “Reconfigurable quadripolarization diversity aperture-coupled patch antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 1009–1012, Mar. 2007. [24] W. K. Toh, Z. N. Chen, X. Qing, and T. See, “A planar UWB diversity antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 11, pp. 3467–3473, Nov. 2009. [25] D. Deslandes and K. Wu, “Integrated microstrip and rectangular waveguide in planar form,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 2, pp. 68–70, Feb. 2001. [26] L. Yan, W. Hong, G. Hua, J. X. Chen, K. Wu, and T. J. Cui, “Simulation and experiment on SIW slot array antennas,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 9, pp. 446–449, Sep. 2004. [27] W. Hong, K. Wu, H. Tang, J. Chen, P. Chen, Y. Cheng, and J. Xu, “SIW-like guided wave structures and applications,” IEICE Trans. Electron., vol. E92-C, no. 9, pp. 1111–1123, Sep. 2009. [28] T. M. Shen, C. F. Chen, T. Y. Huang, and R. B. Wu, “Design of vertically stacked waveguide filters in LTCC,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 8, pp. 1771–1779, Aug. 2008. [29] Y. Dong, T. Yang, and T. Itoh, “Substrate integrated waveguide loaded by complementary split-ring resonators and its applications to miniaturized waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 9, pp. 2211–2223, Sep. 2009.

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[30] X. P. Chen, K. Wu, L. Han, and F. H. , “Low-cost high gain planar antenna array for 60-GHz band applications,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 2126–2129, Jun. 2010. [31] S. Park, Y. Okajima, J. Hirokawa, and M. Ando, “A slotted post-wall waveguide array with interdigital structure for 45 linear and dual polarization,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2865–2871, Sep. 2005. [32] Y. Cheng, W. Hong, and K. Wu, “Millimeter-wave half mode substrate integrated waveguide frequency scanning antenna with quadri-polarization,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 1848–1855, Jun. 2010. [33] Y. Dong and T. Itoh, “Realization of a composite right/left-handed leaky-wave antenna with circular polarization,” in Proc. Asia-Pacific Microw. Conf., Yokohama, Japan, Dec. 2010, pp. 865–868. [34] Z. J. Chen, W. Hong, Z. Kuai, J. Chen, and K. Wu, “Circularly polarized slot array antenna based on substrate integrated waveguide,” in Proc. Int. Conf. Microw. Millimeter Wave Technol., Nanjing, China, Apr. 2008, vol. 3, pp. 1066–1069. [35] P. Chen, W. Hong, Z. Kuai, and J. Xu, “A substrate integrated waveguide circular polarized slot radiator and its linear array,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 120–123, 2009. [36] D. Kim, J. W. Lee, C. S. Cho, and T. K. Lee, “X-band circular ringslot antenna embedded in single-layered SIW for circular polarization,” Electron. Lett., vol. 45, no. 13, pp. 668–669, Jun. 2009. [37] J. S. Gomez-Diaz;, A. Alvarez-Melcon, and T. Bertuch, “An iteratively refined circuital model of CRLH leaky-wave antennas derived from the mushroom structure,” in Proc. IEEE Antennas Propag. Soc. Int. Symp. Dig., Jul. 2010, pp. 1–4. [38] D. M. Pozar, “Microwave filters,” in Microwave Engineering, 3rd ed. Hoboken, NJ: Wiley, 2005, ch. 8. [39] “Left-Handed Metamaterial Design Guide,” Ansoft Corporation, 2007. [40] M. Caillet, M. Clenet, A. Sharaiha, and Y. Antar, “A compact wideband rat-race hybrid using microstrip lines,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 4, pp. 191–193, Apr. 2009. [41] B. Liu, W. Hong, Y. Wang, Q. Lai, and K. Wu, “Half mode substrate integrated waveguide (HMSIW) 3 dB coupler,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 1, pp. 22–24, Jan. 2007. Yuandan Dong (S’09) received the B.S. and M.S. degrees in radio engineering from Southeast University, Nanjing, China, in 2006 and 2008, respectively, and is currently pursuing the Ph.D. degree in electrical engineering at the University of California at Los Angeles (UCLA). From 2005 to 2008, he was studying in the State Key Laboratory of Millimeter Waves, Southeast University. Since 2008, he has been a Graduate Student Researcher with the Microwave Electronics Laboratory, UCLA. He is serving as a reviewer for several

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IEEE and IET journals including the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and IEEE TRANSACTION ON ANTENNAS AND PROPAGATION. His research interests include the characterization and development of RF and microwave components, circuits, antennas, and metamaterials. Mr. Dong was the recipient of the Best Student Paper Award in 2010 from the Asia Pacific Microwave Conference, Yokohama, Japan. He has authored more than 20 journal and conference papers.

Tatsuo Itoh (S’69–M’69–SM’74–F’82–LF’06) received the Ph.D. degree in electrical engineering from the University of Illinois, Urbana, in 1969. After working for the University of Illinois, SRI International in Menlo Park, and University of Kentucky, Lexington, he joined the faculty at The University of Texas at Austin in 1978, where he became a Professor of Electrical Engineering in 1981. In 1983, he was selected to hold the Hayden Head Centennial Professorship of Engineering at The University of Texas at Austin. In 1991, he joined the University of California, Los Angeles, as Professor of Electrical Engineering and Holder of the TRW Endowed Chair in Microwave and Millimeter Wave Electronics (currently Northrop Grumman Endowed Chair). Dr. Itoh is a member of the Institute of Electronics and Communication Engineers of Japan, and Commissions B and D of USNC/URSI. He was Editor of IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES from 1983 to 1985 and was President of the Microwave Theory and Techniques (MTT) Society in 1990. He was the Editor-in-Chief of IEEE MICROWAVE AND GUIDED WAVE LETTERS from 1991 to 1994. He was elected as an Honorary Life Member of the MTT Society in 1994. He was the Chairman of Commission D of International URSI for 1993–1996. He serves on the advisory boards and committees of a number of organizations. He served as Distinguished Microwave Lecturer on Microwave Applications of Metamaterial Structures of the IEEE MTT-S for 2004–2006. He received a number of awards, including IEEE Third Millennium Medal in 2000 and the IEEE MTT Distinguished Educator Award in 2000. He was elected to member of the National Academy of Engineering in 2003. He has many journal publications and refereed conference presentations. He has also written many books/book chapters in the area of microwaves, millimeter waves, antennas, and numerical electromagnetics. He has graduated 70 Ph.D. students.

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Design and Characterization of Miniaturized Patch Antennas Loaded With Complementary Split-Ring Resonators Yuandan Dong, Student Member, IEEE, Hiroshi Toyao, and Tatsuo Itoh, Life Fellow, IEEE

Abstract—An investigation into the design of compact patch antennas loaded with complementary split-ring resonators (CSRRs) and reactive impedance surface (RIS) is presented in this study. The CSRR is incorporated on the patch as a shunt LC resonator providing a low resonance frequency and the RIS is realized using the two-dimensional metallic patches printed on a metal-grounded substrate. Both the meta-resonator (CSRR) and the meta-surface (RIS) are able to miniaturize the antenna size. By changing the configuration of the CSRRs, multi-band operation with varied polarization states can be obtained. An equivalent circuit has been developed for the CSRR-loaded patch antennas to illustrate their working principles. Six antennas with different features are designed and compared, including a circularly-polarized antenna, which validate their versatility for practical applications. These antennas are fabricated and tested. The measured results are in good agreement with the simulation. Index Terms—Circular polarization, complementary split-ring resonator (CSRR), flexible polarization, metamaterial, microstrip antennas, miniaturized antennas, multi-frequency antennas, reactive-impedance surface (RIS).

I. INTRODUCTION

E

LECTROMAGNETIC metamaterials have been a field of intense research activity with remarkable progress witnessed over the past decade [1]–[4]. Use of metamaterials for antennas is one of the most important applications currently being investigated, including both the resonant-type small antennas and the transmission-line type leaky-wave antennas [5]–[16]. Split-ring resonator (SRR) and its dual, complementary split-ring resonator (CSRR), have been the popular resonators which are widely used to synthesize metamaterials [4], [17]–[21]. CSRRs, originally introduced by Falcone et al. in 2004, have been proven to exhibit negative permittivity [21]. Their applications to miniaturize microwave devices and various antennas were widely investigated and presented [22]–[30].

Manuscript received November 02, 2010; revised April 05, 2011; accepted July 20, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. Y. Dong and T. Itoh are with the Electrical Engineering Department, 63-129 ENGR-IV, University of California at Los Angeles, Los Angeles, CA 90095 USA (e-mail: [email protected]; [email protected]). H. Toyao is with the System Jisso Research Laboratories, NEC Corporation, Sagamihara Kanagawa 211-8666, Japan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173120

As a sub-field of metamaterials, meta-surface has also drawn increasing attention in recent years, finding widespread applications in microwave circuits and antennas [31]–[38]. There are several different types of meta-surfaces, such as the reactive impedance surface (RIS) composed of periodic metallic patches [31]–[34], the mushroom-like high-impedance or artificial magnetic surface (AMS) [35], [36], and the UC-PBG surface [37], [38]. The AMS is able to suppress the surface wave as well as to provide a zero reflection phase. It is shown in [31] that the RIS can be utilized to miniaturize the antenna size and improve the radiation performance. This paper presents a comprehensive investigation into the patch antennas loaded by CSRRs over an RIS based on some preliminary research shown in [30]. The CSRR is embedded on factor resonator which can the top surface as a high-quality couple the field to the antenna patch and make it radiate. The structure of the adopted CSRR and its equivalent-circuit model are depicted in Fig. 1. The CSRR is modeled as a shunt LC resonator tank [19] which can be excited by the orthogonal electric field. It can be equivalent to an electric dipole placed along the ring axis [19]. As a dipole it essentially generates wave propagating along the plane of ring surface and relies on the edges of patch for radiation. The coupling between the CSRR and patch mainly comes from the capacitive coupling through the ring slot and the magnetic coupling through the split of the outer ring. By properly feeding the antenna, the inherent half-wavelength patch resonant mode can still be well excited. It is interesting to note that the interaction between the CSRR-inspired resonance and the patch resonance is very weak when they are orthogonally polarized. Under this condition circular polarization (CP) is attainable when they share the same operating frequency with a 90 phase delay in excitation. In addition the interaction is strong when they are polarized in the same plane, which gives rise to two mixed modes. The RIS is employed to further decrease the resonance frequency and improve the antenna radiation performance. When it works as an inductive surface it is able to store the magnetic energy and increases the inductance value of the patch type resonance. The resonance frequency of the patch, which is inherently a parallel RLC resonator [39], is shifted down in this way resulting in the antenna miniaturization. It is shown that the antenna polarization state can be easily changed by altering the configuration of the CSRRs. Dual- and triple-band operations can also be achieved by appropriately exciting the CSRRs and the microstrip patch. A circularly-polarized antenna has also been developed by exciting two orthogonally-polarized modes with a 90 phase difference. Six different

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Fig. 1. (a) Topology and (b) its equivalent circuit model of the CSRR [19]. Gray zone represents the metallization.

antennas are fabricated and measured to verify the simulation based on the multi-layer PCB process. The proposed antennas show advantages in terms of the compact size, low-fabrication cost, low-cross polarization level and the multi-band operation with flexible polarization states. In the following parts the six antennas will be studied and presented in each different section with a detailed analysis provided to the first antenna in Section II. Here the simulation is performed using Ansoft’s High Frequency Structure Simulator (HFSS) software package. II. PATCH ANTENNA LOADED WITH CSRRs AND RIS In this section, the characters of the CSRR-loaded patch antenna over an RIS will be investigated and discussed in detail using an antenna model loaded with two CSRRs which are face-to-back oriented. An equivalent circuit for the proposed structure is derived to gain an insight into the working principle. The design of CSRRs is summarized. The features and influence of the RIS is also presented. It is noted that the major results presented in this section have already been published in [30]. A. Configuration Fig. 2 shows the geometrical layout of the proposed antenna with two CSRRs face-to-back oriented with respect to the direction of the ring split. This configuration is chosen here since it is simpler than the side-by-side configuration which will be discussed later. A coaxial probe-feeding is utilized and placed in the center of the microstrip patch. Due to this center feeding no patch resonances can be excited. Also the alternative cases with two CSRRs either face-to-face or back-to-back oriented could not radiate well due to a symmetrical structure which cancels all the radiation from the patch edges. Under those conditions the main radiation should come from the ring slot of the CSRR. However, as explained later the CSRR is a high- resonator instead of a good radiator. To verify this conclusion we also simulated these two cases and found that the resonance could still be excited at a little higher frequency but the radiation efficiency for both of the two cases is below 0.7%. Face-to-back case is a good option since the patch can radiates well. The RIS, which is composed of a periodic array of metallic square patches printed on a metal-backed dielectric substrate, is introduced below the top surface. It is a three-layer structure where the top and bottom dielectric substrate is “MEGTRON 6” with a relative permittivity of 4.02 and a measured loss tangent of 0.009 at 2.4 GHz. B. Equivalent Circuit Model Fig. 3(a) shows the circuit model of a conventional probe-fed microstrip patch antenna. The input impedance of the patch antenna is modeled as an RLC resonator near its resonance fre-

Fig. 2. Configurations of the proposed CSRR-loaded patch antenna over an RIS. The CSRRs are face-to-back oriented and the feeding probe is in the center. (a) Perspective view, (b) top view, and (c) side view.

quency [39], [40]. The series inductor represents an inductive probe feeding. The structure of Fig. 2(a) can be roughly represented by the circuit model shown in Fig. 3(b). The CSRR is modeled as a high- shunt-connected RLC resonator tank ( , and ) which has been designed to exhibit a lower resonance frequency compared with microstrip patch. denotes the losses including both the conductor and dielectric losses.

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Fig. 3. Equivalent circuit model for (a) conventional probe-fed patch antenna, and (b) proposed patch antenna loaded with CSRRs as shown in Fig. 1.

The probe inductance is represented by . Due to the equivalence to an electric dipole, the wave generated by the CSRR is mainly propagating along the plane inside the substrate and radiates when it arrives at the edges of the patch. The CSRR itself has little radiation since the radiation from the ring slot would cancel itself. To verify this viewpoint we have created a model with the CSRR etched on a rectangular cavity. Unlike other cavity-backed slot antennas, little radiation through the ring slot is detected and most of power is dissipated by the loss, which demonstrates that the CSRR itself is not a good radiator. The field can be coupled from the CSRR to the patch and radiates away using the radiation resistance of the microstrip patch. It is noted that this coupling is a combination of the inductive and the capacitive couplings, where the former comes from the connection through the split of the outer ring and the latter comes from slot coupling. The same situation applies to the coupling between the feeding probe and the CSRR. Here it is also pointed out that this circuit model is just a simplified approximation where we have neglected the direct coupling between the two CSRRs on the two sides. Since the structure is not symmetrical, the coupling for the two CSRRs is different. The series inductance is relatively larger when the ring split is far away from the feeding probe on the other side. Since the value of the coupling elements is different, the resonance frequency looking into circuit also varies. Therefore, two resonances could be observed on the input impedance which will be shown later in Fig. 11(a). C. CSRR It would be helpful to know the characters and design methodology for the CSRRs while designing the proposed CSRR-loaded patch antennas. The CSRR can be represented by an LC resonator tank as shown in Fig. 1 when the loss is neglected. Its inherent resonance frequency is determined by (1) where the capacitance of the CSRR is approximately equal to that corresponding to a metallic disk surrounded and backed by the ground plane [19]. Here the inductance can be calculated based on a CPW structure with an equivalent perimeter of the

Fig. 4. An investigation into the features of the CSRR inherent resonance frequency based on HFSS eigen-mode simulation. The initial CSRR is etched on the surface of a rectangular cavity and resonates at 3.075 GHz. (a) Field distribution at the CSRR resonance frequency; (b) the resonance frequency versus the substrate thickness ; (c) the change of the resonance frequency with difis fixed here); (d) the variation of resoferent CSRR geometries ( nance frequency for different slot width . (e) Simulated resonance frequency mm, mm, versus the strip width . The parameters are: mm, mm and mm.

CSRR, strip width , and slot width . The detailed properties for the CSRR is presented in [19], including the analytical calculation of the resonance frequency. However those equations are lengthy and calculation would become extremely difficult for irregular CSRR structures. Here we adopted an approach based on the eigen-mode simulation to quickly obtain the resonance frequencies. In this setup, the CSRR is etched on the surface of a dielectric-filled rectangular cavity. Note that cavity itself is resonating at a much higher frequency. First we simulated an example given in [19] and found a good agreement. Then based on this method we extracted the resonance frequency for the rectangular CSRRs used in our antennas. Fig. 4(a) shows the field distribution from eigen-mode simulation for the CSRR used in this design. A series of simulations has been carried out by changing the parameters to investigate the properties of the CSRR. The results are displayed in Figs. 4(b) to (e). Fig. 4(b) shows the influence of the substrate thickness to the resonance frequency. The decrease of thickness corresponds to an enhancement of the capacitance which leads to a decrease of the resonance frequency. Fig. 4(c) shows how the variation of the geometry affects the resonance frequency. The perimeter is set to be fixed. It is seen that the CSRR

DONG et al.: DESIGN AND CHARACTERIZATION OF MINIATURIZED PATCH ANTENNAS

resonance frequency has a very small variation when the geometry changes. It has a minimum value for a square shape. This is predictable since this shape gives the maximum area resulting in a maximum and finally a minimum resonance frequency. It is worth noting that for the CSRR-loaded patch, the coupling can be adjusted by the change of the CSRR geometry since the CSRR frequency is almost not affected. When slot width is reduced, the capacitance is increased and the resonance frequency is decreased. Fig. 4(d) verifies this conclusion. Fig. 4(e) shows the increase of inductance , achieved by reducing the strip width g, could also decrease the resonance frequency. Other parameters, such as the dielectric constant and CSRR size, may be used to control the frequency as well. Note the above results are obtained without the consideration of RIS.

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Fig. 5. Unit-cell and its equivalent circuit of the RIS bounded with PEC and PMC walls and illuminated by a normal incident plane wave.

D. RIS The RIS is first proposed and studied in [31]. Here a brief investigation about the features is presented. As shown in Fig. 2 it is composed by two dimensional periodic metallic patches printed on a grounded substrate. The periodicity of the metallic patches is much smaller than the wavelength. Considering a single cell illuminated with a TEM plane wave, PEC and PMC boundaries can be established around the cell as shown in Fig. 5 [31]. The resulting structure can be modeled as a parallel LC circuit displayed in Fig. 5. The edge coupling of the square patch provides a shunt capacitor and the short-circuited dielectric loaded transmission line can be modeled as a shunt inductor. The impedance then can be obtained as (2) where (3) The calculation of and has been detailed in [31]. The variation of the patch size and slot width mainly changes the capacitor value while the substrate thickness and dielectric constant mainly affects the inductance value, all of which can be used to control the resonance frequency. Either an inductive RIS (below the PMC surface frequency) or a capacitive RIS (above the PMC surface frequency) can be obtained depending on the geometry and the operating frequency [31]. Note that since the near field generated by the patch antenna is not a uniform plane wave and the meta-surface is size-limited far from being periodic, the thinking or design of a radiating patch over the meta-surface (RIS) using the unit-cell analysis shown in Fig. 5 is just an approximation to qualitatively explain its working principle. To better explain the role played by RIS an analysis into the near field interaction would be more meaningful and essential. As demonstrated in [31] and [36], due to the matching difficulty and loss problem, PMC surface is not a proper choice. An inductive RIS is able to store the magnetic energy which thus increases the inductance of the circuit. Therefore, it can be used to miniaturize the size of a patch type antenna which is essentially an RLC parallel resonator. At the same time it is shown in [31] the inductive RIS is also able to provide a

Fig. 6. A parameter study on the RIS for the proposed antenna shown in Fig. 2. while It shows the different simulated reflection coefficients by (a) varying fixed (3 mm). (b) Varying while keeping mm. keeping All the other parameters are the same as shown later in Fig. 10.

wider matching bandwidth therefore it is more suitable for antenna application. E. Simulated and Measured Results Based on HFSS commercial software package, this antenna loaded by CSRR and RIS as shown in Fig. 2 is designed and optimized at a working frequency of 2.4 GHz. To have more design information about the RIS, a parametric study is first performed by changing the slot width between the RIS patches and the RIS thickness. The result is shown in Fig. 6. Note that here in all these simulations the ground size is assumed to be 34 mm 34 mm and the patch size is 12.4 mm 19.2 mm. It is seen that the resonance frequency can be pushed down by either increasing the equivalent capacitance of the RIS or increasing the equivalent inductance. To provide a better understanding about the

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Fig. 7. A comparison of for the proposed antenna with different RIS unit-cell numbers. All other parameters remain unchanged in the simulation.

function of the RIS, Fig. 7 shows the variation of antenna resonance frequency by changing of the RIS unit-cell numbers. It is seen that for the case without the RIS, the resonance frequency is 2.75 GHz shown by both the simulation and measurement. By adding the RIS, the resonance frequency has been moved down to 2.4 GHz. It is important to note that as long as the RIS covers the mircostrip patch size (minimum 3 5 unit-cells in this case), the increase of the unit-cell number only poses a very weak influence on the resonance frequency. The reason is that outside the microstrip patch region the field is weak. It is also observed that further reducing the unit-cell number could change the resonance frequency distinctly. Fig. 8 shows a parameter analysis on the size of the CSRR and the microstrip patch. It is seen that the resonance frequency is mainly determined by the CSRR while the patch size also affects it but in a weaker manner. Fig. 9 shows a photograph of the fabricated antenna. The patch size is around , which is very compact. The sizes of the RIS and ground are and , respectively. The reflection coefficient of the antenna is measured and plotted in Fig. 10, compared with the results from the circuit simulation and full-wave simulation using both HFSS and CST. A good agreement between them is observed. It is also important to bear in mind that since the circuit simulation cannot include the radiation for an antenna and it is simplified for the couplings and the losses, this is just a rough approximation used to explain its working principle. The values for those lumped elements are listed in the caption. Two resonance frequencies are observed which are exactly from the two circuit branches shown in Fig. 3(b). The designed resonance is found to be at 2.406 GHz in the measurement and it exhibits an impedance match better than 20 dB. The measured relative bandwidth is 1.04%. The second resonance at 2.84 GHz is not matched as predicted by the input impedance shown in Fig. 11(a). The simulated radiation pattern at this frequency indicates that this second resonance is also polarized in plane. The field distribution at the two resonance frequencies is plotted in Fig. 11(b). It is clearly seen that the field is coupled to the microstrip patch and radiated from the two patch edges. The CSRRs are resonating separately at each of the resonance frequencies. Due to the poor matching for the second resonance the field is only strong along the lower part of the patch and it is not well radiating. It should be pointed out that the two

Fig. 8. Simulated reflection coefficients by (a) varying while keeping the (the size of patch). other parameters unchanged, and (b) varying

Fig. 9. Photograph of the fabricated patch antenna loaded with face-to-back CSRRs and the RIS.

CSRRs have the same inherent resonance frequency. However, the appeared resonance frequencies here are not from the CSRR only but from the whole circuit. Because of the different orientation, the couplings are also different which leads to different resonance frequencies shown in Fig. 10. The equivalent circuit shown in Fig. 3(b) is developed in order to provide a clear picture about this working mechanism. The radiation patterns are measured in a far-field anechoic chamber and are plotted in Fig. 12. It is noted that the antenna is polarized in plane as shown in Fig. 11(b). The measured gain and front-to-back ratio at 2.406 GHz are 2.02 dBi and 6.5 dB, respectively, which agree with the simulation. Since the

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Fig. 12. Measured and simulated far-field patterns in -plane ( plane) plane) at the resonance frequency. The scale is 5 dB per and -plane ( division. Fig. 10. The measured reflection coefficient compared with the results from circuit simulation and full-wave simulation using HFSS and CST. The equivalent circuit parameter values are: nH, , nH, , nH, , nH, , nH, , nH, nH, , , The geometrical parammm, mm, mm, mm, eters are: mm, mm, mm, mm, mm, mm, mm and mm.

with the antenna without the RIS, the patch antenna with RIS but without the CSRR, and the simple patch antenna without the RIS or CSRR. For the patch antennas without the CSRR we excited the direction resonance by changing probe position, which means that they have the same polarization and patch size. Table I shows the comparison on the resonance frequency, simulated gain and radiation efficiency. III. DUAL-BAND DUALLY-POLARIZED ANTENNA WITH FACE-TO-BACK CSRRs

Fig. 11. (a) Input impedance from HFSS full-wave simulation and equivalent circuit model shown in Fig. 3, and (b) electric field distribution at the two resplane which is onance frequencies. The two resonances are polarized in indicated by their radiation patterns.

antenna is electrically small a relatively low gain is expected. The measured and simulated radiation efficiency is 22.54% and 25.5%, respectively. Here all the measured radiation efficiency is obtained using the gain/directivity (G/D) method. And the antenna directivity is measured in the near-field chamber in our department. The relatively low radiation efficiency is also due to the high dielectric loss of the substrate. Finally, it is interesting to give a comparison for the proposed CSRR-loaded antennas

The antenna shown in the previous section is excited by a probe fed in the center and is polarized in plane. By moving the probe off the center along -direction, the original 2.4 GHz resonance still exists only with the matching a little affected, while a conventional microstrip patch resonance can be excited simultaneously, which is polarized in plane. Note that when the antenna is fed in the center no half-wavelength patch resonance can be excited due to a symmetrical structure requiring a symmetrical field distribution. Also since this patch resonance is orthogonally polarized compared with the CSRRinspired resonance, their interaction is relatively small. Fig. 13 presents the detailed structure and a photograph of the fabricated antenna. It is noted that this antenna has exactly the same dimensions as the first antenna except the difference in the probe feeding position. Fig. 14 shows the simulated and measured reflection coefficient of the antenna. The two bands are measured at 2.41 GHz and 3.82 GHz. Fig. 15 shows the electric field distribution at the two resonance frequencies. The simulated and measured radiation patterns are plotted in Fig. 16. As expected the two resonances have orthogonal polarizations. The first resonance is polarized in plane while the second one is polarized in plane. The measured gain at the two resonance frequencies is 2.13 dBi and 5.04 dBi, respectively. The measured bandwidth for the two bands is 0.91% and 1.76%. The front-to-back ratio is 5.17 dB at the first frequency and 12.87 dB at the second frequency. The corresponding radiation efficiencies are 25.3% and 80.2% in the simulation, and 22.8% and 74.5% in the measurement. The low efficiency at the first band is due to the high conductor and dielectric losses. The CSRR is essentially a high- resonator which induces strong current and field along the ring position. This strong current would take away considerable power resulting in a reduced efficiency. By

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TABLE I A COMPARISON FOR THE ANTENNAS WITH THE SAME SIZE BUT WITH DIFFERENT LOADINGS

Fig. 13. (a) Perspective view and (b) a photograph of the proposed dual-band dually-polarized antenna with face-to-back CSRRs. Parameter value are the mm. same as the first one,

Fig. 16. Measured and Simulated far-field patterns at (a) 2.4 GHz, and plane for the first band and plane (b) 3.79 GHz. The -plane is for the second band. The display scale is 5 dB per division. Fig. 14. Measured and simulated reflection coefficient for the dual-band antenna loaded with face-to-back CSRRs.

Fig. 15. Electric field distribution at the two operating frequencies: 2.4 GHz and 3.79 GHz.

employing low-loss material the efficiency can be substantially increased. IV. DUAL-BAND EQUALLY-POLARIZED ANTENNA From the antenna discussed in Section II, it is seen that the polarization of the antenna resonance excited by the CSRRs is mainly determined by their orientations. In this section a dualband antenna with the same polarization is designed by utilizing

this feature. Two CSRRs are side-by-side equally placed on the patch as shown in Fig. 17. It is worth noting that all the antennas designed here use the same substrate as the first antenna indicated by Fig. 2(c). The CSRRs are embedded in the middle of the patch along the -direction. A photograph of the fabricated antenna is displayed in Fig. 17. In terms of the wavelength of the first resonance frequency, the patch size is around , and the sizes of the RIS and ground are and , respectively. Fig. 18 shows the measured and simulated reflection coefficient. It is seen that two resonances are excited with a good impedance matching. The resonance frequencies are simulated at 2.37 GHz and 2.93 GHz, and are measured to be 2.386 GHz and 2.958 GHz. The measured 10 dB bandwidth is 1.32% for the first band and 2.68% for the second band. It is also noted that the patch resonance by removing the CSRR occurs at 2.88 GHz. The initial patch and the CSRRs couple to each other generating two mixed modes polarized in the same direction. The coupling is through both the electric and magnetic couplings. Since the orientation of the CSRR coincides with the patch antenna polarization plane, which facilitates the interaction between them, the coupling is substantially enhanced. Fig. 19 shows the field distribution for the two resonances. It is observed that the field

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Fig. 19. Electric field distribution at the two operating frequencies: 2.37 GHz and 2.93 GHz.

Fig. 17. (a) Perspective view, (b) a photograph, and (c) top view of the proposed dual-band equally-polarized antenna with side-by-side CSRRs. The gemm, mm, mm, ometrical parameters are: mm, mm, mm mm, mm, mm, mm and mm.

Fig. 20. Simulated by (a) varying where , and (b) varying (the length of the patch). Other parameters remain the same. Fig. 18. Measured and simulated reflection coefficient for the dual-band equally-polarized antenna loaded with side-by-side CSRRs.

is strong along the left edge of the patch for the first resonance, meaning a strong slot coupling and a large coupling inductor (weak inductive coupling due to a large distance). While for the second resonance the field is strong along the right edge, meaning a smaller coupling inductor due to a small distance. Fig. 20 gives a parametric study on the size of the CSRR and the patch. It is seen that both of them can be used to control the antenna resonance frequencies, which indicates that the patch and the CSRR are mixed giving rise to two resonating modes. Fig. 21 shows the measured and simulated radiation patterns for the antenna. The cross polarization level is too low to be observed in the plot. The gain at the two resonance frequencies is simulated to be 0.1 dBi and 2.99 dBi, which correspond

to 40.7% and 63.8% simulated radiation efficiencies. The measured gain is 0.21 dBi and 3.13 dBi, corresponding 38.5% and 59.3% measured radiation efficiencies. The measured front-toback ratio is 8.16 dB and 14 dB, respectively. V. DUAL-BAND DUALLY-POLARIZED ANTENNA WITH SIDE-BY-SIDE CSRRs In this section we show that dual-band antenna with orthogonal polarizations can also be obtained by side-by-side reversing the orientation of the CSRRs. The structure and photograph of the fabricated antenna is shown in Fig. 22. In terms of the wavelength of the first resonance frequency, the patch size is around , and the sizes of the RIS and ground are and , respectively. The CSRRs are side-by-side reversely placed in the center of the patch. It is to be noted that due to the

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Fig. 23. Electric field distribution at the two operating frequencies: 2.27 GHz and 2.78 GHz.

Fig. 21. Measured and Simulated far-field patterns at (a) the first resonance plane frequency, and (b) the second resonance frequency. The -plane is for both of the two resonances. The display scale is 5 dB per division.

Fig. 24. Simulated reflection coefficients by (a) varying , and (b) varying (the patch size) where where

Fig. 22. (a) Perspective view, (b) a photograph, and (c) top view of the proposed dual-band, orthogonally-polarized antenna with side-by-side CSRRs. The mm, mm, mm, geometrical parameters are: mm, mm, mm mm, mm, mm, mm and mm.

configuration the two CSRRs reach the electric field maximum with a phase difference of 180 , which means that the positive maximum for one CSRR corresponds to the negative maximum for the other one. This is also indicated by the field distribution shown in Fig. 23. Unlike the symmetrical configuration of the

(the CSRR size) .

previous antenna, for this case the field can be coupled from one CSRR to the other one directly. The coupling between two CSRRs has also been studied in [26]. The resonance generated here by the CSRRs is polarized in which is along the diagonal line of the square patch. Another resonance, which is the inherent patch resonance, is excited along the perpendicular direction as shown in Fig. 23, which is similar to the corner-truncated or corner-fed dually polarized patch antennas [41]. This pair of resonances is generated independently with little interference. Fig. 24 shows a parametric study by changing the patch and the CSRR. It is obviously seen that the first resonance is mainly determined by the CSRRs while the second one is mainly controlled by the microstrip patch. Fig. 25 shows measured and simulated reflection coefficient, where the two resonances are observed at 2.27 GHz and 2.78 GHz in the simulation, and 2.31 GHz and 2.83 GHz in the measurement. A

DONG et al.: DESIGN AND CHARACTERIZATION OF MINIATURIZED PATCH ANTENNAS

Fig. 25. The measured and simulated reflection coefficients for the dual-band, orthogonally-polarized antenna loaded with side-by-side CSRRs.

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Fig. 27. (a) The perspective view and (b) a photograph of the proposed and fabricated circularly-polarized antenna with side-by-side CSRRs. The geometrical mm, mm, mm, mm, parameters are: mm, mm mm, mm, mm, mm, mm and mm.

Fig. 28. The measured and simulated reflection coefficients for the circularlypolarized antenna loaded with side-by-side CSRRs.

is 24.45% and 71.8%, respectively. The discrepancy is mainly caused by the shift of the resonance frequency and probably a smaller loss tangent for the real material. VI. CIRCULARLY-POLARIZED ANTENNA Fig. 26. Measured and simulated far-field patterns at (a) the first resonance frequency, and (b) the second resonance frequency. The -plane is at for the first band and for the second band.

little frequency shift is observed which is probably due to the change of the dielectric constant and the fabrication error. The measured 10 dB bandwidth for the two bands is 1.38% and 3.29%. It should be pointed out that there is another resonance around 2.97 GHz which is brought by the CSRRs. However it is not matched and appears to be very weak. Fig. 26 shows the measured and simulated radiation patterns. The measured front-to-back ratio for the two bands is 8.33 dB and 14.4 dB, respectively. The gain is simulated to be 3.14 dBi and measured as 2.09 dBi for the lower resonance. For the second one it is simulated as 3.13 dBi and measured to be 3.85 dBi. Considering the patch size is only, the antenna efficiency is relatively low, which is simulated to be 22% for the first resonance and 69.2% for the second resonance. The measured antenna radiation efficiency for the two bands

Based on the dual-frequency, orthogonally-polarized antenna proposed in the above section, a circularly-polarized antenna is designed here. The principle is to overlap the two working frequencies and excite these two resonances with a 90 phase difference. Since the probe feed is in the center and the wave goes to the two diagonal lines oppositely with 45 phase delay, the 90 phase difference can be automatically introduced. The reason is that at the resonance frequency, the wave travels from one edge to the opposite edge with 180 phase delay. Now the wave propagates from the probe to the diagonal plane covering an angle of only 45 , which give rise to 45 phase delay. Here only the impedance matching needs to be improved, which is the -position of the probe feed. The size of the CSRRs is scaled down in order to push up its resonance frequency. The final structure, as well as a photograph of the fabricated antenna, is shown in Fig. 27. This antenna has a patch size of , an RIS size of , and a ground size of . The measured and simulated reflection coefficient is shown in Fig. 28. The center frequency is 2.8 GHz and the 10 dB bandwidth is 5.03% in the simulation. And in the measurement they are 2.824 GHz and 4.9%.

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Fig. 29. Measured and simulated far-field patterns at the center frequency in plane and plane. The display scale is 5 dB per division.

Fig. 32. Three-dimensional (a) radiation pattern and (b) AR measured in a spherical near field chamber at the center frequency.

Fig. 30. Measured and simulated AR and the realized gain for the CP antenna.

Fig. 33. (a) Perspective view, (b) a photograph, and (c) top view of the proposed triple-band antenna with different polarizations. The geometrical parammm, mm, mm, mm, eters are: mm, mm, mm, mm, mm, mm, mm, mm, mm and mm.

Fig. 31. Simulated AR at the center frequency in two different planes.

The radiation characters of the CP antenna are tested in the UCLA spherical chamber. The measured and simulated radiation patterns in and plane are shown in Fig. 29. The discrepancy is mainly due to the interference of the testing equipment. The measured and simulated gain and axial ratio (AR) are shown in Fig. 30. The measured and simulated AR at center frequency in plane and plane is also provided in Fig. 31. The bandwidth for AR less than 3 dB is observed as 1.60% in the simulation and 1.68% in the measurement. Fig. 32 shows the measured three-dimensional AR and radiation patterns. It is seen that a CP radiation is retained in a very wide region and the pattern is very similar to a traditional patch antenna. This antenna radiation efficiency is observed to be 80% in the simulation and 74.1% in the measurement.

VII. TRIPLE-BAND ANTENNA WITH VARIED POLARIZATIONS In this section a triple-band antenna with different polarization states is developed. The structure, as shown in Fig. 33, is similar to the previous two antennas. Two CSRRs are side-byside reversely embedded on the top surface. They are shifted from the patch center by . Also the patch itself is not a square patch. This structure is able to generate three resonances at same time with a proper feeding. Two of them, the first and the third one, come from the CSRRs and the second one is mainly excited by the microstrip patch. This is justified by comparing it with the inherent patch resonance frequency and checking the field distribution. We also found that compared with the other two resonances the CSRR size is not very influential to the second resonance frequency. Fig. 33(b) shows a photograph of the fabricated antenna. “MEGTRON 6” with a relative permittivity of 4.02 is also used here as the substrate. In terms of the wavelength of the first resonance frequency, this antenna exhibits a

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Fig. 34. Measured and simulated reflection coefficient for the triple-band antenna loaded with side-by-side CSRRs.

Fig. 35. Electric field distribution at the three operating frequencies: 2.4 GHz, 2.8 GHz and 3.4 GHz.

patch size of , an RIS size of , and a ground size of . Fig. 34 shows the simulated and measured reflection coefficient. As seen the three bands are well matched to be below 22 dB. The measured 10 dB bandwidth for the three bands is 1.61%, 3.27%, and 3.08%, respectively. Fig. 35 shows the field distribution at the three resonance frequencies. Clearly they have different polarization angles. The CSRRs are strongly resonating in the first and third resonances. The second resonance should mainly come from the patch itself. To verify this we calculated and found that the inherent patch resonance (without CSRRs) in -direction occurs at 2.78 GHz, which is very close to the resonance frequency of this second mode. Their -planes are located at , 54 , and 160 , respectively. To confirm this, a near field measurement was performed in the spherical near field chamber which directly verifies the polarization angles from simulation. Fig. 36 shows the measured pattern from a far-field measurement, compared with the simulated data. It is noted that they are measured in their own -plane and -planes independently. The measured front-to-back ratio for the three bands is 6.5 dBi, 8.3 dBi and 13.0 dBi. The gain measured at these three resonance frequencies is 0.27 dBi, 3.31 dBi, and 4.45 dBi, respectively. Their corresponding measured efficiencies are 43.7%, 69.8% and 75.5%, which are very close to the simulated efficiencies: 41.9%, 68.5% and 77.62%. Finally, it is interesting to note that their principle polarization angles can be steered simply by changing the position of the CSRRs ( : the x-distance between the patch center and the CSRR center). Fig. 37 shows the simulated -plane angle variations for all the three modes. It is seen they are rotated together as the CSRRs move along the -direction. The reason for this

Fig. 36. Measured and simulated far-field patterns in their principle -plane and -plane at (a) 2.426 GHz, (b) 2.845 GHz, and (c) 3.373 GHz. The display scale is 5 dB per division.

Fig. 37. The variation of the principle -plane angles by changing sition of the CSRRs). Other parameters remain unchanged.

( -po-

variation is that the split of the CSRRs plays an important role in the wave coupling which affects the polarization angle. The position of the CSRRs also affects the patch resonance since they block the original wave propagation and lead to the modification of the field pathway. Table II summarizes the influencing factors on polarization for the different CSRR-loaded antennas.

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TABLE II POLARIZATION INFLUENCING FACTORS FOR A CSRR-LOADED PATCH ANTENNA WITH FIXED CSRR AND PATCH CONFIGURATION

Note that besides the above factors, the patch geometry and feeding position are also capable of changing the polarization of the resonances.

VIII. CONCLUSION A comprehensive study on the CSRR-loaded patch antennas over an RIS has been presented in this paper. The employment of the meta-resonator and the meta-surface enables the antenna miniaturization. These antennas can be designed to perform specified functions, such as the circularly-polarized radiation, dual- and triple-band operation with the same or different polarizations. The antenna radiation characteristics can be easily controlled by changing the configuration of the CSRRs. Wideband operation by combining different resonances is also feasible which is under our current investigation. These CSRRloaded patch antennas are compact, simple in structure, but versatile in applications. REFERENCES [1] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. New York: Wiley-IEEE Press, 2005. [2] G. Eleftheriades and K. Balmain, Negative Refraction Metamaterials: Fundamental Principles and Applications. New York: Wiley-IEEE Press, 2005. [3] N. Engheta and R. W. Ziolkowski, Electromagnetic Metamaterials: Physics and Engineering Explorations. New York: Wiley-IEEE Press, 2006. [4] R. Marques, F. Martin, and M. Sorolla, Metamaterials With Negative Parameters: Theory, Design and Microwave Applications. New York: Wiley, 2008. [5] H. R. Stuart and A. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1644–1653, Jun. 2006. [6] R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 2113–2130, Jul. 2006. [7] C. Lee, K. M. Leong, and T. Itoh, “Composite right/left-handed transmission line based compact resonant antennas for RF module integration,” IEEE Trans. Antennas Propag., vol. 54, no. 8, pp. 2283–2291, 2006. [8] A. Alu, F. Bilotti, N. Engheta, and L. Vegni, “Subwavelength, compact, resonant patch antennas loaded with metamaterials,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 13–25, Jan. 2007. [9] J. H. Park, Y. H. Ryu, J. G. Lee, and J. H. Lee, “Epsilon negative zerothorder resonator antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3710–3712, Dec. 2007. [10] M. Antoniades and G. V. Eleftheriades, “A folded-monopole model for electrically small NRI-TL metamaterial antennas,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 425–428, 2008. [11] Y. Dong and T. Itoh, “Miniaturized substrate integrated waveguide slot antennas based on negative order resonance,” IEEE Trans. Antennas Propag., vol. 58, no. 12, 2010.

[12] T. Kokkinos, C. D. Sarris, and G. V. Eleftheriades, “Periodic FDTD analysis of leaky-wave structures and applications to the analysis of negative-refractive-index leaky-wave antennas,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1619–1630, Jun. 2006. [13] T. Ueda, N. Michishita, M. Akiyama, and T. Itoh, “Dielectric resonator based composite right/left-handed transmission lines and their application to leaky wave antenna,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 10, pp. 2259–2268, Oct. 2008. [14] S. Paulotto, P. Baccarelli, F. Frezza, and D. Jackson, “Full-wave modal dispersion analysis and broadside optimization for a class of microstrip CRLH leaky-wave antennas,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 2826–2837, Dec. 2008. [15] T. Kodera and C. Caloz, “Uniform ferrite-loaded open waveguide structure with CRLH response and its application to a novel backfire-to-endfire leaky-wave antenna,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 4, pp. 784–795, Apr. 2009. [16] Y. Dong and T. Itoh, “Composite right/left-handed substrate integrated waveguide and half mode substrate integrated waveguide leaky-wave structures,” IEEE Trans. Antennas Propag., vol. 59, no. 3, pp. 767–775, Mar. 2011. [17] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [18] S. Hrabar, J. Bartolic, and Z. Sipus, “Waveguide miniaturization using uniaxial negative permeability metamaterial,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 110–119, Jan. 2005. [19] J. D. Baena, J. Bonache, F. Martin, R. Marques, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia, I. Gil, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1451–1461, Apr. 2005. [20] T. Decoopman, A. Marteau, E. Lheurette, O. Vanbesien, and D. Lippens, “Left-handed electromagnetic properties of split-ring resonator and wire loaded transmission line in a fin-line technology,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1451–1457, Apr. 2006. [21] F. Falcone, T. Lopetegi, J. D. Baena, R. Marques, F. Martin, and M. Sorolla, “Effective negative-epsilon stopband microstrip lines based on complementary split ring resonators,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 14, pp. 280–282, Jun. 2004. [22] J. Bonache, I. Gil, J. Garcia, and F. Martin, “Complementary split ring resonators for microstrip diplexer design,” Electron. Lett., vol. 41, no. 14, Jul. 2005. [23] J. Niu and X. Zhou, “A novel dual-band branch line coupler based on strip-shaped complementary split ring resonators,” Microw. Opt. Technol. Lett., vol. 49, no. 11, pp. 2859–2862, Nov. 2007. [24] Y. Zhang, W. Hong, C. Yu, Z. Kuai, Y. Dong, and J. Zhou, “Planar ultrawideband antennas with multiple notched bands based on etched slots on the patch and/or split ring resonators on the feed Line,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 3063–3068, Sep. 2008. [25] H. Zhang, Y. Q. Li, X. Chen, Y. Q. Fu, and N. C. Yuan, “Design of circular polarization microstrip patch antennas with complementary split ring resonator,” IET Microw. Antennas Propag., vol. 3, no. 8, pp. 1186–1190, Aug. 2009. [26] Y. Dong, T. Yang, and T. Itoh, “Substrate integrated waveguide loaded by complementary split-ring resonators and its applications to miniaturized waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 9, pp. 2211–2223, Sep. 2009. [27] S. Eggermont, R. Platteborze, and I. Huynen, “Investigation of metamaterial leaky wave antenna based on complementary split ring resonators,” in Proc. Eur. Microw. Conf., Rome, Italy, Sep. 2009, pp. 209–212. [28] N. Ortiz, F. Falcone, and M. Sorolla, “Dual band patch antenna based on complementary rectangular split-ring resonators,” in Proc. Asia-Pacific Microw. Conf. (APMC’2009), Singapore, Dec. 2009, pp. 2762–2765. [29] H. Zhang, Y. Q. Li, X. Chen, Y. Q. Fu, and N. C. Yuan, “Design of circular/dual-frequency linear polarization antennas based on the anisotropic complementary split ring resonator,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3352–3355, Oct. 2009.

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[30] Y. Dong and T. Itoh, “Miniaturized patch antenna loaded with complementary split-ring resonators and reactive impedance surface,” in Proc. Eur. Conf. Antennas Propag. 2011 (Eucap 2011), Rome, Apr. 2011, submitted for publication. [31] H. Mosallaei and K. Sarabandi, “Antenna miniaturization and bandwidth enhancement using a reactive impedance substrate,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2403–2414, Jun. 2004. [32] G. Goussettis, A. P. Feresidis, and J. C. Vardaxoglou, “Tailoring the AMC and EBG characteristics of periodic metallic arrays printed on grounded dielectric substrate,” IEEE Trans. Antennas Propag., vol. 54, pp. 82–89, 2006. [33] F. Yang, Y. Rahmat-Samii, and A. Kishk, “Low-profile patch-fed surface wave antenna with a monopole-like radiation pattern,” IET Microw., Antennas Propag., vol. 1, no. 1, pp. 261–266, Feb. 2007. [34] A. Al-Zoubi, F. Yang, and A. Kishk, “A low-profile dual band surface wave antenna with a monopole-like pattern,” IEEE Trans. Antennas Propag., vol. 55, pp. 3404–3412, Dec. 2007. [35] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopolus, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2059–2074, 1999. [36] F. Yang and Y. Rahmat-Samii, “Reflection phase characterizations of the EBG ground plane for low profile wire antenna applications,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2691–2703, Oct. 2003. [37] F. Yang, K. Ma, Y. Qian, and T. Itoh, “A uniplanar compact photonic-bandgap (UC-PBG) structure and its applications for microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 8, pp. 1509–1514, 1999. [38] A. Lamminen, A. R. Vimpari, and J. Saily, “UC-EBG on LTCC for 60-GHz frequency band antenna applications,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 10, pp. 2904–2912, Oct. 2009. [39] C. A. Balanis, Antenna Theory: Analysis and Design. New York: Wiley, 2005. [40] M. Manteghi, “Analytical calculation of impedance matching for probe-fed microstrip patch,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3972–3975, Dec. 2009. [41] K. L. Wong, Compact and Broadband Microstrip Antennas. New York: Wiley, 2002. Yuandan Dong (S’09) received the B.S. and M.S. degrees from Southeast University, Nanjing, China, in 2006 and 2008, respectively. He is currently working toward the Ph.D. degree in the department of electrical engineering, University of California at Los Angeles (UCLA). From September 2005 to August 2008, he was studying in the State Key Lab. of Millimeter Waves in Southeast University. Since September 2008, he has been a Graduate Student Researcher with the Microwave Electronics Laboratory in UCLA. He has authored more than 20 journal and conference papers. His research interests include the characterization and development of RF and microwave components, circuits, antennas and metamaterials. Mr. Dong is serving as a reviewer for several IEEE and IET journals including the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and IEEE TRANSACTION ON ANTENNAS AND PROPAGATION. He is the recipient of the Best Student Paper award from 2010 Asia Pacific Microwave Conference, Yokohama, Japan.

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Hiroshi Toyao received the B.S. and M.S. degrees in physics from Tokyo Institute of Technology, Tokyo, Japan, in 2000 and 2006, respectively. In 2006, he joined NEC Corporation, Japan, where he has been engaged in research on high-speed interconnection and electromagnetic compatibility. His current interests include EBG structures, metamaterials and antennas.

Tatsuo Itoh (S’69–M’69–SM’74–F’82–LF’06) received the Ph.D. degree in electrical engineering from the University of Illinois, Urbana, in 1969. After working for University of Illinois, SRI and University of Kentucky, He joined the faculty at The University of Texas at Austin in 1978, where he became a Professor of Electrical Engineering in 1981. In September 1983, he was selected to hold the Hayden Head Centennial Professorship of Engineering at The University of Texas. In January 1991, he joined the University of California, Los Angeles as Professor of Electrical Engineering and holder of the TRW Endowed Chair in Microwave and Millimeter Wave Electronics (currently Northrop Grumman Endowed Chair). He has 400 journal publications, 820 refereed conference presentations and has written 48 books/book chapters in the area of microwaves, millimeter-waves, antennas and numerical electromagnetics. He generated 70 Ph.D. students. Dr. Itoh received a number of awards including IEEE Third Millennium Medal in 2000, and IEEE MTT Distinguished Educator Award in 2000. He was elected to a member of National Academy of Engineering in 2003. He is a Fellow of the IEEE, a member of the Institute of Electronics and Communication Engineers of Japan, and Commissions B and D of USNC/URSI. He served as the Editor of IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES for 1983–1985. He was President of the Microwave Theory and Techniques Society in 1990. He was the Editor-in-Chief of IEEE MICROWAVE AND GUIDED WAVE LETTERS from 1991 through 1994. He was elected as an Honorary Life Member of MTT Society in 1994. He was the Chairman of Commission D of International URSI for 1993–1996, the Chairman of Commission D of International URSI for 1993–1996. He serves on advisory boards and committees of a number of organizations. He served as Distinguished Microwave Lecturer on Microwave Applications of Metamaterial Structures of IEEE MTT-S for 2004–2006.

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Dual-Band Circularly Polarized Microstrip RFID Reader Antenna Using Metamaterial Branch-Line Coupler Youn-Kwon Jung and Bomson Lee, Member, IEEE

Abstract—A dual-band circularly polarized aperture coupled microstrip RFID reader antenna using a metamaterial (MTM) branch-line coupler has been designed, fabricated, and measured. The proposed antenna is fabricated on a FR-4 substrate with relative permittivity of 4.6 and thickness of 1.6 mm. The MTM coupler is designed employing the provided explicit closed-form formulas. The dual-band (UHF and ISM) circularly-polarized RFID reader antenna with separate Tx and Rx ports is connected to the designed metamaterial (MTM) branch-line coupler. The maximum measured LHCP antenna gain is 6.6 dBic at 920 MHz (UHF) and RHCP gain is 7.9 dBic at 2.45 GHz (ISM). The cross-polar CP gains near broadside of the RFID reader antenna are approximately less than compared with the mentioned co-polar CP gains in both bands. The isolations between the two ports are about 25 dB and 38 dB, at 920 MHz and 2.45 GHz, respectively. The measured axial ratios are less than 0.7 dB in the UHF band (917–923 MHz) and 1.5 dB in the ISM band (2.4–2.48 GHz). Index Terms—Circular polarization, dual-band, metamaterial, RFID reader antenna.

I. INTRODUCTION

T

HE Radio-frequency identification (RFID) is a technology that uses communication via radio waves to exchange data between a reader and an electronic tag attached to an object, for the purpose of identification and tracking. The current RFID technology is employed in the fields of medicine, security, transportation, logistics, defense, and so on. The frequency bands assigned to RFID are 125–135 kHz (ISO 18000-2), 13.56 MHz (ISO 18000-3), 433.92 MHz (ISO 18000-7), UHF 860–960 MHz (ISO 18000-6), and ISM 2.45 GHz (ISO 18000-4). There is an increasing need to develop a dual-band RFID reader antenna and tag to accommodate two frequency bands in one structure. Circular polarization (CP) is also an important aspect of the RFID reader antenna. It is for signal reception regardless of the physical orientation of the tag. One more requirement for the RFID reader antenna is sufficient isolation between the Tx and Rx ports. Although there have been some commercial RFID reader Manuscript received March 03, 2011; revised June 08, 2011; accepted July 15, 2011. Date of publication September 15, 2011; date of current version February 03, 2012. This work was supported by a Mid-career Researcher Program through a National Research Foundation of Korea (NRF) Grant (No. 20100027006) funded by the Korea government (MEST). The authors are with the Department of Electronics and Radio Engineering, College of Electronics and Information, Kyung Hee University, Yong-in, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2167943

with separate Tx and Rx antennas, a two-port Tx/Rx one-body reader antenna is certainly a favorite. To realize a dual-band circularly-polarized RFID reader antenna with separate Tx and Rx ports is not an easy task. Research papers for this goal is rare, although there have been some partial efforts. In [1], a linearly-polarized aperture-coupled two-layered dual-band RFID reader antenna was designed with a 2:1 VSWR bandwidth of 17% in the UHF band 31% in the ISM band. Circularly-polarized RFID reader antennas for a single UHF band have been extensively studied [2]–[6] and there are already some commercial products from various manufacturers. A dual-band (1227 MHz and 1575 MHz) and circularly-polarized stacked microstrip antenna was first proposed in [7] employing an aperture-coupled type feeding. It is fed by a sequentially rotated feed line through crossed slots on the ground plane. Its drawback is narrow bandwidth due to the difference of the electrical lengths of the feed line at each band. Besides, the frequency ratio of the two bands must not be very far for a proper operation of the stacked patches. In [8], a dual-band circularly polarized RFID reader antenna was proposed with separate Wilkinson power divider networks for the two small and large stacked patches. This reader antenna shows a fair performance. However, it can be used for the Tx or Rx only operation. In [9], another dual-band circularly polarized RFID reader antenna was realized by feeding a cross-shaped patch with a dual-band branch-line coupler [10] for UHF (900 MHz) and ISM (2.45 GHz) bands. The cross-shaped patch is connected with the coupler outputs using the two vertical pins. Its 10 dB return losses and port isolations are reported to be reasonable: 20.1% (805–985 MHz), 8.9% (2.290–2.504 GHz), 22 dB (885 MHz), and 38 dB (2.46 GHz), respectively. The 3 dB axial ratio (AR) bandwidth performance about the ISM center frequency of 2.45 GHz is very poor. This poor AR performance at the ISM band seems to come from the effects of the radiating patch resonance occurring at the third harmonic frequency of 2700 MHz. The use of the vertical pins for feeding the radiating patch is also an inconvenience for mass production. The antenna gains are not reported in [9]. The use of the dual-band branch-line coupler [10] requires more space compared with the conventional coupler since four open quarter-wavelength or short half-wavelength stubs must be employed additionally. In this work, a metamaterial (MTM) branch-line coupler [11]–[14] is adopted for a compact design. The MTM technology has drawn much attention since it can be used to show unusual characteristics such as negative phase constants. Series inductance and shunt inductance of a conventional transmission line, typically in the positive region, can now be effectively

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Fig. 2. Geometry of branch-line coupler. (Unit: mm).

Fig. 1. Conceptual schematics of dual-band branch-line coupler. (a) UHF band. (b) ISM band.

achieved and further controlled in the negative region. The expansion of the region for effective series inductance, or permeability, and shunt capacitance, or permittivity, has lead to many new applications. In this paper, a dual-band (UHF: 917–923 MHz, ISM: 2.4–2.48 GHz, in Korea) circularly-polarized RFID planar reader antenna with separate Tx and Rx ports is proposed. A single rectangular patch is fed by the two output lines of the designed meta-structured branch-line coupler through the near orthogonally positioned slots. The near orthogonally positioned slots significantly help to maximize isolation between the Tx and Rx ports [14]. In II, based on the methodology in [13] for general dual-band operations, more explicit design formulas are provided for convenience of users. Then, a metamaterial-based dual-band branch-line coupler for 920 MHz and 2.45 GHz is designed using the provided formulas. Finally, a dual-band CP antenna is designed and connected to the coupler. In III, the metamaterial-based coupler is evaluated in terms of S-parameters and phase balance. Besides, the antenna combined with the coupler with separate Tx and Rx ports is evaluated in terms of its impedance bandwidth, isolation, antenna gains, axial ratios, and so on. The paper is concluded in IV.

composite right- and left-handed transmission line (RLH-TL) can be constructed [12]. The requirements for a segment consisting of N unit cells with length are given by [12], [13]

(1)

(2) (3) (4) , is the charwhere is the angular frequency acteristic impedance of a conventional host transmission line, is the speed of light, and is the relative effective permittivity of the line. Simultaneously solving the five equations given by [1]–[4], we obtain the closed-form solutions expresses as (5) (6) (7)

II. THEORY AND REALIZATION OF METAMATERIAL-BASED DUAL-BAND CP RFID ANTENNA The required characteristic impedances and phase shifts of a dual-band branch-line coupler are shown in Fig. 1(a) and (b). The reference impedance of the four ports is . A conventional branch-line coupler with four quarter-wavelength segments is well known to show the phase shifts of and at the fundamental and third harmonic frequencies, respectively, as seen in Fig. 1. Thus, it is basically a dual-band device. The phase shifts of and of the segment at arbitrarily chosen two frequencies ( and ) can be realized using the metamaterial techniques in [12], [13]. The conventional transmission lines, which usually support TEM waves and follow the right-hand rule, have been characterized by the distributed series inductance (given in unit of H/m) and shunt capacitance (given in unit of F/m). By adding a lumped-type series capacitance and shunt inductance periodically in a unit cell with its size much smaller than the wavelength, the

(8) (9) where (10) (11) The microstrip line on an FR-4 substrate with a relative permittivity of 4.6 and a height of 1.6 mm has the relative effective permittivity of . The required dual frequencies are , and . For and , , , , , and . For and , , , , , and .

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Fig. 3. Fabricated dual-band branch-line coupler.

TABLE I CALCULATED AND EM-SIMULATED VALUES

Fig. 2 shows the geometry of the metamaterial dual-band coupler. It is constructed using microstrip lines on a FR-4 substrate. The diameter of the via for the shunt inductors is 0.5 mm. The coupler is designed by EM simulation based on the calculated values using (5)–(9). The total size of the coupler is approximately 50 55 mm. The other dimensions are as seen in Fig. 2. We summarize the calculated values (obtained using (5), (6) and (7)) and the actual values adjusted by EM simulation in Table I. Fig. 3 shows the fabricated MTM dual-band branch-line coupler. Fig. 4 shows the schematics of the proposed dual-band (UHF and ISM) circularly polarized aperture-coupled RFID microstrip patch antenna. The radiating patch antenna (a) is fed by the two output lines of the designed dual-band MTM branch-line coupler through two slots (b). A cross slot is implemented at the center of the radiating patch to enhance isolation. An aperture coupled antenna is usually constructed with a single slot. However, the proposed antenna is specifically constructed with two slots. The reason why the two slots are employed will be explained shortly. The slots on the ground plane of the antenna are positioned in a T-shape in order to increase the isolation between them [12]. When the port 1 is excited, a left-handed circular polarization (LHCP) is generated in the UHF band and right-handed CP (RHCP), in the ISM band. When the port 2 is excited, the opposite is true. The dimensions of the designed antenna are summarized in Table II. III. RESULTS AND DISCUSSION Fig. 5 shows the S-parameters of the coupler. The S-parameters are compared among the circuit, EM, and measured results; the magnitudes of the S-parameters are compared in (a) and (b), and the phase balance as a function of frequency is compared in (c). The results are shown to be reasonably consistent. The phase difference between and is shown to

Fig. 4. Schematics of proposed antenna. (a) Radiating patch. (b) Feeding circuit. (c) Side view.

TABLE II DIMENSIONS OF DESIGNED ANTENNA (UNIT: mm)

be 90 in the UHF band and 270 in the ISM band. Based on the phase difference of and magnitude difference of , the bandwidth of the coupler is 896 to 934 MHz (4.1%) and 2380 to 2480 MHz (4.1%). In Fig. 6, ’s are compared for different numbers of slots (N) and distances between the two slots (See Fig. 4). With one slot, the resonances are observed at the fundamental frequency of 920 MHz, its second, and third harmonic frequency of about 2.9 GHz. With increasing ’s ( , 31 mm, and 36 mm) in case of two slots, the third harmonic frequency of 2.9 GHz is shown to shift to a lower one. With , the required dual frequency of 920 MHz (UHF) and 2.45 GHz (ISM) is obtained. In Fig. 7, the effect of the cross slot on the patch is shown in terms of the isolation . It is observed that the patch antenna

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Fig. 7. Simulated isolation of the antenna with cross slot on the patch and conventional antenna (without cross slot on the patch).

Fig. 8. Fabricated dual-band circularly polarized antenna. (a) Top view. (b) Bottom view.

Fig. 5. S-parameters of dual-band branch-line coupler. (a) Magnitudes of and . (b) Magnitudes of and . (c) .

Fig. 6. Comparison of

with different number of slots.

without a cross slot presents isolations of 18 dB in each band, whereas the antenna with a cross slot on the patch presents isolation of 25 dB at 920 MHz and isolation of 38 dB at 2.45 GHz. The cross slot is useful for more desirable current flows in two perpendicular directions. Fig. 8 shows the top and bottom views of the fabricated antenna.

Fig. 9. S-parameters of proposed antenna. (a) UHF band. (b) ISM band.

The S-parameters of the antenna are shown in Fig. 9. The Sparameters are compared between the results of EM-simulation

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Fig. 11. Axial ratios of proposed antenna at UHF and ISM bands.

IV. CONCLUSIONS

Fig. 10. CP Gain patterns of proposed antenna in YZ plane. (a) At 920 MHz (UHF band). (b) At 2.45 GHz (ISM band).

and actual measurements. The magnitudes of and are presented in UHF band (a) and in ISM band (b). The results are shown to be fairly consistent. The measured 10 dB bandwidth of the antenna is 3.4% (908–939 MHz) at 920 MHz and 7.0% (2.37–2.54 GHz) at 2.45 GHz. Fig. 10(a) and (b) show the CP gain patterns of the proposed antenna in the YZ plane at 920 MHz and 2.45 GHz, respectively, based on EM simulations and measurements when the port 1 is excited. The proposed antenna is shown to generate a LHCP at 920 MHz and RHCP at 2.45 GHz. The measured gains are shown to reasonably agree with the EM-simulated ones. The maximum measured CP gain is 6.6 dBic at 920 MHz (UHF) and 7.9 dBic at 2.45 GHz (ISM). The cross-polar gains near the broadside of the RFID reader antenna are observed to be approximately less than compared with the co-polar ones. Due to the symmetry of the structure, the antenna gains in the XZ plane are similar to those in the YZ plane and thus are not shown here. Fig. 11 shows the axial ratios (AR) of the antenna in the UHF and ISM bands, respectively. The EM-simulated and measured AR’s are maintained simultaneously below 3 dB from 0.87 GHz up to 0.99 GHz (4.4%), and from 2.38 GHz up to 2.58 GHz (8.2%). The measured AR’s are about 0.5 dB at the two design frequencies of the proposed antenna. The measured AR’s are shown to reasonably agree with the EM-simulated ones.

A dual-band, circularly polarized, aperture-coupled microstrip RFID reader antenna with separate Tx and Rx ports has been proposed with a dual-band metamaterial branch-line coupler. Both the EM simulation and actual measurement results of the antenna have been provided. The measured performance has been found to be fairly consistent with the simulated performance. The proposed antenna presents isolation of 25 dB at UHF frequency and isolation of 38 dB at ISM frequency. The maximum measured LHCP gain is 6.6 dBic at 920 MHz (UHF) and RHCP gain is 7.9 dBic at 2.45 GHz (ISM). The cross-polar gains near the broadside of the proposed RFID reader antenna are approximately less than compared with the co-polar gains. Besides, the measured AR’s are less than 0.7 dB in the UHF band (917–923 MHz) and 1.5 dB in the ISM band (2.4–2.48 GHz). The proposed antenna is a good candidate for a dual-band RFID reader for both UHF and ISM band applications. REFERENCES [1] Z. Xu and X. Li, “Aperture coupling two-layered dual-band RFID reader antenna design,” Proc. IEEE ICMMT, pp. 1218–1221, Feb. 2008. [2] D. Yu, Y. Ma, Z. Zhang, and R. Sun, “Circularly polarized aperturecoupled patch antenna for RFID reader,” in Proc. Wireless Communications, Networking and Mobile Computing, 2008, pp. 1–3. [3] Nasimudin, Z. N. Chen, and X. Qing, “Asymmetric-circular shaped slotted microstrip antennas for circular polarization and RFID applications,” IEEE Trans. Antennas Propag., vol. 58, no. 12, pp. 3821–3828, Dec. 2010. [4] Z. N. Chen, X. Qing, and H. L. Chung, “A universal UHF RFID reader antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1275–1282, May 2009. [5] Z. Wang, S. Fang, and S. Fu, “A lowcost miniaturized circularly polarized antenna for UHF radio frequency identification reader applications,” Microw. Opt.Tech. Lett, vol. 51, no. 10, pp. 2382–2384, Oct. 2009. [6] S. Maddio, A. Cidronali, and G. Manes, “A new design method for single-feed circular polarization microstrip antenna with an arbitrary impedance matching condition,” IEEE Trans. Antennas Propag., vol. 59, no. 2, pp. 379–389, Feb. 2011. [7] D. M. Pozar and S. M. Duffy, “A dual-band circularly polarized aperture-coupled stacked microstrip antenna for global positioning satellite,” IEEE Trans. Antennas Propag., vol. 45, no. 11, pp. 1618–1625, Nov. 1997. [8] D. Shin, P. Park, W. Seong, and J. Choi, “A novel dual-band circularly polarized antenna using a feeding configuration for RFID reader,” in Proc. IEEE ICEAA, China, 2007, pp. 511–514. [9] H.-Y. A. Yim, C.-P. Kong, and K.-K. M. Cheng, “Compact circularly polarized microstrip antenna design for dual-band applications,” IEE Electron. Lett, vol. 42, no. 7, pp. 380–381, Mar. 2006.

JUNG AND LEE: DUAL-BAND CIRCULARLY POLARIZED MICROSTRIP RFID READER ANTENNA

[10] K. Cheng and F. L. Wong, “A novel approach to the design and implementation of dual-band compact 90 branch-line coupler,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 11, pp. 2458–2463, Nov. 2004. [11] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2172–2702, Dec. 2002. [12] T. Kim and B. Lee, “Modelling and analysis of radiation effects for one-dimensional metamaterial-based transmission lines,” IET Microw. Antennas Propag., vol. 4, no. 3, pp. 278–295, Mar. 2010. [13] I.-H. Lin, M. DeVincentis, C. Caloz, and T. Itoh, “Arbitrary dual-band components using composite right/left-handed transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1142–1149, Apr. 2004. [14] X. Q. Lin, R. P. Liu, X. M. Yang, J. X. Chen, X. X. Yin, Q. Cheng, and T. J. Cui, “Arbitrarily dual-band components using simplified structures of conventional CRLH TLs,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 7, pp. 2902–2909, Jul. 2006. [15] B. Lee, S. Kwon, and J. Choi, “Polarization diversity microstrip base station antenna at 2 GHz using T-shaped aperture-coupled feeds,” IEE Proc. Microw. Antennas Propag., vol. 148, no. 5, pp. 334–338, Oct. 2001.

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Youn-Kwon Jung (M’09) received the B.S. degree in radio communication engineering and the M.S. degree in electronics and radio engineering from Kyung Hee University, Yong-in, Korea, in 2008 and 2011, respectively, where he is currently working toward the Ph.D. degree. His fields of research include small antennas, passive devices, RFID reader antennas, wireless power transmission, and metamaterials.

Bomson Lee (M’96) received the B.S. degree in electrical engineering from Seoul National University, Seoul, Korea, in 1982, and the M.S. and Ph.D. degrees in electrical engineering from the University of Nebraska, Lincoln, in 1991 and 1995, respectively. From 1982 to 1988, He was with the Hyundai Engineering Company Ltd., Seoul, Korea. In 1995, he joined the faculty at Kyung Hee University, where he is currently a Professor with the Department of Electronics and Radio Engineering. He was an Editor-in-Chief of the Journal of the Korean Institute of Electromagnetic Engineering and Science in 2010. He is an Executive Director (Project) in the Korea Institute of Electromagnetic Engineering & Science (KIEES). His research activities include microwave antennas, RF identification (RFID) tags, microwave passive devices, wireless power transmission and metamaterials.

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Small-Size Shielded Metallic Stacked Fabry–Perot Cavity Antennas With Large Bandwidth for Space Applications Shoaib Anwar Muhammad, Ronan Sauleau, Senior Member, IEEE, and Hervé Legay

Abstract—New configurations of small-size shielded metallic Fabry–Perot (FP) antennas with improved performance over a large frequency band are presented in -band for space missions. The bandwidth enlargement is obtained by stacking two FP cavities of different size, each of them presenting a low quality factor. Their radiating apertures measure around and 2 , respectively. Concentric corrugations are also introduced between both cavities to control the higher-order modes that are excited systematically in shielded small-size FP antennas due to lateral resonances. The obtained results are compared to those of a single-stage FP cavity antenna with the same aperture size. Several prototypes have been fabricated and measured. An aperture efficiency higher than 70%, a reflection coefficient smaller than 15 dB, and sidelobe levels lower than 20 dB have been obtained experimentally, over a wide frequency band (2.4–2.66 GHz). These characteristics make stacked FP cavity antennas very attractive to replace global coverage horn antennas, or to be used in feed clusters of multiple-beam antennas, especially in - and -bands, where they lead to more compact and less bulky solutions compared to classical feed horns. Index Terms—Compact feeds for space applications, corrugations, Fabry–Perot (FP) antennas, horn antennas, partially reflecting surfaces (PRSs).

I. INTRODUCTION

H

ORN antennas are widely used for global coverage antenna systems [1], [2] for telemetry, tracking, and control (TT&C), and as feed clusters for multiple-beam antennas (MBAs) [3], [4] for space missions. The main requirements for these antennas are the following: high aperture efficiency, low return loss, low sidelobe level (SLL) and cross-polarization level, purely metallic structure (no dielectrics) especially for power applications, stable and axisymmetrical radiation patterns, and small aperture size. These requirements are defined to ensure an effective illumination of the main dish and a high beam overlapping level for multibeam applications. For global coverage antennas, the emphasis Manuscript received January 05, 2011, revised June 28, 2011; accepted August 03, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was performed using HPC resources from GENCIIDRIS under Grant 2010-050779. This work was carried out under a Ph.D. thesis at the Institut d’Electronique et de Télécommunications de Rennes, IETR France, under a project supported by CNRS and Thales Alenia Space, France. S. A. Muhammad and R. Sauleau are with the IETR, UMR CNRS 6164, University of Rennes 1, 35042 Rennes, France (e-mail: shoaib.muhammad; ronan. [email protected]). H. Legay is with the Thales Alenia Space, France, 31037 Toulouse Cedex 1, France (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2173133

is especially to obtain high aperture efficiency over a large bandwidth. Traditional horn antennas provide excellent performance for the above-mentioned applications (e.g., [5]–[9]), but they often lead to heavy and bulky structures. Several alternative solutions with radiating aperture size comprised between 1.5 and 2.5 have been proposed in the literature to bypass these limitations. A patch antenna loaded with parasitic elements printed on a superstrate [10] can provide equivalent performances with a considerable reduction of size and weight, but its use is limited in terms of maximum size and power-handling capability due to the presence of dielectric materials. More recently, it has been shown that short backfire antennas [11], [12] with parasitic wires can also produce axially symmetric radiation patterns. However, these antennas do not provide good performance for radiating apertures larger than 2 . After several seminal studies published in the late 1990s (e.g., [13]–[15]), there has been a very strong renewed interest since 2001 in electromagnetic band-gap (EBG) resonator and Fabry–Perot (FP) cavity antennas [15]. In this frame, we can identify two possible FP solutions for feed clusters of MBA. 1) Use of an EBG material or a frequency selective surface (FSS) over an array of interlaced multiple feeds [17]–[20]. This technique enables to reduce the spillover loss. The superstrate is often oversized and is not shielded. 2) Use of small-size individual elements without interlacing. In these cases, each element is shielded [21]; it can be designed independently and is isolated from the neighboring elements. The major problems encountered up to now are the following: narrow operational bandwidth, excitation of parasitic modes, and strong deterioration of the radiation pattern quality (no axial symmetry, high SLL). In this paper, we focus our attention on the second solution and propose new techniques to improve its performance in terms of beam axis symmetry, aperture efficiency , impedance bandwidth, and radiation bandwidth. These issues are discussed step by step, where we restrict our review to FP/EBG antenna structures. A standard FP cavity antenna [15], [22] consists of a partially reflecting surface (PRS) placed at a distance of about [13], [23] over a metallic PEC ground plane. The antenna can be fed by a patch element (e.g., [24] and [25]) or a waveguide opening (e.g., [15] and [26]). These antennas can provide high gains with low-profile structures, but inherently suffer

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MUHAMMAD et al.: SMALL-SIZE SHIELDED METALLIC STACKED FP CAVITY ANTENNAS WITH LARGE BANDWIDTH

from narrow operational bandwidths and possible excitation of parasitic cavity modes. Several studies have been carried out to improve their performance in terms of: 1) profile reduction; 2) improvement; and 3) bandwidth enlargement techniques [25]. One attractive solution to reduce their height down to subwavelength profiles has been proposed in [26]–[30]. The idea consists in using the negative phase value of the reflection coefficient of a capacitive FSS [31] in order to ensure that the resonance condition [15], [22] is satisfied for a physical height much smaller than . Although attractive, this solution cannot be used to reach high aperture efficiencies because the energy is not distributed uniformly inside the entire cavity [26]. Nonuniform FSSs (or PRSs) [32], [33] have been proposed to improve the aperture efficiency of FP antennas. This approach is very powerful, but suffers from several weak points: 1) the antenna bandwidth is reduced significantly [33]; 2) it is difficult to obtain a good impedance matching level; and 3) high sidelobe levels might be observed [33]. Another promising solution to reach high values consists in designing quasi-TEM FP resonators using either artificial magnetic conductor side walls [34] or hard walls [35]. These techniques enable to achieve a better energy distribution over the antenna aperture, but lead again to narrowband FP antennas. To improve the bandwidth of FP antennas excited by a single feed, two-layer FSS structures have been introduced in [36] and [37]. This technique has been applied only to large-size overdimensioned resonators, and its applicability is questionable when dealing with small-size FP antennas due to the impact of higher-order modes [38]. An innovative approach combining the EBG/FP structure with the transmitarray and reflectarray concepts is reported in [39] for an antenna aperture of around 2.7 . A combined dB radiation and dB impedance bandwidth of about 8% has been measured, but the radiation pattern quality is not suitable for space applications. To our knowledge, only a few studies have been carried out on metallic shielded FP cavity antennas with small aperture sizes , e.g., [21]. In this work, and in contrast to large, directive FP antennas, the authors showed that, for a given antenna size, there exists an optimum value of FSS reflectivity (defined as the square of the magnitude of the reflection coefficient) leading to the largest bandwidth and the maximum gain level. Design guidelines were also proposed. However, such antennas suffer from a narrow bandwidth and a poor impedance matching at frequencies where the antenna directivity reaches its maximum value. In this paper, we describe new configurations of FP antennas with improved characteristics in terms of radiation and impedance bandwidth, level, and radiation pattern quality. The proposed concept consists of stacking FP resonators with different -factors. It is very different from the solutions previously proposed in [36] and [39] because the antenna configurations studied here have a small aperture size and are shielded, which requires to accurately control the cavity modes and find solutions to generate axisymmetrical beams. To this end, we propose several simple techniques to obtain a good impedance matching and axially symmetric radiation

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TABLE I SPECIFICATIONS OF FEED ANTENNAS FOR MBA APPLICATIONS IN -BAND

patterns over the entire bandwidth. All numerical results have been obtained using HFSS (ver. 12). The antenna specifications in -band are provided in Table I. They have been defined by Thales Alenia Space, France, for global coverage applications. Our objective is to satisfy these performance parameters by proposing a compact metallic solution replacing large-size horns. In this paper, the global bandwidth is defined as the frequency band (in gigahertz or in percent) over which the directivity (or gain) variation is less than 1 dB, the aperture efficiency is greater than a given value (70%), and the reflection coefficient and the SLL are lower than predefined values ( 15 and 20 dB, respectively). This paper is organized as follows. We study in Section II a single-stage small-size FP cavity antenna in order to quantify its performance and highlight its limitations both numerically and experimentally. Section III deals with the study of stacked FP cavity antennas. We describe the antenna geometry and characteristics, and we provide design guidelines. An optimized configuration is proposed and validated experimentally. Moreover, we show in Section IV that the performance of stacked configurations can be improved further using concentric corrugations. These results are also validated experimentally. Finally, conclusions are drawn in Section V. II. SINGLE-STAGE FP CAVITY ANTENNA A. Geometry The geometry of a typical square FP cavity antenna is presented in Fig. 1. A 2-D inductive grid [40] is placed over a shielded cavity at a distance above the ground plane. The mesh period and strip width are labeled and , respectively. The inductive FSS grid was chosen over the complementary capacitive FSS [31], [41] principally to avoid the use of dielectric substrates, as required by Thales Alenia Space. A cylindrical waveguide with diameter is used to excite the antenna. This choice is made due to the possible use of an orthomode transducer (OMT) to excite the antenna. The impedance matching system is based on the combination of a waveguide penetration and a circular iris. The lateral size of the FP resonator is about 2 . The other antenna dimensions are defined in Fig. 1(b). B. Impact and Control of Higher-Order Modes The impact of higher-order cavity modes, especially lateral resonances, has almost never been investigated in the literature since the wide majority of papers on FP resonators deal with nonshielded oversized cavities, e.g., [15]. Two solutions

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Fig. 2. Control of the cavity modes by choosing the optimum lateral size for mm . Black solid line: the antenna. Gray dashed line: mm .

aperture efficiency defined as follows: Fig. 1. Single-stage small-size FP cavity antenna. (a) 3-D view. (b) Cross-section view.

have been proposed to control them: use of metallic bars on the ground plane of an FP cavity [42], selection of the optimum cavity size [33]. As the FSS mirror is highly reflective in practice [13]–[15], the cutoff frequency of a given mode can be approximated assuming the FP cavity behaves as a closed PEC resonator

(1) where is the speed of light in vacuum. This relation can be used to determine which modes can be excited within the antenna bandwidth. Therefore, the value of can be slightly changed to reject these modes outside the operational bandwidth. The impact of lateral resonant modes is illustrated in Fig. 2, where the directivity curves of two single-stage FP antennas are compared. The antenna dimensions are the same in both cases, except their lateral size , which is equal either to 223 or to 244 mm. We can see in Fig. 2 that the cavity mode excited at 2.42 GHz is shifted to higher frequencies (around 2.63 GHz), when is reduced from 244 to 223 mm. For a cavity size of 223 223 mm , we observe a pure mode inside the cavity over the 1-dB radiation band (2.4–2.56 GHz). Therefore, we select this cavity size in the following in order to prevent parasitic modes from spoiling the radiation band. C. Optimization of the Antenna Dimensions For a given lateral size, the design procedure of single-stage small-size FP antennas consists in two steps. First, the optimum FSS power reflectivity is determined by tuning the grid parameters and [13], [40] in order to obtain the largest possible 1-dB radiation bandwidth with a sufficiently high

. In this paper, the aperture efficiency is

(2) is the operating frequency, is the antenna where directivity, is the maximum directivity of a uniform radiating aperture of same area , and is the wavelength in free space. The second step consists in optimizing the impedance matching level over the antenna radiation bandwidth. As explained below, and in contrast to the well-established design rules for large and directive FP cavities and antennas (e.g., [15] and [43]), parametric studies based on full-wave simulations need to be carried out for the design of small-size shielded FP antennas. Indeed, the standard resonance condition [13], [22] (3) cannot be applied in a straightforward way because the phase values and (at the resonance frequency ) of the reflection coefficients of the upper (FSS) and lower (ground) reflecting mirrors are significantly affected by two major effects: 1) the finite size of the FSS, and 2) the size of the waveguide aperture , which cannot be neglected compared to the ground plane size , respectively. In practice, for a given cavity size and for a predefined value of FSS reflectivity (i.e., a given pair ), the cavity height must be tuned to make the FP antenna resonate at . This design procedure is then much more time-consuming than for classical FP antennas for which (3) provides immediately very accurate results (e.g., [13]). Here, for mm at GHz , we have found that the optimum value of the FSS reflectivity is 74%. The dimensions of the metallic mesh are given in Table II. Note that standard tabulated values and equivalent circuit models (e.g., [40] and [44]) cannot be used directly here because of the nonnegligible grid thickness mm .

MUHAMMAD et al.: SMALL-SIZE SHIELDED METALLIC STACKED FP CAVITY ANTENNAS WITH LARGE BANDWIDTH

Fig. 3. Antenna performance (directivity and reflection coefficient ) before and after the introduction of the impedance matching system. Black dashed line: directivity before matching. Black solid line: Directivity after matching. Gray before matching. Gray solid line: after matching. dashed line:

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Fig. 4. Measured gain (black solid line) and reflection coefficient (gray solid line) for the single-stage FP cavity antenna. Comparison with the simulation results (dashed lines).

TABLE II OPTIMIZED DIMENSIONS FOR THE SINGLE-STAGE FP CAVITY ANTENNA

The other dimensions of the optimized prototype are also given in Table II, including the values of the iris width mm and penetration of the waveguide within the cavity mm . Each conductor thickness has been chosen with mechanical engineers to ensure a good mechanical rigidity of all antenna parts. The impact of the matching system upon the antenna directivity and reflection coefficient is represented in Fig. 3. In presence of an iris penetrating inside the cavity, a better impedance matching is achieved over the entire radiation bandwidth. However, the latter is reduced from 190 MHz (before matching) to only 90 MHz (after matching). The maximum directivity level is also decreased by about 0.4 dB. This comes from the slight difference between the field distributions inside the cavity for the two configurations (not shown here for brevity). D. Experimental Results A prototype has been fabricated and measured in -band. The measured gain and reflection coefficient are plotted in Fig. 4. The gain has been determined with the comparison method using a 12-dBi standard horn. The simulation results are also included for comparison. We obtain a close agreement between the simulation and measured data. A slight frequency shift is observed, which is principally due to the manufacturing errors and problem of alignment for the FSS over the FP cavity. A maximum gain of 14.8 dB has been obtained at 2.48 GHz. The 1-dB radiation bandwidth and 15-dB reflection coefficient bandwidth equal 112 MHz (2.4–2.512 GHz) and only 32 MHz (2.474–2.506 GHz), respectively. Therefore, by adopting the definition given in Section I, the global bandwidth equals only 32 MHz, i.e., 1.28% of the center frequency

Fig. 5. Measured radiation pattern for the single-stage FP cavity antenna at 2.49 . Gray solid line: . Dashed line: GHz. Black solid line: . Dash-dotted line: X-pol at 45 .

(2.49 GHz). Over this frequency band, the aperture efficiency varies between 60% and 70%. The radiation patterns at the central frequency are shown in Fig. 5. High sidelobe levels dB are observed in the E-plane cut-plane . The cross-polarization component in the diagonal plane is also presented in Fig. 5. Its level remains below 18 dB. We do not provide the measured values in the and planes since they are much lower. To conclude, single-stage small-size shielded FP cavities can be used only for narrowband applications bandwidth . To increase their bandwidth, less reflective FSSs are required (well-known). However, this leads to lower aperture efficiencies because the energy is not spread over the entire cavity surface. On the contrary, to achieve acceptable gain or aperture efficiency, FSS grids with higher reflectivity should be used. However, this results in the possible excitation of higher-order parasitic modes due to lateral resonances between opposite cavity side walls, which reduces the antenna bandwidth. Therefore, single-stage FP cavities cannot be employed to meet the specifications of Table I. The solution proposed to overcome these limitations is described in Section III.

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Fig. 7. Flowchart describing the design procedure of small-size stacked FP bandwidth . cavity antennas Fig. 6. Stacked FP cavity antenna. (a) 3-D view. (b) Cross-section view.

III. STACKED FP CAVITY ANTENNAS A. Geometry and Operating Principle Two square FP cavity antennas with different lateral and are stacked on top of each other [45] sizes (Fig. 6). The idea is that the smaller cavity placed at the bottom serves as an impedance matching intermediate stage that excites a larger cavity on the top behaving as a compact radiating element. The objective behind this configuration is to permit us to use two cavities, each of them exhibiting a low -factor. In this way, we expect to achieve larger radiation bands with sufficient gain levels and simultaneously minimize the effects of higher-order modes that degrade cavity bandwidth (Section II-D). In addition, compared to single-stage antennas, we can observe that there is no iris in the stacked configuration. The reason for this is that the iris is a narrowband impedance-matching solution that is eligible only for single FP cavity antennas because of the presence of a highly reflecting grid (Section II). When designing stacked configurations, it has been observed that no iris is required, and only a simple waveguide penetration is sufficient to match the antenna over a large bandwidth. B. Design and Performance One of the key design steps consists in finding the best pair of power reflectivities for both FSS, i.e., the grid dimensions , 1 and 2. By analogy with single-stage FP cavity antennas (Section II and [21]), for each pair , the cavity heights and must be optimized through a

full-wave parametric study, so as to achieve the largest possible bandwidth. In this end, two main constraints must be taken into account: 1) the resonance frequency shift due to the cumulative effects of the cavity walls and the small aperture size [21]; 2) the hybrid nature of the bottom planes for both cavities. Indeed, the phase of their reflection coefficient (3) is tricky to define because: a) the ground plane of cavity #2 is a combination of a small-size FSS and a PEC frame surrounding ; and b) the bottom plane of cavity #1 consists of a large circular aperture (the waveguide diameter) in a small-size ground plane. For these reasons, an iterative design procedure is necessary in contrast to stacked FP antennas of larger size [36], [37] whose design rules are straightforward. The proposed flowchart is represented in Fig. 7. The lateral size for both cavities is chosen to avoid exciting parasitic modes inside the frequency band of interest (Section II-B). For a given reflectivity pair, the cavity heights are optimized one after the other in order to achieve a 10% radiation bandwidth. The conclusion of this parametric study is that the largest radiation bandwidth is obtained for and . The dispersion relation for the optimized case has also been studied. An in-house code has been developed (based on the work in [20]) in order to track the leaky-wave poles for an infinite two-layer antenna with the optimized FSS reflectivity values . The results (not shown here for brevity) confirmed the existence of several strongly attenuated TE/TM leaky waves inside the antenna operating close to the split-off condition. These results tend to demonstrate that the proposed stacked configuration can be considered as an FP

MUHAMMAD et al.: SMALL-SIZE SHIELDED METALLIC STACKED FP CAVITY ANTENNAS WITH LARGE BANDWIDTH

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Fig. 9. Purely metallic stacked FP cavity antenna for operation in -band. Fig. 8. Stacked FP cavity antenna. Simulation results for the directivity and reflection coefficient before and after the introduction of the impedance matching system. Black dashed line: Directivity before matching. Black solid line: Direcbefore matching. Gray solid line: tivity after matching. Gray dashed line: after matching.

TABLE III OPTIMIZED DIMENSIONS FOR THE STACKED FP CAVITY ANTENNA

Fig. 10. Stacked FP cavity antenna. Measured gain (black solid line) and reflection coefficient (gray solid line). The measured gain of the single-stage FP antenna is represented in black dashed line.

antenna and not a three-stage waveguide. However, a detailed study should be made to understand the contribution of each leaky pole to the radiation mechanism of this antenna, as done in [20]. After having identified the best FSS reflectivity pairs, the final stage before prototyping is to optimize the impedance matching bandwidth. To this end, a simple impedance matching system is implemented, namely a waveguide penetration inside the lower cavity [Fig. 6(b)]. The final optimized dimensions are provided in Table III. The upper cavity has of course the same size as the single-stage cavity mm . The upper FSS is thicker mm than the lower one mm to avoid any grid bending after manufacturing. It should be noted that the initial design was first based on zero FSS thickness mm, for 1 or 2). Then, these FSSs were replaced by thick ones, and their width and period were changed in order to keep the same reflection and transmission coefficients. The simulated antenna directivity and reflection coefficient, before and after impedance matching, are plotted in Fig. 8. Compared to Fig. 3, we can now notice that the 15-dB bandwidth (2.3–2.675 GHz) has been enlarged by more than 90%, and that it coincides now with the radiation bandwidth. This was not the case for the single-stage FP cavity as shown in Fig. 4. The antenna directivity varies very slightly with frequency, and the 1-dB radiation bandwidth spans from 2.35 to 2.675 GHz (after matching), which corresponds to a relative bandwidth of 12.1%.

To conclude, the use of stacked FP cavities and the introduction of a simple impedance-matching structure enable to improve the antenna performance, both in terms of directivity levels and impedance-matching bandwidth. This is opposed to single-FP-cavity configurations where the performance of the antenna has to be degraded (by using less reflective mirrors) in order to obtain better impedance-matching levels. C. Experimental Results and Discussions A prototype has been fabricated (Fig. 9). The total antenna height, without the feed waveguide, is around , i.e., twice the height of the single-stage counterpart. The measured gain and reflection coefficient are represented in Fig. 10 in solid lines. The gain curve measured for the single-stage cavity is also plotted (dashed line) for comparison purposes. A 1-dB radiation bandwidth of 11.48% is obtained for a maximum gain of 15.5 dB. The global bandwidth equals 10.28% (2.4–2.66 GHz). Over this band, the aperture efficiency (2) varies between 63% (at 2.66 GHz) and 81% (at 2.4 GHz). The radiation patterns measured over the global bandwidth are shown in Fig. 11 in three vertical cut-planes ( 0 , 45 , and 90 ) to better appreciate the pattern quality. Nearly axially symmetric beams with SLL lower than 17 dB are obtained over nearly the entire frequency band, except in the lower part of the global bandwidth [Fig. 11(a)]. In addition, slight shouldering effects are observed in the E-plane in

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Fig. 12. Concentric corrugations. (a) Top view. (b) Cross-section view.

Fig. 13. Stacked FP cavity antenna with corrugations. For clarity, only one corrugation is represented here.

present in the upper cavity. We show in Section IV that the antenna performance can be further improved using circular corrugations. IV. IMPROVEMENT OF THE ANTENNA CHARACTERISTICS USING CIRCULAR CORRUGATIONS A. Concept and Background

Fig. 11. Measured radiation patterns for the stacked FP cavity antenna at . Gray three frequency points and in three cut planes. Black solid line: . Dashed line: . Dash-dotted line: X-pol at 45 . solid line: (a) 2.4 GHz. (b) 2.53 GHz. (c) 2.66 GHz.

Fig. 11(b) and (c). The cross-polarization level remains below 18 dB over the entire bandwidth. As a summary, small-size stacked FP cavities exhibit much better performance than their single-stage counterpart. The proposed design nearly fulfills all specifications (Table I), except the aperture efficiency threshold of 70%, the SLL, and the beam symmetry at the lower edge of the antenna bandwidth. These remaining defects are due to higher-order resonant modes still

Circular corrugations (Fig. 12) have been widely used (e.g., [46] and [47]) to stop the propagation of surface waves and reduce the sidelobe level and backward radiation of monopole or horn antennas. Their typical size is given by , , and . Here, we propose to introduce corrugations in the antenna design to better control the energy distribution in cavity #2. The corresponding geometry of the modified stacked FP cavity is represented in Fig. 13. The corrugations are introduced only in the top cavity for two main reasons: 1) this cavity is the radiating element; 2) it is more sensitive to parasitic modes due to its larger lateral size. In addition, integrating the corrugations in the available space between cavities #1 and #2 does not increase the total antenna size, which is crucial for -band space antennas. The corrugation dimensions ( , , and ), their number, and their location have been optimized using HFSS. It is important to note here that only the corrugation parameters were varied; the other antenna dimensions are fixed (Table III). The numerical results have shown that only one corrugation is necessary. Its dimensions are the following: mm, mm, and mm.

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Fig. 14. Purely metallic stacked FP cavity antenna with concentric corrugation.

Fig. 15. (a) Stacked FP cavity antenna with corrugation. Measured gain (black solid line) and reflection coefficient (gray solid line). The measured gain and reflection coefficient of the stacked FP cavity antenna without corrugation are represented in dashed black and grey lines, respectively. (b) Comparison between the aperture efficiency before (gray solid line) and after (black solid line) the introduction of the corrugation. The aperture efficiency of an equivalent pyramidal horn (dash-dotted line) is also shown for comparison.

B. Numerical and Experimental Results The new antenna configuration described in Section IV-A has been fabricated (Fig. 14). It has the same dimensions as the prototype without corrugation (Table III). The measured gain and reflection coefficient are provided in Fig. 15(a). The experimental results obtained without corrugation are also shown in dashed lines. By comparing the gain curves of both configurations, we can notice a substantial enlargement of the radiation bandwidth with a slight gain increase beyond 2.5 GHz. Although the impedance matching is slightly degraded, it remains

Fig. 16. Measured radiation pattern of the stacked FP cavity antenna with corrugation at three frequency points and in three cut planes. Black solid line: . Gray solid line: . Dashed line: . Dash-dotted line: X-pol at 45 . (a) 2.40 GHz. (b) 2.53 GHz. (c) 2.66 GHz.

below 15 dB over a large frequency band. With the corrugation, the antenna gain is maximum at 2.6 GHz (15.9 dB). The measured global bandwidth (10.28%) is the same as for the case without corrugation. The aperture efficiency [Fig. 15(b)] has been improved over the whole bandwidth ( without corrugation, and with the corrugation). In addition, the aperture efficiency of an equivalent pyramidal horn antenna is also compared. This horn has the same aperture size and height as that of the proposed stacked FP cavity configuration. We can see that the horn antenna has a maximum aperture efficiency of only 61%. Additional results have shown that the pattern quality of the equivalent horn are significantly degraded (no axisymmetry, higher sidelobes and cross polarization). This is principally due to the fact that the E-field magnitude is higher at the horn corners, while in the case of the FP

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Fig. 18. -parameters for the 2 1 array (HFSS simulations). One of the elements is excited, and the other is terminated with a matched load. Comparison with a single element.

TABLE IV COMPARISON AMONG THE MEASURED PERFORMANCE OF THE SINGLE-STAGE FP CAVITY, STACKED FP CAVITY (WITH AND WITHOUT CORRUGATIONS), AND AN EQUIVALENT PYRAMIDAL HORN ANTENNA (SIMULATIONS)

Fig. 17. Comparison between the measured radiation patterns (E-plane only) over the global bandwidth for the stacked FP cavity antenna. Black solid line: 2.4 GHz. Gray solid line: 2.53 GHz. Dashed line 2.66 GHz. (a) With corrugation. (b) Without corrugation.

cavity the E-field is concentrated at the center and negligible at the edges of the antenna. These results further demonstrate the relevance of the stacked configuration. The radiation patterns measured in three cut-planes are plotted in Fig. 16. We can now notice that the SLL remains below 20 dB for all the observation planes over the entire frequency band. The measured peak cross-polarization level is below 20 dB. The radiation patterns in E-plane over the entire bandwidth for both stacked configurations, with and without corrugations, are represented in Fig. 17(a) and (b), respectively. We can see that the shouldering effect present over the entire frequency band for the case without corrugation [Fig. 17(b)] is completely removed by introducing one corrugation inside the top cavity [Fig. 17(a)]. In addition, with corrugation, the SLL remains below 20 dB, and the radiation patterns are very stable over a 10% frequency band. C. Mutual Coupling Between Stacked Cavities To evaluate the mutual coupling between stacked FP cavity antennas used as a feed cluster for focal arrays, a simple configuration, namely a 2 1 array in the E-plane has been simulated. In this case, only one cavity is excited, the other one being loaded by a matched load. The numerical results (not given) show that the single element and the 2 1 array present a similar directivity level. Their -parameters are compared in Fig. 18. We can see that the array and single element present almost the

same reflection coefficients ( and , respectively). In the array case, the mutual coupling in copolarization and cross polarization is very low dB . The reis lower than flection coefficient in cross-polarization 24 dB over the operational bandwidth. These results demonstrate low coupling coefficient in an array configuration. V. CONCLUSION In this paper, we have proposed new configurations of small-size shielded metallic FP antennas with improved characteristics compatible with feed specifications of global coverage antennas and multiple-beam antennas in -band for space missions. In this context, we have shown that classical small-size single-stage FP cavity antennas cannot be used because reaching high aperture efficiency values requires using highly reflecting mirrors; this leads to narrowband resonators whose performance is strongly affected by higher-order

MUHAMMAD et al.: SMALL-SIZE SHIELDED METALLIC STACKED FP CAVITY ANTENNAS WITH LARGE BANDWIDTH

cavity modes. A modified configuration consisting of two stacked FP cavities with different lateral sizes and -factors has been proposed. The lower cavity behaves as a smooth transition between the feed waveguide and the upper radiating cavity. This configuration exhibits much better performance, especially in terms of radiation and impedance bandwidth, since it allows using much less reflecting FSS. The aperture efficiency and radiation pattern quality, especially its axisymmetry, have been further enhanced using a circular corrugation in the top cavity. Good results in an array configuration with low coupling have also been shown. All numerical results have been validated experimentally. Table IV summarizes the main experimental results obtained with the single-stage FP antenna and both stacked configurations. The performance obtained with an equivalent horn antenna of same size is also provided. This table shows that only the stacked antenna with corrugations fulfils the requirements specified in Table I. REFERENCES [1] F. Croq, E. Vourch, M. Reynaud, B. Lejay, B. Ch, A. Couarraze, M. Soudet, P. Carati, J. Vicentini, and G. Mannocchi, “The GLOBALSTAR 2 antenna sub-system,” in Proc. EuCAP, Berlin, Germany, Mar. 2009, pp. 598–602. [2] T. Bird and C. Granet, “Fabrication and qualifying a lightweight corrugated horn with low side-lobes for global-earth coverage,” IEEE Antennas Propag. Mag., vol. 50, no. 1, pp. 80–86, Feb. 2008. [3] P. O. Iverson, L. J. Ricardi, and W. P. Faust, “A comparison between 1-, 3-, and 7-horn feeds for a 37-beam MBA,” IEEE Trans. Antennas Propag., vol. 42, no. 1, pp. 1–8, Jan. 1994. [4] P. S. Kildal and E. Lier, “Hard horns improve cluster feeds of satellite antennas,” Electron. Lett., vol. 24, no. 8, pp. 491–492, Apr. 1998. [5] K. K. Chan and K. S. Rao, “Design of high efficiency circular horn feeds for multibeam reflector applications,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 253–258, Jan. 2008. [6] A. K. Bhattacharyya and G. Goyette, “A novel horn radiator with high aperture efficiency and low cross-polarization and applications in arrays and multibeam reflector antennas,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 2850–2858, Nov. 2004. [7] O. Sotoudeh, P.-S. Kildal, P. Ingvarson, and S. P. Skobelev, “Singleand dual-band multimode hard horn antennas with partly corrugated walls,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 330–338, Feb. 2006. [8] C. Granet, G. L. James, R. Bolton, and G. Moorey, “A smooth-walled spline-profile horn as an alternative to the corrugated horn for wide band millimeter-wave applications,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 848–854, Mar. 2004. [9] Ph. Lepeltier, J. Maurel, C. Labourdette, F. Croq, G. Navarre, and J. F. David, “Thales Alenia Space France antennas: Recent achievements and future trends for telecommunications,” in Proc. 30th ESA Workshop Antennas Earth Observ. Sci., Telecommun. Space Missions, Noordwijk, The Netherlands, May 2008. [10] H. Legay and L. Shafai, “A new stacked microstrip antenna with large bandwidth and high gain,” Inst. Elect. Eng. Proc., Microw., Antennas Propag., vol. 141, no. 3, pp. 199–204, Jun. 1994. [11] D. Gray and H. Tsuji, “Short back-fire antenna with microstrip Clavin feed,” Microw. Antennas Propag., vol. 3, no. 8, pp. 1211–1217, Dec. 2009. [12] D. Gray and L. Shafai, “Optimization of large diameter short back-fire antenna by cavity juncture curvature,” Electron. Lett., vol. 37, no. 11, pp. 675–676, May 2001. [13] R. Sauleau, Ph. Coquet, J.-P. Daniel, T. Matsui, and N. Hirose, “Study of Fabry–Perot cavities with metal mesh mirrors using equivalent circuit models. Comparison with experimental results in the 60 GHz band,” Int. J. Infrared Millim. Waves, vol. 19, no. 12, pp. 1693–1710, Dec. 1998. [14] R. Sauleau, Ph. Coquet, J.-P. Daniel, T. Matsui, and N. Hirose, “Analysis of millimeter wave Fabry–Perot cavities using the FDTD technique,” IEEE Microw. Guided Waves Lett., vol. 9, no. 5, pp. 189–191, May 1999.

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[15] Ph. Coquet, R. Sauleau, D. Thouroude, J.-P. Daniel, and T. Matsui, “A 57 GHz Gaussian beam antenna for wireless broadband communications,” Electron. Lett., vol. 36, no. 7, pp. 594–596, Mar. 2000. [16] R. Sauleau, “Fabry Perot resonators,” in Encyclopedia of RF and Microwave Engineering, K. Chang, Ed. Hoboken, NJ: Wiley, 2005, vol. 2, pp. 1381–1401. [17] N. Llombart, A. Neto, G. Gerini, M. Bonnedal, and P. D. Maagt, “Leaky wave enhanced feed arrays for the improvement of the edge of coverage gain in multibeam reflector antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 5, pp. 1280–1291, May 2008. [18] C. Menudier, R. Chantalat, E. Arnaud, M. Thèvenot, T. Monédière, and P. Dumon, “EBG focal feed improvements for Ka-band multi-beam space applications,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 611–615, 2009. [19] N. Llombart, A. Neto, G. Gerini, M. Bonnedal, and P. de Maagt, “Impact of mutual coupling in leaky wave enhanced imaging arrays,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 1201–1206, Apr. 2008. [20] A. Neto, N. Lombart, G. Gerini, M. D. Bonnedal, and P. de Maagt, “EBG enhanced feeds for improvement of the aperture efficiency of reflector antennas,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2185–2193, Aug. 2007. [21] O. Roncière, B. A. Arcos, R. Sauleau, K. Mahdjoubi, and H. Legay, “Radiation performance of purely metallic waveguide fed compact Fabry Perot antennas for space applications,” Microw. Opt. Technol. Lett., vol. 49, no. 9, pp. 2216–2221, Sep. 2007. [22] G. V. Trentini, “Partially reflecting sheet arrays,” IRE Trans. Antennas Propag., vol. AP-4, no. 4, pp. 666–671, Oct. 1956. [23] R. Sauleau, Ph. Coquet, D. Thouroude, J.-P. Daniel, H. Yuzawa, N. Hirose, and T. Matsui, “FDTD analysis and experiment of Fabry–Perot cavities at 60 GHz,” IEICE Trans. Electron., vol. E82-C, pp. 1139–1147, Jul. 1999. [24] R. Sauleau, Ph. Coquet, and T. Matsui, “Low-profile directive quasiplanar antennas based on millimeter wave Fabry–Perot cavities,” Inst. Elect. Eng. Proc., Microw., Antennas Propag., vol. 150, no. 4, pp. 274–278, Aug. 2003. [25] A. Pirhadi, M. Hakkak, and F. Keshmiri, “Using electromagnetic bandgap superstrate to enhance the bandwidth of probe-fed microstrip antenna,” Prog. Electromagn. Res., vol. 61, pp. 215–230, 2006. [26] R. Sauleau, Ph. Coquet, D. Thouroude, J.-P. Daniel, and T. Matsui, “Radiation characteristics and performance of millimeter wave horn-fed gaussian beam antennas,” IEEE Trans. Antennas Propag., vol. 51, no. 3, pp. 378–387, Mar. 2003. [27] A. P. Feresidis, G. Goussetis, S. Wang, and J. C. Vadaxoglou, “Artificial magnetic conductor surfaces and their application to low profile high-gain planar antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 209–215, Jan. 2005. [28] L. Zhou, H. Li, Y. Qin, Z. Wei, and C. T. Chan, “Directive emissions from subwavelength metamaterial-based cavities,” Appl. Phys. Lett., vol. 86, pp. 101101–101101, Feb. 2005. [29] J. R. Kelly and A. P. Feresidis, “Low-profile high-gain sub-wavelength resonant cavity antennas for WiMax applications,” in Proc. EuCAP, Edinburgh, U.K., Nov. 2007, pp. 1–5. [30] N. Tentillier, F. Krasinski, R. Sauleau, B. Splingart, H. Lhermite, and Ph. Coquet, “A liquid crystal tunable ultra-thin Fabry–Perot resonator in Ka-band,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 701–704, 2009. [31] R. Sauleau and N. Falola, “Ultra-wideband wave reactances of capacitive grids,” Int. J. Infrared Millim. Waves, vol. 25, no. 10, pp. 1401–1421, Oct. 2004. [32] R. Sauleau, Ph. Coquet, T. Matsui, and J.-P. Daniel, “A new concept of focusing antennas using plane-parallel Fabry–Perot cavities with nonuniform mirrors,” IEEE Trans. Antennas Propag., vol. 51, no. 11, pp. 3171–3175, Nov. 2003. [33] G. Benelli, “Development of the fictitious sources method for stratified media and design of resonant cavities antennas,” Ph.D. dissertation, Univ. Paul Cezanne Aix-Marseille III, Marseille, France, 2007. [34] O. Roncière, R. Sauleau, K. Mahdjoubi, and H. Legay, “Quasi-TEM Fabry–Perot cavities with high aperture efficiency,” in Proc. 29th ESA Antenna Workshop Multiple Beams Reconfig. Antennas, Innov. Challenges, Noordwijk, The Netherlands, Apr. 2007. [35] H. Legay, “Dispositif Rayonnant à Cavité(s) Résonante(s) à air à Fort Rendement de Surface, Pour Une Antenne Réseau,” Patent FR 06 51717, 2006. [36] A. P. Feresidis, G. Goussetis, and J. C. Vardaxoglou, “A broadband high gain resonant cavity antenna with single feed,” in Proc. EuCAP, Nice, France, Nov. 2006, pp. 1–5.

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[37] L. Moustafa, B. Jecko, M. Thèvenot, T. Monédière, and R. Gonzalo, “EBG antenna performance enhancement using conducting element FSS,” in Proc. EuCAP, Edinburgh, UK, Nov. 2007, pp. 1–4. [38] S. A. Muhammad, “Etude et conception d’antennes à résonateur de Perot–Fabry compacts pour des applications spatiales,” Ph.D. dissertation, Université de Rennes 1, Rennes, France, 2010. [39] X. He, W. X. Zhang, and D. L. Fu, “A broadband compound printed air-fed array antenna,” in Proc. Int. Conf. Electromagn. Adv. Appl., Turin, Italy, Sep. 2007, pp. 1054–1057. [40] R. Sauleau, Ph. Coquet, and J.-P. Daniel, “Validity and accuracy of equivalent circuit models of passive inductive meshes. Definition of a novel model for 2D grids,” Int. J. Infrared Millim. Waves, vol. 23, no. 3, pp. 475–498, Mar. 2002. [41] A. Foroozesh and L. Shafai, “Investigation into the effects of the reflection phase characteristics of highly-reflective superstrates on resonant cavity antennas,” IEEE Trans. Antennas Propag., vol. 58, no. 10, pp. 3392–3396, Oct. 2010. [42] M. Thévenot, J. Drouet, R. Chantalat, E. Arnaud, T. Monédière, and B. Jecko, “Improvements for the EBG resonator antenna technology,” in Proc. EuCAP, Edinburgh, U.K., Nov. 2007, pp. 1–6. [43] R. Sauleau, G. L. Ray, and Ph. Coquet, “Parametric study and synthesis of 60-GHz Fabry–Perot resonators,” Microw. Opt. Technol. Lett., vol. 34, no. 4, pp. 247–252, Aug. 2002. [44] O. Luukkonen, C. Simovski, G. Granet, G. Goussetis, D. Lioubtchenko, A. V. Raisanen, and S. A. Tretyakov, “Simple and accurate analytical model of planar grids and high-impedance surfaces comprising metal strips or patches,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1624–1632, Jun. 2008. [45] S. A. Muhammad, R. Sauleau, and H. Legay, “Overview of metallic small-size Fabry–Perot cavity antennas for space applications,” in Proc. 32nd ESA Workshop Antennas Space Appl., Noodwijk, The Netherlands, 2010. [46] Z. Ying, P. S. Kildal, and A. A. Kishk, “Study of different realizations and calculation models for soft surfaces by using a vertical monopole on a soft disk as a test bed,” IEEE Trans. Antennas Propag., vol. 44, no. 11, pp. 1474–1481, Nov. 1996. [47] P. J. B. Clarricoats and A. D. Olver, Corrugated Horns for Microwave Antennas, ser. IEE Electromagnetic Waves Series 39. London, U.K.: IEE, 1994. Shoaib Anwar Muhammad received the Bachelor of Electrical Engineering degree from the National University of Sciences and Technology, Islamabad, Pakistan, in 2005, the Master’s degree in telecommunications, RF, and microelectronics from Université de Nice, Sophia Antipolis, France, in 2007, and the Ph.D. degree in signal processing and telecommunications from the IETR, Université de Rennes 1, Rennes, France, in 2010. Currently, he is working as a Post-doc Fellow with the IETR. His research interests include electromagnetic band-gap antennas, Fabry–Perot cavity antennas, frequency selective surfaces, and leaky-wave antennas for space applications.

Ronan Sauleau (M’04–SM’06) graduated in electrical engineering and radio communications from the Institut National des Sciences Appliquées, Rennes, France, in 1995. He received the Agrégation degree from the Ecole Normale Supérieure de Cachan, Cachan, France, in 1996, and the Doctoral degree in signal processing and telecommunications and the “Habilitation à Diriger des Recherche” degree from the University of Rennes 1, Rennes, France, in 1999 and 2005, respectively. He was an Assistant Professor and Associate Professor with the University of Rennes 1 between September 2000 and November 2005 and between December 2005 and October 2009. He has been a Full Professor with the same University since November 2009. He has received four patents and is the author or coauthor of 95 journal papers and more than 220 contributions to national and international conferences and workshops. His current research fields are numerical modeling (mainly FDTD), millimeter-wave printed and reconfigurable (MEMS) antennas, lens-based focusing devices, periodic and nonperiodic structures (electromagnetic band-gap materials, metamaterials, reflectarrays, and transmitarrays), and biological effects of millimeter waves. Prof. Sauleau received the 2004 ISAP Conference Young Researcher Scientist Fellowship (Japan) and the first Young Researcher Prize in Brittany, France, in 2001 for his research work on gain-enhanced Fabry–Perot antennas. In September 2007, he was elevated to Junior member of the “Institut Universitaire de France”. He was awarded the Bronze medal by CNRS in 2008.

Hervé Legay was born in 1965. He received the Electrical Engineering Degree and the Ph.D. degree from the National Institute of Applied Sciences (INSA), Rennes, France, in 1988 and 1991, respectively. For two years, he was a Postdoctoral Fellow with the University of Manitoba, Winnipeg, MB. Canada, where he developed innovating planar antennas. He joined Alcatel Space, Toulouse, France, in 1994, which is now Thales Alenia Space. He initially conducted studies in the areas of military telecommunication advanced antennas and antenna processing. He designed the architecture and the antijamming process of the Syracuse 3 antenna. He is first author of 21 patents. He currently leads research projects in integrated front ends and reflectarray antennas and coordinates the collaborations with academic and research partners in the area of antennas. Dr. Legay is a co-prize-winner of the 2007 Schelkunoff prize paper award. He received the Gold Thales Awards in 2008, rewarding the best innovations in the group Thales.

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A Simple Technique for the Dispersion Analysis of Fabry-Perot Cavity Leaky-Wave Antennas Carolina Mateo-Segura, Member, IEEE, Maria García-Vigueras, Student Member, IEEE, George Goussetis, Member, IEEE, Alexandros P. Feresidis, Senior Member, IEEE, and Jose Luis Gómez-Tornero, Member, IEEE

Abstract—A simple analysis technique to extract the complex dispersion characteristics of thin periodic 2-D Fabry-Pérot leaky wave antennas (LWA) is presented. The analysis is based on a two-stage process that dispenses with the need for root-finding in the complex plane. Firstly, full-wave MoM together with reciprocity is employed for the estimation of the LWA radiation patterns at different frequencies from which the phase constant is calculated. Employing array theory the phase constant is subsequently used to estimate the radiation patterns for different values of the leakage rate. The correct value for the leakage rate is identified by matching the corresponding radiation pattern to that obtained using the full-wave method. To demonstrate this technique, we present results for half-wavelength and sub-wavelength profile LWAs. Unlike the transverse equivalent network method, the proposed technique maintains its accuracy even for antennas with low profile. Index Terms—Frequency selective surface (FSS), leaky-wave antenna (LWA), periodic structures, resonant cavities.

H

I. INTRODUCTION

IGH gain antennas consisting of a 2-D periodic metallodielectric array suspended above a ground plane at a distance of approximately half-wavelength have been presented in the past [1] and have recently received increased attention [2]–[7]. They offer a simple solution for achieving highly directive patterns from a single low-directivity source. To a first approximation, their operation can be modeled by a Fabry-Pérot resonant cavity formed between the periodic array acting as a partially reflective surface (PRS) and the fully reflective ground plane [2]. The resonance condition ensures that the radiation emitted by a point source inside the cavity is converted into a Manuscript received February 23, 2010; revised March 22, 2011; accepted June 02, 2011. Date of publication September 15, 2011; date of current version February 03, 2012. The work of G. Goussetis was supported by the Royal Academy of Engineering under a five-year research fellowship. C. Mateo-Segura is with the School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, U.K. (e-mail: c.mateo@ieee. org). M. García-Vigueras and J. L. Gómez-Tornero are with the Departamento de Tecnologías de la Información y las Comunicaciones, Technical University of Cartagena, Antiguo Hospital de Marina, Cartagena 30202, Spain (e-mail: maria. [email protected]; [email protected]). G. Goussetis was with the Institute for Integrated Systems, Heriot-Watt University, Edinburgh EH14 4AS, U.K. He is now with the Institute of Electronics Communications and Information Technology (ECIT), Queen’s University Belfast, Northern Ireland, BT3 9DT, U.K. (e-mail: [email protected]). A. P. Feresidis is with the Wireless Communications Research Group, Department of Electronic and Electrical Engineering, Loughborough University, Leicestershire LE11 3TU, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2167900

Fig. 1. a) Layout of the resonant cavity leaky-wave antenna formed by metallodielectric PRS and AMC with excitation source inside the cavity b) Unit cell of a square patch PRS array and c) AMC array.

directive beam on the other side of the PRS. Antennas of this type have been also realized using stacks of uniform dielectric layers of different thickness [8]–[10]. However, periodic metallodielectric PRS, which are compatible with commonly employed printed circuit techniques, minimize the number of required layers and offer increased design flexibility. More recently, planar 2-D periodic metallic arrays printed on a grounded dielectric substrate have been presented as artificial magnetic conductors (AMC). Such structures exhibit a high surface impedance for incident plane waves within a specific frequency range [11]–[14], so that the average tangential magnetic field is small and the electric field large along the surface [15]. Due to this unusual boundary condition, AMC structures reflect incident plane waves in-phase to the incident field and can be used as ground planes for low-profile antennas. Employing this type of ground plane (Fig. 1), Fabry-Pérot type LWAs with quarter wavelength [16] and sub-wavelength [17] profiles have been reported. Several techniques have been proposed for the analysis and design of infinite Fabry-Pérot Leaky Wave Antennas (LWAs).

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An approximate ray-optics model was employed in [1] to extract the radiation pattern and the resonance condition. In [8] a transmission line model was introduced in order to predict the radiation characteristics and resonance conditions of antennas formed using multiple layers of dielectrics. More recently, the radiation patterns of Fabry-Pérot antennas formed by 2-D periodic metallodielectric arrays as PRS have been extracted using rigorous full-wave Method of Moments (MoM) and invoking reciprocity [4], [18]; however, this technique stops short of obtaining the complex propagation constant, which is useful for the design of antennas with tailored radiation patterns. The radiation characteristics of infinite LWAs can also be obtained by the complex wavenumber of the associated leaky mode [7], [9]. The wavenumber dispersion allows estimation of the antenna radiating aperture profile, which in turn can be used to obtain the far-field radiation patterns, their beamwidths and associated bandwidths, as well as the variation of the antenna pointing angle with frequency [19]. Knowledge of the complex dispersion relation is also helpful in the synthesis of practical LWAs. For example, the leakage rate allows estimation of the power radiated within a finite antenna length, which is essential in designing finite LWAs with high radiation efficiency. The complex wavenumber is also required for the systematic design of a non-uniform LWA, which can produce tapered illumination patterns that avoid phase aberration [19]–[21], leading to far-field patterns with reduced side-lobes and antenna systems prone to reduce interference. The complex dispersion of Fabry-Pérot LWAs with a PRS consisting of 2-D periodic metallodielectric arrays was first extracted in [22] employing a Transverse Equivalent Network (TEN) and a pole-zero method to estimate the equivalent impedance of the array. Since a single mode TEN is employed, the accuracy of this technique is reduced for sub-wavelength profile antennas. Although it is possible to produce multiport TEN [23] and other formulations of the eigenvalue problem to obtain the complex dispersion of bound and leaky modes of 2D periodic structures using full-wave techniques, such as MoM [22], the associated eigenvalue equations, zeros of the impedance matrix equation, typically take non-canonical form [24], [25], which is cumbersome to solve numerically in the complex plane. Techniques based on the Finite-Difference Time-Domain (FDTD) method have also been developed in order to extract the dispersion of the complex wavenumber for this type of antennas [26], [27]. These techniques can be time consuming and, particularly for very small or large values of the leakage rate, have limited accuracy. In this paper, we propose a new simple technique for the estimation of the complex dispersion of thin periodic 2-D LWAs in the leaky wave region. The technique combines for the first time array theory as well as periodic MoM with reciprocity. An overview of the method is given in Section II. Subsequently, the technique is applied in Section III in order to study three different antenna designs, namely a half-wavelength, a quarterwavelength and a sub-wavelength profile 2-D LWAs. The radiation patterns and the complex dispersion are derived employing the proposed method and compared with those obtained using a TEN.

II. DISPERSION OF FABRY-PEROT LEAKY-WAVE ANTENNAS The complex wavenumber, , of a leaky-mode in general takes the form: (1) where is the phase constant and is the leakage rate. The complex nature of expresses the decrease of the amplitude of the leaky wave as it propagates due to radiation. In the absence of other sources of radiation, the phase constant, , determines the pointing angle, , of the antenna’s main lobe and the leakage rate, , determines the illumination of the radiating aperture. Significantly, the radiation pattern of a LWA can be obtained analytically for a uniform LWA with a given complex wavenumber [19], [28]. The method that we propose here is based on the following procedure; the radiation pattern of a particular infinite-size LWA is initially obtained using full-wave periodic MoM and invoking reciprocity [4]. Subsequently an iterative procedure is employed based on array theory [28], [29] in order to reproduce this pattern from pairs of and . Since the calculations involved in this iterative process are analytical, and since prior knowledge of the propagation constant, , can be obtained by the angle of maximum radiation, the proposed technique is fast and computationally efficient. In the following we present the method and the analytical expressions involved in the calculation of the radiation patterns. A. Spectral Domain Periodic MoM and Reciprocity Reciprocity suggests that the far-field radiated at a certain direction by an antenna fed by a point source is proportional to the relative excitation of the near fields at an observation point upon plane wave incidence from the same direction. Hence, by scanning the relative field strength at an observation point inside the antenna cavity for plane waves incident with all possible angles at a fixed frequency, the radiation pattern of the antenna at this frequency can be obtained [4], [18]. This method can be efficiently applied employing the spectral domain periodic MoM for the full-wave modeling of LWAs such as the one depicted in Fig. 1. The Electric Field Integral Equation (EFIE) is determined by applying the boundary condition on the metallic elements that compose the array (here assumed perfect conductors), and subsequently solved using the Galerkin MoM. For simple array element geometries, such as the one shown in Fig. 1, the currents can be modelled using zero-ended entire domain sinusoidal basis functions [30], [31], yielding fast and accurate results. The details of this method are described elsewhere and therefore not repeated here [4], [30]. B. Array Factor Approach The array factor (AF) approach serves as an alternative method to calculate the radiation characteristics of periodic LWAs [28], [29]. The array factor for a 2-D planar array is given by the following expression [29]:

(2)

MATEO-SEGURA et al.: A SIMPLE TECHNIQUE FOR THE DISPERSION ANALYSIS OF FABRY-PEROT CAVITY LEAKY-WAVE ANTENNAS

where is the periodicity and is the number of unit cells along the - -axis respectively. For an infinitely long antenna there is no contribution to the radiation by edge effects. The phase in (2) represents the relative phase shift of the excitation for the order element referenced to the element at the origin. Assuming that all higher Floquet space harmonic (FSHs) are evanescent and only the fundamental can radiate, then the relative phase shift is determined by the propagation constant, , of the fundamental FSH in the - - directions:

(3) The relevant excitation strength of the array element, in (2), can be obtained from the attenuation rate, , due to the leakage, as well as the magnitude of the reference element, . Since for a uniformly periodic array the leakage rate, , is constant along the antenna, the excitation strength drops exponentially for elements away from the excitation point. To a step approximation, we can therefore write for the element along the - -axis:

(4) The radiation pattern of the antenna under consideration can be obtained as the product of the array factor with the radiation pattern of the PRS array element. In this example we assume a free-standing PRS consisting of square patches with edge (Fig. 1) whose radiation intensity, , at every can be obtained using Babinet’s principle from that of a rectangular aperture [29]:

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Fig. 2. Estimation of the propagation constant from the angle of maximum gain and the E-plane for a LWA. at the H-plane

where is the free-space wavenumber. In the following we assume that the antenna of Fig. 1 is excited by a Hertzian dipole polarized along the -axis. In this case, correspond to the phase constants along the H-/E-plane and can be obtained by varying the angle of the incident wave along the -planes respectively. Subsequently, the AF approach is employed to obtain the dispersion of the leakage rate, . This can be estimated using an inverse and iterative procedure. For each frequency point, we use the corresponding value of the propagation constant, , and the radiation pattern is successively estimated according to (2) for different values of the leakage rate, . For each value of , the corresponding radiation pattern is compared with the one derived using full wave MoM [30] and reciprocity [4]. This is done by calculating an error function which is expressed as the meansquare error between the two normalized patterns. The value of for which the error function is minimized corresponds to the actual value of the leakage rate at the particular frequency. For a new frequency point, estimations of the leakage rate at nearby frequencies can be used as starting values, also considering that higher frequencies typically produce lower leakage rates. Since the calculations involved in the iterative procedure are analytical, the proposed method is fast and efficient. III. NUMERICAL RESULTS

(5) where is the free space wavenumber, is the intrinsic impedance and is a constant. By combining (2) and (5), the radiation pattern of a LWA such as the one depicted in Fig. 1 can be analytically obtained for a given wavenumber . C. Derivation of the Complex Propagation Constant As shown above, the estimation of the radiation pattern following an array factor approach requires prior knowledge of both the real, , and imaginary, , part of the wavenumber, . In order to reduce the complexity of the problem, the former can be obtained by tracking the angle of maximum directivity in the full-wave radiation pattern. In particular, in order to extract the dispersion of the propagation constant in a particular direction the radiation pattern if the antenna under consideration is obtained at different frequencies. The angle, , corresponding to the direction of maximum directivity for each frequency is then related with by means of simple trigonometry [19] (Fig. 2): (6)

In this section we initially demonstrate the application of the proposed method in working examples of resonant cavity antennas with a single periodic array (PRS) and half-wavelength profile. This refers to the structure shown in Fig. 1 where a Perfect Electric Conductor (PEC) at is used instead of an AMC. Subsequently, we extend this technique to the case of antennas with two periodic arrays (PRS and AMC) and sub-wavelength profile; structure shown in Fig. 1 where is either or . The results from the proposed technique are compared with those from a TEN, where a pole and zero method is employed for the estimation of the effective impedance of the arrays. The latter is very well described in [32], [33] and therefore applied here directly. A. Half-Wavelength Antennas The structure under consideration involves a PRS consisting of square patches with edge 8.0 mm arranged in a square lattice with periodicity 9.0 mm and printed on a dielectric slab of thickness, , equal to 1.5 mm and relative permittivity 2.55. This PRS is located at a distance, , equal to 9.82 mm above a ground plane. This corresponds to approximately half-wavelength at 14 GHz, where the antenna produces a broadside pat-

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Fig. 3. Radiation pattern a) H-plane and b) E-plane of the LWA formed with a , , , square patch PRS with dimensions (in mm) and .

tern. The excitation is assumed to be a Hertzian dipole polarized along y and placed in the middle of the cavity (e.g., ). Periodic MoM in the spectral domain is employed to obtain the y-polarized fields at the center of the unit cell and (observation point). On the calculation of the near fields an optimized number of 40 FSH is considered for convergence better than 1%, [34]. The H- and E- plane radiation patterns for this LWA are obtained by the full-wave method discussed in Section II-A for a range of frequencies between 14 GHz and 16.5 GHz for the H-plane and between 14 GHz and 18 GHz for the E-plane. Some examples of these results are presented in Fig. 3. Tracking the angle of maximum, , and using (6), the dispersion of the phase constant is readily obtained. In agreement with previous studies of 2-D LWA Fig. 3 shows that at broadside a pencil beam is produced with equal 3 dB beamwidth in the H- and E-plane [35]. The patterns in the Eand H-plane are increasingly different at higher angles towards endfire. Further observation of this figure shows that as the beam angle increases, the peak field amplitude increases in the H-plane. The opposite is happening in the E-plane. These observations are in agreement with [35]. Furthermore, the inset in Fig. 3 shows the presence of grating lobes that correspond to the from 14.4 GHz onwards in the H- and the E-plane. The dispersion diagrams as obtained from the patterns of Fig. 3 in the frequency range studied are shown in Fig. 4. Based on (6), the H-plane pattern provides the phase constant of a TE mode along , and the E-plane pattern gives corresponding to a TM mode along y [32]. This figure also shows

Fig. 4. Normalized wavenumber versus frequency for a) the TE mode along (H-plane) and b) the TM mode along (E-plane) as obtained by the proposed technique and a Transverse Equivalent Network for the LWA with dimensions as in Fig. 3.

Fig. 5. a) H- and b) E-plane radiation pattern at 14.4 GHz for the half-wavelength antenna of Fig. 3 as obtained by full-wave Method of Moments and Array Factor theory.

superimposed the phase constant values as obtained from a TEN model [33]. A comparison of the values for indicates a very good agreement between the two techniques. Fig. 5 shows the radiation pattern calculated according to full-wave MoM together with that estimated using the AF approach assuming an infinitely long antenna with the obtained

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Fig. 7. Normalized wavenumber versus frequency for the H- and E-plane as obtained by the proposed technique and a Transverse Equivalent Network for , the sub-wavelength antenna of Fig. 1, with dimensions (in mm) , for the PRS: square patches , and and for the AMC: square patches , and operating at 14 GHz.

Fig. 6. Normalized leakage rate versus frequency, a) H-plane and b) E-plane as obtained by the proposed technique and a Transverse Equivalent Network for the LWA with dimensions as in Fig. 3.

leakage rate, . Since the AF calculation is based on the assumption of a single radiating Floquet space harmonic, it cannot predict the side lobes that emerge as a result of higher Floquet space harmonics in Fig. 5(b). Therefore, in the calculation of , the error function will only include the portion of the radiation pattern that is occupied by the main lobe, neglecting the higher values of , which correspond to side lobes. The computed values of the normalized leakage rate, , at the H- and E-plane as calculated using the proposed technique as well as the TEN model are shown in Fig. 6. The agreement between both techniques is good for the given range of frequencies. As common with LWAs [19], the normalized leakage rate, , decreases towards endfire direction. The interference of the side lobes with the main lobe limits the applicability of the proposed technique at higher frequencies. B. Quarter Wavelength Antennas Antennas with sub-wavelength profile can be produced introducing a second periodic array in close proximity to the ground plane [15], [16]. To a ray optics approximation, this can be attributed to the reduced reflection phase of the AMC ground plane. Here we employ a working example of an antenna such as the one shown in Fig. 1. The PRS employed previously is now located at a distance above an AMC array, which consists of patches with edge, and is printed

on a dielectric slab of thickness, and relative permittivity 2.2. The height of the cavity, , has been designed for the antenna to produce a broadside pattern at 14 GHz. A similar study as the one performed for the half-wavelength antenna is carried out. In order to apply the periodic MoM commensurate periodicities are assumed ensuring that the set of FSH is suitable for expanding the fields at both arrays. For thinner cavities, higher order evanescent FSH can increasingly interact and therefore become increasingly important in the calculation of the near fields and thus in the estimation of the patterns. For this example 60 FSH in the - and - direction are considered for a convergence better than 1%, [34]. The dispersion of the phase constants are shown in Fig. 7 for a range of frequencies between 14 GHz and 15.6 GHz for the H- and between 14 GHz and 16.5 GHz for the E-plane. This figure also shows superimposed the phase constant values as obtained from a TEN. The computed values of the leakage rate at the H- and E-plane are also depicted in Fig. 8 between 14 GHz and 15 GHz for the H- and E-plane. The TEN utilized in the calculations only accounts for a single mode, therefore when the antenna profile decreases the accuracy of the method is also reduced. Consequently, as is evident in Figs. 7 and 8, the agreement between the two methods for this antenna is reduced compared to the half-wavelength antenna, particularly in the H-plane. C. Thin Antennas Thin antennas with sub-wavelength profile can be produced employing an AMC with reflection phase lower than 0 in the configuration of Fig. 1. Due to the low profile, the interaction of higher order evanescent modes between the two arrays significantly increases. The accuracy of the single mode transverse equivalent network model gradually reduces compared to the half-wavelength profile LWA. The technique proposed here can be directly applied for thin antennas without loss of accuracy. Here we demonstrate this by means of an example involving an antenna with profile . The PRS is the same as in the previous studies and is located at a distance above an AMC array, which consists of

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Fig. 8. Normalized leakage rate versus frequency for the H-plane and E-plane as obtained by the proposed technique and a Transverse Equivalent Network for the sub-wavelength antenna of Fig. 7.

Fig. 10. Normalized wavenumber versus frequency for the H-plane and E-plane as obtained by the proposed technique and a Transverse Equivalent Network for the LWA of Fig. 9.

Fig. 11. Normalized leakage rate versus frequency for the H-plane and E-plane as obtained by the proposed technique and a Transverse Equivalent Network for the LWA of Fig. 9.

Fig. 9. Radiation pattern a) H-plane and b) E-plane of the LWA with dimen, , for the PRS: square patches , sions (in mm) and and for the AMC: square patches , and operating at 14 GHz as obtained using full-wave MoM and array factor procedure.

at broadside at the frequency of 14 GHz. However, a narrower beamwidth is obtained in the E-plane attributed to a lower value of the leakage rate at this plane. The dispersion diagrams for the H- and E-plane are shown in Fig. 10 for frequencies between 14–14.6 GHz and 14–16.5 GHz, respectively. The interference of the side-lobes with the main lobe impedes the application of the proposed technique in this case beyond 14.6 GHz for the H-plane and beyond 16.5 GHz for the E-plane. The computed values of the leakage rate are also depicted in Fig. 11 between 14–14.6 GHz and 14–15.1 GHz for the H- and E-plane, respectively. In both figures, the resulting dispersion parameters as obtained using TEN are depicted clearly showing how the accuracy of the method has reduced even more for the antenna, particularly at the H-plane. IV. DISCUSSION AND CONCLUSION

patches with printed on a dielectric slab of thickness, and relative permittivity 2.2. The H- and E- plane radiation patterns of the LWA at different frequencies are determined in order to extract the phase constant in either plane. A total number of 120 FSH in each direction has been taken into account for convergence better than 1%. The radiation patterns at both planes as obtained using MoM as well as an AF approach at different frequencies are presented in Fig. 9 showing a good agreement between both techniques that validate the AF model accuracy. A pencil beam is obtained

The antennas under investigation have been designed to operate at 14 GHz using the same PRS but different AMCs, so that the profile of the antenna reduces as the dimension of the lower array (AMC) is increased. By observation of Figs. 6, 8 and 11 one can conclude that higher values of the leakage rate are obtained for antennas with reduced profile and the same PRS. This leads to less directive radiation patterns for thinner antennas. For angles away from broadside (frequency higher than 14 GHz) the difference between the values of the leakage rate for antennas with different profiles becomes smaller. Figs. 5, 7

MATEO-SEGURA et al.: A SIMPLE TECHNIQUE FOR THE DISPERSION ANALYSIS OF FABRY-PEROT CAVITY LEAKY-WAVE ANTENNAS

and 10 further demonstrate that as the profile reduces, the phase constant varies more rapidly with frequency in both the Hand and E- plane. Moreover, the phase constant at the HE-plane take similar values for half-wavelength antenna (Fig. 5(a) and (b)). However, for thinner antennas the values of the phase constant for angles away from broadside (frequency higher than 14 GHz) increasingly differ being always larger at the H-plane, than at the E-plane, . In conclusion, a simple technique for the dispersion analysis of high-gain planar leaky-wave antennas employing either one or two periodic surfaces (PRS and AMC) has been presented. MoM together with reciprocity as well as an array factor approach have been used to estimate the complex propagation constant of these antennas. The proposed technique was firstly applied to the analysis of a LWA with half-wavelength profile and subsequently extended to antennas with lower profile. The radiation patterns for the E- and H- plane at different frequencies were obtained using MoM in order to extract the phase constant. The produced dispersion diagrams were in good agreement with those derived by a TEN. Reactive interaction between adjacent layers due to evanescent higher-order Floquet harmonics limits the validity of the single mode TEN, which is based on equivalent impedances of the PRS and AMC arrays and single mode circuits. The proposed technique overcomes this problem, so that low-profile LWAs can be accurately and efficiently analyzed. The main limitation of the proposed technique is due to the appearance of grating lobes, which limit the applicability of the technique. REFERENCES [1] G. V. Trentini, “Partially reflecting sheet arrays,” IRE Trans. Antennas Propag, vol. 4, pp. 666–671, 1956. [2] A. P. Feresidis and J. C. Vardaxoglou, “High-gain planar antenna using optimized partially reflective surfaces,” IEE Proc. Microw. Antennas Propag., vol. 148, no. 6, Feb. 2001. [3] Y. J. Lee, J. Yeo, R. Mittra, and W. S. Park, “Design of a high-directivity electromagnetic band gap (EBG) resonator antenna using a frequency selective surface (FSS) superstrate,” Microwave and Optical Tech. Lett., vol. 43, pp. 462–467, Dec. 2004. [4] T. Zhao, D. R. Jackson, J. T. Williams, D. Hung-Yu, and A. A. Oliner, “2-D periodic leaky-wave antennas-part I: Metal patch design,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3515–3524, Nov. 2005. [5] T. Zhao, D. R. Jackson, and J. T. Williams, “2-D periodic leaky-wave antennas-part II: Slot design,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3515–3524, Nov. 2005. [6] R. Gardelli, G. Donzelli, M. Albani, and F. Capolino, “Design of patch antennas and thinned array of patches in a Fabry-Perot cavity covered by a partially reflective surface,” in Proc. Eur. Conf. on Antennas and Propag. (EuCAP), Nice, France, Nov. 2006, pp. 6–10. [7] P. Kosmas, A. P. Feresidis, and G. Goussetis, “Periodic FDTD analysis of a 2-D leaky-wave planar antenna based on dipole frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 55, no. 7, pp. 2006–2012, Jul. 2007. [8] D. R. Jackson and N. G. Alexopoulos, “Gain enhancement methods for printed circuit antennas,” IEEE Trans. Antennas Propag., vol. AP-33, no. 9, Sep. 1985. [9] D. R. Jackson, A. A. Oliner, and A. Ip, “Leaky-Wave propagation and radiation for a narrow-beam multiple-layer dielectric structure,” IEEE Trans. Antennas Propag., vol. 41, no. 3, Mar. 1993. [10] R. Gardelli, M. Albani, and F. Capolino, “Array thinning by using antennas in a Fabry-Perot cavity for gain enhancement,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 1979–1990, Jul. 2006. [11] D. Sievenpiper, Z. Lijun, R. F. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [12] G. Goussetis, A. P. Feresidis, and J. C. Vardaxoglou, “Tailoring the AMC and EBG characteristics of periodic metallic arrays printed on grounded dielectric substrate,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 82–89, Jan. 2006.

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[13] O. Luukkonen, C. Simovski, G. Granet, G. Goussetis, D. Lioubtchenko, A. V. Räisänen, and S. A. Tretyakov, “Simple and accurate analytical model of planar grids and high-impedance surfaces comprising metal strips or patches,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1624–1632, Jun. 2008. [14] O. Luukkonen, P. Alitalo, C. R. Simovski, and S. A. Tretyakov, “Experimental verification of an analytical model for high-impedance surfaces,” Electron. Lett., vol. 45, no. 14, pp. 720–721, 2009. [15] G. Goussetis, A. P. Feresidis, and R. Cheung, “Quality factor assessment of subwavelength cavities at FIR frequencies,” J. Opt. A, vol. 9, pp. s355–s360, Aug. 2007. [16] A. P. Feresidis, G. Goussetis, S. Wang, and J. C. Vardaxoglou, “Artificial magnetic conductor surfaces and their application to low-profile high- gain planar antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 209–215, Jan. 2005. [17] S. Wang, A. P. Feresidis, G. Goussetis, and J. C. Vardaxoglou, “HighGain subwavelength resonant cavity antennas based on metamaterial ground planes,” IEE Proc. Antennas Propag., vol. 153, no. 1, pp. 1–6, Feb. 2006. [18] C. Mateo-Segura, G. Goussetis, and A. P. Feresidis, “Sub-wavelength profile 2-D leaky-wave antennas with two periodic layers,” IEEE Trans. Antennas Propag., vol. 59, no. 2, pp. 416–424, Feb. 2011. [19] A. Oliner, “Leaky-wave antennas,” in Antenna Engineering Handbook, R. C. Johnson, Ed., 3rd ed. : McGraw Hill, 1993. [20] J. L. Gómez, D. Cañete, and A. Álvarez-Melcón, “Printed-Circuit leaky-wave antenna with pointing and illumination flexibility,” IEEE Microwave Wireless Compon. Lett., vol. 15, no. 8, pp. 536–538, Aug. 2005. [21] J. L. Gómez, G. Goussetis, A. Feresidis, and A. A. Melcón, “Control of leaky-mode propagation and radiation properties in hybrid dielectric-waveguide printed-circuit technology: Experimental results,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3383–3390, Nov. 2006. [22] S. Maci, M. Casaletti, M. Caiazzo, and C. Boffa, “Dispersion properties of periodic grounded structures via equivalent network synthesis,” in Proc. IEEE Antennas Propag. Int. Symp., Jun. 22–27, 2003, vol. 1, pp. 493–496. [23] J. L. Gómez-Tornero, G. Goussetis, D. Cañete-Rebenaque, F. Quesada-Pereira, and A. Álvarez-Melcón, “Simple and accurate transverse equivalent network to model radiation from hybrid leaky-wave antennas with control of the polarization,” in Proc. IEEE Antennas Propag. Society Int. Symp., Jul. 5–11, 2008, pp. 1–4. [24] G. Goussetis, A. P. Feresidis, and P. Kosmas, “Efficient analysis, design, and filter applications of EBG waveguide with periodic resonant loads,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 11, Nov. 2006. [25] P. Baccarelli, S. Paulotto, and C. Di Nallo, “Full-wave analysis of bound and leaky modes propagating along 2D periodic printed structures with arbitrary metallisation in the unit cell,” IET Microw. Antennas Propag., vol. 1, no. 1, pp. 217–225, 2007. [26] T. Kokkinos, C. D. Sarris, and G. V. Eleftheriades, “Periodic finite-difference time-domain modeling of leaky-wave structures applied to the analysis of negative-refractive-index metamaterial-based leaky-wave antennas,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 1619–1630, Apr. 2006. [27] J. R. Kelly, T. Kokkinos, and A. P. Feresidis, “Analysis and design of sub-wavelength resonant cavity type 2-D leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2817–2825, Sep. 2008. [28] C. Caloz and T. Itoh, “Array factor approach of leaky-wave antennas and application to 1D/2D composite right/left-handed (CRLH) structures,” IEEE Microwave Wireless Comp. Lett., vol. 14, no. 6, pp. 274–276, Jul. 2006. [29] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997. [30] R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces-a review,” Proc. IEEE, vol. 76, pp. 593–615, Dec. 1988. [31] J. C. Vardaxoglou, Frequency Selective Surfaces Analysis and Design. New York: Wiley, 1997. [32] S. Maci, M. Caiazzo, A. Cucini, and M. Casaletti, “A pole-zero matching method for EBG surfaces composed of a dipole FSS printed on a grounded dielectric slab,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 70–81, Jan. 2005. [33] M. García-Vigueras, J. L. Gómez-Tornero, G. Goussetis, J. S. GómezDiaz, and A. Álvarez-Melcón, “A modified pole-zero technique for the synthesis of waveguide leaky-wave antennas loaded with dipole-based FSS,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 1971–1979, Jun. 2010. [34] C. Mateo-Segura, G. Goussetis, and A. P. Feresidis, “Resonant effects and near field enhancement in perturbed arrays of metal dipoles,” IEEE Trans. Antennas Propag, vol. 58, no. 8, pp. 2523–2530, Aug. 2010.

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[35] T. Zhao, D. R. Jackson, J. T. Williams, and A. A. Oliner, “General formulas for 2-D leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3515–3524, Nov. 2005.

C. Mateo-Segura (S’08–M’10) was born in Valencia, Spain, in 1981. She received the M.Sc. degree in telecommunications engineering from the Polytechnic University of Valencia, Valencia, Spain, in 2006 and the Ph.D. degree, jointly awarded between the University of Edinburgh and Heriot-Watt University, Edinburgh, U.K., in 2010. In 2006, she joined the Security and Defence Department of Indra Systems, Madrid, Spain, as a Junior Engineer. In 2009, she was a Research Associate in the Wireless Communications Research Group, Loughborough University, Leicestershire, U.K. In December 2010, she joined the Antennas & Electromagnetics Research Group, Queen Mary, University of London, as a Research Associate where she worked on the electromagnetic modelling and design of novel metamaterial antennas for high power applications. She is currently with the School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, U.K. Her research interests include the analysis and design of frequency selective surfaces, artificial periodic electromagnetic structures with applications on high-gain array antennas and medical imaging systems. Dr. Mateo-Segura was awarded a prize studentship from the Edinburgh Research Partnership and the Joint Research Institute for Integrated Systems to join the RF and Microwave group at Heriot-Watt University, Edinburgh, Scotland, U.K in 2006.

Maria García-Vigueras (S’09) was born in Murcia, Spain, in 1984. She received the Telecommunications Engineer degree from the Technical University of Cartagena (UPCT), Spain, in 2007, where she is currently working towards the Ph.D. degree. In 2008, she joined the Department of Communication and Information Technologies, UPCT, as a Research Assistant. She has been a visiting Ph.D. student at Heriot-Watt University in Edinburgh (Scotland, United Kingdom), at the University of Seville (Spain) and in Queen’s University of Belfast (Northern Ireland, United Kingdom). Her research interests focus on the development of equivalent circuits to characterize periodic surfaces, with application to the analysis and design of leaky-wave antennas.

George Goussetis (SM’99–M’02) graduated from the Electrical and Computer Engineering School, National Technical University of Athens, Greece, in 1998, and received the B.Sc. degree in physics (first class) from University College London (UCL), U.K. and the Ph.D. degree from the University of Westminster, London, U.K., both in 2002. In 1998, he joined the Space Engineering, Rome, Italy, as an F Engineer and in 1999 the Wireless Communications Research Group, University of Westminster, U.K., as a Research Assistant. Between 2002 and 2006, he was a Senior Research Fellow at Loughborough University, U.K. Between 2006 and 2009, he was a Lecturer (Assistant Professor) with the School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, U.K. He joined the Institute of Electronics Communications and Information Technology, Queen’s University Belfast, U.K, in September 2009, as a Reader (Associate Professor). In 2010, he was a Visiting Professor at UPCT, Spain. He has authored or coauthored over 100 peer-reviewed papers three book chapters and two patents. His research interests include the modelling and design of microwave filters, frequency-selective surfaces and periodic structures, leaky wave antennas, microwave heating as well numerical techniques for electromagnetics. Dr. Goussetis received the Onassis foundation scholarship in 2001. In October 2006 he was awarded a five-year research fellowship by the Royal Academy of Engineering, UK.

Alexandros P. Feresidis (S’98–M’01–SM’08) was born in Thessaloniki, Greece, in 1975. He received the Physics degree from Aristotle University of Thessaloniki, Greece, in 1997, the M.Sc.(Eng) in radio communications and high frequency engineering from the University of Leeds, U.K, in 1998, and the Ph.D. degree in electronic and electrical engineering from Loughborough University, U.K., in 2002. During the first half of 2002, he was a Research Associate and in the same year he was appointed Lecturer in Wireless Communications in the Department of Electronic and Electrical Engineering, Loughborough University, UK, where, in 2006, he was promoted to Senior Lecturer. He has published more than 100 papers in peer reviewed international journals and conference proceedings and has coauthored three book chapters. His research interests include analysis and design of artificial periodic metamaterials, electromagnetic band gap (EBG) structures and frequency selective surfaces (FSS), array antennas, small/compact antennas, numerical techniques for electromagnetics and passive microwave/mm-wave circuits.

José Luis Gómez-Tornero (M’06) was born in Murcia, Spain, in 1977. He received the Telecommunications Engineer degree from the Polytechnic University of Valencia (UPV), Valencia, Spain, in 2001, and the Ph.D. degree (laurea cum laude) in telecommunication engineering from the Technical University of Cartagena (UPCT), Cartagena, Spain, in 2005. In 1999, he joined the Radio Communications Department, UPV, as a research student, where he was involved in the development of analytical and numerical tools for the automated design of microwave filters in waveguide technology for space applications. In 2000, he joined the Radio Frequency Division, Industry Alcatel Espacio, Madrid, Spain, where he was involved with the development of microwave active circuits for telemetry, tracking and control (TTC) transponders for space applications. In 2001, he joined the Technical University of Cartagena (UPCT), Spain, as an Assistant Professor. From October 2005 to February 2009, he held de position of Vice Dean for Students and Lectures affairs in the Telecommunication Engineering Faculty at the UPCT. Since 2008, he has been an Associate Professor at the Department of Communication and Information Technologies, UPCT. In February 2010, he was appointed CSIRO Distinguished Visiting Scientist by the CSIRO ICT Centre, Sydney. His current research interests include analysis and design of leaky-wave antennas and the development of numerical methods for the analysis of novel passive radiating structures in planar and waveguide technologies. Prof. Gómez Tornero received the national award from the foundation EPSON-Ibérica to the “best Ph.D. project in the field of technology of information and communications (TIC),” in July 2004 and the Vodafone foundation-COIT/AEIT (Colegio Oficial de Ingenieros de Telecomunicación) award to the best Spanish Ph.D. thesis in the area of “advanced mobile communications technologies,” in June 2006. This thesis was also awarded the “best thesis in the area of electrical engineering,” by the Technical University of Cartagena, in December 2006.

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Analyzing the Complexity and Reliability of Switch-Frequency-Reconfigurable Antennas Using Graph Models Joseph Costantine, Member, IEEE, Youssef Tawk, Member, IEEE, Christos G. Christodoulou, Fellow, IEEE, James C. Lyke, Senior Member, IEEE, Franco De Flaviis, Senior Member, IEEE, Alfred Grau Besoli, Member, IEEE, and Silvio E. Barbin, Senior Member, IEEE

Abstract—This paper addresses the functional reliability and the complexity of reconfigurable antennas using graph models. The correlation between complexity and reliability for any given reconfigurable antenna is defined. Two methods are proposed to reduce failures and improve the reliability of reconfigurable antennas. The failures are caused by the reconfiguration technique or by the surrounding environment. These failure reduction methods proposed are tested and examples are given which verify these methods. Index Terms—Complexity, graph theory, reconfigurable antennas, reliability, switches.

I. INTRODUCTION

T

HE incorporation of switches into reconfigurable antenna structures increases their complexity which in turn diminishes the reliability of the antennas. In particular, the reliability of reconfigurable antennas is of upmost importance in unknown and unpredictable environments. The design of switching elements is highly dependent on environmental conditions. For instance, if the reconfigurable antennas were deployed in space, the environment is unpredictable and the antenna structure is difficult to access. Various publications discuss certain environmental effects on different types of switches which are used in antennas to achieve reconfiguration. In [1], RF MEMS capacitive switches are built on microwave-laminate printed circuit boards (PCBs). The proposed technology promises further monolithic integration of switches and antennas on PCBs to form reconfigurable Manuscript received June 09, 2010; revised June 14, 2011; accepted August 22, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. J. Costantine is with the Electrical Engineering Department, California State University Fullerton, Fullerton, CA 92834 USA. Y. Tawk and C. G. Christodoulou are with the Electrical and Computer Engineering Department, University of New Mexico, Albuquerque, NM 87131 USA. J. C. Lyke is with the Air Force Research Laboratory, Kirtland Air Force Base, Albuquerque, NM 87117 USA. F. De Flaviis is with the Department of Electrical Engineering and Computer Science, University of California at Irvine, Irvine, CA 92697 USA. A. Grau Besoli is with Broadcom Corporation, Irvine, CA 92617 USA. S. E. Barbin is with the Telecommunications and Control Engineering Department of the Polytechnic School, University of São Paulo, SP 05508-900 Brazil. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173104

antennas without the mismatch problems associated with the use of discrete switching elements. In [2] the authors explain that the one-switch membrane topology of RF MEMS switches used in most designs is limiting for highly dynamic applications. Such applications require a great deal of reconfigurability. Three sets of electrostatic actuated RF MEMS switches with different actuation voltages are used to sequentially activate and deactivate parts of a Sierpinski fractal antenna. This allows direct actuation of the MEMS switches through the RF single feed without the need for individual DC bias lines. The antenna is fabricated on liquid crystal polymer substrate and constitutes the first integrated RF MEMS reconfigurable antenna on a flexible organic polymer substrate. Air-bridged RF-MEMS switches in single pole singlethrough transmission (SPST) configuration are proposed in [3] for antenna applications. In [4], tunable RF MEMS are proposed for the development of reconfigurable antennas fabricated on sapphire substrate with a barium strontium titanate dielectric. The problem of integrating commercially packaged RF MEMS into a reconfigurable antenna is discussed in [5], [6], not only the insertion loss and isolation behavior of the switches are addressed, but also their impact on the radiation characteristics of the antenna. In [7] the effect of carbon contamination on the reliability of RF MEMS is considered. It is shown that the use of RF MEMS in many commercial and military applications is limited by poor reliability [7]. Most publications in this area do not reflect the reliability of systems relying on switches, and few designers investigate the environmental effects, as in [7], on the good operation of the system. The fundamentals of improving systems reliability is first addressed by Shannon and Moore, they propose using redundancy to increase reliability [8]. Another publication [9] discusses reliability, where the fundamental mathematics of fault-tolerant circuit-switching networks is illustrated. These publications [8], [9] emphasize and recommend redundancy to improve the reliability of any switching circuit. This work [8], [9] is done on electronic circuits without considering any electromagnetic aspects. Finally in [10], [11] a complexity reduction approach for switch-reconfigurable antennas is developed. This approach reduces the number of unnecessary switches.

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Fig. 2. The antenna in [16] with its graph model. Fig. 1. Example of an undirected graph, and a directed graph with weighted edges.

In this paper we base our discussion on finding a trade-off between the reduction in redundancy and the improvement in reliability. The analysis in [8] is used to study the effect of the complexity reduction approach [11] on the reliability of a particular antenna. Even though the number of switches decreases after the implementation of the approach in [11], the number of equivalent configurations is proven to be sufficient. The electromagnetic behavior is sufficient for reliability while the number of switches is decreased. We also show that the frequency dependent reliability is inversely proportional to the complexity. We propose two methods to increase the robustness of reconfigurable antennas and present a methodology that ensures the reliability of a reconfigurable antenna system. In the next section graphs are presented as a modeling tool for reconfigurable antennas and equivalent configurations are discussed. II. GRAPH MODELING OF RECONFIGURABLE ANTENNAS AND EQUIVALENT ANTENNA CONFIGURATIONS Graph modeling is a useful tool for analyzing reconfigurable antennas as discussed in [10]–[13]. A graph is defined as a collection of vertices connected by lines called edges [10]–[13]. A graph may be either directed or undirected. In a directed graph, the edges have a determined direction, while in an undirected graph edges may be traversed in either direction. Fig. 1 shows examples of a directed and an undirected graph. The vertices represent physical entities and the edges indicate the existence of functions relating these entities. In a graph model of antennas, an edge is created between two vertices whenever their physical connection represents a meaningful antenna function [13]. Edges can have weights associated with them in order to represent costs or benefits that are to be minimized or maximized. The directed graph in Fig. 1 is an example of a weighted graph. A path is an ensemble of edges connecting two vertices, and its weight is defined as the sum of the weights of its constituent edges. For example, if a switch connects two parts of an antenna, then a weight represents the connection’s distinctive direction. There are several ways to graph model reconfigurable antennas. Rules for graph modeling switch-reconfigurable antennas are discussed in [10]. Graph modeling a reconfigurable

antenna translates this antenna from a bulky device into a software accessible device. The use of graph models allows designers to use their algorithms for control and automation [14]. Graphs also allow a swifter and better implementation of cognitive radio applications [15] and a reduction in redundant configurations. Thus, graphs improve the antenna efficiency and reduce cost and losses. Graph models are also used to represent equivalent antenna configurations. In a switch frequency reconfigurable antenna, many switching configurations yield the same antenna frequency behavior without affecting the other radiation properties. Equivalent configurations are obtained from simulations. These equivalent configurations constitute back-up configurations for maintaining the same antenna performance at a certain frequency. For example, the antenna shown in Fig. 2 [16], and its graph model, resonates at 5 GHz for eight different switch configurations. These eight configurations are shown in Table I and a comparative S11 plot is shown in Fig. 3. These eight configurations allow us options for achieving the desired resonance frequency at 5 GHz while each configuration has different multiband characteristics. It is true that reducing the number of switches decreases the number of equivalent antenna configurations at each resonant frequency [10], [11]. However, the number of remaining configurations is sufficient for reliable antenna operation and decreases the total “cost” of the antenna. To demonstrate reduced redundancy without loss of reliability, a previously optimized antenna (in [12], [13]) is studied below. The optimized antenna is shown in Fig. 4. Some of the antenna configurations for different antenna resonances are shown in Table II. Even after optimization, this antenna has several equivalent configurations for each resonant frequency. The optimization technique reduced the number of switches and hence cost without reducing the reliability of the system at operating frequencies. It is also important to point out that certain resonant frequencies are only achievable with a single configuration. There is a need to develop some methods to improve the efficiency and insure continuous antenna operation. In the next section the complexity and reliability of reconfigurable antennas are formulated and methods for improving the antenna reliability are proposed.

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TABLE I THE DIFFERENT CONFIGURATIONS OF THE ANTENNA IN [15] THAT LEAD TO OPERATION AT 5 GHZ

Fig. 4. Optimized antenna [12], [13].

III. RELIABILITY FORMULATIONS FOR FREQUENCY RECONFIGURABLE ANTENNAS

Fig. 3. S11 plot for the antenna in [16] for the configurations presented in Table I. (a) Zoomed out and (b) zoomed in at 5 GHz.

According to Shanon and Moore [8] a switch failure occurs when: 1) a switch is originally OFF and fails to switch ON upon request; 2) a switch is originally ON and fails to switch OFF upon request. A switching failure heavily affects the reliability of a switch reconfigurable antenna. These failures are due to the environment of operation, the aging and corrosion process, and the frequency of operation. Thus, the reliability of reconfigurable antennas depends on all the previously mentioned factors. With graphs, we can calculate the reliability using models which represent the different antenna configurations. The reliability is dependent on the number of antenna configurations at a certain frequency and the probability to achieve these configurations. However, it is inversely proportional to the number of edges needed to create these configurations. The solution is to

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TABLE II SOME ANTENNA CONFIGURATIONS FOR DIFFERENT RESONANCES (ALL FREQUENCIES ARE IN GHZ)

design reconfigurable antennas with several alternative configurations but only a small number of connections. This relationship is shown by (1):

(1)

Assuming the probability that each edge exists in a given configuration is equal to 0.98, this is the probability of success, then according to (1) (see the equation at the bottom of the page). Example 2: Let us now consider the same antenna shown in Fig. 4 but at 1.7 GHz. Let us assume that the probability of switching success with switch 1 is 0.999, the probability of success with switch 2 is 0.998, and the probability of success with switch 3 is 0.900. According to (1), the reliability at 1.7 GHz is

where: reconfigurable antenna reliability at a particular frequency ; number of configurations achieving the frequency ; number of edges for different configurations at the frequency ; probability of achieving the edge switch failing).

(a

Example 1: Let us consider the antenna shown in Fig. 4. Assume we want to calculate the antenna’s reliability at 2.9 GHz. According to Table II, at 2.9 GHz the antenna has three equivalent configurations which resonate at this particular frequency.

%

The variation of the reliability for different probability values at a particular frequency is linear and Fig. 5 shows this variation at GHz. Here, if no switches are used for achieving a certain reconfiguration, then the reliability is 100%. An example of

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Fig. 5. The variation of the reliability for different values of the probabilities GHz. at Fig. 6. The antenna in [17].

such a 100% reliable antenna is the antenna in Fig. 4 at 5.2 GHz. One of the configurations which resonate at this frequency does not use any switches. IV. GENERAL COMPLEXITY OF RECONFIGURABLE ANTENNAS Increasing the number of edges in a reconfigurable antenna graph model adds to the complexity of the system. The complexity is based on the size of the graph; i.e., the number of edges for all possible connections in that graph. (2) where NE represents the number of edges for all possible connections. This definition of complexity is different from other definitions of complexity, such as computational complexity. Removal of redundant elements results in reduction of the general complexity of the hardware used as well as simplification of software analysis employed to control the reconfiguration technique. We show below some examples where complexity is decreased using the optimization technique [10], [11]. Example 3: In [17], a reconfigurable pixeled antenna is proposed and is shown in Fig. 6. A discussion of the reliability issues and redundancy minimization is presented in [10]. This antenna exhibits five different modes of operation for any frequency between 6 and 7 GHz [17]. Graph models showing the different antenna configurations based on different switch activation status are shown in Fig. 7. The different sections of the optimized antenna are shown in Fig. 8. The antenna is optimized while preserving its core function and its topology [11]. One notes that some of the switches are not needed to achieve the required functions. This reduction in switches reduces the complexity of the antenna. The general complexity of this antenna before optimization is , and after optimization C is reduced to .

Fig. 7. Graph models for the required antenna configurations (five different modes). Parts in green indicate parts where switches are ON; parts in red indicate switches are OFF.

Example 4: The general complexity of the unoptimized structure of Fig. 4 is according to (2) whereas the general complexity of the optimized topology discussed in [13] is . V. CORRELATION OF COMPLEXITY AND RELIABILITY OF RECONFIGURABLE ANTENNAS The redundancy reduction technique presented in [10] and [11] can reduce the general complexity of reconfigurable antenna systems. However, since an antenna can have several configurations at different frequencies of operation, we must define complexity at each particular frequency. Equation (3) defines this frequency-dependent complexity. (3)

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Table I. The complexity of this antenna at 5 GHz is calculated by (3) as

The complexity of the optimized antenna shown in Fig. 4 at 5.2 GHz is calculated by (3) as

The correlation of the complexity of an antenna at a frequency and its reliability at that same frequency can be derived using (2), as shown in (4):

(4)

Fig. 8. Different antenna sections as discussed in [10].

where: where:

C(f) represents the complexity of the antenna system at a frequency ;

NC(f) represents the number of equivalent configurations at a frequency ; represents the number of edges at the configuration for a frequency . As an example, let us take the antenna shown in Fig. 2. The different configurations of this antenna at 5 GHz are shown in

is calculated in (3);

N C(f) is the number of equivalent configurations at a frequency without the configuration with maximum edges. From (4) we can deduce that the reliability of a reconfigurable antenna at a frequency is inversely proportional to the complexity of that antenna’s structure at the same frequency . Example 5: Taking the same antenna from example 1 and recalculating the reliability according to (4) reveals (see the equations at the bottom of the page).

COSTANTINE et al.: ANALYZING THE COMPLEXITY AND RELIABILITY OF SWITCH-FREQUENCY-RECONFIGURABLE ANTENNAS

VI. INCREASING

Fig. 9. Antenna using backup switch for 2.05 GHz equivalent configurations.

Fig. 10. Graph model of the equivalent configuration in Fig. 9.

Fig. 11. The antenna’s reflection coefficient with backup switch S2 and S3 activated showing clear operation at 2.05 GHz.

THE

RELIABILITY ANTENNAS

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OF

RECONFIGURABLE

In this section we propose two methods to improve the reliability of a reconfigurable antenna system. These methods are based on the presence of antenna redundant configurations even after the implementation of the redundancy reduction approach [10], [11]. These redundant configurations are a manifestation of the antenna electromagnetic behavior under the remaining switch states. These methods utilize redundant configurations to improve the reliability of the reconfigurable antenna. The first method suggests that one should organize the desired frequencies starting from high priority to low priority. If the first method is not sufficient to increase the reliability of an antenna at a certain desired frequency then the second method is applied. This method, based on adding a back-up switch, utilizes the analysis in [8] to improve the reliability of such antenna as explained below. Method 1: The No-Switch Configuration: The first method advises the designer to first prioritize the frequencies needed. The frequency with the highest priority should have more than one equivalent configuration. If we look at Table II, we can deduce that the frequency with the largest number of equivalent configurations is GHz. It has seven equivalent configurations, including the no-switch configuration (all switches off). A good design approach is to design the antenna to operate at the most important frequency or frequencies with all switches off. In that case, under the worst possible scenario of all switches breaking down at the same time, the most important frequency is always achievable. Method 2: The Back-Up Switch: This method proposes installing a back-up switch. The back-up switch can be installed at any place in the antenna system as long as its presence achieves the desired frequency. This method is used when a certain frequency is needed at all times; and the design does not include enough back-up configurations. Many factors come in play when installing a back-up switch. We assume that the probability of failure of a switch remains constant in time and does not change. Thus, the back-up switch method can be used if and only if, it satisfies the following constraints: 1) Its probability of failure is lower than or equal to the lowest probability of failure among all switches. 2) The sum of probabilities of success in the back-up configuration is higher than the sum of probabilities of success in the original configuration. Example 6: The optimized antenna in Fig. 4 operates at 2.05 GHz for only one configuration (S1 ON) as shown in Table II. Installing a back-up switch as shown in Fig. 9 and activating switches S2 and S3 constitutes a back-up configuration. The graph model of this system is represented in Fig. 10 where P0 is replaced by P’0 since by adding the back-up switch new vertices appear and the topology of the antenna section represented by P0 has changed. As stated, the placement of the switch is up to the designer as long as the presence of that switch achieves the desired function. The reflection coefficient plot is shown in Fig. 11. The reliability of this antenna can be increased by applying either or both of the two methods proposed in this section.

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This antenna originally has only one configuration that achieves 2.05 GHz, that configuration requires only the activation of S1. Assuming the probability of switch activation success is 0.98 for switch 1, 0.985 for switch 2, and 0.999 for switch 3, then according to (1):

The back-up switch configuration adds a new configuration possibility to achieve 2.05 GHz by switching ON S2, S3 and BUS (back-up switch), so calculating the reliability with the back-up switch ON according to (1) gives us

Step 2: Identify desired frequency, where the desired frequency is the resonance at which the antenna operation is required. Step 3: In the library table, create a pointer at row i corresponding to the desired frequency. Step 4: In the library table, create a pointer at column j corresponding to the defected configuration. Step 5: Move the pointer j to the placement . Step 6: Search for a possible edge representing a connection from the defected switch. Step 7: If no connection is found, use configuration in the column . Step 8: If a connection is found repeat step 5 and 6. Step 9: If no solution is found, declare frequency unachievable. VII. CONCLUSION In this paper we use graph models to study the reliability and complexity of reconfigurable antenna systems. We analyze the different reliability and complexity aspects and present methods for improving the reliability of switch-reconfigurable antennas. We correlate the complexity and reliability parameters of reconfigurable antennas that are proven to be inversely proportional. Examples are presented and discussed to demonstrate the validity of the proposed concept. Finally, the continuous functioning of reconfigurable antennas is studied and a methodology is presented to ensure the reliability of such systems under different conditions. REFERENCES

thus proving that adding the back-up switch not only assures the continuous functionality of this antenna but also improves its reliability at that particular frequency. To overcome a switch failure and thus restore a lost resonance, the following methodology is proposed. This methodology identifies the defected switch, specifies the desired frequency and changes the antenna topology to restore the desired resonance based on the equivalent configuration. Before applying this approach, the designer creates a library similar to Table II in which all possible configurations for all desired frequencies are identified. The designer installs switches or RF components on the antenna structure for the specific purpose of tuning it to a certain frequency or achieving a certain antenna reconfiguration function. Neural networks (NN) can also be used as in [6] to determine the library of equivalent configurations. The use of NN speeds up the library assembly process for large structures. The designer should include in the library the backup switch configuration if such configuration exists for specific frequencies. The algorithm is described below: Step 1: Identify defected switch.

[1] H. Chang, J. Quian, B. A. Cetiner, F. De Flaviis, M. Bachman, and G. P. Li, “RF MEMS switches fabricated on microwave-laminate printed circuit boards,” IEEE Electron Dev. Lett., vol. 24, no. 4, pp. 227–229, Apr. 2003. [2] N. Kingsley, D. E. Anagnostou, M. Tentzeris, and J. Papapolymerou, “RF MEMS sequentially reconfigurable Sierpinski antenna on a flexible organic substrate with novel DC-biasing technique,” J. Microelectromechan. Syst., vol. 16, no. 5, pp. 1185–1192, Oct. 2007. [3] C. W. Jung and F. De Flaviis, “RF-MEMS capacitive series of CPW&MSL configurations for reconfigurable antenna application,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 2005, vol. 2A, pp. 425–428. [4] G. Wang, T. Polley, A. Hunt, and J. Papapolymerou, “A high performance tunable RF MEMS switch using barium strontium titanate (BST) dielectrics for reconfigurable antennas and phased arrays,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 217–220, 2005. [5] G. H. Huff and J. T. Bernhard, “Integration of packaged RF MEMS switches with radiation pattern reconfigurable square spiral microstrip antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pt. 1, pp. 464–469, Feb. 2006. [6] D. E. Anagnostou, G. Zheng, M. Chryssomallis, J. Lyke, G. Ponchak, J. Papapolymerou, and C. G. Christodoulou, “Design, fabrication and measurements of an RF-MEMS-based self-similar reconfigurable antenna,” IEEE Trans. Antennas Propag., Special Issue on Multifunction Antennas and Antenna Systems, vol. 54, no. 2, pt. 1, pp. 422–432, Feb. 2006. [7] A. Carton, C. G. Christodoulou, C. Dyck, and C. Nordquist, “Investigating the impact of carbon contamination on RF MEMS reliability,” in Proc. IEEE Antennas and Propagation Int. Symp., Jul. 2006, pp. 193–196. [8] E. F. Moore and C. E. Shannon, Reliable Circuits Using Less Reliable Relays, vol. 262, no. 3, pt. I and II, pp. 191–208, Sep. 1956, J. Franklin Inst., No. 4 (Oct. 1956), 281–297.

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[9] N. Pippenger and G. Lin, “Fault-tolerant circuit-switching networks,” SIAM J. Discrete Math., vol. 7, no. 1, pp. 108–118, June 1994. [10] J. Costantine, “Design, optimization and analysis of reconfigurable antennas,” Ph.D. dissertation, Univ. New Mexico, Dept. Electr. Comput. Eng., Dec. 2009. [11] J. Costantine, S. Al-Saffar, C. G. Christodoulou, and C. T. Abdallah, “Reducing redundancies in reconfigurable antenna structures using graph models,” IEEE Trans. Antennas Propag., vol. 59, no. 3, pp. 793–801, 2011. [12] J. Costantine and C. G. Christodoulou, “Analyzing reconfigurable antenna structure redundancy using graph models,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2009, pp. 1–4. [13] J. Costantine, C. G. Christodoulou, C. T. Abdallah, and S. E. Barbin, “Optimization and complexity reduction of switch-reconfigurable antennas using graph models,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1072–1075, 2009. [14] J. Costantine, S. Al-Saffar, C. G. Christodoulou, K. Y. Kabalan, and A. El-Hajj, “The analysis of a reconfiguring antenna with a rotating feed using graph models,” IEEE Antennas Wireless Propag. Lett., 2009, accepted for publication. [15] Y. Tawk and C. G. Christodoulou, “A new reconfigurable antenna design for cognitive radio,” IEEE Antennas Wireless Propag. Lett., pp. 1378–1381, 2009. [16] J. Costantine, C. G. Christodoulou, and S. E. Barbin, “A new reconfigurable multi-band patch antenna,” in Proc. IEEE IMOC Conf., Salvador, Brazil, Oct. 2007, pp. 75–78. [17] A. Grau, L. Ming-Jer, J. Romeu, H. Jafarkhani, L. Jofre, and F. De Flaviis, “A multifunctional MEMS-reconfigurable pixel antenna for narrowband MIMO communications,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2007, pp. 489–492.

Joseph Costantine (M’10) is an Assistant Professor at the Electrical Engineering Department in California State University Fullerton. He received the Bachelor’s degree in electrical, electronics, computer and communications engineering from the second branch of the Faculty of Engineering in the Lebanese University in 2004. He received the Masters in computer and communications engineering from the American University of Beirut in 2006, during which he was awarded a 6 months research scholarship at Munich University of Technology (TUM) as part of the TEMPUS program. In 2009 he received the Ph.D. degree in electrical and computer engineering from the Electrical and Computer Engineering Department at University of New Mexico, where he also completed his Post-Doc Fellowship in July 2010. His research interests are in the areas of reconfigurable systems and antennas, antennas in wireless communications, deployable antennas, electromagnetic fields, RF Electronic Design and communication systems. Dr. Costantine received many awards during his studies and career. He has also published many research papers and is a co-author of an upcoming book on reconfigurable antennas.

Youssef Tawk (M’10) is a Post-Doc fellow at the University of New Mexico, where he received the Ph.D. degree from the Electrical and Computer Engineering Department in May 2011. He received the Master degree in computer and communications engineering from the American University of Beirut in 2008 and the Bachelor degree in computer and communications engineering from Notre Dame University, Louaize, Lebanon, in 2006. His research areas include reconfigurable antenna systems, cognitive radio, RF electronic design and photonics. He has received many awards during his studies. He has published several journal and conference papers.

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Christos G. Christodoulou (F’02) received the Ph.D. degree in electrical engineering from North Carolina State University in 1985. He served as a faculty member in the University of Central Florida, Orlando, from 1985 to 1998. In 1999, he joined the faculty of the Electrical and Computer Engineering Department of the University of New Mexico, where he served as the Chair of the Department from 1999 to 2005. He is a Fellow member of IEEE and a member of Commission B of URSI. He served as the general Chair of the IEEE Antennas and Propagation Society/URSI 1999 Symposium in Orlando, Florida and the co-technical chair for the IEEE Antennas and Propagation Society/URSI 2006 Symposium in Albuquerque. Currently he is the Director of the Aerospace Institute at the University of New Mexico, and the chief research officer for COSMIAC (Configurable Space Microsystems Innovations & Applications Center at UNM). He was appointed as an IEEE AP-S Distinguished Lecturer (2007–2010) and elected as the President for the Albuquerque IEEE Section in 2008. He served as a associate editor for the IEEE Transaction on antennas and Propagation for six years, as a guest editor for a special issue on “Applications of Neural Networks in Electromagnetics” in the Applied Computational Electromagnetics Society (ACES) journal, and as the co-editor of a the IEEE Antennas and Propagation Special issue on “Synthesis and Optimization Techniques in Electromagnetics and Antenna System Design” (March 2007). He is the recipient of the 2010 IEEE John Krauss Antenna Award for his work on reconfigurable fractal antennas using MEMS switches, the Lawton-Ellis Award and the Gardner Zemke Professorship at the University of New Mexico. He has published about 400 papers in journals and conferences, written 14 book chapters and co-authored four books. His research interests are in the areas of modeling of electromagnetic systems, reconfigurable antenna systems, cognitive radio, and smart RF/photonics.

James C. Lyke (SM’06) received the B.S.E.E. degree from the University of Tennessee, the M.S.E.E. degree from the Air Force Institute of Technology, and the Ph.D. degree from the University of New Mexico. He is technical advisor to the Air Force Research Laboratory’s Space Electronics Branch (Space Vehicles Directorate) and an AFRL Fellow. He has led over 100 in-house and contract research efforts involving advanced packaging, radiation-hardened circuits, and scalable, reconfigurable architectures, with recent emphasis on rapid formation of complex systems (“plug-and-play”). He has authored over 100 publications, four receiving best paper awards, and has been awarded eleven U.S. patents. He is a senior member of IEEE and associate fellow of AIAA.

Franco De Flaviis (SM’07) was born in Teramo, Italy, in 1963. He received the Italy degree (Laurea) in electronics engineering from the University of Ancona, Italy), in 1990. In 1991 he was an engineer employee at Alcatel as a researcher specialized in the area of microwave mixer design. In 1992 he was a Visiting Researcher at the University of California at Los Angeles (UCLA) working on low-intermodulation mixers. He received the M.S. and Ph.D. degrees in electrical engineering from the Department of Electrical Engineering at UCLA in 1994 and 1997, respectively. He is currently a Professor with the Department of Electrical Engineering and Computer Science at the University of California Irvine. He has authored and coauthored over 100 papers in reference journals and conference proceedings, filed several international patents and authored one book and three book chapters. He is a member of the URSI Commission B. His research interest include the development of microelectromechanical systems (MEMS) for RF applications fabricated on unconventional substrates such as printed circuit board and microwave laminates with particular emphasis on reconfigurable antenna systems. He also is active in the research field of highly integrated packaging for RF and wireless applications.

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Alfred Grau Besoli (M’07) was born in Barcelona, Spain, in 1977. He received the Telecommunications Engineering degree from the Universitat Politècnica de Catalunya (UPC), Barcelona, in 2001. He received the M.S. and Ph.D. degrees in electrical engineering from the Department of Electrical Engineering and Computer Science at the University of California at Irvine (UCI) in 2004 and 2007, respectively. He is currently a senior scientist with Broadcom Corporation, Irvine, CA. His interests are in the field of reconfigurable antennas and software defined antennas, cross-layer design of channel coding techniques for reconfigurable antennas, miniature and integrated on-chip antennas, multi-port antennas and MIMO wireless communication systems, microelectromechanical systems (MEMS) for RF applications, metamaterials, reconfigurable electromagnetics devices and materials, and computer-aided electromagnetics.

Silvio E. Barbin (SM’04) was born in Campinas, Brazil, in 1952. He received the B.S. degree in electrical engineering from Escola Politécnica da Universidade de São Paulo (USP), Brazil, in 1974 and the M.S. and Ph.D. degrees from the same institution. He worked for AEG-Telefunken in Germany and Brazil and served as CTO of Microline RF-Multiplexers and Deputy Director of Center for Information Technology Renato Archer from the Ministery of Science and Technology in Brazil. He was a research scholar at the University of California, Los Angeles, CA and a research professor at the University of New Mexico, Albuquerque, NM. He has published more than 90 papers in conferences and journals. In 1987 he joined the Telecommunications and Control Engineering Department at University of Sao Paulo, where he is a Professor in electromagnetics and other related subjects. His research interests are in the areas of cognitive radio, reconfigurable and smart antennas, microwave circuits, and electromagnetic modeling. Dr. Barbin is a co-founder of the Brazilian Microwave Society and a member of several other scientific societies. He was honored professor for a number of times at his university. He is a member-at-large of the Products Services Publications Board and an Associated Editor for Antennas Wireless and Propagation Letters, among other functions at the IEEE.

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Free Space Radiation Pattern Reconstruction from Non-Anechoic Measurements Using an Impulse Response of the Environment Jinhwan Koh, Arijit De, Student Member, IEEE, Tapan K. Sarkar, Fellow, IEEE, Hongsik Moon, Weixin Zhao, and Magdalena Salazar-Palma, Senior Member, IEEE

Abstract—The objective of this paper is to investigate a methodology, which can extract approximate results for the free space radiation pattern from non-anechoic measurements. Using an impulse response both in the time and angular domains of the nonanechoic measurement environment, the free space pattern of the device under test is estimated. The purpose of this paper is, as opposed to what has been stated in some papers, to show that a deconvolution based technique is feasible for reflection compensation in non-anechoic measurements. The proposed method can also be applied at a single frequency as illustrated in this paper. Simulated data has been used to illustrate the applicability of this new technique and its improved performance over the conventional FFTbased methods. Index Terms—Angular domain impulse response, antenna measurement, non-anechoic measurement, pattern reconstruction.

I. INTRODUCTION

M

EASUREMENTS of antenna patterns are usually carried out in anechoic chambers to eliminate any reflected signal component. The purpose of employing an anechoic chamber is to eliminate the reflected field components emanating from the measurement enclosures so that the measured radiation pattern corresponds to the case when similar data can be generated for the antenna in free space conditions. The reduction of reflected signal can be accomplished using appropriate absorbing materials. However, the cost of a good anechoic chamber is proportional to the quality of the absorbers used for the construction of the chamber. Hence, on a fixed budget it is not possible to have a high quality anechoic chamber. The radiation patterns measured in reverberant or semi-anechoic chambers, or even open-area test sites, thus provides a viable alternative. However, presence of undesired components will result in inaccuracies in the measured patterns.

Manuscript received October 19, 2010; revised April 04, 2011; accepted July 02, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. J. Koh is with the Engineering Research Institute, Gyeongsang National University, Jinju 660-701, Korea (e-mail: jikoh@ gnu.ac.kr). A. De, T. K. Sarkar, H. Moon, and W. Zhao are with Syracuse University, Syracuse, NY 13214 USA (e-mail: [email protected]). M. Salazar-Palma is with the Department of Signal Theory and Communications, Universidad Carlos III de Madrid, Leganés-Madrid 28903, Spain (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173117

The objective of this paper is to study methodologies which may eliminate the undesired reflections measured in non-anechoic conditions through further processing of the measured data. Such a procedure would make antenna measurements much easier and cheaper. In the past, several research projects attempted to reduce undesired reflections and diffraction contributions from the walls and objects present inside the chamber [1]–[10]. The traditional methodology to achieve the reduction of reflections and diffractions is the FFT (Fast Fourier Transform)based method, which describes the impulse response of the reverberant chamber from its frequency response by using the Inverse Fourier Transform [1]. In the time domain, the direct contribution is detected and gated, thus eliminating undesired late-time echoes. Applying the Fourier Transform to this new gated time response helps us to obtain only the direct contribution present in the radiation pattern at the frequency of interest. However, a major drawback of this methodology is that we need to determine the amount of time delay, which is the minimum travel time of the beam from AUT (Antenna Under Test) to the measurement probe. This is not an easy task especially when multiple reflectors are present in the measurement site close to the direct path. The matrix pencil method essentially achieves similar goals as the FFT-based method but requires less bandwidth [2]–[4]. B. Fourestie et al. modeled the measured signal using the Matrix Pencil method and detected resonances in anechoic chambers. P. S. H. Leather and D. Parson described a method to determine an angle by angle equalization of the non-anechoic environment where an antenna is to be tested [5]. But there was neither detailed mathematical formulation supporting the theory nor examples. The purpose of this paper is, as opposed to what has been stated in some papers [6], to show that a deconvolution-based technique is feasible for reflection compensation in non-anechoic measurements. Other researchers have presented techniques of accounting for room scattering in the frequency-domain using a test zone field (TZF) compensation method [6]–[10]. The single-frequency techniques compensate for any reflected fields entering the test zone where the AUT is located. The proposed approach is computationally more efficient than the TZF method as the proposed method is a deconvolution based technique. It has been claimed in [6] that a deconvolution based technique may not be the appropriate approach to compensate for the reflections in a non-anechoic environment. However, here we illustrate that a deconvolution

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Fig. 1. Multiple reflections of the signal back to the probe.

based method is indeed feasible as the fields are obtained by convolving the current with the Green’s function [11]. In this paper, we introduce the concept of an impulse response both in the time and angular domain of the non-anechoic measurement environment. By now performing a deconvolution of the beam pattern in the angular domain using the beam pattern of a calibrated device followed by a second deconvolution in the time domain, one may provide an estimate of the pattern in an anechoic environment. As shown in Fig. 1, the beam patexperiences reflection from Object 1 tern at azimuth angle and Object 2 located in the far-field of the antenna (assumed for simplicity in the explanation), and multiple reflections between the two objects. This process can be characterized in the time domain by an impulse response and will be a unique feature with respect to every angle . Therefore the non-anechoic environment can be modeled by a set of impulse responses in the time domain as a function of the spatial angles. These responses can easily be estimated by performing deconvolutions from first carrying out measurements in the environment of interest using two antennas whose ideal patterns are known. Once we get an estimate of the reference impulse response in time and angular domains, we can apply the reference responses to estimate patterns in anechoic environment of the antenna of interest by carrying out measurements in the same non-anechoic environment. Section II describes the method used to obtain the data. In Section III the classical FFT-based method to process the data is reviewed. In Section IV the new methodology is proposed for a single frequency of operation and numerical examples are presented in Section V followed by conclusions. II. DESCRIPTION OF THE SYSTEM All the simulations presented in this paper have been carried out using a spherical-range measurement system without probe compensation as shown in Fig. 2. The probe and the AUT are separated from each other by 2 m for the entire series of simulations. The azimuth angle of the AUT was varied from to 90 with a 5 step. A full-wave Method of Moments based EM software [12] was used to compute the various electromagnetic interactions. The frequency under consideration was varied from 6 GHz to 12 GHz with a step of 0.05 GHz, to characterize the impulse response of the non-anechoic environment. A metal plate 0.5 m 0.2 m in size centered with respect to the probe and the AUT was introduced in the far-field of the measurement environment. This plate will introduce additional contributions due to the reflected and diffracted components of

Fig. 2. Antenna measurement system with the reflecting plate (AUT is a helical antenna, probing antenna is a horn antenna).

Fig. 3. Dimensions of the horn antenna (PROBE).

the fields in the measurements. This plate is located at a distance of 0.5 m from the line of sight between the AUT and the measurement probe. First we carry out the simulation of this non-anechoic environment using a transmitter/receiver system whose characteristics we know under ideal conditions. The transmitting and receiving antennas we consider to be standard gain horn antennas, whose dimensions are shown in Fig. 3. The feeding probe of length 9.4 mm and diameter 1 mm is located at a distance of 7.5 mm from the end wall of the horn. We measure between these two known horn antennas in the non-anechoic environment. We then introduce the AUT, which is a helical antenna, to be characterized. The dimensions of the helical antenna are shown in Fig. 4. The helical antenna is fed at the junction of the wire helix and the backing metal plate. The diameter of the wire constituting the helix is 1 mm and it is backed by a finite circular PEC backplane. The simulation was carried out in the frequency domain, by measuring of this transmitter/receiver system which consists of the helical and the horn antenna. We generated 2 sets of data. One set of data was generated without the metal plate shown in Fig. 2, the same as would occur in a true anechoic condition called the reference data. The other data was generated with the presence of the metal plate. Because of the presence of the metal plate, the measured radiation pattern differs from the true one. The problem of interest is whether one can do some post-processing to recover the original radiation pattern from the measured one in the non-anechoic environment. III. PROCESSING USING THE CLASSICAL FFT-BASED METHOD It is apparent that in an ideal situation, the unique path that exists between the AUT and the probe is the direct path of propagation. The channel in the frequency domain is then characterized by a constant amplitude response, independent of the frequency and with a linear phase. However, the above is not true anymore when multi-paths are present due to the metal plate, indicating

KOH et al.: FREE SPACE RADIATION PATTERN RECONSTRUCTION FROM NON-ANECHOIC MEASUREMENTS

Fig. 4. Dimensions of the helical Antenna (AUT).

the influence of the echo contributions both in the amplitude and phase of the antenna response. The main idea of the FFT-based method is that there would not be any desired direct path contribution after some time duration. This time duration can be calculated based on the shortest path from the AUT to the probe in the presence of the reflector. This has been described in detail in [1]. Here, the various steps of the FFT-based method are summarized for convenience: 1) Measure the coherent (amplitude/phase) frequency response covering the bandwidth of 6 GHz to 12 GHz, i.e., between the two antennas in the presence of the metal plate as shown in Fig. 2. The dimensions of the AUT and the probing antenna are described by Figs. 3 and 4 respectively. The various reflected and diffracted fields for the non-anechoic condition are primarily located in the region covering azimuth angles ranging from . 2) Apply the Inverse FFT with respect to the frequency to , which is shown in Fig. 5, to obtain the temporal response as shown in Fig. 6. 3) Once in the time domain, the direct ray contribution between the transmitting and the receiving antenna is approximately retained and the unwanted signals can be approximately gated by truncating the waveform beyond a time interval of 7.45 nsec as illustrated in Fig. 7. The problem with the FFT-based method is that there is no clear demarcation between the direct path and the reflected rays when the bandwidth used in the measurement is small. In addition one needs to have an estimate of this delay time. This implies that the environment of measurement is known in detail. 4) Upon transforming the truncated temporal data to the frequency domain by applying the FFT, one obtains the processed frequency domain response as illustrated in Fig. 8. When compared with Fig. 5, one can observe that most of the reflected and diffracted fields in the azimuth angle range of have been reduced. The amplitude and the phase of the proposed pattern will be described and discussed later in comparison with the proposed method.

Fig. 5.

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between the helical antenna and the probe.

Fig. 6. Time domain response of Fig. 5 using the FFT-based method.

Fig. 7. The shortest path would be 2.2361 m which is equivalent to 7.45 nsec.

A problem with the FFT-based method is that one needs to determine the time delay which is the minimum travel time of the beam from AUT to the probe via the reflector. This is not an easy task especially when multiple paths are present. Because of the decimation of the data in the time domain, the reconstructed pattern sometimes has less power than the original signal. An additional normalization process should be ap-

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Fig. 8. Magnitude response of the helical antenna in the frequency domain for the FFT-based method.

Fig. 9. Reference

of the horn without a reflector.

plied to reduce this problem. The determination of this normalizing factor is a problem by itself. Furthermore, the quality of the result depends on the available bandwidth of the measurement system. For these reasons, we introduce a new method as described next. IV. PATTERN RECONSTRUCTION USING RESPONSE CONCEPT

THE IMPULSE

Here, we consider an AUT which can generate a very sharp pencil beam. As shown in Fig. 1, the beam at angle experiences reflection from Object 1 and Object 2, and multiple reflections from the two objects. Note that we assume the objects do not change in time (time invariant). The receiving time domain signal at the probing antenna, when the AUT is at an angle of , is unique with respect to the other angles and will not be related to the response of AUT at . Therefore the time domain response at angle can be described as an impulse response at or a spatial signature of . The measured signal at the probing antenna along with the corrupting reflections can be represented as a convolution in time with the ideal signal without any reflection and the impulse response of the AUT at an angle of can be written as (1) or (2) denotes a time convolution, and time domain signal at the probing antenna from the AUT in the presence of reflections for the angle , frequency domain signal at the probing antenna from the AUT in the presence of reflections for the angle , ideal time domain signal without any reflection from the AUT for the angle , ideal frequency domain signal without any reflection from the AUT for the angle ,

where

Fig. 10.

of the horn in presence of the reflector.

impulse response of the environment with objects present when the AUT has a pencil beam pointing along the angle , frequency domain response of the environment with objects present when the AUT has a pencil beam pointing along the angle . Note that represents the contribution from the reflective stationary environments along the angle independent of the particular AUT. In a real situation, the AUT would have a non-ideal beam pattern which may not be as sharp as a pencil beam. We define the impulse response of the environment with respect to such a non-ideal beam pattern of the AUT along the angle , as . Then using (2) we have (3) = frequency domain response with a non-ideal Here, beam pattern of the AUT at the angle of . Now, contains information of the beam pattern of the AUT (here the reference antenna) as well as the spatial signature characterizing the environment along . We need the

KOH et al.: FREE SPACE RADIATION PATTERN RECONSTRUCTION FROM NON-ANECHOIC MEASUREMENTS

Fig. 11. (a) of the horn in dB scale at 9 GHz. (b) Phase of of horn in radian at 9 GHz.

impulse response, , which is independent of the beam pattern of the AUT. The impulse response with the non-ideal beam pattern of the AUT can be considered as a convolution in the angular domain of the normalized beam pattern and the true impulse response when the AUT has the ideal pencil beam. That is, (4) Here, is the ideal signal without any reflection from the environment at the frequency , and is the convolution operator in the angular domain. Therefore, the beam pattern of the AUT in the presence of reflections can be considered as a convolution in the angular domain of the beam pattern of the AUT and the impulse response of the environment. Using (3) and (4) we have (5) For a general angle of , we have (6)

Fig. 12.

in dB scale.

Fig. 13.

in dB scale.

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Thus the impulse response can easily be calculated by taking the Inverse Fourier Transform of (6). Once is calculated from a reference measurement, the ideal beam pattern, can be obtained for any antenna measured in the same environment, using (6). Observe that (6) can also be applied at a single frequency. The FFT based approaches and its variants require broadband characterization while the present method requires a single frequency measurement, and is independent of the bandwidth of the measurement. The steps of the proposed method are summarized as follows: i) Measure the reference antenna response, in an environment with reflection. Also obtain the reference antenna response . ii) Calculate using (6). iii) Measure the AUT response in the same non-anechoic environment as described in step (i). Let be the result. iv) Obtain the ideal response of AUT, , through deconvolution using

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Fig. 14.

Fig. 15.

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in dB scale at 9 GHz.

of the helical antenna without a reflector.

In the next section, we simulate three antenna responses, a horn antenna, a helical antenna, and a Yagi antenna, in the frequency and angle domains. The response of the horn antenna was set to be the reference response. We tried to recover the beam pattern of the helical antenna and the Yagi antenna from the antenna responses corrupted by the presence of the reflectors. V. NUMERICAL EXAMPLES Several assumptions are made for the numerical simulations: i) There is only one signal source which is from the AUT. ii) The reflectors (environment) do not vary with time. iii) We only consider 2D pattern measurements (azimuth angle only, not elevation angle), even though the numerical calculations have been done in 3D. Under these assumptions, we build an antenna simulation system as shown in Fig. 2 with the AUT being a horn antenna instead of a helix. Thus, we use a horn antenna as a reference, which has a wide radiation pattern, described in Fig. 4, both as a probe as well as an AUT. The distance between the AUT and the probing antenna was 2 m. The azimuth angle of the AUT was varied from 90 to 90 with a step of 5 . We calculate

Fig. 16. (a) Amplitude pattern of the helical antenna in dB scale at 7 GHz. (b) Phase of helical antenna in radian at 7 GHz.

, of the probing antenna in frequency domain without any reflector. The frequency response from 6 GHz to 12 GHz with a step of 0.05 GHz, generates 121 data points. Therefore, the center frequency corresponds to the measurement at the frequency of interest of 9 GHz. We have 37 by 121 data points for the horn as the first reference data. Fig. 9 shows the contour plot of , without the reflector. Fig. 10 shows the contour plot in the presence of reflected echoes. Fig. 11(a) and (b) are the comparison of the magnitude and phase of , at the center frequency of 9 GHz with and without the reflector. The red-line termed non-ideal is the result in the presence of the echoes. One can observe that between the angles of 30 to 70 there is a large difference due to the reflection from the reflector. Magnitude of calculated from (6) is shown in Fig. 12. The difference is not limited to the 22 34 regions, which is due to the wide beam pattern of the AUT. If we use a very narrow pencil beam, the range of angle will be reduced to 22 34 . Using the deconvolution equation (6), which can eliminate the effect of the beam pattern, we can get an estimate of the impulse response from . Fig. 13 represents the

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Fig. 18. Estimate of the reconstruction error for the helix in frequency.

Fig. 19. Antenna measurement system with the reflector (AUT is a Yagi antenna, probing antenna is a horn antenna).

Fig. 17. (a) Amplitude pattern of helical antenna in dB scale at 8.6 GHz. (b) Phase of helical antenna in radian at 8.6 GHz.

plot of . One can observe that some responses occur between the angles of 22 34 where the reflector exists. Fig. 14 is the magnitude of when the frequency is 9 GHz. The deep null is caused by the interference between the direct wave and the large reflections occurring between 22 34 . Once the impulse response is obtained, one can apply this to estimate the pattern of a different antenna in a similar environment. First, we use a helical antenna as shown in Fig. 4 as an AUT in the antenna measurement system of Fig. 2. Figs. 15 and 5 are the magnitude of at the probe without the reflector and with the reflector, respectively. We want to reconstruct the pattern from the data with the reflector using two approaches: the FFT-based method and the proposed impulse response-based method. It is important to point out that the FFT-based method requires data over a sizeable bandwidth. The proposed method can be implemented at a single frequency, however we have also applied it over a frequency band as used in the FFT-based method so that we can compare the performances between the two methods. Fig. 6 corresponds to the time domain response of Fig. 5.

Figs. 16 and 17 plot the comparisons for the various methods and the reference pattern. Fig. 16(a) represents the magnitude of the reconstructions at 7 GHz. The proposed method gives almost identical results to the reference pattern. The FFT-based method has less power than the others because it decimates the signal in the time domain. Fig. 16(b) shows the comparison of the corresponding phases. Fig. 17(a) represents the magnitude of the reconstructions at 8.6 GHz and both the proposed method and the FFT-based method give acceptable results. Fig. 17(b) provides the corresponding phase comparison. It is important to note that the FFT-based method requires additional information about the first reflection in the time domain and so one has to know the placement of the reflectors, whereas this method does require a non-parametric description of the environment through the impulse response. Fig. 18 represents the error in the reconstruction of the two methods with respect to the frequency. The error is defined by:

(7)

represents the ideal pattern of the AUT and represents the processed pattern of AUT. The proposed method performs better than the FFT-based method in most of the frequency regions. where

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Fig. 20. Dimension of the Yagi antenna (AUT). Fig. 22.

Fig. 21.

of the Yagi antenna with a reflector.

of the Yagi antenna without a reflector.

The second example to be considered is a Yagi antenna. The Yagi antenna has the center frequency of operation at 9 GHz. Fig. 19 describes the antenna measurement system. Fig. 20 shows the dimensions of the Yagi antenna. The six-element Yagi antenna consists of one reflector, a feed-element and 4 equi-spaced directors. They are made of PEC wires of diameter 1 mm. The Yagi is center fed at the feed element. The probing antenna is a horn as shown in Fig. 3. Figs. 21 and 22 are the magnitude of at the probe without the reflector and with the reflector, respectively. We want to reconstruct the ideal pattern from the data with the reflector using both the FFT-based method and the proposed impulse response-based method. Fig. 23(a) compares the magnitudes of the reconstructions at 9 GHz. The proposed method gives good results to the reference pattern. Fig. 23(b) is the comparison of their corresponding phases. Fig. 24 represents the error in the reconstruction of the two methods with respect to the frequency. The proposed method outperforms the FFT-based method at most of the frequency bands. For a more realistic example, we put 4 PEC walls surrounding the measurement setup as shown in Fig. 25 to introduce more reflections. The AUT is a helical antenna described in Fig. 4.

Fig. 23. (a) Amplitude pattern of the Yagi antenna in dB scale at 9 GHz. (b) Phase of the Yagi antenna in dB scale at 9 GHz.

The wall is located 1.25 m away from AUT. The azimuth angle of the AUT was varied from 180 to 180 with a step of

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Fig. 24. Estimate of the reconstruction error for the Yagi antenna in frequency.

Fig. 25. Antenna measurement system with the 4 metal reflecting walls (AUT is a helical antenna to the right, probing antenna is a horn antenna to the left).

1 . The frequency response at 7 GHz generates 361 data points. Fig. 26(a) compares the magnitudes of the reconstructions at 7 GHz. The proposed method gives reasonable results to the reference pattern. It is important to note that in the back lobe of the actual signal, the true response is 15 dB lower than the nonideal one. Fig. 26(b) is the comparison of their corresponding phases. The proposed method generates acceptable results even though there exist multiple reflections. VI. CONCLUSION The objective of this paper is to investigate a deconvolution methodology which can extract approximate results from non-anechoic measurements. Using the concept of an impulse response both in time and angular domains, the non-anechoic environment has been successfully described and has a better performance than the FFT-based methods. However, as illustrated, the proposed method can also be applied at a single frequency. Simulated data has been used to illustrate the applicability of this new methodology and its improved performance over the conventional FFT-based methods in the presence of multiple reflections. Work is in progress to apply the proposed method to 3D measurements, where deconvolution needs to be carried out simultaneously in azimuth and elevation angles.

Fig. 26. (a) Amplitude pattern of the helical antenna in dB scale at 7 GHz. (b) Phase of the helical antenna in radian at 7 GHz.

ACKNOWLEDGMENT Grateful acknowledgement is made to all the reviewers, the associate editor and the editor for suggesting ways to improve the readability of the manuscript. REFERENCES [1] S. Loredo, M. R. Pino, F. L. Heras, and T. K. Sarkar, “Echo identification and cancellation techniques for antenna measurement in non-anechoic test sites,” IEEE Antennas Propag. Mag., vol. 46, no. 1, pp. 100–107, Feb. 2004. [2] B. Fourestie, Z. Altman, and M. Kanda, “Anechoic chamber evaluation using the matrix pencil method,” IEEE Trans. Electromagn. Compatibil., vol. 41, no. 3, pp. 169–174, Aug. 1999. [3] B. Fourestie, Z. Altman, and M. Kanda, “Efficient detection of resonances in anechoic chambers using the matrix pencil method,” IEEE Trans. Electromagn. Compatibil., vol. 42, no. 1, pp. 1–5, Feb. 2000. [4] B. Fourestie and Z. Altman, “Gabor schemes for analyzing antenna measurements,” IEEE Trans. Antennas Propag., vol. 49, no. 9, pp. 1245–1253, Sep. 2001. [5] P. S. H. Leather and D. Parson, “Equalization for antenna-pattern measurements: Established technique—New application,” IEEE Antennas Propag. Mag., vol. 45, no. 2, pp. 154–161, Apr. 2003.

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[6] D. N. Black and E. B. Joy, “Test zone field compensation,” IEEE Trans. Antennas Propag., vol. 43, no. 4, pp. 362–368, 1995. [7] J. T. Toivanen, T. A. Laitinen, S. Pivnenko, and L. Nyberg, “Calibration of multi-probe antenna measurement system using test zone field compensation,” in Proc. 3rd European Conf. Antennas and Propagation (EuCAP’09), Berlin, Germany, 2009, pp. 2916–2920. [8] J. T. Toivanen, T. A. Laitinen, and P. Vainikainen, “Modified test zone field compensation for small antenna measurements,” IEEE Trans. Antennas Propag., vol. 58, no. 11, pp. 3471–3479, 2010. [9] R. Pogorzelski, “Extended Probe Instrument Calibration (EPIC) for accurate spherical near-field antenna measurements,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3366–3371, 2009. [10] R. Pogorzelski, “Experimental demonstration of the Extended Probe Instrument Calibration (EPIC) technique,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 2093–2097, 2010. [11] Y. Lopez, F. Las-Heras, F. Andres, M. R. Pino, and T. K. Sarkar, “An improved super-resolution source reconstruction method,” IEEE Trans. Instrum. Measur., vol. 58, no. 11, pp. 3855–3866, 2009. [12] Y. Zhang and T. K. Sarkar, Parallel Solution of Integral EquationBased EM Problems in the Frequency Domain. New York: WileyIEEE Press, 2009. Jinhwan Koh received the B.S. degree in electronics from Inha University, Incheon, Korea, and the M.S. and Ph.D. degrees in electrical engineering from Syracuse University, Syracuse, NY. He is now a Professor with the Department of Electronic Engineering, Engineering Research Institute, Gyeongsang National University, Jinju, Korea. His current research interests include radar signal processing and electromagnetic measurement.

Arijit De (S’04) received the B.Tech. degree (with honors) in electronics and electrical communication engineering and a minor in computer science and engineering, from Indian Institute of Technology, Kharagpur, India, in 2004. He is currently working towards the Ph.D. degree at Syracuse University, Syracuse, NY. In Summer 2003, he was a summer intern with Center of Excellence Embedded DSP, Tata Consultancy Services, where he was involved with the design and implementation of 802.11g Wireless LAN. From 2004 to 2005 he was a Research Consultant in the Advanced VLSI Design Lab., IIT Kharagpur, working on development of next generation Analog CAD tools for National Semiconductor Corporation, Santa Clara. Since 2005, he has been a Graduate Research Assistant in the Computational Electromagnetics Group, Syracuse University. His research interests are in the field of Computational Techniques applied to electromagnetics and signal processing. Currently, he is an Assistant Professor at the Indian Institute of Technology, Kharagpur, India. Tapan K. Sarkar (F’92) received the B.Tech. degree from the Indian Institute of Technology, Kharagpur, India, in 1969, the M.Sc.E. degree from the University of New Brunswick, Fredericton, NB, Canada, in 1971, and the M.S. and Ph.D. degrees from Syracuse University, Syracuse, NY, in 1975. From 1975 to 1976, he was with the TACO Division of the General Instruments Corporation. He was with the Rochester Institute of Technology, Rochester, NY, from 1976 to 1985. He was a Research Fellow at the Gordon McKay Laboratory, Harvard University, Cambridge, MA, from 1977 to 1978. He is now a Professor in the Department of Electrical and Computer Engineering, Syracuse University. His current research interests deal with numerical solutions of operator equations arising in electromagnetics and signal processing with application to system design. He has authored or coauthored more than 300 journal articles and numerous conference papers and 32 chapters in books and 15 books, including his most recent ones, Iterative and Self Adaptive Finite-Elements in Electromagnetic Modeling (Artech House, 1998), Wavelet Applications in Electromagnetics and Signal Processing (Artech House, 2002), Smart Antennas (IEEE Press and Wiley, 2003), History of Wireless (IEEE Press

and Wiley, 2005), Physics of Multiantenna Systems and Broadband Adaptive Processing (Wiley, 2007), Parallel Solution of Integral Equation-Based EM Problems in the Frequency Domain (IEEE Press and Wiley, 2009), and Time and Frequency Domain Solutions of EM Problems Using Integral Equations and a Hybrid Methodology (IEEE Press and Wiley, 2010). Dr. Sarkar is a Registered Professional Engineer in the State of New York. He obtained one of the “Best Solution” awards in May 1977 at the Rome Air Development Center (RADC) Spectral Estimation Workshop. He received the Best Paper Award of the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY in 1979 and in the 1997 National Radar Conference. He received the College of Engineering Research Award in 1996 and the Chancellor’s Citation for Excellence in Research in 1998 at Syracuse University. He was an Associate Editor for feature articles of the IEEE Antennas and Propagation Society Newsletter (1986–1988), Associate Editor for the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY (1986–1989), Chairman of the Inter-commission Working Group of International URSI on Time Domain Metrology (1990–1996), Distinguished Lecturer for the Antennas and Propagation Society from (2000–2003) and (2011-present), Member of Antennas and Propagation Society ADCOM (2004–2007), on the Board of Directors of ACES (2000–2006), Vice President of the Applied Computational Electromagnetics Society (ACES), and a member of the IEEE Electromagnetics Award board (2004–2007), and an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION (2004–2010). He is also on the editorial board of Digital Signal Processing—A Review Journal, Journal of Electromagnetic Waves and Applications, and Microwave and Optical Technology Letters. He is the Chair of the International Conference Technical Committee of IEEE Microwave Theory and Techniques Society # 1 on Field Theory and Guided Waves. He is listed by ISI among the top 250 of most referenced authors in this field. He received Docteur Honoris Causa both from Universite Blaise Pascal, Clermont Ferrand, France in 1998 and from Politechnic University of Madrid, Madrid, Spain in 2004. He received the Medal of the Friend of the City of Clermont Ferrand, France, in 2000. His website is http://lcs.syr.edu/faculty/sarkar/ Hongsik Moon received the B.S. degree from Kyungpook National University, Daegu, South Korea, in 2004, and the M.S. degree from Syracuse University, Syracuse, NY, in 2007, both in electronics and electrical engineering, where he is currently working toward the Ph.D. degree.. Since 2006, he has been a Graduate Research Assistant in the Computational Electromagnetics Group at Syracuse University. His research interests are in UWB sensors, scattering scenes and computational electromagnetic. Weixin Zhao was born in Wuxi, China. He received the B.S. degree in electrical engineering and information science from the University of Science and Technology of China, Hefei, China, in 2007. He is currently working toward the Ph.D. degree in the Department of Electrical Engineering at Syracuse University, Syracuse, NY. He has been a Research Assistant at Syracuse University since 2007. His current research interest is antenna design and optimization.

Magdalena Salazar-Palma (SM’01) was born in Granada, Spain. She received the M.S. and Ph.D. degrees in Ingeniero de Telecomunicación (Electrical and Electronic Engineer) from the Universidad Politécnica de Madrid (UPM), Spain. She has been Profesor Colaborador and Profesor Titular de Universidad at the Department of Signals, Systems and Radiocommunications, UPM. Since 2004, she is with the Department of Signal Theory and Communications (TSC), College of Engineering, Universidad Carlos III de Madrid (UC3M), Spain, where she is Full Professor, Co-Director of the Radiofrequency Research Group and served for three years as Chairperson of the TSC Department. She has been member of numerous University Committees, both at UPM and UC3M. She has developed her research in the areas of electromagnetic field theory; advanced computational and numerical methods for microwave and millimeter

KOH et al.: FREE SPACE RADIATION PATTERN RECONSTRUCTION FROM NON-ANECHOIC MEASUREMENTS

wave passive components and antennas analysis and design; advanced network and filter theory and design; antenna arrays design and smart antennas; use of novel materials and metamaterials for the implementation of devices and antennas with improved performance (multiband, miniaturization, and so on) for the new generation of communication systems; design, simulation, optimization, implementation, and measurement of microwave circuits both in waveguide and integrated (hybrid and monolithic) technologies; millimeter, submillimeter and THz frequency bands technologies; and history of telecommunications. She has authored a total of 490 scientific publications: six books and 23 contributions (chapters or articles) for books published by international editorial companies, 13 contributions for academic books and notes, 62 papers in scientific journals, 234 papers in international conferences, symposiums, and workshops, 71 papers in national conferences and 81 project reports, short course notes, and other publications. She has coauthored two European and USA patents which have been extended to other countries and several software packages for the analysis and design of microwave and millimeter wave passive components, antennas and antenna arrays, as well as computer aided design (CAD) of advanced filters and multiplexers for space applications which are been used by multinational companies. She has delivered numerous invited presentations and seminars. She has lectured in more than 50 short courses, some of them in the frame of Programs of the European Community and others in conjunction with IEEE (The Institute of Electrical and Electronic Engineers) International AP-S (Antennas and Propagation Society) Symposium and IEEE MTT-S (Microwave Theory and Techniques Society) Symposium and others IEEE Symposia. She has participated at different levels (principal investigator or researcher) in a total of 81 research projects and contracts, financed by international, European, and national institutions and companies, among them: the National Science Foundation, USA; the European Office of Aerospace Research and Development of the Air Force Office of Scientific Research (one of the Air Force Research Laboratory Directorates), USA; the European Union; Spain Inter-ministry Commission of Science and Technology (CICYT), Spain

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Ministry of Education and Culture, Spain Ministry of Science and Innovation, and Council of Education of the Regional Government of Madrid. Prof. Salazar-Palma is a Registered Engineer in Spain. She has received two individual research awards. She has assisted the Spain National Agency of Evaluation and Prospective and the Spain CICYT in the evaluation of projects, research grants applications, and so on. She is member of the Accreditation Committee of Full Professors in the field of Engineering and Architecture of the Spanish Agency of Quality Evaluation and Accreditation (ANECA). She has also served on several evaluation panels of the Commission of the European Communities. She has been a member of the editorial board of three scientific journals. She has been associated editor of several scientific journals, among them, the European Microwave Association Proceedings and IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS. She has been a member of the Technical Program Committees of many international and national symposiums and reviewer for different international scientific journals, symposiums, and editorial companies. Since 1989, she has served IEEE under different volunteer positions: Vice Chairperson and Chairperson of IEEE Spain Section AP-S/MTT-S Joint Chapter, President of IEEE Spain Section, Membership Development Officer of IEEE Spain Section, member of IEEE Region 8 Committee, member of IEEE Region 8 Nominations and Appointments Subcommittee, Chairperson of IEEE Region 8 Conference Coordination Subcommittee, member of IEEE Women in Engineering (WIE) Committee, liaison between IEEE WIE Committee and IEEE Regional Activities Board, Chairperson of IEEE WIE Committee, member of IEEE Ethics and Member Conduct Committee, member of IEEE History Committee, member of IEEE MGAB (Member and Geographic Activities Board) Geographic Unit Operations Support Committee, and member of IEEE AP-S Administrative Committee. Presently she is serving as member of IEEE Spain Section Executive Committee (officer for Professional Development), member of IEEE MTT-S Subcommittee # 15, and member of IEEE AP-S Transnational Committee. In December 2009 she was elected 2011 President of IEEE AP-S Society, acting as Vice President during 2010.

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Electric Field Amplification inside a Porous Spherical Cavity Resonator Excited by an External Plane Wave Paul A. Bernhardt, Fellow, IEEE, and Richard F. Fernsler

Abstract—A spherical polyhedron constructed from open surface polygons is an electromagnetic wave resonator that can be excited by an external plane wave. The resonant frequencies of the porous sphere depend on the radius of the sphere and the area of the openings in the surface of the sphere. The strength of the internal electric fields varies with the width of the conducting edges that comprise the polyhedron frame. At the optimum edge width, the external EM wave field excites the strongest internal field amplitudes. The WIPL-D EM simulation model is used to determine the optimum porous resonator for polyhedrons with 180 and 960 vertices. All of the cavity modes for a solid spherical cavity resonator can be excited in the porous spherical cavity resonator (PSCR). With a high resonator Q, an EM plane-wave of 1 V/m can excite an internal electric field of over 1000 V/m that takes finite time for fields to build up. The spherical cavity modes provide a variety of electric field distributions at the interior of the PSCR. The PSCR may be used to greatly increase the electric fields of a high power radio beam in order to produce isolated plasma clouds by neutral gas breakdown. Index Terms—Externally excited spherical resonant cavity, gas breakdown, large electric fields.

I. INTRODUCTION

A

large spherical polyhedron with open polygon surfaces has been examined by Bernhardt et al. [1], [2] for use as an HF radar calibration satellite. The radius of this sphere was 5-m and the number of vertices on the polyhedron ranged from 60 to 1900. The polyhedron surface is composed of polygons, edges, and vertices. The general formula for the number of vertices in a polyhedron is where u and v are integers. The number of edges is . Each polyhedron has 12 pentagons and hexagons [1]. In this paper, the resonant properties of a 5-m radius sphere are examined for polyhedrons with 180 and 960 vertices. The properties for other sized spherical polyhedra can be obtained by scaling the frequencies with the inverse of the sphere radius. The electromagnetic properties of a spherical conducting polyhedron are determined by waves external and internal to the spherical conductive mesh. The internal wave is excited by an incident plane wave impinging on the sphere. Interactions between the external and internal waves yield: (1) enhanced radar scatter relative to a solid sphere for

Manuscript received January 01, 2011; revised June 15, 2011; accepted July 21, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported by the Office of Naval Research. The authors are with the Plasma Physics Division, Naval Research Laboratory, Washington, DC 20375 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2173132

TABLE I SPHERICAL CAVITY RESONATOR MODES

most frequencies; (2) greatly reduced backscatter at selected frequencies from cancellation of the incident and reflected waves; and (3) excitation of large internal electric fields at resonant frequencies. The spherical polyhedron mesh, which behaves as a high-Q cavity resonator, can produce localized high power electric fields when the sphere is imbedded in an external plane wave. Under suitable conditions, the porous spherical cavity resonator (PSCR) can lead to the breakdown of a neutral gas inside the sphere, thereby generating localized plasma clouds and the emission of light. The theory of spherical cavity resonators formed by a hollow inside a conducting metal block is well known [3], [4]. For an infinite conducting shell around a spherical region of radius , the resonant oscillations are determined for the magnetic (transverse electric or TE) and electric (transverse magnetic or TM) modes, respectively, from the equations (1) is the wave number in the cavity with where and , is the resonant frequency, dielectric constants , and is the the prime is the derivative with respect to spherical Bessel function. The index “s” denotes which zero is found for the solution to (1). and where the The modes are designated by indices (s, m, n) denote the variations in the spherical r, , coordinates, r is radius from the center of the sphere, is the azimuth angle from the x-coordinate and is the meridian angle from the z-coordinate. The lowest order solutions to (1) are given in Table I. Also listed are the resonant frequencies for a . All EM comspherical vacuum cavity with a radius putations are based on this radius and the resonant frequencies for other diameter spheres can be scaled by the inverse of the cavity radius. Spherical cavity resonators require excitation through a hole in the side of the cavity. Coupling to the cavity through this hole may involve: (1) a conducting probe or loop fed by a transmission line; (2) a pulsating electron beam passing through the gap in the resonator; or (3) attachment of a waveguide to the

U.S. Government work not protected by U.S. copyright.

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hole. The orientation of the internal electric field patterns is determined by the excitation process. The porous cavity resonator uses a regular pattern of holes in the surface of the sphere for both excitation and observations. An external plane wave enters the sphere through the surface mesh and any optical emissions may be imaged by light passing through the holes. The excitation process is examined in the next section. II. MODELING OF THE POROUS CAVITY RESONATOR A. Computing the Near Electric and Magnetic Fields To compute the fields inside the spherical polyhedron mesh, the method-of-moments solution of the electric field integral equations is solved for a conducting structure. The polyhedron sphere is designed in Mathematica V7.0 using the geometric algorithm described by Bernhardt et al. [1], [2]. The nodes and wire map is imported into WIPL-D [5] as a three dimensional object with constant radius, conducting edges. At each frequency of a range of frequencies, the electric and magnetic fields are computed in a plane containing the electromagnetic wave normal and one of the orthogonal axes. The magnitude of the peak electric field is computed from the complex electric field vector by

(2) With this definition, the peak magnitudes of either linearly or circularly polarized electric fields are correctly computed for all linear or circular polarized plane waves. Both perfect electrical conductor (PEC) and finite electrical conductor materials are considered for the polyhedron edges. The numerical computations are sorted by largest electric field amplitudes. With a fixed edge radius, the peak electric field amplitudes are found at discrete resonant-mode frequencies. The size of the polygons in the surface mesh was chosen to be much smaller than the radio wavelength. A 960 vertex polyhedron sphere (V960) with 10 mm radius edges and 5 m internal radius was found to have sufficient surface tessellation to have a radar cross section (RCS) that is almost independent of orientation for frequencies up to 60 MHz [1], [2]. In all test cases, the spherical mesh is excited by a unit amplitude wave with right-hand circular polarization (RHCP). B. Tuning the Openings in the Sphere for Maximum Resonance Electric Fields Adjusting both the radii of the edges and the transmitted frequency provides an optimization of the cavity resonance. Using a 1 V/m RHCP wave, the magnitude of the internal electric fields is computed numerically using WIPL-D. Fig. 1 illustrates the effects of increasing the edge radius and reducing the size of the openings into the sphere for a polyhedron with 180 vertices (V180). The rotational orientation of the V180 polyhedron has negligible influence on the RCS up to 36 MHz [2]. The resonant frequency for each edge radius is defined as the frequency that produces the maximum internal electric field. The optimum

Fig. 1. V180 polyhedron sphere used as a porous cavity resonator with an external plane wave with right-hand-circular polarization and wave amplitude of 1 V/m. The mechanical diagram overlay shows that the conducting edges confine the internal electric fields and control the resonant mode dimensions.

porous cavity resonator mode frequency, , is the resonant frequency that produces the largest overall internal electric field. The excitation of the porous cavity resonator with an external plane wave involves leakage of the external field into the interior and leakage of the internally excited field to the exterior. If the edge radius is too small (Fig. 1(a)), the spherical resonator is not able to prevent the internal fields from passing through the mesh and the internal electric fields are small. If the edge radius is too large as in Fig. 1(c), the incident plane wave cannot efficiently couple through the surface mesh to excite the interior. The optimum design for the like-mode (Fig. 1(b)) has the most efficient excitation of the internal electromagnetic wave and the minimum loss through the surface. In this case, the maximum internal electric field is 82 V/m. This mode will be designated as for the porous spherical cavity resonator (Fig. 1(b)) because it is related in both frequency and electric field pattern to the same mode for the spherical cavity. The influence of edge radius and EM excitation frequency is further illustrated in Fig. 2. The goal of the adjustments is to yield maximum internal electric field (Fig. 2(a) peak). Precise construction of the sphere is required to obtain large internal electric fields. The maximum internal field drops by a factor of two if the edge radius differs from the optimum value by 0.8 mm (0.36%). As the edge radius is increased, the resonant frequency is also increased, ultimately approaching the spherical cavity limit (Fig. 2(b)). The location of the maximum electric field shifts from the near side through the center at resonance to the far side of the sphere (Fig. 2(c)). Once the edges are adjusted for optimum resonance, the Q of the porous cavity sphere is large (Fig. 2(d)) and an external wave with 1 V/m is amplified to an internal field greater than 1000 V/m The electric field patterns in the lowest frequency PSCR modes excited by a circularly polarized plane wave are shown in Fig. 3 for spherical meshes with 960 vertices (V960). The TM101, TM103, TM201, and TE101 mode patterns resemble the corresponding spherical modes for a solid cavity [3]. The differences in the cases of the TM102 and TE102 modes result from a breaking of the meridian plane symmetry present in the

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Fig. 2. Edge and frequency tuning of the TM101 porous cavity resonator. The optimum edge radius is 225 mm for this mode.

Fig. 3. Electric field magnitudes of the first 6 modes in the externally excited porous cavity resonator with 960 vertices and optimized edge radii.

spherical cavity modes but not in the externally driven porous cavity modes. Other modes of V180 and V960 PCSRs were investigated with a combination of Mathematica V7.0 and WIPL-D to find optimum designs and maximum internal electric fields. In each case, the resonant frequency for the PSCR was slightly less than that of the spherical cavity resonator. The electric field patterns were similar for both spherical cavities, but the PSCR had spatial modes that extended outside the surface of the sphere. This increase in mode size or wavelength is responsible for the decrease in the corresponding resonant mode frequency. Leakage of radiated fields outside the spherical mesh also limits the maximum Q of the resonator.

The mechanical and electrical design parameters for these spheres are given in Table II. The wave-number times radius product in each case is smaller than the corresponding mode values in Table I for a solid-shell, spherical cavity resonator. The wavelength for the porous resonant cavity mode is slightly larger than the wavelength for the solid spherical resonators because of leakage of the electric fields through the porous surface. The cavity-mode frequencies for the V180 PCSR are always lower than the frequencies for the V960 conducting polygon. The optimum edge radius is roughly one-fifth to one-fourth the edge length. The fraction of non-conducting, free-space surface on each sphere is then about 50 to 60 percent of the total surface area. The electric and magnetic fields for a high order (TM104) mode are illustrated in Fig. 4(a). The electric fields are confined by the mesh boundary but significant E and H fields leak to the exterior of the sphere. The high order meridian modes have weak fields in the center of the sphere and large fields near the surface. The previous calculations of the internal electric fields were made assuming that the edges of the polyhedron sphere have infinite conductivity. In practice, the actual edges are metallic and the tangential electric field along these edges is not zero. The computed Q is the resonant curve frequency width at one-half peak amplitude divided by the resonant frequency. Using the method of moments code technique with conductivities of representative of aluminum, only small changes in the Q of the resonators or of the strength of the internal electric fields are found (Fig. 4(b)). The results of the simulations hold up using the WIPL-D model for a wide range of surface conductivities. With a highly conductive surface, the Q of the porous

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TABLE II 5-METER CAVITY RESONANCES: V180 AND V960 SPHERES

Fig. 4. Computed electric and magnetic fields for the TM104 mode of the externally excited spherical mesh resonator (a) excited by a circularly polarized wave. The computed peak internal electric fields (b) are only slightly affected using a metallic conductor such as aluminum (red curve) instead of an idealized PEC material (blue curve).

resonator is primarily determined by the wave leakage through the polygon holes in the surface of the conducting spherical shell. III. ELECTRIC FIELDS IN THE POROUS CAVITY RESONATOR The structure of the internal electric fields is governed by the cavity mode and the polarization of the external driving wave. The spherical cavity modes for the lowest azimuthal order have an axis of symmetry. Inside the porous cavity, this axis is defined by the direction of electric or magnetic field for a linearly polarized pump wave. Circular or elliptical polarization can be decomposed into two orthogonal linearly polarized waves with a 90 degree phase shift as given by where and are real electric field amplitudes for a wave propagating in the x-coordinate direction. The internal electric field for a combination of linearly polarized plane waves is the sum of the natural modes associated with each linear polarization. The internal electric field modes will be illustrated for the polyhedrons listed in Table II. The magnitude of the internal electric fields may differ from those computed in

Fig. 5. The PSCR is excited along the -x axis shown in green and with linear (red) or circularly polarized (red and blue) electric field vectors. TM101 mode excited inside a V960 polyhedron by a 1 V/m plane wave at 25.595 MHz propagating along the -x direction with linear polarization along the (a) y-axis, (b) the z-axis, and with (c) circular polarization. In all cases, the maximum fields are located at the center of the 5-m radius sphere. Each electric field magnitude contour is shown at 70% of the maximum internal electric field.

the previous sections depending on the how close the selected frequency is to the resonance for each mode. The effects of polarization on the lowest order TM101 mode are illustrated in Fig. 5 for a V960 polyhedron with 95.9 mm radius edges. All the driving plane waves propagate along the minus x-axis toward the origin. The electric fields inside the sphere form a contour elongated in the direction of the initial linear polarization, which is either along the y-axis vector (Fig. 5(a)) or the z-axis vector (Fig. 5(b)). The induced electric field forms a spheroid that is rotationally symmetric around each axis centered at the origin. A circular polarized wave produces a flattened spheroid that is rotationally symmetric around the x-axis (Fig. 5(c)). The strength of the internal electric fields is about the same for all three cases. The next example is the TM102 mode excited inside a V960 polyhedron with 103.9 mm radius edges. An external linear polarization excites a 2 2 matrix of maxima in the plane of the incident electric field (Fig. 6(a) and (b)). The conducting mesh

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Fig. 6. TM102 mode excited inside a V960 polyhedron by a 1 V/m plane wave at 36.617 MHz propagating along the -x direction with linear polarization along the (a) y-axis, (b) the z-axis, and with (c) circular polarization. For plane wave excitation (a and b), the electric field maxima are located at the four vertices of a rectangle in the plane of the incident electric field. The electric field magnitude contour is shown at 70% of the maximum internal electric field.

Fig. 7. TE101 mode excited inside a V960 polyhedron by a 1 V/m plane wave at 42.539 MHz propagating along the -x direction with linear polarization along the (a) y-axis, (b) the z-axis, and with (c) circular polarization. For excitation by linear polarization (a and b), the maximum electric fields are located in a ring around an axis through the center of sphere perpendicular to both the wavenormal direction and the direction of the incident electric field. For excitation by a circularly polarized wave, maximum electric fields are found on a spherical shell centered inside the spherical cavity. The electric field magnitude contour is shown at 70% of the maximum internal electric field.

modulates the electric fields on the boundary of the polyhedron sphere. The electric field contour for the RHCP excitation (Fig. 6(c)) shows two rings around the wave-normal axis. The electric fields interact with the mesh through a radial electric field component near the spherical mesh surface. As the orientation of the V960 polyhedron is changed relative to the incident wave direction, the internal electric field amplitudes vary by as much as 30%, but the basic shape of the structures is maintained. The higher order polygons like the V960 in Fig. 5 are less sensitive to changes in orientation than the lower order polygons such as the V180 in Fig. 1. The effect of orientation is most pronounced for electric fields near the surface of the polyhedron. The transverse electric (TE) modes produce electric fields that are isolated from the surrounding spherical mesh. The simplest TE101 mode forms an electric field ring aligned with the electric field vector for incident plane wave excitation (Fig. 7(a) and (b)). With a circularly polarized pump wave, the rings are combined into a shell with a local minimum in electric field at the center of the V960 mesh. The next highest order transverse electric field mode (TE102) yields four local maxima when excited by a linearly polarized wave (Fig. 8(a) and (b)). These electric field structures are elongated along the coordinate direction of the incident electric field and form a ring around the center of the sphere. A circular polarized wave causes these structures to combine into a ring around

Fig. 8. TE102 mode excited by a wave at 54.565 MHz propagating along the -x direction with linear polarization along the (a) y-axis, (b) the z-axis, and with (c) circular polarization. For excitation by linear polarization (a and b), the maximum electric fields are located in four regions. For excitation by a circularly polarized wave these four regions combine into a ring and two isolated patches aligned with the incident wave propagation direction. The electric field magnitude contour is shown at 70% of the maximum internal electric field.

Fig. 9. TM201 mode excited by a wave at 57.902 MHz propagating along the -x direction with linear polarization along the (a) y-axis, (b) the z-axis, and with (c) circular polarization. In all cases, the region of maximum electric fields is located near the center of the spherical resonator mesh. The electric field magnitude contour is shown at 70% of the maximum internal electric field.

the direction of propagation along with two caps on either side of the ring (Fig. 8(c)). The composite structure is symmetric about the incident x-axis. The last example shows the effects of increasing the order of the radial variations. The TM201 mode (Fig. 9) resembles the TM101 mode calculations with the electric fields confined to a smaller region at the center of the polyhedron. As shown in Fig. 2, all spherical cavity modes become compressed toward the center of the sphere as the radial order index “s” is increased. All of the enhanced electric field patterns take a finite time to build up inside the porous cavity resonator. This build up time is the inverse of the resonance band width illustrated in Figs. 2(d) and 4(b). This time delay is discussed in the next section. IV. TIME DEPENDENCE FOR EXCITATION Because of the narrow bandwidth (high-Q) of the PSCR, substantial time may be required for the generation of the internal electric fields. As an example, a linearly polarized plane wave propagating along the -x direction is used to excite a V180 sphere designed for the TM101 mode. The excitation frequency is at the center of the resonance curve shown in Fig. 2(d). The half amplitude width of this curve is 1.33 kHz so the time constant for excitation of the resonator should be on the order of the inverse of this frequency width or 3/4 ms. The Fourier transform of the computed frequency response is used determine the envelope of the resonant oscillations at the

BERNHARDT AND FERNSLER: ELECTRIC FIELD AMPLIFICATION INSIDE A PSCR EXCITED BY AN EXTERNAL PLANE WAVE

Fig. 10. Numerically computed decay response of the V180 sphere with edges tuned for the TM101 mode.

center of the sphere (Fig. 10). The computed decay time for this curve is . A symbolic representation of this response is (3) is the initial amplitude and U(t) is the unit step at where . The Fourier transform of (5) gives the frequency response for the resonator as (4) This represents a single pole filter for the incident plane waves. The magnitude of (6) provides an excellent match for computed frequency responses, such as shown in Figs. 2(d) and 4, of the PSCR. The decay time constant, , depends how well the polyhedron edge radius is tuned for a given cavity mode. The temporal response of the PSCR to an EM-wave, excitation pulse is computed by multiplying the frequency response of the TM101 resonator with the Fourier transform of an incident unit amplitude pulse and then computing the inverse Fourier transform of this product. The initial pulse waveform and the resultant envelope of the electric field at the center of the sphere are illustrated in Fig. 11. The incident pulse is 4.51 ms duration at a frequency of (Fig. 11(a)). Then as the pulse impacts the sphere, the internal electric fields grow to 1500 V/m with a time constant of . After the pulse passes the sphere, the internal electric fields decay back to zero with the same time constant. The internal wave is a TM101 source field that is approximately isotropic at the center of the sphere. Consequently, the reradiated wave also tends to be isotropic. The porous spherical cavity resonator has the remarkable property of radiating a scattered signal long after the incident signal has passed. As a radar target, the 5-m radius target could resemble a thick object with a physical size given by where c is the speed of light. V. APPLICATIONS FOR GAS BREAKDOWN Measurements of the internal electric fields are difficult with probes that can disturb the resonance conditions. However, optical indications of the electric field patterns can be obtained

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Fig. 11. Time history of the electric field at the center of the V180 sphere tuned to the TM101 mode. The center of incident pulse (a) intersects the center of the . The build-up and decay of the internal electric fields occurs sphere at with a computed time constant of 0.4157 ms. Thus, the PSCR is unusual as both a device to amplify electric fields and as an extremely high Q radar target.

using glowing gas discharges. The strength of the electric fields can be estimated from the intensity and locations of the glow patterns. Breakdown of gases to form plasma clouds occurs when the electric field of a wave becomes sufficiently strong to cause an electron avalanche [6]. In discharges, electrons are created by gas ionization and destroyed by processes like attachment, recombination and diffusion. Breakdown occurs when the ionization rate exceeds the total destruction rate. Since the ionization rate depends far more strongly on the electron temperature than does the destruction rate, breakdown usually occurs once exceeds a well-defined threshold. The threshold temperature in air, for example, is 1.8 eV, nearly independent of experimental conditions. The electron temperature is determined by a balance between electron heating from the applied fields and electron cooling from inelastic collisions (ionization, excitation, etc.). Like the ionization rate, the cooling rate is a strong function of , whereas the heating rate is determined by the time-average of ; here is the applied electric field and is the electron velocity. Temporal oscillations in produce a magnetic field, but that field has little effect on the heating rate to order . Of greater importance is a strong DC magnetic field transverse to . To illustrate, assume is along the x-direction and is along the y-direction. The equations of motion along the x and z directions are then given by

(5) and

(6) respectively. Here is the electron-neutral collision frequency and is the (fixed) electron cyclotron frequency. If oscillates at frequency (7) and (8) reduce to

(7)

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and (8) Accordingly,

(9) is the rms electric field and m is the electron mass. where The electron temperature is obtained by setting (10) is the reduced collisional where is the gas density and cooling rate. Similarly , where is the reduced electron-neutral collision frequency. varies strongly with , but g varies only as where . The functions and can be computed either from first principles (using known cross sections) or more easily from swarm data. These relationships then give as a function of , , , and . The behavior of DC discharges is well known from swarm data in many gases. To utilize this data for RF discharges, use (11) to define an effective electric field,

(11) Note that

when

. Similarly, (12)

if

, and (13)

if

. is an important discharge The DC breakdown strength parameter, and fortunately it is nearly constant for the reasons mentioned earlier. For example, in air , and at this field strength . Noble gases like argon contain few low-lying excited states, so is smaller. Similarly, is higher in gases like which contain many low-lying excited states. (The electron attachment rate in SF6 is also much higher, but this alone does not explain the increase in .) To determine the RF electric field needed for breakdown, set in (13). The electron energy distribution is invariably nonMaxwellian when the degree of ionization is low, and therefore does not have a unique value or definition. In this work is taken to be the characteristic electron energy, a well-defined experimental parameter in swarm data. Ionization derives only

from the high-energy tail of the distribution, and the shape of the distribution changes somewhat with . Fortunately, those changes and others are automatically incorporated in the above model when using swarm data. The momentum-transfer collision frequency is an exception, however, since it involves an average over the energy distribution. The method of averaging depends on the magnitude of relative to and , but because the collision frequency varies relatively weakly with energy, the different methods usually give similar values to within 20% or so. Finally, because the loss rate varies with the experimental conditions, the breakdown strength is not truly constant in a given gas. Tabulations of swarm data therefore give the ionization rate and various loss rates, but not per se. Nevertheless, experimental values for are easily found [6], [7], or the values can be computed from the ionization and loss rates. The values reported for typically vary by less than 10% in a given gas. The porous cavity resonator excited by an external electromagnetic wave provides a new technique for plasma formation. The intense electric fields reside inside the spherical mesh so an isolated region of plasma will be formed where this field exceeds the breakdown field. The TMs01 and all of the TE modes produce intense electric fields that do not come in contact with the spherical mesh (Figs. 5, 7, 8 and 9). With such excitation, optical displays of glowing plasma with controllable spatial configurations may be produced for viewing through the polygon holes in the spherical mesh. The sizes of the plasma discharge clouds are determined by the boundary where the effective electric field equals the breakdown threshold. According to (13), is controlled by adjusting the strengths of the EM pump wave , the neutral density N, or the magnetic field B in the experimental apparatus. When a high power EM wave is propagated to a high-Q PSCR, the internal electric fields will build up until a discharge occurs. If the internal refractive index for the wave is sufficiently changed, the resonance conditions may be disrupted and the discharge could shut itself off. In addition, the Q of the system will decrease from the absorption of energy by the electron collisions in the plasma. After the breakdown plasma has dissipated, the internal electric field could build up again, triggering another discharge. This process may repeat as relaxation oscillations. If an equilibrium is established with the plasma inside the cavity resonator, a self-sustained discharge may be formed. Gas breakdown within a PSCR will be addressed with future computational work. A single PSCR with fixed edges can be scanned in frequency to find resonances. The 5-m radius sphere has multiple high-Q resonances with an edge radius near 89 mm (Table II). The peak electric field for each mode is dependent on how close the excitation frequency is to the actual resonant frequency. The spectrum of peak internal fields (Fig. 12) shows the four strong spherical resonator modes TE101, TE102, TM201, and TE103 all excited in the same polyhedron. The spectrum also shows three weak modes at 46.75, 56.74, and 66.64 MHz that are predicted by the spherical cavity resonator theory. The internal glow structures with the excitation threshold set to 70% of

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shape of the glow plasma discharge displays will indicate the excited mode. In addition, a model of gas breakdown is being constructed which includes the effects of the plasma on the EM fields. These effects include absorption by the plasma ball to reduce the Q of the resonator and changing of the refractive index inside the sphere to change the resonant frequency. REFERENCES

Fig. 12. Computed frequency response and simulated glow patterns inside the 960 vertex polyhedron with a radius of 5 meters and an edge radius of 89 mm. The excited modes are TE101 at 42.55 MHz with peak field strength of 193 V/m, TE102 at 54.56 MHz with a peak field strength of 212 V/m, TM201 at 57.88 MHz with a peak field strength of 155 V/m and TE103 at 66.15 MHz with a peak field strength of 193 V/m all driven by a circularly polarized plane wave with 1 V/m amplitude. The polyhedron mesh is included to illustrate the optical glow might be viewed from outside the sphere.

the peak field intensities are illustrated by the contour surfaces in Fig. 12. Cavity resonators fed by waveguides driven by high power radio sources have been used to study the effects of pressure on threshold electric fields for breakdown [7], [8]. Such measurements may also be provided by the PSCR with the added advantage of being able to observe the ionization process from a wide variety of viewing angles and being able to transport the internally generated plasma through the mesh of holes in the surface of the sphere. The magnetic confinement and the stability of isolated plasma clouds can be studied using gas discharges inside the porous cavity resonator excited by an external EM pump. VI. CONCLUSIONS In summary, iterative use of Mathematica V7.0 for mechanical structure and WILP-D for electromagnetic fields provides a powerful tool for simulating the production of large amplitude electric fields inside a porous cavity resonator. The EM calculations have been tested at NRL by looking for light emission caused by gas breakdown in the field-enhanced regions inside the resonator. Argon plasma glow was produced by inside at TM101 copper plated PSCR excited by an external electromagnetic field at 2.45 GHz. The sphere was placed inside a evacuated chamber with a background argon pressure of 50 mT and an excitation power of less than 50 W. Without the sphere, the microwave excitation can be as much as 5 kW without gas , breakdown. The spherical resonator had a measured Q over thereby amplifying the internal electric fields by more than at the resonant frequency. This test provided strong validation of the theory in this paper, and other tests are in preparation. A future experimental paper will provide a complete examination of the resonant amplifications with a series of porous spherical resonators each designed for a specific cavity mode. The

[1] P. A. Bernhardt, “Radar backscatter from conducting polyhedral spheres,” IEEE Antennas Propag. Mag., vol. 52, no. 3, pp. 52–70, 2010. [2] P. A. Bernhardt, C. L. Siefring, J. F. Thomson, S. P. Rodriquez, A. C. Nicholas, S. M. Koss, M. Nurnberger, C. Hoberman, M. Davis, D. L. Hysell, and M. C. Kelley, “The design and applications of a versatile HF radar calibration target in low earth orbit,” Radio Sci., vol. 43, p. RS1010, 2008, 10.1029/2007RS003692. [3] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941, pp. 554–563. [4] S. Ramo, J. R. Winnery, and T. Van Duzer, Fields and Waves in Communication Electronics. New York: John Wiley, 1965, pp. 552–558. [5] X. B. Kolundzija, J. Ognjanovic, M. Tasic, D. Olcan, D. Sumic, M. Bozic, M. Kostic, and M. Pavlovic, WIPL-D Software Users Manual. Belgrade: WIPL-D d.o.o., 2010. [6] H. L. Rowland, R. F. Fernsler, and P. A. Bernhardt, “Breakdown of the neutral atmosphere in the D region due to lightning driven electromagnetic pulses,” J. Geophys. Res., vol. 101, pp. 7935–7945, 1996. [7] S. C. Brown, Basic Data of Plasma Physics. College Park, MD: American Inst. Phys. Press, 1994, pp. 302–309. [8] P. Y. Raizer, Gas Discharge Physics. Berlin: Springer, 1997, pp. 138–166. Paul A. Bernhardt (M’98–SM’01–F’06) received the B.S. EE degree from the University of California, Santa Barbara, in 1971, and the M.S. EE and Ph.D. EE degrees from Stanford University, Stanford, CA, in 1972 and 1976, respectively. He is the Head of the Space Use and Plasma Environment Research Section, Plasma Physics Division, Naval Research Laboratory. His primary area of research is remote sensing of the upper atmosphere using radio techniques including: (1) computerized ionospheric tomography (CIT); (2) optical excitation by high power radio waves; (3) radar diagnostics Space Shuttle engine burns. He has been Principal Investigator on a number of NASA and DoD sponsored experiments. His theoretical interests include modeling of non-linear interactions of high-power radio waves in the ionosphere, numerical solutions of partial differential equations for fluids and waves, and reconstruction algorithms for tomographic imaging. He has published over 120 papers in refereed journals. Dr. Bernhardt is past Chairman (1994–1997) for Commission H of the United States National Committee of the International Union of Radio Science (URSI), Former Chairman of Subcommission C4/D4 on Active Experiments of COSPAR Experiments (1998–2004), a member and previous books-board editor of the American Geophysical Union (AGU), Associate Editor for Radio Science and a Fellow of the American Physical Society (APS).

Richard F. Fernsler graduated from Haverford College, Haverford, PA, in 1966 with a BA in engineering-physics. He received the Ph.D. degree in plasma physics from the University of Maryland, College Park, in 1976. He was a National Research Council Postdoctoral Fellow at the Naval Research Laboratory. His principle research areas are plasma sources and plasma processing, electron beam physics, electrical and laser breakdown, plasma electrodynamics, plasma diagnostics, and air chemistry. Dr. Fernsler is a member of the American Physical Society and Sigma Xi.

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A 76 GHz Multi-Layered Phased Array Antenna Using a Non-Metal Contact Metamaterial Waveguide Hideki Kirino, Member, IEEE, and Koichi Ogawa, Senior Member, IEEE

Abstract—A 76 GHz phased array antenna (PAA) using waffleiron ridge waveguides with non-metal contacts has been developed. The non-metal contact technology has the advantage of avoiding losses due to imperfect metal contacts, and also facilitates the fabrication of small-size and multi-layer stacked structures. The principle and results of the developed phase shifter, radiator and feed network are presented. For the feed network, it is shown that the phase differences between adjacent radiators are the same, which confirms the validity of the fundamental operation of the PAA. A PAA combining the feed network and 16 radiators with a size of 62 mm 62 mm 25 mm was realized. The characteristic of the PAA was evaluated by calculating the directivity using measured data from the radiator and the feed network. From these considerations, the PAA was found to have the capability of providing a beam tilt angle of 18 while maintaining a gain of more than 32 dBi. Index Terms—Antennas, metamaterial, millimetre wave radar, phased arrays.

I. INTRODUCTION

P

REVIOUSLY we have presented a phase shifter, a radiator and a basic phased array antenna (PAA) using a movable waffle-iron metal plate [1]–[3] and a waffle-iron ridge waveguide (WIRWG) [4], [5]. The WIRWG is categorized as a kind of “metamaterial” technology, in which an artificial magnetic wall is formed on the surface of the waveguide. During the period in which our work was published, P. -S. Kildal et al. presented a similar structure to the WIRWG, which they called a “Gap Waveguide”, and for which they evaluated the dispersion characteristics [6], [7]. Since it is expected that this newly proposed waveguide will have various derivative applications, progressive research work undertaken so far has been presented [8]–[15]. A 76 GHz PAA for use in vehicular radar is one of the unique applications that utilizes the excellent features of this structure, for the following reasons. It has been asserted in many papers [5]–[15] that no metal contact is needed between the upper and lower conductors in the WIRWG, so that the losses due to imperfect metal contacts can be permanently avoided. This advantage implies that the WIRWG has some tolerance to high and low temperature heat cycles and vibration in long-term vehicle Manuscript received September 18, 2010; revised April 29, 2011; accepted July 12, 2011. Date of publication October 24, 2011; date of current version February 03, 2012. H. Kirino is with the Panasonic Healthcare Co., Ltd., Ehime, Japan. K. Ogawa is with the Toyama University, Toyama, Japan. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173112

applications. Moreover a structure with no metal contacts is suitable for millimetre wave circuits and small-size multi-layer stacked structures. To utilize these benefits for vehicular radar, a structure that is small in size, with an appropriate beam tilt angle and gain is required. For the realization of such a PAA, the phase difference between each pair of adjacent radiators should be the same, which is an unusual condition. This condition is difficult to realize using a straight-shaped phase shifter as described in [4], and hence we have proposed a novel structure comprising a movable waffle-iron metal plate formed in the shape of a circular disk with phase shifters consisting of concentric arcs [13], [16]. This paper presents a 76 GHz PAA incorporating a waffleiron ridge waveguide. Particular emphasis is placed on detailed descriptions of the operating principles and the theoretical and empirical results of the developed phase shifter, radiator and feed network. For the feed network, it is shown that the phase differences between all adjacent radiators are identical, which confirms the validity of the fundamental operation of the PAA. From the measured phase differences of the feed network and the measured radiation patterns of the radiator, the overall characteristics of the PAA, such as its beam scanning capabilities over a specified solid angle in front of a car, are given. Apart from the input port, which uses a WR10 rectangular waveguide and a base-plate, the size of the PAA, including a radome, is 62 mm 62 mm 25 mm. With this construction we demonstrate that the developed PAA has unique features that make it particularly suitable for vehicular radar applications because of its well-shaped radiation pattern featuring high directivity. This paper is organized as follows. First, we explain the configuration and principles of the WIRWG. The basic components, including a phase shifter and a radiator, are expanded on in Section II. The 76 GHz 16-column PAA for vehicular radar is shown in Section III. Finally, the conclusion is given in Section IV. II. BASIC COMPONENTS A. Waffle-Iron Ridge Waveguide Fig. 1 shows the structure of the WIRWG. As described in [5]–[15], the structure contains waffle-iron conductor rods that are a quarter-wavelength in height, which allows the top surfaces of the rods to form an Equivalent Magnetic Boundary (EMB). Since there is no electric field normal to the EMB,

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Fig. 3 are chosen to be the depth of the grooves when the surface of the ridge is at the same height as the tops of the waffle-iron conductor rods. It is found from Fig. 3 that the relationship between the groove depth and the wavelength is similar to that of an ordinary ridge waveguide when the structure of the WIRWG is replaced with that of an ordinary ridge waveguide as illustrated on the right hand side of Fig. 3, resulting in a similar shape for the dispersion curves in each case. This fact is supported by the following considerations; from the dispersion curves shown in Fig. 3 and the fact that no tangential electrical fields exist in the small gap that separates the electrical walls between the upper conductor and the ridge, it is suggested that, in the case of the WIRWG, the energy propagating along the waveguide is carried mainly by the TE wave. It should be noted, however, that there is no cut-off phenomenon in the case of the WIRWG as observed in a usual metal waveguide supporting the TE mode, since at low frequencies, transmission waves spread out in two-dimensions due to the inability of the conductor rods to confine the electromagnetic energy to the ridge region. Hereinafter in this paper, the structural parameters are chosen , and , where such that is the wavelength at 76.5 GHz. Under these conditions, the . wavelength of the WIRWG is calculated to be

Fig. 1. Structure of the WIRWG.

Fig. 2. Current distribution on the WIRWG.

B. Fundamental Characteristics of the Waveguide

Fig. 3. Dispersion diagram of the WIRWG for L2 =

g=4.

the transverse electromagnetic (TEM) mode propagating between the parallel metal plates is suppressed. On the ridge, however, the condition for the electric boundary is satisfied, so that electromagnetic waves propagating between the ridge and the upper metal plate are excited, resulting in the whole structure becoming a practical waveguide. Fig. 2 shows the current component in the z-direction of the coordinate system shown in Fig. 1 for different groove depths L1 beside the ridge. As shown in the figure, the distance between the wave-tops of the current on the ridge, which is equal to the wavelength of the WIRWG, becomes shorter when the depth L1 is increased from 1/4 to 3/8 of the wavelength in free space. This means that the relationship between L1 and the wavelength is similar to that between the height of the ridge and the wavelength in an ordinary ridge waveguide, shown in the diagram on the right hand side of Fig. 3. The wavelength of the WIRWG can be obtained from the phase difference between the transmission waves arising on two waveguides of different length, which can be calculated from an EM-simulator [17], [18]. Fig. 3 shows the dispersion diagram of the WIRWG as a function of wave number. The parameters in

As shown in Fig. 1, the WIRWG has a number of waffle-iron rods that form a discontinuous and periodic structure. This might be the cause of fluctuations in transmission factors, such as changes in the phase constant. A smooth change in the phase constant is one of the essential requirements for realizing a phase shifter, as will be mentioned in Section II, and thus, before applying the waveguide to actual components, the effect of discontinuities in the waveguide on phase change should be clarified. Fig. 4 shows the simulated results of the phase change caused by a two-way transmission reflected by a short-circuit wall, where all the metal parts in the simulation model are set to be Perfect Electrical Conductors (PEC). A schematic diagram depicting the simulation model is shown in the figure. In Fig. 4, L3 depicted at the input port is the lateral length measured from the side wall to the centre of the gap between the ridge and the waffle-iron rods. L3 is selected to be a quarter-wavelength in order to minimize the matching condition of the input port. Adopting this port configuration, a matching condition of less for the frequency band to 1.15 was than achieved. As shown in Fig. 4, when the position of the short-circuit wall moves from 0.625 to 1.125, which are the values normalized to , the phase of the reflected wave changes without significant fluctuations. This result means that the WIRWG can be used as an ordinary waveguide such as a rectangular waveguide or a micro-strip-line with a uniform phase constant independent of the position of the periodic structure.

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C. Phase Shifter

Fig. 4. Phase change caused by two-way transmission reflected by a short-circuit wall.

Fig. 5. Calculated loss characteristics of the WIRWG at 76.5 GHz.

Fig. 5 shows the calculated transmission loss characteristics of a WIRWG with the configuration at the input port shown in . Fig. 4, which keeps the matching condition to less than As shown in Fig. 5, the transmission losses for the typical metals copper and aluminium, whose conductivities are 5.8 and 3.8 , are calculated to be around 0.01 dB/mm. The actual transmission loss can be measured from the phase shifter, and this is described in Section II-C. Fig. 6 shows the calculated isolation characteristics of a WIRWG arranged in a parallel configuration. As shown in the figure, a pair of WIRWGs are isolated using several lines of waffle-iron rods, where the number of lines is indicated by N. The isolation is defined as the ratio of the energy of the escaped wave to that of the incident wave in the TEM field on the ridge, where the direction of the incident wave is normal to the ridge as shown in the figure on the right hand side of Fig. 5. It can be seen from the figure that if the desirable isolation is defined for as the number of lines that gives an isolation of , for , for , and for , a bandwidth of to 1.1 can be obtained.

Fig. 7 shows the structure of the phase shifter. Fig. 7(a) shows the fundamental configuration, which includes the input and output ports on the upper and lower metal plates. For port A on the upper metal plate, there is a plurality of short-ended choke holes with a depth of a quarter-wavelength separated by a quarter-wavelength from the port-edge. The input impedance of the short-ended choke holes is equivalent to an open-circuit, so that the edge of port A is equivalent to a short-circuit, whereas for port B on the lower metal plate, there is an open ended quarter-wavelength choke ridge with a plurality of waffle-iron rods. The impedance at the end of the choke ridge is equivalent to an open-circuit, so that the edge of port B is equivalent to a short-circuit. Due to the effects of these chokes, the transmission line between the two ports consequently becomes a single line without any branches. Therefore, the phase of this transmission line can be varied by changing the distance between the two ports by sliding the two plates relative to each other. Fig. 7(b) shows a more practical version of the phase shifter. This practical phase shifter has two fundamental phase shifters with their backs mounted together, which creates a new structure with a mid-plate placed in between the two plates. This structure permits the mid-plate to slide between the two plates while fixing the position of the two input ports A and A’, creating a so-called ’trombone shaped phase shifter’, which yields double phase changes with respect to the fundamental configuration. In Fig. 7(b), L is the distance between port A (A’) and port B (B’), when the inner edges of the two ports coincide in which with each other; L is used as a variable in the horizontal axes in Figs. 11–13. Figs. 8 and 9 show the configurations and calculated matching characteristics of the port. The best way to widen the frequency characteristics of the input impedance matching of the phase shifter is to minimize the individual port-matching of ports A and B, because the conjugate matching condition is not stable when the distance between the two ports varies. We now give an explanation of the steps that are required to widen the matching of each port. Fig. 8(a)–(d) show the steps needed to improve the matching of port A. Fig. 8(a) shows a cross-sectional view of a suitable initial structure, since this structure has a similar field distribution to the WIRWG. Fig. 8(b) shows an improved version, in which the fields connect gradually to both WIRWGs in opposite directions beyond the ports. Fig. 8(c) shows a further-improved version, in which the inner corners are rounded accommodating the shaven form of the milling process. Fig. 8(d) shows the final configuration, in which some depth and width have been added in order to widen the bandwidth. Fig. 8(e) and (f) show the steps used to improve the matching of port B. Fig. 8(e) shows a shape employing an ordinary rightangled corner. Fig. 8(f) shows an improved version, in which one corner is cut in a stair-like structure to widen the bandwidth. Fig. 9 shows the improvements in the characteristics as a result of implementing the above modifications. It can be seen from

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Fig. 6. Calculated isolation of the WIRWG as a function of the number of lines of rods.

Fig. 7. Structure of the phase shifter.

Fig. 10. Photograph of the phase shifter.

Fig. 8. Modification of the port configuration to improve the impedance matching.

Fig. 9. Calculated matching characteristics of the ports.

the figure that bandwidths of 3.9 GHz for port A and 5.2 GHz . for port B are obtained for a reflection coefficient of

Fig. 10 shows a photograph of the developed phase shifter with the structure shown in Fig. 7(b) and the optimized port configuration shown in Fig. 8. Figs. 11–13 show the measured phase change, the insertion loss and the impedance characteristics. As can be seen in the figures, the phase shifter exhibits phase changes of more than 1800 and an insertion loss of less than 1.5 dB at 76 GHz when the mid-plate moves by 12 mm. are obtained beImpedance characteristics of less than tween 75 and 76.5 GHz. As can be seen from Figs. 12 and 13, the insertion loss and impedance characteristics have the best performance at 75 GHz, which is slightly lower than the target frequency of 76.5 GHz. A possible reason for this result is due to the automatic and coarse meshing functions of the FDTD-based EM-simulator used in the simulation, since it is well known that an appreciable discrepancy for the optimized frequency might occur due to the

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Fig. 14. Structure of the radiator.

Fig. 11. Measured phase change of the phase shifter.

Fig. 15. Photograph of the radiator.

average gradient is found to be 0.25 dB/12 mm, giving a transmission loss of 0.02 dB/mm. Fig. 12. Measured insertion loss of the phase shifter.

Fig. 13. Measured impedance characteristics of the phase shifter.

meshing function. The verification of this assumption and optimization of the port configuration are left for our future work. The actual transmission loss can be measured from Fig. 12. However, there are some ripples in the insertion loss in Fig. 12 which are caused by multi-reflection effects due to impedance mismatching at both ends of the WIRWG. The insertion loss is considered to be the sum of the mismatch at the ports, conductor losses, and the energy loss due to leakage of EM-waves out of the waveguide passing though the waffle-iron rods. Since the loss due to mismatching is constant, the total loss is proportional to the length of the transmission line. Therefore, the average gradient of the insertion loss indicates the sum of the conductor and leakage losses in the transmission line. The actual transmission loss for aluminium at 76.5 GHz is calculated from the average gradient indicated by the dashed line in Fig. 12. The

D. Radiator Fig. 14 shows the structure of the radiator. Fig. 15 shows a photograph of the developed radiator. The actual PAA for vehicular radars has plural vertical radiators aligned in columns in order to sharpen the beam width in the vertical direction for beam scanning operation. Conventional slot arrays often adopt a feed network creating a travelling wave along the waveguide [19]. In this paper, however, a guided feed based on a resonant transmission line with both terminals open ended, creating a standing wave in the waveguide, is employed. Furthermore, resonant slots inclined by 45 are located at the position of maximum current density of the standing wave as shown in Fig. 14. On the feed line employing standing waves, all of the slots are excited in phase with the same magnitude, because the slots cut current of the same phase and magnitude at the position of maximum current on the transmission line. Therefore, the main beam is directed toward the bore-site direction with no beam tilt, which is advantageous compared to conventional slot arrays adopting a feed line of travelling waves, where the phase and magnitude of the currents are not the same in all positions on the line, resulting in a tilted beam [19]. Travelling wave feed lines have the advantage that the impedance bandwidth is wider than that of standing-wave feed lines because there is no resonant current on the feed structure. More specifically, the standing wave feed line has a drawback in that the impedance bandwidth is narrow. However, the required bandwidth, including guard bands for vehicular radar at 76 GHz, is as small as 1.3%, which is much smaller than the calculated bandwidth of 3.9% for a criterion of reflection coefficient when using the radiator shown in Fig. 14.

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Fig. 16. Gain of a single slot with a DR as a function of the height of the DR.

In Fig. 14, the length of the WIRWG is and the number of slots is 10. As described in Section II-A, the wavelength on the WIRWG is longer than that in free space. Furthermore, the spacing between the slots is equal to the wavelength on the WIRWG, enforcing the in-phase excitation of a standing wave. These features mean that the spacing between the slots is longer than the wavelength in free space. Hence, grating lobes of a sig. In nificant level can be created over an angular range of order to suppress these grating lobes and to increase the main lobe intensity, a dielectric rod (DR) is loaded above the slots [20], [21]. The DR is fabricated as an integral part of the radome, which facilitates the assembly of the PAA. Fig. 16 shows the calculated gain of a single slot loaded with a DR as a function of the height of the DR. A plan view and a cross-sectional view of the DR are illustrated on the right hand side of Fig. 16. The dimensions of the DR in the plan view, D and W, and the angle of the DR, , are fixed to be 1.2 mm, 2.4 mm and 45 respectively. As shown in Fig. 16, the gain of a single slot changes when the height of the DR changes [20]–[22]. In order that the total antenna thickness is not too large, an appropriate value for the height, , of the DR is 4 mm, since any increase in gain is substantially saturated beyond this value. Fig. 17 shows the calculated directivities Gmax, the 1st side lobe levels L1 and the grating lobe levels Lg of the radiator shown in Fig. 14 for different DR geometries, where all the metal parts in the simulation model are set to be PECs and the loss tangent of the DR is set to be zero. In Fig. 17, the upper figure shows the directivity under the condition that , and , whereas the lower table shows the results for various combinations of different DR parameters. The figure shows that changes in D and W have a greater effect on L1 and Lg than on Gmax. It is also found from Fig. 17 that the maximum Gmax, and the minimum L1 and Lg , and . are obtained with However, under these conditions, adjacent DRs overlap each other, and thus it is impossible to place DRs using these geometrical parameters. Taking these geometrical restrictions into consideration, a possible and practical configuration that provides the maximum Gmax, and the minimum L1 and Lg was , and . Furdetermined to be thermore, Gmax increases in accordance with the height of the

Fig. 17. Calculated directivity of the radiator as a function of DR shape.

DR, , up to 3.5 mm (not shown in the figure), but does not show an appreciable change beyond that value, which agrees with the analytical results for a single slot shown in Fig. 16. From these considerations, the size of the DR was chosen to be W 2.0 mm D 2.0 mm H 4.0 mm. To confirm proper excitation in the feed line when using standing waves, a near-field measurement was conducted using a waveguide proving method. Fig. 18 shows the phase and magnitude of the fields picked up by a waveguide probe located 2 mm above a slot without the DR. It can be seen from the figure that the deviations in phase and magnitude are less than 7 dB and 70 respectively, indicating that standing wave feeding is properly realized with some deviation. A study on the reason for the deviation is left for future work. Fig. 19 shows the radiation patterns measured in an anechoic chamber. Fig. 19(a) shows the radiation pattern for the co-polar component in the zx-plane with and without the DR. As can be seen from the figure, enhancement of the main lobe level and suppression of the grating lobe level are achieved due to the presence of the DR. The main lobe, the grating lobe, and the 1st side lobe level with respect to the main lobe are found to be and respectively. 22.6 dBi,

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Fig. 18. Measured phase and magnitude of the fields above the slot without a DR using a near-field measurement.

Fig. 19(b) shows the radiation patterns for the co-polar component in the yz-plane with and without the DR. As can be seen in the figure, the DR is effective in increasing the gain in the bore site direction. The figure also indicates that the DR unexpectdirections. This phenomedly increases the gain in the enon may be due to the effect of the radome and investigation of this is left for further studies. Fig. 19(c) shows the radiation patterns for the cross-polar component in the zx-plane with and without the DR. Comparing the main lobe level achieved in Fig. 19(c) with those in Fig. 19(a) and (b), the cross-polar discrimination is found to be more than 20 dB and the DR does not affect this value. Furthermore, it shows that the DR causes little deterioration of the grating lobes. It should be noted from Fig. 19(a) that the first side lobe level with respect to the main lobe indicating that each is slot is excited at the same magnitude. Furthermore, Fig. 19(a) shows that the main lobe is directed in the bore-sight direction, meaning that each slot is excited in phase. It is confirmed from these results that a feeding method to create standing waves was successfully realized. In vehicular radar applications, all instances of unexpected lobes must be adequately suppressed over the entire angular region corresponding to the requirement of particular radar systems. However, this observation is difficult to achieve from evaluations using ordinary principal cut-plane measurements (which are commonly executed in an anechoic chamber) such as the measured data shown in Fig. 19. Hence, we have attempted to obtain a three-dimensional radiation pattern over a hemispherical region in front of the car by employing a Fourier transformation of the fields measured by near-field probes on the plane 5 mm above the surface of the radome. Fig. 20 shows a three-dimensional picture of the radiation pattern for the radiator with a radome. Fig. 20 indicates that there are no undesirable lobes in the hemispherical region, suggesting that the developed antenna can be successfully used for vehicular radar applications. III. APPLICATION TO A 76 GHZ 16-COLUMN PAA A. Overall Structure of Feed Network and PAA

Fig. 19. Measured radiation patterns of the radiator at 76.5 GHz. (a) zx plane (co-polar) (b) yz plane (co-polar) (b) yz plane (co-polar).

Fig. 21 shows the complete structure of a 16-column PAA that uses WIRWGs for all the components, including the divider, the phase shifter and the radiator. As shown in the figure, the PAA has one radome with DRs, five fixed conductor plates #1 to #5 and one rotational plate. All components are formed using WIRWGs between each of the conductor plates. The DRs are an integral part of the radome, and they are fabricated simultaneously. This structure improves the accuracy of locating the DRs with respect to the slots fabricated on plate #1, and facilitates the assembly of the PAA. The functional block comprising the divider and the phase shifter is hereinafter called the “PAA feed network”. The signal applied to the antenna port is guided up to the radiator though the WIRWGs and via-holes fabricated on each

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Fig. 20. Radiation pattern calculated from measured near-field data.

Fig. 21. Overall Structure of the 16-Column PAA. Fig. 22. Cross-sectional view of the phase shifter.

plate. The structure of the via-holes is similar to that of the port depicted in Fig. 7(b). The PAA feed network divides the signal into 16 fragments and controls the phases to realize a 16-column PAA. To fold a complicated feed network into a smaller size, the signal path is designed to pass along the route shown in Fig. 21;

in such a way that the signal passes through each phase shifter three times. Signal turnarounds, for example, are achieved by passing through the via-hole on plate #3, the WIRWG in Divider2 and the via-hole on plate #3 again. B. Principle The principle of operation is explained in the following. As shown in Fig. 21, there are plural concentric shaped phase shifters on both sides of the rotational plate. With this configuration, the length of the transmission line between the input and the output ports of the phase shifter is proportional to the angle of rotation of the rotational plate and the radial position of the phase shifter. In other words, the phase changes between

signals passing through different phase shifters are proportional to each radius of the phase shifter. This suggests that if all adjoining phase shifters have the same difference in radius, the signals delivered to all adjacent columns of the radiators also have the same difference in phase change. However, in an actual feed network in the PAA, the phase shifter cannot be located at the centre of the rotational plate because the socket of the motor shaft occupies that space. This restriction causes a problem in that the phase changes between all the adjacent columns of the radiator are not the same. To solve this problem, three types of phase shifters are employed in the actual feed network in the PAA. The solution is described in detail with reference to Figs. 22–24 in the following. Fig. 22 shows a cross-sectional view of the phase shifter on a rotational plate fabricated on two fixed plates #3 and #4 in a concentric configuration. Fig. 22(a) shows the first type of phase shifter, where ’R’ denotes the radial position of the phase denotes the rotation angle of the rotational plate. shifter and The second type of phase shifter, which is not shown in the figure, is symmetrical to the one shown in Fig. 22(a). The first and second types of phase shifters cause phase changes with opposite signs to each other when the rotational plate rotates in the same direction. Fig. 22(b) shows the third type of phase

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Fig. 25. Layer structure of the feed network. Fig. 23. Allocation of the phase shifters on the rotational plate.

PAA is designed such that all signal paths are in-phase when the rotational plate is in the centre position, the signal phase on each radiator varies in the ratio according to each radiator position. From this principle, the condition required for the PAA can be satisfied. The ratio of the differential phase change between adjacent paths to the rotation angle of the rotational plate is (1) , and the difference in radii between the where is set as shown in Fig. 23. For our application phase shifters and at 76.5 to vehicular radar, given in Section II-A, the GHz are chosen. From (1) and is obtained. The beam tilting angle around the ratio , is calculated from bore site direction in PAA operation, Fig. 24. Connections between the phase shifters.

(2) shifter, which has no phase change when the rotational plate is rotating. Fig. 23 is an allocation diagram of the phase shifters on the rotational plate, in which the dividers and the radiators are shown together. As shown in the figure, there are hairpin and crankshaped bold lines that indicate the three types of phase shifter corresponding to the phase shifters illustrated in Fig. 22. In Fig. 23, the thin dashed lines indicate Divider 1 and the thin solid lines indicate Divider 2, which are shown in Fig. 21. The detailed arrangement of the phase shifters is shown in the inset in Fig. 23 (which corresponds to half the area covered by the desixteen symmetrically-arranged phase shifters), where notes the difference in radius between adjacent phase shifters. Fig. 24 shows the connection diagram for the phase shifters. The equations printed in the blocks in the figure represent the phase changes obtained when the rotational plate is turned . The dashed and solid lines indicate Divider clockwise by 1 and Divider 2 respectively, the same as in Fig. 23. In Fig. 24, the total phase changes from the input port to each radiator are calculated by summing the equations in the boxes in each path. Thus, the phase change differences between adjacent signal from Fig. 24, and this rule can be paths are found to be applied to all adjacent paths. Thereby, if the feed network of a

where this paper,

and

is the distance between radiators. In and are selected, so that is obtained from (1) and (2). In an actual system, it is desirable that the rotational plate rotates in one direction in order to achieve low battery consumption and a high scanning speed, which can be realized by using a uni-directional motor mounted behind fixed plate #5. In such a structure, the beam scanning is also uni-directional and intermittent rather than the bidirectional beam scanning realized by a bidirectional motor. C. Layer and Port Structure in Feed Network Fig. 25 shows the layer structure for Divider 1, the Phase Shifter and Divider 2, depicted in Fig. 21, where the figure shows a cross sectional view of a typical layer structure focusing on the port configuration. As shown in Fig. 25, the feed network has three types of port structure A’’, B’’ and C. Ports A’’ and B’’ are derived from ports A and B in Fig. 7, where the sizes are adjusted for packing a number of phase shifters into a limited space on the rotational plate. Figs. 26 and 27 show detailed diagrams illustrating the sizes of the ports and the impedance matching characteristics corresponding to the type of port structure, A’’,

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Fig. 27. Matching characteristics of the ports in Fig. 26.

vehicular radar applications, whereas type A’’ has a narrower bandwidth, which must be adjusted carefully to the centre frequency of the actual radar system. To accomplish a wider bandwidth for type A’’ will be the subject of future work. D. Other Components of the Feed Network For the realization of a 16-column PAA, some other components such as Branches, Bends and Dividers must be prepared. Fig. 28 shows diagrams in a perspective view illustrating the structure of a T-Branch, a Bend, a Two-Port Divider, and a Four-Port Divider, respectively. The figures show additional shapes for matching with the sizes in millimetres. On the top right hand sides of Fig. 28(a)–(d), plan views are depicted, and cross sectional views through the line T-T in the plan views are shown below these. Fig. 29 shows the matching characteristics of the components when all the metal parts in the simulation model are set to be PECs. As shown in Fig. 29, the Four-Port Divider has a rather matching condition that must narrow bandwidth for a be adjusted carefully to the centre frequency in an actual radar system. E. Layout of Waveguides

Fig. 26. Port structures of the feed network. (a) Sizes of Port A’’ in Fig. 25. (b) Sizes of Port B’’ in Fig. 25. (c) Sizes of Port C in Fig. 25.

B’’ or C, shown in Fig. 25, where all metal parts in the simulation model are set to be PECs. In Fig. 26, the coordinate system agrees with that in Fig. 25. The plan view is shown at the top of the figure, while the cross sectional view at the line T-T in the plan view is shown at the bottom. The units used in Fig. 26 are millimetres. As shown in Fig. 26, there are some additional shapes and changes from Fig. 8 for each port to give matching improvement under the restriction of limited packing space. It is found from Fig. 27 that types B’’ and C have a wide bandmatching condition, which is sufficient for width for a

Fig. 30 shows the waveguide layout on plates #3, #4 and the rotational plate. In Fig. 30(a)–(d) (corresponds to the layouts on plate #4 facing plate #5, on the rotational plate facing plate #4, on the rotational plate facing plate #3, and on plate #3 facing plate #2, respectively, where each figure corresponds to the position indicated on the right hand side of Fig. 25. As shown in Fig. 30, a number of WIRWGs are arranged in a confined space, where the waffle-iron rods between ridges are shared not only by WIRWGs with a parallel configuration but also WIRWGs located on the same radius. Fig. 30(a) shows the arrangement of each component indicated with dashed lines in Figs. 23 and 24. At the top of the figure, there is a T-Branch that divides the input signal from WR10 into two signals, and Bends that change the directions of the waveguides. In the lower part of the figure, there are the Four-Port Dividers with Bends that divide the four signals into sixteen. At the end of each waveguide, a type B’’ port is

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Fig. 28. Structures of other components. (a) T-Branch. (b) Bend. (c) Two-Port Divider. (d) Four-Port Divider.

Fig. 29. Matching characteristics of the feed network components.

mounted, connecting the layer shown in Fig. 30(a) to the phase shifter layer shown in Fig. 30(b). Fig. 30(b) and (c) show the arrangements of the phase shifters indicated with bold lines in Fig. 23. As shown in Fig. 22, each phase shifter is configured with a pair of waveguides on each

side of the rotational plate, and arranged in the shape of concentric arcs. In Fig. 30(b) and (c), Ports A’’ and B’’ shown in Fig. 25 that correspond to Ports A, A’, B, B’ in Fig. 7(b) are also mounted. Fig. 30(d) shows the arrangement of each component indicated by the solid lines shown in Figs. 23 and 24. At the top of the figure, there are the Two-Port Dividers that divide two signals into four and the Bends that change the direction of the waveguides. The bottom of the figure shows the waveguides that conduct the sixteen signals to each radiator. A type B’’ port is mounted at the end of each waveguide, connecting the layer shown in Fig. 30(d) to the phase shifter layer shown in Fig. 30(c), and type C ports are mounted, connecting the layer shown in Fig. 30(d) to the radiator layer. As shown in Fig. 30(b) and (c), the phase shifters are designed using cylindrical coordinates in order to facilitate the concentric arrangement. Therefore, the port shapes, designed previously in Section III-C using Cartesian coordinates, were adjusted to accommodate the cylindrical coordinate axes. This adjustment

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Employing cylindrical coordinates for the phase shifters also affects the shape of the waffle-iron rods at the sides of the ridges. In particular, when this modification is applied to a number of parallel ridges, the shape of all the rods must agree with the radial axes, making the arc length of the rods located in the outer region too long for a proper transmission factor. In order to prevent such an undesirable result, the shapes of the rods were adjusted appropriately based on the sizes described in Fig. 1. Although those adjustments and approximations mentioned above can cause perturbations of the centre frequency, re-optimization from the values determined in Sections III-C and III-D was not undertaken because some robustness in the design of the proposed PAA can be expected due to its large bandwidth compared to the system requirements for vehicular radar applications. F. Measurement

Fig. 30. Waveguide layouts on the plates. (a) Waveguide layout on plate #4 facing plate #5. (b) Waveguide layout on the rotational plate facing plate #4. (c) Waveguide layout on the rotational plate facing plate #3. (d) Waveguide layout on plate #3 facing plate #2.

also restricts the configuration of port B’’ on the layers in Fig. 30(a) and (d). As shown in Fig. 30(a) and (d), the configurations are modified under this restriction.

Fig. 31 shows a photograph of a 16-column PAA including a radome, while the photograph on the right hand side shows the components of each plate unmounted. The size of the PAA without the base plate and the input port using a WR10 rectangular waveguide is 62 mm 62 mm 25 mm. Each plate has the same shape as that shown in Fig. 21. The metal plates #2, #3 and #4 are formed in aluminium by a milling process, whereas the metal plates #1 and #5 are formed by an etching process in 0.5 mm thick stainless-steel of. The reason for using stainless-steel for plates #1 and #5 is that it is difficult to maintain a flat surface with aluminium. However, the metal conduction losses of stainless-steel are larger than those of aluminium, which causes a transmission loss in the waveguide and the radiation efficiency of the slot antenna to deteriorate. These failings might be eliminated by coating the stainless-steel with a high conductivity metal, such as gold or silver; this, however, is left for a future study. The radome is fabricated from polypropylene using a milling process. Fig. 32 shows the measured phase changes of the feed network. The plots are shown to coincide with the origin when the rotational plate is located at the centre position. Some irregular behaviour in the lines is observed. The reason for this phenomenon is that the PAA was detached from and re-attached to the measurement instrument each time the angle of the rotational plate was changed. In this measurement, sufficient field intensity was detected on slots A1 to A4 and A13 to A16 in Figs. 23 and 24, whereas the field intensity detected for slots A5 to A12 was insufficient to evaluate the phase change operation. A possible reason for this phenomenon is that there may be some mismatch in the small radius region near the centre in the feed network since re-optimization was not undertaken as mentioned in Section III-E. As the structure of the phase shifter is symmetrical, symmetrical characteristics can be anticipated, and thus the phase changes on slots A1 to A4 were measured and are plotted in Fig. 32. The figure clearly shows that the phase changes are proportional to the angle of the rotational plate, despite the compli-

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Fig. 31. Photograph of the 16-Column PAA.

Fig. 32. Measured phase changes of A1-A4 Radiators at 76.5 GHz.

cated signal paths. Furthermore, the slope of each line changes in accordance with the position of the phase shifter for each radiator. It is confirmed from the measurement that the fundamental requirement for a PAA, in that the phase change between adjacent radiators should be the same, is satisfied. defined in (1) is found to be from The ratio Fig. 32, which is in good agreement with the designed value of . This result suggests that a beam tilt angle of can be anticipated using the proposed PAA system. Fig. 33 shows the characteristics of the PAA calculated from the radiation pattern shown in Fig. 19(b) and the phase changes for radiators in Fig. 32. In the calculation, a value of A5 to A12 and A13 to A16 was used under the assumption that all of the radiating elements work properly. In Fig. 33, the radiation patterns are plotted when the rotato 10 . The figure tional plate rotates over angles from shows that the peak of the main lobe traces the radiation pattern of the radiator shown in Fig. 19(b) while maintaining a to 18 . The gain of more than 32 dBi, and changes from infigure also shows that the grating lobes emerging at crease as the main lobe is tilted. This phenomenon can be understood from the fact that an element spacing of more than half a wavelength generates grating lobes in this direction, and the element pattern created by the DR, as mentioned in Section II-D, does not provide a sufficiently narrow beamwidth to suppress

Fig. 33. Directivity of the PAA calculated from the measured element and array factors.

the grating lobes. In order to overcome this difficulty, it is necessary to optimize the shape of the DR and the radome, and this will be left for future work. IV. CONCLUSION A 76 GHz multi-layered phased array antenna using nonmetal contact metamaterial waveguides has been presented. The fundamental characteristics of the waveguide, phase shifter, radiator, and other components, such as a T-Branch, a Bend, a Two-Port Divider, and a Four-Port Divider, required for constructing a 16-column PAA are explained in detail. A description of the layout of the waveguides in the feed network is also given to illustrate the design details. The phase shifter demonstrates the low loss characteristics inherent in the nature of the WIRWG. Furthermore, good linearity and a wide range of phase changes are confirmed. The radiator exhibits a high gain characteristic while suppressing the grating lobes. The feed network of the 16-column PAA demonstrates the possibility of achieving the phase change characteristics required for the PAA. The developed PAA has a beam-tilt angle

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estimated at while maintaining a gain greater than 32 dBi, which is sufficient for vehicular radar applications. It should be emphasized that a 16-column PAA that includes a radome was successfully constructed with a size of 62 mm 62 mm 25 mm, demonstrating that a small sized PAA with many columns of radiators can be realized using WIRWG technology. The measured results show the excellent features of the proposed structure in the high frequency band at 76 GHz. The non-metal contact structure of the waveguide ensures high resistance to heat cycling over a wide temperature range and longterm operation under the severe vibrations that are encountered in vehicle applications. Furthermore, the simple manufacturing process is conducive to mass production for consumer use. As mentioned in [6], [7], all of the field energy exists only in air. Therefore, it is possible to form the WIRWG using metal plated resin. This means that much lighter microwave and millimetre wave systems can be constructed using the WIRWG than those made with conventional hollow waveguides. The WIRWG is a newly proposed waveguide. Hence there may be various derivative structures to be considered. It is expected that considerable research will soon advance the WIRWG technology, which will promote the emergence of new applications. REFERENCES [1] H. Kirino, K. Ogawa, and T. Ohno, “A variable phase shifter using a movable waffle iron metal and its applications to phased array antennas,” in Proc. IEICE ISAP Intl. Symp., Aug. 2007, vol. 4B3-2. [2] H. Kirino, K. Ogawa, and T. Ohno, “A variable phase shifter using a movable waffle iron metal plate and its applications to phased array antennas,” IEICE Trans. Commun., vol. E91-B, no. 6, Jun. 2008. [3] H. Kirino, K. Ogawa, and T. Ohno, “A variable phase shifter using movable waffle iron metal plate for applications to vehicle millimeterwave radar,” Panasonic Tech. J., vol. 54, no. 2, Jul. 2008. [4] H. Kirino and K. Ogawa, “A ridge waveguide phase shifter using waffle-iron structure for a 76 GHz slot array,” in Proc. IEICE General Conf., Mar. 2009, vol. B-1-94. [5] H. Kirino and K. Ogawa, “A 76 GHz dielectric loaded slot array antenna fed by a ridge waveguide using waffle-iron structure,” in Proc. IEICE General Conf., Mar. 2009, vol. B-1-171. [6] P.-S. Kildal, E. Alfonso, A. Valero, and E. Rajo, “Local metamaterialbased waveguides in gaps between parallel metal plates,” IEEE Trans. Antennas Propag. Lett., vol. 8, pp. 84–87, 9, Sep. 2009. [7] P.-S. Kildal, E. Rajo, E. Alfonso, A. Valero, and A. U. Zaman, “Wideband, lowloss, low-cost, quasi-TEM metamaterial-based local waveguides in air gaps between parallel metal plates,” in ICEAA2009, Torino, Italy, Sep. 2009. [8] E. Alfonso, M. Baquero, A. Valero-Nogueira, J. I. Herranz, and P.-S. Kildal, “Power divider in ridge gap waveguide technology,” in EuCAP2010, Barcelona, Spain, Apr. 2010, vol. C32P1-2. [9] M. Bosiljevac, Z. Sipus, and P.-S. Kildal, “Efficient spectral domain Green’s function analysis of novel metamaterial bandgap guiding structures,” in EuCAP2010, Barcelona, Spain, Apr. 2010, vol. C32P1-3. [10] A. Polemi, S. Maci, and P.-S. Kildal, “Approximated closed form characteristic impedance for the bed of nails-based gap waveguide,” in EuCAP2010, Barcelona, Spain, Apr. 2010, vol. C32P1-4. [11] E. Pucci, A. U. Zaman, E. Rajo-Iglesias, P.-S. Kildal, and A. Kishk, “Losses in ridge gap waveguide compared with rectangular waveguides and microstrip transmission lines,” in EuCAP2010, Barcelona, Spain, Apr. 2010, vol. C32P1-5. [12] A. Kishk and P.-S. Kildal, “Quasi-TEM H-plane horns with wideband open hard sidewalls,” in EuCAP2010, Barcelona, Spain, Apr. 2010, vol. C32P2-1.

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[13] H. Kirino and K. Ogawa, “A 76 GHz phased array antenna using a waffle-iron ridge waveguide,” in EuCAP2010, C32P2-2, Barcelona, Spain, Apr. 2010. [14] A. U. Zaman, P.-S. Kildal, M. Ferndahl, and A. Kishk, “Validation of ridge gap waveguide performance using in-house TRL calibration kit,” in EuCAP2010, Barcelona, Spain, Apr. 2010, vol. C32P2-3. [15] E. Rajo-Iglesias and P.-S. Kildal, “Groove gap waveguide: A rectangular waveguide between contactless metal plates enabled by parallel-plate cut-off,” in EuCAP2010, Barcelona, Spain, Apr. 2010, vol. C32P2-4. [16] J P. Patent applied 2008, PA2008-277969 (PCT/JP09/005087). [17] MW-Studio [Online]. Available: http://www.cst.com/ [18] Femtet [Online]. Available: http://www.muratasoftware.com/ [19] K. Sakakibara, J. Hirokawa, M. Ando, and N. Goto, “A linearly-polarized slotted waveguide array using reflection-cancelling slot pairs,” IEICE Trans. Commun., vol. E77-B, no. 4, Apr. 1994. [20] T. Tsugawa and Y. Sugio, “Experimental study on high efficiency dielectric loaded antenna,” IEICE Trans., vol. E73, no. 1, Jan. 1990. [21] S. Kobayashi, R. Mittra, and R. Lampe, “Dielectric tapered rod antennas for millimeter-wave applications,” IEEE Trans. on Antennas and Propag., vol. 30, no. 1, Jan. 1982. [22] T. Ohno and K. Ogawa, “A sector array using dielectric loaded antennas for indoor high-speed wireless LAN applications at 60 GHz,” IEICE Trans. Commun., vol. J88-B, no. 9, Sep. 2005.

Hideki Kirino (M’10) was born in Ehime, Japan, on April 25, 1961. He received the B.S. degrees in electronic engineering from the University of Electro-Communications, Tokyo, Japan, in 1985. In 1985, he joined Panasonic Shikoku Electronic Co., Ltd., Ehime, Japan, where he has been engaged in research and development on microwave and millimetre wave devices. From 1988 to 1998, he was the Research Student in the University of Electro-Communications in order to research the waveguides. Mr. Kirino received the Paper Award from the International Symposium on Antenna and Propagation 2007, and also received the Best Paper Award from the Institute of Electronics, Information and Communication Engineers (IEICE) Transactions of Japan, in Sep., 2009, both of which are based on accomplishments and contributions to the phase shifter and the phased array antenna technologies.

Koichi Ogawa (M’89–SM’06) was born in Kyoto on May 28, 1955. He received the B.S. and M.S. degrees in electrical engineering from Shizuoka University, Shizuoka, Japan, in 1979 and 1981, respectively. He received the Ph.D. degree in electrical engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 2000. He joined Matsushita Electric Industrial Co., Ltd., Osaka in 1981. Dr. Ogawa is currently a Professor of the Toyama University, Toyama, Japan. rom 2003 Dr. Ogawa has been engaged as a Guest Professor at the Center for Frontier Medical Engineering, Chiba University, Chiba, Japan. In 2005 he was also a Visiting Professor with the Antennas and Propagation Division, Department of Communication Technology, Aalborg University, Denmark.His research interests include compact antennas, diversity, adaptive, and MIMO antennas for mobile communication systems, electromagnetic interaction between antennas and the human body. His research also includes millimeter-wave circuitry and other related areas of radio propagation. He received the OHM Technology Award from the Promotion Foundation for Electrical Science and Engineering in 1990, based on accomplishments and contributions to millimeter-wave technologies. He also received the TELECOM System Technology Award from the Telecommunications Advancement Foundation (TAF) in 2001, based on accomplishments and contributions to portable handset antenna technologies. He also received the Best Paper Award from the Institute of Electronics, Information and Communication Engineers (IEICE) Transactions of Japan, in Sep., 2009. He is listed in Who’s Who in the World.

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Beam Switching Reflectarray Monolithically Integrated With RF MEMS Switches Omer Bayraktar, Ozlem Aydin Civi, Senior Member, IEEE, and Tayfun Akin, Member, IEEE

Abstract—A reflectarray antenna monolithically integrated with 90 RF MEMS switches has been designed and fabricated to achieve switching of the main beam. Aperture coupled microstrip patch antenna (ACMPA) elements are used to form a 10 10 element reconfigurable reflectarray antenna operating at 26.5 GHz. The change in the progressive phase shift between the elements is obtained by adjusting the length of the open ended transmission lines in the elements with the RF MEMS switches. The reconfigurable reflectarray is monolithically fabricated with the RF MEMS switches in an area of 42.46 cm using an in-house surface micromachining and wafer bonding process. The measurement results show that the main beam can be switched between broadside and 40 in the H-plane at 26.5 GHz. Index Terms—Reflectarray antennas, reconfigurable antennas, micro-electro-mechanical systems (MEMS) switches, microstrip antennas.

I. INTRODUCTION

R

EFLECTARRAYS are mostly planar printed surfaces that direct the incident electromagnetic field radiated from a feed horn antenna to a desired direction. Microstrip reflectarrays have many advantages compared to parabolic reflectors and electronically scanned phased array antennas. Microstrip reflectarrays have lower weight and smaller size compared to parabolic reflector antennas; furthermore, they allow electronic beam scanning. Reflectarrays do not contain a complex feed system as in phased array antennas; they employ feeding through free space which eliminates the losses of a microstrip feed network that limits the performance of high-gain millimeter wave arrays [1]. In reflectarrays, the phase of the reflected field from each element is adjusted so that the main beam can be directed to a desired direction. In the literature, there are several configurations proposed to control the reflection phase [2], such as patch antennas with variable-length stubs [3], variable-length Manuscript received March 19, 2010; revised May 18, 2011; accepted August 08, 2011. Date of publication October 20, 2011; date of current version February 03, 2012. This work was supported in part by the Scientific and Technological Research Council of Turkey (TUBITAK-EEEAG-104E041), by the Turkish State Planning Organization (DPT), and by the AMICOM (Advanced MEMS For RF and Millimeter Wave Communications) Network of Excellence under the 6th Framework Program of the European Union. O. Bayraktar and O. Aydin Civi are with the Department of Electrical and Electronics Engineering, Middle East Technical University, Ankara 06800, Turkey (e-mail: [email protected]; [email protected]). T. Akin is with the Department of Electrical and Electronics Engineering, Middle East Technical University, Ankara 06800, Turkey, and also with the METU-MEMS Center, Ankara 06800, Turkey (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173099

cross dipoles [4], and patch antennas with variable size [5]. Most of the reflectarrays available in the literature have fixed beams. Recently, there is a growing interest to design and implement beam steering reflectarrays. The electronically beam scanning reflectarrays are obtained by using reconfigurable components and materials to control the reflection phase difference between the antenna elements, such as tunable dielectrics [6], [7], varactor diodes [8]–[12], PIN diodes [13], [14], and various micro-electro-mechanical systems (MEMS) structures (such as micro-motors), or RF MEMS switches [15], [16], [18]–[26]. In [6] and [7], the dielectric constant of the nematic liquid crystal under each patch antenna in the reflectarray is changed by applying a DC voltage to steer the beam. Although there is no need for a complex biasing network for such a reflectarray, the response time of a liquid crystal is very slow, limiting its applications. The phase of the reflected field can dynamically be adjusted using semiconductor varactor diodes that are placed in various configurations, such as to control the slot susceptance of patches [8], to control the surface impedance [9], to load a transmission line stub in aperture coupled patches [10], to obtain capacitive loading of hollow patches [11], and to adjust the resonant frequency of microstrip patches [12]. The phase of the reflected field can also be adjusted using PIN diodes to control the length of a short circuited stub [13] or using both varactor and PIN diodes to change the current distribution on a cross shaped microstrip loop [14]. Recent reconfigurable reflectarrays [15], [16], [18]–[26], and lens arrays [27] prefer RF MEMS components (such as switches, varactors, and phase shifters), since electrostatically actuated RF MEMS components provide almost zero DC power consumption, low insertion loss, high isolation, and linear characteristics compared to solid state switches. Although RF MEMS switches and other components have drawbacks in terms of reliability and low switching speed, as presented in a detailed discussion on performance comparison of different switch technologies in [28], they provide several advantages in mm-wave reconfigurable array applications. The most important advantage is that RF MEMS switches and other components can be easily manufactured monolithically with antennas on the same substrate. The monolithic integration is very important in the realization of reconfigurable antenna and array applications especially at mm-wave frequencies, because hybrid integration would be very complicated due to the size limitations at the mm-wave frequencies. Furthermore, losses increase due to the use of several connecting wire bonds in the case of hybrid integration. Most of the MEMS reconfigurable reflectarray studies in the literature are limited by design and implementation of unit cell

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BAYRAKTAR et al.: BEAM SWITCHING REFLECTARRAY MONOLITHICALLY INTEGRATED WITH RF MEMS SWITCHES

structures. Some examples of unit cells are a stub loaded patch antenna rotated by micro-machined motors [16], series of half dipoles connected to the periphery of a circular metal layer by means of diodes [17] or MEMS switches [18], [19], split ring elements with MEMS switches to obtain a phase shift by rotation of elements for circular polarization applications, [20], variable-length dipoles using electrically [21] or optically [22] actuated MEMS switches, ring elements loaded with MEMS capacitors [23], and patches loaded by MEMS varactors [24]. In [24], it has been demonstrated that, by using MEMS varactors instead of semiconductor counterparts to load patches in [12], losses can be reduced significantly, and nonlinear effects due to semiconductor diodes can be eliminated. There are a few monolithically fabricated MEMS reflectarray studies in which whole reflectarray structures have been designed but their prototypes have been fabricated either without MEMS switches, or with frozen MEMS switches [25], [26]. To the authors’ knowledge, there are no monolithically integrated MEMS reconfigurable reflectarrays presented in the literature. Thus, this study presents the first monolithically fabricated reconfigurable reflectarray employing a large number of functional RF MEMS switches distributed over a large wafer area. The reflectarray in this study is composed of aperture coupled microstrip patch antenna (ACMPA) elements, and reconfigurability in the main beam direction is obtained with series RF MEMS switches placed on open-ended transmission lines of the ACMPA elements. Section II presents details of the reflectarray design. Section III explains design of the series RF MEMS switch structure in the reflectarray and examines effects of bias lines on the performance. Section IV describes fabrication steps of a 10 10 reconfigurable reflectarray antenna monolithically produced with the RF MEMS switches and discusses the fabrication challenges and imperfections. Finally, Section V gives simulation and measurement results. II. RECONFIGURABLE REFLECTARRAY ANTENNA STRUCTURE AND DESIGN PROCEDURE The ACMPA elements shown in Fig. 1 are linearly spaced with half a free space wavelength, , in both directions to form a 10 10 reflectarray at 26.5 GHz. Reconfigurability is achieved using series RF MEMS switches monolithically integrated with the transmission lines of the ACMPA elements. Then, phase center of the pyramidal feed horn antenna having aperture dimensions 2.212 cm 2.212 cm is positioned at cm, cm, and cm with respect to the center of the reflectarray. Using the procedure described in [1], required reflection phase values from the elements of the reflectarray are calculated to direct the main beam toward broadside and 40 in the H-plane. For the th element, the reflection phase values to direct the main beam to the broadside and 40 are denoted as and , respectively. Note that, for the elements in the first column, because the first column is taken as a reference in the calculation of the progressive phase shifts. Two mostly used approaches to achieve required phase of reflected field from reflectarray cells in linearly polarized applications are (i) tuning of resonance of elements either by manipulating dimensions or by reactive loading of the elements [12], [23], [29] and (ii) using a separate phase shifter, as in the

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Fig. 1. (a) Backside and (b) cross-sectional views of the aperture coupled microstrip patch antenna element used in the reconfigurable reflectarray.

reflectarray presented in this paper [15], [25]. The reflectarray cells having a phase shift control mechanism based on resonance tuning possess very low losses in the operation band except around the resonance frequencies of structures [12]. However, the reflectarray cells with separate phase shifters have relatively higher losses due to the insertion loss of a phase shifter, which is generally larger than the insertion loss of a switch or a capacitor. On the other hand, the use of a separate phase shifter simplifies the design and analysis of a reflectarray, since the radiating structures of all cells are identical. The ACMPA reflectarray configuration has many advantages as well as drawbacks over the configuration where the phase shifter and antenna are on the same layer. One of the main advantages is that the microstrip transmission line and patch antenna are printed on different substrates separated by a common ground plane, and hence, a large space on the microstrip line side is obtained to place bias lines for MEMS switches and/or some active components if needed. Furthermore, the length of each transmission line in the reflectarray can be extended to obtain several multiples of 360 phase delay to eliminate the bandwidth limitation due to differential spatial phase delays [30] so that the bandwidth of the reflectarray is determined by the element bandwidth. Another advantage is the flexibility of choosing two separate substrates for the patch antenna and transmission line, for example, a high dielectric substrate for the

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phase shifters can be chosen to obtain large phase shifts, while the microstrip patch antenna can be printed on a low dielectric substrate in order to increase the element bandwidth, radiation efficiency, and steering range without scan blindness. One other advantage is that the spurious radiation due to the transmission line, RF MEMS switch, and bias lines is backwards and does not disturb the radiation pattern. Besides, the spurious radiation, and hence the power loss, can be eliminated by placing an additional ground plane at a specific distance from the microstrip lines at the back. The final advantage of the multilayer structure over the single layer structure is the simplicity of the design as far as the biasing scheme is concerned, as the effects of the bias lines and RF MEMS switches on the radiation pattern in the single layer structure should be taken into account in the design stage. On the other hand, the main drawback of multilayered reflectarray structures is the fabrication complexity and cost, which can be tolerated considering their advantages. The principle of operation of the ACMPA element in the reflectarray is as follows. A patch antenna printed on an antenna substrate receives linearly polarized electromagnetic wave. Then, the electromagnetic wave couples to the microstrip line printed on a feed substrate by means of an aperture on the ground plane between two substrates. Since the microstrip transmission line is open ended, the wave reflects back and couples to the patch antenna using the aperture on the ground plane. The distance that the wave propagates on the transmission line determines the phase of the reflected field. Hence, two sets of transmission line lengths are needed for each element in the reflectarray to switch the main beam between the broadside and 40 , respectively. Therefore, there is one RF MEMS switch per each element, which corresponds to 1-bit of control resulting in two beam states. The number of beam states can be increased by placing 3 or 5 bit MEMS phase shifters on the transmission lines. This will increase the complexity of the biasing scheme and the loss; however, the concept of having reconfigurability using MEMS will still remain valid. To determine the transmission line length of the element for a given reflection phase value, the graph that relates the reflection phase to the transmission line length, namely, the phase design curve must be obtained. To calculate the phase design curve, mutual couplings with the neighboring elements are taken into account by the infinite array assumption. The unit cell has been simulated as an infinite array using the periodic boundary conditions in the Ansoft High Frequency Structure Simulator (HFSS). The element spacing is half a free space wavelength ( mm) in both directions, thus the unit cell has a dimension of 5.66 mm 5.66 mm. A glass substrate of thickness mm, dielectric constant and is used for both the patch and microstrip line substrates. Initial dimensions of the reflectarray element are determined by considering the element as a single radiating antenna, i.e., by matching the input impedance of a patch to the characteristic impedance of the transmission line. Then, the reflectarray element dimensions are optimized to have a linear phase design curve at 26.5 GHz by exciting the unit cell with a y-polarized plane wave normally incident to the array surface and by calculating the phase and magnitude of the field reflected from the unit cell using HFSS for each value of L incremented in the direction shown in Fig. 1(a). The ACMPA dimensions are de-

Fig. 2. The phase and magnitude curves of the unit cell for different frequencies for the normal incidence, and the ideal phase curve @26.5 GHz.

termined as mm, mm, mm, mm, mm, and mm. Then, for comparison, the microstrip line of width mm on the glass substrate is simulated in HFSS to obtain the ideal phase characteristics of the microstrip line at 26.5 GHz. As seen in Fig. 2, a very good agreement between the phase design curve and the ideal phase characteristics of the transmission line is obtained at 26.5 GHz. The magnitude of the reflected wave changes between 1.42 dB and 3.64 dB at 26.5 GHz as a function of L. Hence, the average value of all losses in the unit cell is 2.53 dB where the conductor losses are 0.4 dB and the dielectric losses are 1.1 dB on average. The remaining 1.03 dB is the back radiation loss. The main loss mechanisms in the unit cell are the dielectric and back radiation losses which can be eliminated by using a lower dielectric constant substrate and by placing an additional ground plane at the back. Both this non-uniform magnitude response and the amplitude variation of the incident field affect the radiation pattern, especially the side lobes. As the frequency deviates from 26.5 GHz, the linearity of the phase design curve is lost, and the range of the magnitude variation increases as shown by the simulation results in Fig. 2. Since the phase curves for different frequencies are not parallel to each other, the operational bandwidth of this reflectarray is narrow. The bandwidth of the reflectarray is discussed in detail in Section V using the measurement results. The electromagnetic wave radiated from the feed antenna does not reach all elements of the reflectarray with the same angle of incidence, and the maximum angle of incidence occurs for the elements at the edges of the reflectarray. The maximum value of the incidence angle to an element on the designed reflectarray surface is 30 , which corresponds to the worst case illumination in both E and H-planes ( and , respectively). In Fig. 3, the phase and magnitude curves are plotted for the worst case incidence angle in both E and H-planes and compared with the ones for the normal incidence. It is observed that the magnitude and phase curves are not affected much with the change in the incidence angle. Hence, the phase curve for the normal incidence is a good approximation for calculating the transmission line lengths.

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Fig. 3. The magnitude and phase responses of a y-polarized plane wave for different angles of [email protected] GHz.

Although all the patches on the feed side have the same size, the lengths of the transmission lines are not identical. Once the transmission line lengths are determined from the phase design curve, series ohmic contact RF MEMS switches of length can be implemented between the transmission lines of length and to switch between two transmission line lengths for all the elements, which enable the main beam to switch between the broadside and 40 . For the th element, the transmission line lengths corresponding to and are denoted as and in Fig. 1(a), respectively. When we consider the phase shifter part in Fig. 1(a), the open ended transmission line of length is connected to the microstrip transmission line of length through the series capacitance introduced by the RF MEMS switch. When the RF MEMS switch is in the up state, the phase shifter has a resonance for some values of due to imperfect isolation of the RF MEMS switch. Hence, for those values of , it is impossible to obtain the required phase shift values by the transmission line of lengths . For this reason, the RF MEMS switches in the columns 4, 7, and 10 are kept in the down state, i.e., the overall lengths of the transmission lines are for those elements, to achieve the phase shift values , while the switches on the other columns are in the up state. The states of the RF MEMS switches are reversed to obtain the phase shift values . III. SERIES RF MEMS SWITCH AND BIAS LINES We considered both shunt and series switch configurations to change the length of the microstrip line [15]. The series switch is preferred due to both size considerations and the fact that the unit cell with the series switch results in better phase design curve characteristics. The series ohmic contact RF MEMS switch used in the reflectarray is the bridge with wings type structure between two transmission line segments as shown in Fig. 4. When the switch is actuated by an applied DC voltage between the actuation pad and the bridge, it connects two physically separated transmission lines pieces named as Tr. Line1 and Tr. Line2. The width of the interconnection region is reduced compared to the transmission line width to improve the isolation characteristics.

Fig. 4. (a) Top and (b) A-A cross-sectional views of the series RF MEMS switch.

Fig. 5. The simulation results of the series RF MEMS switch.

Fig. 5 shows the simulation results of the series RF MEMS switch. The insertion loss of the designed switch is less than 0.5 dB, and the isolation is better than 10 dB, which are acceptable results at the frequency of interest. The switch is fabricated using the process steps given in Section IV. The surface profile measurements on the fabricated switches show that the spacing between the transmission lines and the wings of the bridge is not 2 m as designed but 1.3 m, due to the residual stress of the metal bridge. To see the effect of reduced spacing on isolation, simulations have been performed for a 1.3 m gap height. When the bridge gap becomes a 1.3 m, the isolation is still better than 10 dB between 20–28 GHz frequency band as can be seen in Fig. 5. In order to see the effect of the bridge gap in the design, the radiation pattern simulations of the reflectarray for a 2 m and a 1.3 m gap heights are compared. It is observed that only the side lobe levels are affected by a few dB. When the main beam is directed to the broadside, the largest deviation of about 4 dB occurs in the side lobe around 40 . This is due to the fact that most of the switches in the broadside operation are

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Fig. 7. The effect of the Au bias lines and the RF MEMS switch on both the reflection phase and magnitude curves at 26.5 GHz.

have no significant effect on the phase design curve but the amplitude of the reflected wave decreases for some values of L. The phase design curve given in Fig. 2 is obtained by changing the length of the microstrip line. To calculate the reflection phase more realistically, the unit cell is simulated by including the series RF MEMS switch. Fig. 7 shows the reflection phase values of the reconfigurable unit cell calculated both for different values and for the up or down states of the switch [15], which are seen to be slightly deviating from the phase design curve. Hence, values determined from the phase design curve in Section II are altered for the fine tuning of the phases. Fig. 6. Mask layout of a 10

10 reflectarray with the RF MEMS switches.

in their up states, thus the change in the up state bridge height becomes significant. Fig. 6 shows the layout of the overall reflectarray prototype with bias lines. The bias lines used to actuate the switches have two parts: one is composed of a sputtered gold (Au) layer and the other is composed of a sputtered silicon-chromium (Si-Cr) layer. The actuation mechanism of the series switch can be modeled as a series RC circuit where the Si-Cr layer is modeled as a resistance, and the path between the bridge and actuation pad is modeled as a capacitance. In order to have a reasonable switching time, the time constant should be reduced. So, the entire bias line scheme can be composed of the sputtered gold having high conductivity. But this time, the mutual coupling between the Au bias line and microstrip line increases, and the switch performance is disturbed. In order to avoid these adverse effects, the Au bias lines are connected to the resistive bias lines composed of the sputtered Si-Cr at an average distance of 1500 m before the switch and the transmission line, and the conductivity of the Si-Cr layer is optimized to be 10,000 S/m. To see the effect of the Au bias lines, the phase design curve and the amplitude response are recalculated when there are both vertical and horizontal Au bias lines in the unit cell. Then these results are compared with the ones obtained without a bias line as shown in Fig. 7. As can be seen in Fig. 7, the Au bias lines

IV. FABRICATION OF RECONFIGURABLE REFLECTARRAY The monolithic reconfigurable reflectarray presented in this work is produced using the surface micromachining based process including the wafer bonding step developed at Middle East Technical University MEMS Center (METU-MEMS Center). Fig. 6 shows the layout of the reconfigurable reflectarray. The reflectarray has been fabricated using two 500 m thick glass substrates ( ). Fig. 8 shows a simplified process flow. Fig. 8(a) shows a cross-sectional view of the process which can be obtained after a number of process steps. The process starts by coating each wafer with a 100/8000 thick Ti/Au layer and patterning by wet etching to construct the aperture on the ground plane. Then, both wafers are bonded using gold-to-gold thermal compression bonding at 265 C for 1 hour in a vacuum to construct a common ground plane with the aperture. Next, one side of the bonded glasses is processed to have the microstrip patch antenna, whereas the other side is used to construct the transmission lines with the RF MEMS switches. The microstrip patch antenna is constructed by sputtering and patterning a 100/8000 thick Ti/Au layer. The process at the other side of the bonded glasses starts with a 2000 thick Si-Cr resistive layer deposition by sputtering and then patterning by wet etching. After that, the patch antennas are covered with a 0.8 m thick sputtered Ti layer to protect them while processing the other side of the wafer. The next step is the sputtering of a 100/6000 thick Ti/Au layer on the Si-Cr resistive layer; after wet etching, the

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Fig. 8. The standard process flow developed at METU-MEMS Center for the production of the reconfigurable reflectarray.

transmission lines are formed. Then, again a 100/2500 thick Ti/Au layer is sputtered on the whole wafer area covering the previously formed transmission lines. After wet etching of this layer, the actuation pads are formed, and the height of the transmission lines is increased with respect to the actuation pads, which helps to decrease the contact resistance of the series RF MEMS switch. Then, a 3000 thick Si N layer is coated as a DC isolation layer using the plasma enhanced chemical vapor deposition (PECVD) technique and patterned using the reactive ion etching (RIE) technique, resulting in the structure shown in Fig. 8(a). Fig. 8(b) shows the cross-section after the photo definable polyimide, PI 2737, is spin-coated to form a 2 m thick sacrificial layer and patterned to obtain hollows for the anchor regions, which is followed by sputtering a 1 m thick gold layer on the PI 2737. Fig. 8(c) shows the cross-section after the anchors are strengthened with a 2 m thick electroplated gold layer inside the regions defined by a mold photoresist. Then, the photoresist is removed, the structural layer is patterned, and the Ti layer used to protect the patch antennas is removed by wet etching. Fig. 8(d) shows the final cross-section after the sacrificial layer is removed by wet etching in an EKC-265 solution, and the wafer is rinsed in IPA and dried in a supercritical point dryer. Fig. 9 shows the photographs of the fabricated reflectarray, while Fig. 10 shows the optical and SEM photographs of the series RF MEMS switch monolithically integrated to the reflectarray. Although the individual process steps are easy, it is a challenge to obtain the whole reflectarray on a 4 wafer, as the reflectarray covers nearly the whole area of the 4 wafer and as the monolithic integration requires a high yield of RF MEMS switches distributed over the large wafer area. After a number of trials and process improvements, a high yield process is achieved to obtain a working reflectarray. The size of the reflectarray is 6.75 cm 6.29 cm. It is centered on the wafer. Individual RF MEMS switches are located on the

Fig. 9. (a) Transmission line and (b) patch antenna side of the reconfigurable reflectarray antenna monolithically produced with (c) the series RF MEMS switches.

Fig. 10. (a) Microscopy and (b) SEM views of the series RF MEMS switch monolithically produced with the reflectarray.

rest of the wafer area. The actuation voltage of these switches is measured as 35 V. Since the electroplating thickness is not uniform over the wafer (it increases toward the edge), it is expected that the actuation voltage of the switches in the reflectarray region is less than 35 V. Hence, 35 V is used as the actuation voltage in the measurements. V. SIMULATION AND MEASUREMENT RESULTS The simulations of the full reflectarray, i.e., the reflectarray surface and the feed horn, are carried out in HFSS to compare with the measurements. First of all, up and down states of the series RF MEMS switches in the reflectarray are modeled and

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Fig. 12. The measured radiation patterns in the E-plane when the switches are actuated to direct the main beam to 40 and the simulation result.

Fig. 13. The measurement setup of the reconfigurable reflectarray.

Fig. 11. The measured radiation patterns and simulation results in the H-plane. (a) When the switches are actuated to direct the main beam to the broadside. (b) When the switches are actuated to direct the main beam to 40 .

replaced by series capacitances and inductances. Since the reflectarray together with the feed horn antenna is a very large structure with respect to the wavelength, the problem is divided into two parts: the feed horn region and the reflectarray surface. Then, these two parts are related with an HFSS data-link. First, the near field of the feed horn antenna is calculated by simulations. Then, this near field is taken as the incident field that illuminates the reflecting surface by using the data-link tool. Figs. 11 and 12 show the full wave EM simulation plots of the far fields, in addition to measurement results, as explained later. To prepare the antenna for the measurements, a PCB card is attached to the fabricated reflectarray to apply the DC bias voltage to the switches, as shown in Fig. 9(a). Three wire bonds are taken from the reflectarray to the PCB card to achieve the DC connections to the pads GND, SIGNAL1, SIGNAL2 shown

in Fig. 6. The reflectarray and offset feed horn are assembled on the foam support structure such that the position of the phase center of the feed horn antenna with respect to the center of the reflectarray is as indicated in Section II. Fig. 13 shows a photograph of the complete antenna placed in an anechoic chamber. All the pattern measurements are taken with a 1 angular resolution. When the actuation voltage is applied between GND and SIGNAL2, the main beam of the reflectarray is directed to the broadside as shown in Fig. 11(a); the half power beam width is determined to be 10 . When the actuation voltage is applied between GND and SIGNAL1, the main beam of the reflectarray is switched to 40 in the H-plane as shown in Fig. 11(b) with the half power beam width of 13 . The measured radiation and cross polarization patterns in the E-plane for the 40 operation are presented in Fig. 12. The measured cross polarization level in the E-plane is found to be better than 20 dB. The measurement results given in Fig. 11 show that there are abrupt changes especially in the angular region approximately between 90 and 140 where the field is below 20 dB, due to non-ideal characteristics of the rotary joint of the measurement setup at this frequency range. It is observed from Figs. 11 and 12 that, there is a good agreement between the measurement and simulation results. The positions of the main beam, half power beam width,

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TABLE I LOSS ANALYSIS FOR 40 OPERATION AT 26.5 GHZ

side lobe levels, and back radiation levels in the simulation are nearly the same for both the broadside and 40 operations. The slight deviations in the positions of the side and back lobes and the levels of some side lobes are mainly caused by the differences between the simulation and the measurement setup. The actual interaction of the horn antenna and the reflectarray cannot be fully taken into account in the simulations due to the very large electrical size of the overall antenna. Moreover, the coaxial cable and the connector used to excite the horn antenna are not included in the simulations. The maximum back radiation and side lobe levels are around 12 dB, and 10 dB, respectively. This reflectarray is a proof-of-concept prototype, and the main goal is to demonstrate the beam switching by an RF MEMS switch control. Thus, in the design, no special efforts have been spent to reduce the side lobe levels and back radiation. To reduce the back radiation, a ground plane can be placed at an appropriate distance from the back side of the reflectarray. When we consider both the broadside and 40 operations, the maximum value of the side lobe levels is around 7 dB and the maximum deviation in both the half power beam width (HPBW) values and the main beam directions is 1 within the 26–27 GHz frequency band. These values are acceptable within 26–27 GHz, and hence, the bandwidth is 3.77%. A loss analysis of the reflectarray is performed when the main beam is directed to 40 and tabulated in Table I. The gain of the reflectarray is measured as 11.42 dBi by using a standard gain horn antenna at 26.5 GHz, whereas the gain is calculated as 11.93 dBi using the simulation results. The directivity of a 10 10 uniform array radiating to 40 is calculated numerically as 22.47 dB. Using the measured gain, total loss of the reflectarray is estimated as 11.05 dB which corresponds to an antenna efficiency of 7.85%. Illumination, cross polarization, and element losses are calculated numerically at 26.5 GHz. The main loss source is the spillover loss, which is calculated as 6.51 dB. The taper loss is 0.0064 dB, which is very small, because the illuminating horn antenna has high HPBW values of 34 in the H-plane and 26 in the E-plane, resulting in an almost uniform illumination and a high spillover loss. By optimizing the focal distance/diameter (F/D) ratio, the spillover loss can be reduced. The cross polarization loss is calculated as 0.15 dB. The maximum element reflectivity loss is 3.64 dB at the normal plane wave incidence, but it can be reduced by placing an additional ground plane at

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the back side of the array. The 0.3 dB insertion loss of the series RF MEMS switch also contributes to the total loss of the reflectarray. Calculated losses in Table I add up to 10.61 dB. The difference between the calculated and measured loss (0.44 dB) might be caused by several factors, including small errors in the placement of the phase center of the feed horn at the focal point of the reflectarray. Furthermore, the ohmic losses are ignored in the reflectarray radiation pattern simulations, i.e., all the metals in the full reflectarray structure are assumed to be perfectly conducting and second order interactions of reflecting surface with the horn are not taken into account in the simulations. The good agreement between the simulation and measurement results shows that almost all of the switches on the reflectarray are fully functional, i.e., the yield is very high. The yield is estimated as 88% based on the surface profile measurements of the reflectarray and RF measurements of the individual RF MEMS switches from the same wafer. VI. CONCLUSION Beam switching of a 26.5 GHz 10 10 reconfigurable reflectarray antenna is achieved using 90 RF MEMS switches in the ACMPA elements. The progressive phase shift between the elements is adjusted by the on and off state positions of the series RF MEMS switches inserted in the transmission line of the ACMPA elements. The full reflectarray is produced monolithically with the series RF MEMS switches. Measurement results demonstrate that the main beam of the reflectarray can be switched between the broadside and 40 by the help of the RF MEMS switches. According to the authors’ knowledge, this monolithically integrated MEMS reconfigurable reflectarray is the first functional prototype that employs a large number of RF MEMS switches distributed over a large wafer area, demonstrating the potential of the RF MEMS technology for large scale antennas. ACKNOWLEDGMENT The authors would like to thank METU-MEMS Center staff of Middle East Technical University, Ankara, Turkey, for their support in the fabrication. The authors also thank Dr. Kagan Topalli and Dr. Mehmet Unlu for the development of the process and their supervision in the fabrication. REFERENCES [1] D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarrays,” IEEE Trans. Antennas Propag., vol. 45, no. 2, pp. 287–296, Feb. 1997. [2] J. Huang and J. Encinar, Reflectarray Antennas. Piscataway, NJ: Wiley-IEEE Press, 2007. [3] T. Metzler and D. Schaubert, “Scattering from a stub loaded microstrip antenna,” in IEEE Antennas and Propagation Society Int. Symp., AP-S. Dig., Jun. 1989, pp. 446–449. [4] D. M. Pozar and S. D. Targonski, “A microstrip reflectarray using crossed dipoles,” in IEEE Antennas and Propagation Society Int. Symp., AP-S. Dig., Jun. 1998, pp. 1008–1011. [5] J. A. Encinar, “Design of two-layer printed reflectarrays using patches of variable size,” IEEE Trans. Antennas Propag., vol. 49, no. 10, pp. 1403–14010, Oct. 2001. [6] A. Mössinger, R. Marin, S. Mueller, J. Freese, and R. Jakoby, “Electronically reconfigurable reflectarrays with nematic liquid crystals,” IEE Electron. Lett., vol. 42, pp. 899–900, Aug. 2006. [7] W. Hu, M. Y. Ismail, R. Cahill, J. A. Encinar, V. F. Fusco, H. S. Gamble, R. Dickie, D. Linton, N. Grant, and S. P. Rea, “Electronically reconfigurable monopulse reflectarray antenna with liquid crystal substrate,” in Proc. 2nd EuCAP, Edinburgh, U.K., Nov. 11–16, 2007, pp. 1–6.

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[8] F. Venneri, L. Boccia, G. Angiulli, G. Amendola, and G. Di Massa, “Analysis and design of passive and active microstrip reflectarrays,” Int. J. RF Microw. Comput.-Aid. Eng., vol. 13, pp. 370–377, 2003. [9] D. F. Sievenpiper, J. H. Schaffner, H. J. Song, R. Y. Loo, and G. Tangonan, “Two-dimensional beam steering using an electrically tunable impedance surface,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pt. 1, pp. 2713–272, Oct. 2003. [10] M. Riel and J.-J. Laurin, “Design of an electronically beam scanning reflectarray using aperture-coupled elements,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1260–1266, May 2007. [11] M. Hajian, B. Kuijpers, K. Buisman, A. Akhnoukh, M. Plek, L. C. N. de Vreede, J. Zijdeveld, and L. P. Ligthart, “Active scan-beam reflectarray antenna loaded with tunable capacitor,” in Proc. 3rd EuCAP, Berlin, Germany, Mar. 23–27, 2009, pp. 1158–1161. [12] S. V. Hum, M. Okoniewski, and R. J. Davies, “Modeling and design of electronically tunable reflectarrays,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2200–2210, Aug. 2007. [13] H. Kamoda, T. Iwasaki, J. Tsumochi, and T. Kuki, “60-GHz electrically reconfigurable reflectarray using p-i-n diode,” in IEEE Microwave Symp. Dig., Jun. 2009, pp. 1177–1180. [14] J. Perruisseau-Carrier, “Dual-polarized and polarization-flexible reflective cells with dynamic phase control,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1494–1502, May 2010. [15] O. Bayraktar, K. Topalli, M. Unlu, O. A. Civi, S. Demir, and T. Akin, “Beam switching reflectarray using RF MEMS technology,” in Proc. 2nd EuCAP, Edinburgh, U.K., Nov. 11–16, 2007, pp. 1–6. [16] J. Huang, “Analysis of a Microstrip Reflectarray Antenna for Microspacecraft Applications,” NASA TDA progress report, Feb. 1995, pp. 153–173. [17] R. J. Richards, E. W. Dittrich, O. B. Kesler, and J. M. Grimm, “Microstrip Phase Shifting Reflect Array Antenna,” U.S. Patent 6,020,853, Feb. 1, 2000. [18] R. J. Richards, “Integrated Microelectromechanical Phase Shifting Reflect Array Antenna,” U.S. Patent 6,195,047, Feb. 27, 2001. [19] H. Legay, B. Pinte, M. Charrier, A. Ziaei, E. Girard, and R. Gillard, “A steerable reflectarray antenna with MEMS controls,” in Proc. IEEE Int. Symp. Phased Array Systems and Tech., Oct. 14–17, 2003, pp. 494–499. [20] C. Guclu, J. Perruisseau-Carrier, and O. A. Civi, “Dual frequency reflectarray cell using split-ring elements with RF MEMS switches,” in Proc. IEEE Int. Symp. Antennas and Propagation and USNC/URSI National Radio Science Meeting, Toronto, ON, Canada, Jul. 11–17, 2010, pp. 1–4. [21] R. Gilbert, “Dipole Tunable Reconfigurable Reflector Array,” U.S. Patent 6,426,727, Jul. 30, 2002. [22] H. P. Hsu and T. Y. Hsu, “Optically Controlled RF MEMS Switch Array for Reconfigurable Broadband Reflective Antennas,” U.S. Patent 6,417,807, Jul. 9, 2002. [23] J. Perruisseau-Carrier and A. K. Skrivervik, “Monolithic MEMS-based reflectarray cell digitally reconfigurable over a 360 phase range,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 138–141, 2008. [24] S. V. Hum, G. McFeetors, and M. Okoniewski, “Integrated MEMS reflectarray elements,” in Proc. 1st EuCAP, Nice, France, Nov. 6–10, 2006, pp. 1–6. [25] L. Marcaccioli, B. Mencagli, R. V. Gatti, T. Feger, T. Purtova, H. Schumacher, and R. Sorrentino, “Beam steering MEMS mm-wave reflectarrays,” in Proc. 7th Int. Symp. RF MEMS and RF Microsystems (MEMSWAVE 2006), Orvieto, Italy, Jun. 27–30, 2006. [26] B. Mencagli, R. V. Gatti, L. Marcaccioli, and R. Sorrentino, “Design of large mm-wave beam-scanning reflectarrays,” in Proc. 35th EuMC, Paris, France, Oct. 3–7, 2005. [27] C. C. Cheng, B. Lakshminarayanan, and A. Abbaspour-Tamijani, “A programmable lens array antenna with monolithically integrated MEMS switches,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 8, pp. 1874–1884, Aug. 2009. [28] G. M. Rebeiz, K. Entesari, I. C. Reines, S. J. Park, M. A. El-Tanani, A. Grichener, and A. R. Brown, “Tuning in to RF MEMS,” IEEE Microw. Mag., pp. 55–72, Oct. 2009. [29] H. Salti, E. Fourn, R. Gillard, and H. Legay, “Minimization of MEMS breakdowns effects on the radiation of a MEMS based reconfigurable reflectarray,” IEEE Trans. Antennas Propag., vol. 58, no. 7, pp. 2281–2287, Jul. 2010. [30] J. A. Encinar and J. A. Zornoza, “Broadband design of three-layer printed reflectarrays,” IEEE Trans. Antennas Propag., vol. 51, pp. 1662–1664, Jul. 2003.

Omer Bayraktar was born in Aydin, Turkey, in 1983. He received the B.Sc. and M.Sc. degrees in electrical and electronics engineering from the Middle East Technical University (METU), Ankara, Turkey, in 2005 and 2007, respectively. Since 2005, he has been working as a research assistant at METU in the Department of Electrical and Electronics Engineering. His major research interests include development, characterization, and integration of novel RF MEMS structures such as switches, phase shifters for RF front-ends at microwave and millimeter-wave, reconfigurable antennas, phased arrays, and reflectarrays.

Özlem Aydin Civi (S’90–M’97–SM’05) received the B.Sc., M.Sc., and Ph.D. degrees in electrical and electronics engineering in 1990, 1992, and 1996, respectively, at the Middle East Technical University (METU), Ankara, Turkey. She was a research assistant in METU from 1990 to 1996. In 1997–1998, she was a visiting scientist at the ElectroScience Laboratory, Ohio State University. Since 1998, she has been with the Department of Electrical and Electronics Engineering, METU, where she is currently a Professor. Her research interests include analytical, numerical and hybrid techniques in EMT problems, especially fast asymptotic/hybrid techniques for the analysis of large finite periodic structures, multi-function antenna design, reconfigurable antennas, phased arrays, reflectarrays and RF-MEMS applications. Since 1997, she has been a national delegate of the European actions COST260, COST284, COST-IC0603 on antennas. She is a technical reviewer of the European Community for scientific projects in the fields of antennas and communication. She has published over 100 journal and international conference papers. Dr. Civi was a recipient of the 1994 Prof. Mustafa Parlar Foundation Research and Encouragement award with METU Radar Group and the 1996 URSI Young Scientist Award. She was the chair of the IEEE Turkey Section in 2006 and 2007, and the chair of the IEEE AP/MTT/ED/EMC Chapter between 2004 and 2006. She is a member of the Administrative Committee of the Turkish National Committee of URSI. She is an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

Tayfun Akin (S’90–M’97) was born in Van, Turkey, in 1966. He received the B.S. degree in electrical engineering with high honors from Middle East Technical University, Ankara, Turkey, in 1987, and went to the USA in 1987 for his graduate studies with a graduate fellowship provided by NATO Science Scholarship Program through the Scientific and Technical Research Council of Turkey (TUBITAK). He received the M.S. degree in 1989 and the Ph.D. degree in 1994 in electrical engineering, both from the University of Michigan, Ann Arbor. He became an Assistant Professor in 1995, an Associate Professor in 1998, and Professor in 2004 in the Department of Electrical and Electronics Engineering at Middle East Technical University, Ankara, Turkey. He is also the Director of the METU-MEMS Center which has a 1300 m clean room area for MEMS process and testing. His research interests include MEMS, microsystems technologies, infrared detectors and readout circuits, silicon-based integrated sensors and transducers, and analog and digital integrated circuit design. Dr. Akin has served in various MEMS, EUROSENSORS, and TRANSDUCERS conferences as a Technical Program Committee Member. He was the co-chair of the 19th IEEE International Conference of Micro Electro Mechanical Systems (MEMS 2006) held in Istanbul, and he was the co-chair of the Steering Committee of the IEEE MEMS Conference in 2007. He is the winner of the First Prize in Experienced Analog/Digital Mixed-Signal Design Category at the 1994 Student VLSI Circuit Design Contest organized and sponsored by Mentor Graphics, Texas Instruments, Hewlett-Packard, Sun Microsystems, and Electronic Design Magazine. He is a co-author of the symmetric and decoupled gyroscope project which won the first prize in the operational designs category of the international design contest organized by DATE Conference and CMP in March 2001. He is also the co-author of the gyroscope project which won the third prize of the 3-D MEMS Design Challenge organized by MEMGen (currently Microfabrica).

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Design and Implementation of a Closed Cylindrical BFN-Fed Circular Array Antenna for Multiple-Beam Coverage in Azimuth Nelson Jorge G. Fonseca, Senior Member, IEEE

Abstract—This paper describes a beamforming network well adapted to produce evenly distributed multiple-beam coverage over a full 360 angular range in azimuth. This structure is very simple, flexible, and compatible with low-cost manufacturing processes such as printed technology. The proposed design is based on balanced power dividers and combiners arranged in such a way to feed a sector of a circular array antenna with Gaussian amplitude distribution and in-phase signals. The first characteristic provides control on the radiation pattern shape, including main lobe shape and sidelobe level, while the second ensures stable beam pointing over a wide frequency range. The proposed azimuth beamforming network is described, and a specific design in microstrip technology is presented with a center frequency at 6 GHz. Simulation results are supported by the measurement of several prototypes. In particular, two designs including the circular array antenna are compared to investigate the impact of the circular array antenna radius on radiation patterns. Despite the assumptions, good correlation is found between simulation and measurement results, thus confirming the properties of the proposed beamforming network concept. Integration of phase controls in the feeding network and its impact on the overall antenna efficiency are also discussed. Index Terms—Azimuth distribution, circular and cylindrical array antenna, in-phase Gaussian-like amplitude distribution, multiple beamforming network.

I. INTRODUCTION

M

ULTIPLE-BEAM circular or cylindrical array antennas producing beams evenly distributed in azimuth are of great interest for terrestrial and space communication systems, the azimuth beamforming network (BFN) being eventually combined with an elevation BFN to comply with the application’s needs [1], [2]. In this paper, we focus on the design of the azimuth BFN. The solutions already introduced [1], [2] are based on Butler matrices [3]. More generally, most of the solutions found in the literature are based on elementary components described by a unitary -parameters matrix, typically four-port directional couplers, to achieve theoretically lossless operation (in practice, line losses are introduced by the selected transmission line technology) [4], [5]. Although these solutions

Manuscript received May 05, 2011; revised June 27, 2011; accepted August 22, 2011. Date of publication November 04, 2011; date of current version February 03, 2012. This work was supported by CNES. The author was with the Antenna Department, Centre National d’Etudes Spatiales (CNES), 31400 Toulouse, France. He is now with the Antenna and Submillimetre Wave Section, European Space Agency, 2200 AG Noordwijk, The Netherlands (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2174956

Fig. 1. General topology of a multiple-beam C-BFN.

are attractive because of their low-loss characteristic, the orthogonal excitation laws produced impose strong constraints on the pattern shape [6], requiring careful optimization of the design. An alternative solution based on a much simpler design approach was recently introduced by the author [7], which is a cylindrical adaptation of the planar concept proposed in [8]. This BFN naturally produces an in-phase Gaussian-like amplitude distribution. This paper provides further insight on the performance of this new structure, supported by measurement results. We first review the basic principles of the proposed solution and then describe the experimental results obtained for a BFN alone and BFNs combined with circular array antennas. Impact of the circular array radius on radiation patterns is also investigated. We finally discuss the possibility to integrate phase controls in the BFN and the impact on the overall antenna efficiency. II. CLOSED CYLINDRICAL BEAMFORMING NETWORK CONCEPT DESCRIPTION The concept of Closed Cylindrical Beamforming Network (CC-BFN) is a 3-D evolution of the planar concept known as the Coherently Radiating Periodic Structure Beamforming Network (CORPS-BFN or C-BFN) [8]. The standard C-BFN is illustrated in Fig. 1. It is a connection of successive layers with a specific arrangement of alternating power combiners (C) and power dividers (D). This specific arrangement produces at the output ports (ports on top of the structure in Fig. 2) a Gaussianlike amplitude distribution and in-phase (true time delay) excitation law per input port (ports on the bottom in Fig. 2). The power delivered to each input port propagates within a “triangular” area (a typical electrical path per beam is highlighted in Fig. 1 for the input port 2). This specific arrangement of power combiners and power dividers to produce tapered amplitude is not new as Butler used a similar arrangement (using hybrid couplers with one port loaded) in association with Butler matrices to

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Fig. 2. Proposed closed cylindrical multiple beamforming network.

produce a cosine amplitude distribution [3]. A similar arrangement of hybrid junctions serving as combiners and dividers is used in [9] to feed a linear array with either a cosine amplitude distribution (one layer design) or a cosine-squared amplitude distribution (two layers design). The C-BFN provides a generalization of this solution, investigating beam steering capability of such a structure [8]. C-BFNs described in the literature are limited to planar designs feeding linear array antennas, although this periodic arrangement enables other shapes of both BFN and arrays. Therefore, as illustrated in Fig. 2, we proposed to close the BFN [7], resulting in a cylindrical arrangement. The CC-BFN is enclosed in the dashed area in Fig. 2. The BFN is closed in the sense that the left side of this dashed area is connected to the right side of this same dashed area. This is further illustrated in Fig. 2 by the highlighted electrical paths followed by the signal at input port 1. The portions of signal reaching the left-end side of the structure continue on the right side of it. Obviously, this is a schematic representation. The implementation of this structure will mainly depend on the selected technology. If realized in printed technologies, like the usually planar microstrip BFNs, one may use a soft substrate that can be rolled in a cylindrical shape and then connect the corresponding electrical paths from left and right ends to close the structure. If realized in waveguide or other 3-D technology, the global shape of the BFN is not constrained in the same manner and may differ from a cylinder to make it eventually more compact. Also, the illustration is given with a limited number of ports and layers, but these parameters can obviously be adapted to the need. This specific arrangement avoids the singularity of the edge beams in the planar version: Every beam has two adjacent overlapping beams, overlapping in the sense that output ports are shared between adjacent beams. This peculiarity is well suited for multiple-beam coverage in azimuth. The results reported in [10] about C-BFN mathematical description, efficiency, and possible evolutions to reduce insertion losses can obviously be extended to CC-BFN. Properties of CC-BFN are further investigated through the specific designs and experimental validations reported in Section III. III. CLOSED CYLINDRICAL BEAMFORMING NETWORK PRACTICAL IMPLEMENTATION A. Beamforming Network Design The design of the beamforming network has to be defined taking into account the targeted coverage and the associated

Fig. 3. (a) Schematic of a three-layer CC-BFN with reduced overlap and (b) associated layout in microstrip technology.

circular array antenna. Some of the major parameters influencing the design include the number of beams to be produced and the crossover level between adjacent beams. Interference between beams and insertion losses are other parameters that need to be traded off. To validate the concept of CC-BFN and the possible use of printed technology, we designed a BFN with 7 input ports and 14 output ports. Fig. 3(a) shows a schematic representation of a portion (three input ports, labeled , , and ) of the selected BFN. It is composed of three layers, with a bottom layer simplified according to a modification proposed in [10] to reduce insertion losses. In fact, this modification suppresses the 3-dB loss characteristic of the first layer in a C-BFN (due to unbalanced power combination), but reduces the overlap between adjacent beams. This specific design distributes each beam port signal over four output ports (e.g., distributes the signal toward , , , and ) with an overlap of two output ports between adjacent beams (e.g., and share the output ports and , while and share the output ports and , but and share no common output port). The theoretical amplitude level is 5.51 dB for the two central output ports and 15.05 dB for the two outer output ports. As already mentioned, the four output port signals are in phase. The theoretical losses introduced by this design are equal to 2.04 dB. The elementary component design optimized in [10] to investigate planar C-BFN characteristics has been reused for this study assuming that the effect of the substrate curvature can be neglected. This component is a circular in-phase ring hybrid with wideband frequency performances [11]. The design is centered at 6 GHz. Simulation results were obtained with ADS (from Agilent) using a combination of method-of-moment-based analyses (at elementary component level) and scattering-parameters-based network analyses (at BFN level).

FONSECA: CLOSED CYLINDRICAL BFN-FED CIRCULAR ARRAY ANTENNA FOR MULTIPLE-BEAM COVERAGE IN AZIMUTH

Fig. 4. Manufactured prototype of a three-layer CC-BFN with 7 input ports and 14 output ports.

The substrate (NY9208 from Neltec with a thickness of 0.762 mm and a dielectric constant of 2.08) is soft enough to be rolled and form a cylinder with the proper diameter. To facilitate the subsequent integration of the circular array antenna, a compact design is used as illustrated in Fig. 3(b). Previous results with planar implementations demonstrated that coupling effects between components tend to shift in frequency the overall response of the network (including input matching and transmission coefficients), but due to the wideband operation of the ring hybrid, little impact is observed at center frequency [12]. Radiating elements will be directly connected to the output ports, resulting in an array spacing of at 6 GHz. The BFN diameter is 175 mm. The manufactured prototype is presented in Fig. 4. The manufacturing approach, based on planar printed circuit, requires connecting electrical paths at the junction between the two edges of the printed board. Obviously, the signals passing through these connections (made with soldering in this specific implementation) may be affected in amplitude and/or phase depending on the quality of the connection. Since our purpose is to validate a design concept and not a specific technological implementation or manufacturing process, results associated to the ports having signals passing across the substrate junction were not investigated. To display the results, the ports are numbered clockwise as seen from the bottom (input ports side) starting at the junction. Only ports and are affected by the junction. Measurements were made using a vector network analyzer. Insertion losses were computed per input port summing all the relevant transmission coefficients. Fig. 5 compares measured insertion losses of three of the seven input ports to the simulated insertion losses. As anticipated with previous results in planar realizations [12], an overall shift of the response toward higher frequencies is observed. However, in this design, ripples are observed that were not found in the planar realizations, as well as significantly higher insertion losses. On average for the five measured ports, insertion losses at 6 GHz are around 3.45 dB in measurement, to be compared to 2.37 dB in simulation and 2.04 dB in theory. Considering the wideband performances of the structure and

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Fig. 5. Measured insertion losses of the manufactured three-layer CC-BFN with 7 input ports and 14 output ports.

the better agreement between measurements and simulation for planar realizations, this discrepancy is understood as being mainly due to the effect of the curvature. Fig. 6 provides additional insight on the CC-BFN performance presenting the amplitude and phase of the transmission coefficients associated to input port . Due to their lower level, signals toward the two outer output ports are much more affected by the curvature, with more important ripples. Measured values at 6 GHz are 6.9 and 18.0 dB, respectively, for the two central output ports and the two outer output ports, to be compared to 5.82 and 15.51 dB, respectively, in simulations. Fig. 6(b) confirms that the four output ports are in phase over a wide frequency range. B. Combined Beamforming Network and Array Antenna Design We discuss in this section the performance of circular array antennas fed by a CC-BFN. Taking into account the curvature in the full array antenna design would lead to time-consuming computations. Consequently, only the array element, a square patch antenna, has been designed using a 3-D electromagnetic modeling tool (FEKO from EMSS) to provide a resonance frequency at 6 GHz taking into account the potential effect of the substrate’s curvature. The array antenna patterns were evaluated using a simplified model based on a Gaussian-like radiation pattern for the array element (calibrated using the simulation results) combined with the array factor of a circular array antenna, resulting in the following electromagnetic field expression in the azimuth plane for a circular array of radius comprising equally spaced elements: (1) where and characterize the Gaussian-like pattern of the array element in the azimuth plane, is the angular position of the array element , and the complex signal provided by the BFN to this same element. is the wavenumber. To investigate the impact of the curvature on the antenna performance, two prototypes were manufactured: one with a

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Fig. 7. Manufactured prototype of a three-layer CC-BFN with integrated 14-element circular array antenna.

Fig. 6. (a) Amplitude and (b) relative phase difference with reference to port of the transmission coefficients associated to the input port .

14-element array antenna and one with a 20-element array antenna. The first design is fed by the CC-BFN presented in the previous section. The diameter of this first design remains equal to 175 mm. The second design is fed by a CC-BFN having the same topology but an increased number of ports, i.e. 10 input ports and 20 output ports, resulting in a diameter of 250 mm. The first prototype is presented in Fig. 7. Fig. 8 shows the measured input matching of several ports of the 14-element prototype and compares them to the input matching of a single patch as well as a typical input matching of the CC-BFN alone. As expected, the input matching frequency bandwidth is mainly driven by the narrow bandwidth of the array element. Despite coupling effects and manufacturing errors, the frequency resonance is maintained around 6 GHz with a peak value showing some discrepancy but better than 20 dB for all ports. Similar performances were measured for the 20-element circular array prototype. Radiation patterns evaluated with the simplified model are provided for the two prototypes in Fig. 9. Corresponding measured radiation patterns for some beam ports are also reported in Fig. 9. These measurements were performed at 6 GHz in the Antenna Compact Range available at CNES, Toulouse, France. As

Fig. 8. Measured input matching of the CC-BFN fed 14-element circular array antenna compared to patch-only and CC-BFN-only results.

the CC-BFN topology is the same for the two prototypes, these results enable investigation of the impact of the array radius. As expected, the beamwidth decreases as the radius increases. Interestingly, the crossover level between adjacent beams is quite similar for the two designs, with a value of 3 dB below the peak directivity. Measured patterns are in good agreement with the simplified model in the main lobe region. The sidelobe levels

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Fig. 10. Radiation patterns measured at 6 GHz for the CC-BFN-fed (a) 14-element circular array antenna and (b) 20-element circular array antenna in both cases), and comparison to the simplified model patterns (port and an equivalent circular antenna fed by BFN providing uniform amplitude distribution.

Fig. 9. Computed and measured radiation patterns (at 6 GHz) in normalized field amplitude of the CC-BFN-fed (a) 14-element circular array antenna and (b) 20-element circular array antenna.

tend to be lower in measurements. For better comparison, Cartesian plots comparing the simplified model patterns with measured results are reported in Fig. 10. Theoretically, a four-element linear array with uniform amplitude distribution provides a gain increase of about 6 dB over the array element, while for the Gaussian-like distribution of the selected BFN, the gain increase would be limited to 5.05 dB. The computed gain of the circular array, over the array element, is found to be 1.37 and 3.15 dB for the 14-element and the 20-element array antennas, respectively (the BFN efficiency is not taken into account in these figures). This degradation is due to the uniform phase distribution. In fact, (1) clearly indicates that the phase has to be optimized to achieve maximum gain in the radial angular direction. A discussion on phase control is presented in Section IV. Still, it is

interesting to note that a similar array with uniform amplitude distribution would provide a gain of 0.21 and 3.23 dB for the 14-element and the 20-element design, respectively, over the single-element gain. Logically, a Gaussian-like amplitude distribution is more advantageous for smaller radii, as interference effects due to phase difference are reduced by the predominance of the signals coming out of the central array elements. Also, as the radius increases, the array gain tends toward the linear array case. The antenna gain has been measured to be approximately 7.6 and 8.8 dB for the 14-element and the 20-element array antennas, respectively. The simplified model of (1) has been calibrated ( and ) using the gain measurement of a single-patch antenna, evaluated around 7.4 dB, but this value is believed to be underestimated as coupling effects in an array antenna usually tend to increase the element directivity. Measurement accuracy for the gain might also partially explain the discrepancy. In general, we can say that good agreement is found between measurements and computed results despite the simplicity of the model. The accuracy achieved is sufficient to help in the definition of the BFN topology and size of the array.

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Fig. 11. Schematic of a three-layer CC-BFN with reduced overlap and integrated phase control.

For improved prediction, including sidelobe and cross-polarization levels, a full-wave simulation tool is clearly recommended. Fig. 10 also provides the radiation patterns achieved with an equivalent circular array, but having uniform amplitude distribution instead of Gaussian-like amplitude distribution. We must mention that the BFN efficiency is not taken into account, so absolute values should not be given too much attention. The hypothesis of in-phase uniform amplitude distribution results in nonorthogonal illumination laws; consequently, losses are expected in the BFN [6], but the resulting losses highly depend on the BFN topology. For instance, a parallel feeding network with one power combiner per output port would result in 3-dB insertion loss per beam. Still, it is clear that the proposed CC-BFN provides better main-lobe shaping and reduced sidelobe level, which can be beneficial to limit interference level in a multiple-beam antenna system with frequency reuse.

Fig. 12. Peak comparison for several CC-BFN topologies as a function of the circular array antenna size.

IV. CLOSED CYLINDRICAL BEAMFORMING NETWORK WITH INTEGRATED PHASE CONTROL Finally, it is clear that the main drawback of this structure is that it provides in-phase output signals; consequently, the signal combination is not optimal according to (1). This results in a degraded peak gain when compared to a similar structure with an optimal phase distribution. Accordingly, we investigated the possibility to insert phase controls in the structure to correct the phases so that the respective radiated contributions from each array element associated to one beam port combine in-phase in the radial direction defined as the symmetrical axis of the subset of elements considered. However, it must be mentioned right away that adding phase controls is not simple with this structure for two reasons. First, because adjacent beams partially overlap sharing some common electrical paths, the phase control location has to be selected so that it has the desired effect for all the beams. This means that it has to be located in a nonoverlapping section of the structure, or must be in the axis of symmetry of the overlapping section between two adjacent beams. Second, as the losses occur in the power combiners due to unbalanced power combinations, inserting phase controls will induce additional losses due to unbalanced signals not only in amplitude, but also in phase, as highlighted in [10]. Having in mind these constraints, the first solution investigated is based on the one presented in Section III. The topology considered is illustrated in Fig. 11. This topology does not affect the signal reaching the outer elements, but the power transmitted

Fig. 13. Half-power beamwidth comparison for several CC-BFN topologies as a function of the circular array antenna size.

to the two central elements is lowered. A comparison of performance for different circular array designs using this CC-BFN topology with and without phase compensation is provided in Figs. 12 and 13. These results, derived from the pattern calculation described in Section III-B, were achieved assuming that the distance between radiating elements over the circumference of the array is constant. As a consequence, the array diameter is proportional to the number of elements, and corresponding array physical dimensions can be derived from the two specific designs presented in this paper. Interestingly, it appears that the higher directivity achieved with phase control does not compensate for the additional losses in the CC-BFN, as the gain of a CC-BFN-fed circular array with phase control is always below the gain of the same structure without phase control. As a comparison, a linear array fed by the same BFN topology would have a gain of 10.41 dB. As expected, the peak gain of the two considered structures tends toward this value as the size of the array increases. Actually, the peak gain variation of the structure with phase control mainly comes from the variation of the losses

FONSECA: CLOSED CYLINDRICAL BFN-FED CIRCULAR ARRAY ANTENNA FOR MULTIPLE-BEAM COVERAGE IN AZIMUTH

in the CC-BFN, as detailed in Fig. 12. Without these losses, the peak gain would be quite close to the one of a linear array and mostly independent of the circular array size. As illustrated in Fig. 13, an interesting feature of CC-BFN with in-phase signals lies also on the half-power beamwidth (HPBW): It follows the angular range that needs to cover one beam with a BFN that produces a number of beams that is half the number of array elements, which means that this design naturally produces a good crossover between adjacent beams (around 3 dB). A CC-BFN with phase control produces much narrower beams. Actually, the beamwidth is quite independent of the circular array size, which means that a given crossover level imposes a minimum size for the CC-BFN (i.e., a 3-dB crossover requires at least a 34-element circular array). The fact that both peak gain and HPBW are independent of the circular array size comes mainly from the phase correction that compensates for the curvature effect, this curvature being directly linked to the array size with our design hypothesis. This remark also applies to sidelobe level. Based on these results, we tried to find a BFN topology that would enable phase control without increasing the BFN losses. This can be achieved by removing the top layer in Fig. 11. The resulting BFN distributes the signal toward three elements instead of four, with an overlap between adjacent beams of one output port instead of two. Simulation results for this two-layer CC-BFN topology with and without phase control are compared to the three-layer topology in Figs. 12 and 13. As the insertion losses in the BFN are reduced, the peak gain is significantly improved. In that case, adding a phase control to provide optimal phase combination results as expected in a peak gain that is almost independent of the array size and closer to the linear array case. Fig. 13 indicates that the HPBW converges quickly with the number of array elements. This is consistent with the fact that having a reduced number of active elements per beam attenuates the effect of the array curvature. A design with a crossover of about 3 dB is achieved with a 26-element circular array. The peak gain difference (less than 0.4 dB) with and without phase control is probably not sufficient to justify the added complexity of implementing phase shifters. V. CONCLUSION This paper has demonstrated through practical implementation and measurements that the proposed concept of CC-BFN can be of interest for applications requiring evenly distributed multiple-beam coverage in azimuth. The proposed concept is very simple and requires the optimization of only one component in the case of a design without phase control. Measurements at BFN and array antenna level proved to be in good agreement with simulation results despite the assumptions (substrate curvature not considered at BFN level) to reduce computational time. The proposed design of a three-layer CC-BFN with a modified first layer has an interesting feature as it enables a crossover below peak gain between adjacent beams of about 3 dB for circular arrays ranging from 14 to 26 elements, giving some flexibility in the array design. A discussion supported by simulation results has also demonstrated that the addendum of phase control to achieve optimal signal combination in radiated mode is not always beneficial.

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ACKNOWLEDGMENT The author would like to thank N. Ferrando for his support on simulations, as well as L. Féat, M. Romier, and D. Belot for their support on measurements. REFERENCES [1] P. Chen, W. Hong, Z. Kuai, J. Xu, H. Wang, J. Chen, H. Tang, J. Zhou, and K. Wu, “A multibeam antenna based on substrate integrated waveguide technology for MIMO wireless communications,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1813–1821, Jun. 2009. [2] A. M. Polegre, G. Caille, L. Boyer, and A. Roederer, “Semi-active conformal array for ESA’s GAIA mission,” in Proc. IEEE Ap-S Int. Symp., Jun. 20–25, 2004, vol. 4, pp. 4108–4111. [3] J. Butler and R. Lowe, “Beamforming matrix simplifies design of electronically scanned antennas,” Electron. Design, vol. 9, pp. 170–173, Apr. 1961. [4] G. E. Evans, “Coupling matrix for a circular array microwave antenna,” US Patent no. 5,028,930, Jul. 2, 1991. [5] S. P. Skobelev, “Methods of constructing optimum phased-array antennas for limited field of view,” IEEE Antennas Propag. Mag., vol. 40, no. 2, pp. 39–49, Apr. 1998. [6] J. L. Allen, “A theoretical limitation on the formation of lossless multiple beams in linear arrays,” IEEE Trans. Antennas Propag., vol. AP-9, no. 7, pp. 350–352, Jul. 1961. [7] N. J. G. Fonseca and N. Ferrando, “Design of a closed cylindrical beam forming network fed circular array for space diversity applications,” in Proc. 4th EuCAP, Apr. 12–16, 2010, pp. 1–4. [8] D. Betancourt and C. Del Rio Bocio, “A novel methodology to feed phased array antennas,” IEEE Trans. Antennas Propag., vol. 55, no. 9, pp. 2489–2494, Sep. 2007. [9] W. D. White, “Pattern limitation in multiple-beam antennas,” IRE Trans. Antennas Propag., vol. AP-10, no. 4, pp. 430–436, Jul. 1962. [10] N. Ferrando and N. J. G. Fonseca, “Investigations on the efficiency of array fed coherently radiating periodic structure beam forming networks,” IEEE Trans. Antennas Propag., vol. 59, no. 2, pp. 493–502, Feb. 2011. [11] G. F. Mikucki and A. K. Agrawal, “A broad-band printed circuit hybrid ring power divider,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 1, pp. 112–117, Jan. 1989. [12] N. J. G. Fonseca, “Étude de systèmes micro-ondes d’alimentation d’antennes réseaux pour applications multifaisceaux,” Ph.D. dissertation, INPT, Université de Toulouse, Toulouse, France, Oct. 15, 2010. Nelson Jorge G. Fonseca (M’06–SM’09) was born in Ovar, Portugal, in 1979. He received the Electrical Engineering degree from Ecole Nationale Supérieure d’Electrotechnique, Electronique, Informatique, Hydraulique et Telecommunications (ENSEEIHT), Toulouse, France, in 2003, the Master’s degree from the Ecole Polytechnique de Montreal, Montreal, QC, Canada, in 2003, and the Ph.D. degree from Institut National Polytechnique de Toulouse, Université de Toulouse, Toulouse, France, in 2010. He worked as an Antenna Engineer successively with the Antenna Study Department, Alcatel Alénia Space (now Thalès Alénia Space-France), and in the Antenna Department, French Space Agency (CNES), Toulouse, France, where he completed his Ph.D. degree in parallel of his professional activities. In 2009, he joined his current position with the Antenna and Sub-Millimetre Wave Section, European Space Agency (ESA), Noordwijk, The Netherlands. He has authored or coauthored more than 80 papers in journals and conferences, including two CNES Technical Notes. He holds eight patents and has two patents pending. His interests cover the telecommunication antennas, beamforming network theory and design, as well as new enabling technologies such as fractals, metamaterials, and membranes applied to antenna design. Dr. Fonseca was a member of the 33rd ESA Workshop on Antennas organizing committee in 2011. He served or is currently serving as a technical reviewer for the Journal of Electromagnetic Waves and Applications—Progress in Electromagnetic Research, MIT, the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS and the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS. He received several prizes including the Best Young Engineer Paper Award at the 29th ESA Workshop on Antennas in 2007. He is listed in Who’s Who in the World.

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Rapidly Convergent Representations for Periodic Green’s Functions of a Linear Array in Layered Media Derek Van Orden, Student Member, IEEE, and Vitaliy Lomakin, Senior Member, IEEE

Abstract—Green’s function representations are presented to rapidly compute the fields resulting from a linear (1D) periodic array of dipole current sources on or near a planarly layered medium in 2D and 3D space. The representation is formulated as spectral integral, which accounts for the reflected continuous spectrum of fields, and a series that accounts for the discrete spectrum of guided modes. It is exponentially convergent for observation points on and near the array axis and surface, and for complex phase shifts between periodic unit cells. It can be defined on alternate Riemann sheets with respect to any of the diffraction modes characterizing the array. A complete dyadic Green’s function is derived to fully account for the reflected fields for all source current orientations. This Green’s function representation can greatly accelerate the simulation of printed 1D periodic structures in optics and microwave engineering. Index Terms—Computational Electromagnetics, gratings, Green’s function methods, periodic structures, surface structures.

I. INTRODUCTION

S

TRUCTURES comprising linear periodic arrays of elements near planarly layered surfaces have important applications in microwave engineering and optics, including antennas, waveguiding structures, frequency selective surfaces, and metamaterials [1]–[4]. An efficient way to simulate this class of structures is by solving an integral equation, in which an unknown surface current distribution in a single periodic unit cell is convolved with a Green’s function that accounts for both the structure periodicity and the interactions with the surface. Fast methods to compute such a periodic Green’s function (PGF) are essential for efficient integral equation solvers for periodic structures [5], [6]. Efficient computation of the Green’s function may allow fast analysis of both scattering behavior and the complex dispersion properties of periodic arrays near surfaces. A complete layered media PGF for a linear array can be represented as the sum of a free-space PGF, , which solves for a linear periodic structure in free-space, and a reflected fields Manuscript received November 23, 2010; revised July 09, 2011; accepted August 04, 2011. Date of publication October 24, 2011; date of current version February 03, 2012. This work was supported in part by the DARPA NACHOS program and in part by the NSF ERC CIAN Center. The authors are with the Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173125

PGF, , which accounts for the surface interaction. While several techniques have been proposed for the fast computation of the free-space PGF [7], [8], the problem of fast solvers for the may reflected fields has not been well studied. The PGF be represented as a direct spatial summation of single source (non-periodic) layered media Green’s functions, which may be found via Sommerfeld integrations. This infinite sum, however, is very slowly convergent, and diverges when a complex phase shift is applied between adjacent unit cells. The reflected fields may also be represented as an infinite sum of diffraction (Floquet) modes, each of which is incorporates the reflection coefficient of the surface. This spectral series, however, converges very slowly for the important case of observation points close to the array and the surface. To date, fast methods to accelerate for 3D configurations include a series the computation of representation based on perfectly matched layers [9], and an from the doubly periarray-scanning technique that finds odic layered medium Green’s function [10]. For the 2D case, acceleration schemes include the Ewald method [11], [12] and a method based on perfectly matched layers [13]. In this paper we present a rapidly converging method for for observation points computing the reflected fields PGF close to a 1D array of electric dipole sources placed near a surface. The PGF is formulated in terms of single and double spectral integrals for 2D and 3D configurations, respectively, and a discrete spectrum of guided wave modes supported by the planarly layered medium. The integrals are regularized to make them rapidly convergent for fast numerical evaluation. This representation can seamlessly handle source and observation points directly on the surface and array axis, periodicities defined with complex phase shifts, and is fully applicable to surfaces composed of lossy and gain media. , the three-diThe bulk of this paper is devoted to solving mensional reflected fields Green’s function. Section III.A discusses spectral integral representations for periodic Green’s functions. Section III.B then shows how such representations in terms of the continuous may be used to express spectrum, solved as double integral, and a discrete spectrum of eigenmodes. Sections IV and V are devoted to the fast evaluation of the continuous and discrete spectra, respectively. with respect to Section VI discusses the branch cuts of the phase shift parameter, and the corresponding Riemann sheets. Numerical implementation and results are presented in Sections VII and VIII respectively. Finally, based upon the approaches for the 3D case, Section IX presents a formulation of , along with results. Throughout the 2D Green’s function is assumed. the paper, a time dependence of

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medium. Section III.A briefly describes this procedure for the Floquet series representation of PGF for 3D configurations, which is efficient for observation points far from the surface. Section III.B describes a new formulation that is efficient close to the surface. A. Floquet Series Representation Consider the Floquet series representation of the free-space PGF in terms of an infinite sum of cylindrical waves Fig. 1. A linear array of point dipole sources near a layered medium background. The fields resulting from the array are found in the unit cell centered at the origin.

II. STRUCTURE CONFIGURATION AND OUTLINE Consider a linear, infinitely periodic array of electric dipole sources with spacing oriented along the axis, on or near a planarly layered medium. These are point current sources for 3D configurations, as shown in Fig. 1, and line sources for 2D configurations. The array and observation point are assumed to be on or above the interface, separated by a few wavelengths or less. (Though not presented here, the formulation can be extended to the case of observation and source points inside the slab layers.) The free-space wavelength is and the free-space wavenumber is . The electric dipole sources have dipole moments , where is a (generally complex) phase shift parameter, is an integer counting the sources, and is the dipole moment of the source in the zeroth unit cell of the array, placed at the origin. The top surface of the medium is parallel to the plane, a distance below the array. The surface has known transverse electric (TE) and magnetic (TM) reflection coefficients and , respectively. The reflection coefficients may be expressed as functions of and the spectral parameter , whose ratio uniquely defines the angle of incidence for a plane wave. The electric fields resulting from this system may be expressed as (1) and are the dyadic Green’s functions accounting where for the free-space and reflected electric fields for a linear periodic system. The goal of this paper is to develop an efficient representation of for observation points close to the array and surface. III. INTEGRAL REPRESENTATIONS OF GREEN’S FUNCTIONS A spectral representation of a free-space scalar Green’s function may be used to find the layered medium PGF . The procedure is to expand into plane waves and find the dyadic PGF via a dyadic operator (2) expressed as a plane wave expansion, may be with found as the reflection of each plane wave from the layered

(3) and are the where diffraction mode wavenumber components parallel to and transverse to the array axis, respectively. Each cylindrical diffraction mode may be expanded into plane waves using an integral expansion of the Hankel function. The PGF then becomes

(4) . This representation is well where suited to finding the reflected fields Green’s function. By applying the dyadic operator and inserting the TM reflection coefficient, the component of may be found as

(5) represents the discrete spectrum of guided wave Here, poles. It results from the fact that the complete expansion with respect to the spectral parameter includes not only the continuous (plane wave) spectrum but also the discrete (guided mode) spectrum [14]. The evaluation of the discrete spectrum is discussed in Section V. This Floquet series is efficient for observation points far from the array and surface, e.g., for the far-field radiation from the array. However, it is very slowly convergent for observation points close to the array and surface. B. Alternative Representation The scalar PGF may also be represented as an infinite spatial sum over unit cells of the periodic system (6) Using the Weyl identity

(7)

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with in (6) yields an alternative representation of the scalar, free-space PGF

be significantly improved by explicitly excluding from the expansion a small number of sources in and near the unit cell. The scalar PGF , from which is derived, may be reformulated by expressing the 1D periodic Green’s function as follows:

(8) is the scalar PGF in 1D free-space, which where may be found in closed form as a geometric spatial series. (11)

(9) has poles at It is clear that , corresponding to the diffraction mode wavenumbers. The two terms of in the brackets of (9) represent the contributions of the sources on each side of the origin. By eliminating one of these terms, the Green’s function for a semi-infinite periodic system may easily be formulated. The integral representation (8) may be extended to dyadics and used to find the fields reflected from a surface. For example, the component of may be found from by following the procedure similar to that in [14]:

(10) where symmetry has been used to reduce the integration over to a semi-infinite integral. As in the Floquet series representation, must be included to account for the discrete spectrum of guided wave modes supported by the layered medium. Throughout the paper, equations and derivations are given for the dyadic component to demonstrate this alternative representation. However, this method is readily extended to all of the dyadic components. Their integral representations are given in Appendix A. The integrands in (10) are highly oscillatory, making the integrals difficult to evaluate numerically. The strong variations result primarily from , which contains both rapidly varying exponential terms and poles that may reside close to the integration contour. The reflection coefficient also contains variations that complicate numerical integration. Section IV is therefore devoted to the regularization and fast evaluation of these integrals, while Section V discusses the evaluation of the discrete spectrum.

sources in and where the first term accounts for the closest to the unit cell, while the second term accounts for the remaining infinite sources. For most cases of interest may be chosen as a small number (e.g., 2 or 3), and the contribution of the first summation in (11) can be evaluated rapidly using any fast method [14]–[16]. The second term in (11) may replace in the spectral integrals in (10) to improve convergence. Note that this term has the same poles (and corresponding residues) as , independent of . B. Steepest Descent Path Integration The double spectral integrals used to evaluate the components of may further be made rapidly convergent by deforming both integration contours to the steepest descent path (SDP) in the complex and planes, away from the real axes. This may be accomplished through the following change of variables:

(12) The double integration is then taken along the real axes of the and planes, and the saddle point of this integracomplex tion is located at . With this transformation, the term has exponential decay with respect to both integration variables, and the spectral integral for the component of is expressed as

IV. RAPID EVALUATION OF SPECTRAL INTEGRAL A. Direct Evaluation of Sources Near the Unit Cell When evaluating the fields in the unit cell of a periodic structure, most of the field variations result from those current sources in and near the unit cell of interest. The convergence of the integral expression for the Green’s function may therefore

(13) The term clearly shows that the integrand has Gaussian decay increasing with . As the

VAN ORDEN AND LOMAKIN: RAPIDLY CONVERGENT REPRESENTATIONS FOR PGFs OF A LINEAR ARRAY IN LAYERED MEDIA

periodicity decreases, this decay becomes weaker, requiring larger values of for the same convergence. In the process of deforming the integration contours, one or more poles of may be crossed. It can be shown that this occurs only when has a nonzero imaginary part. In this case, the residues of the integrand at these poles must be accounted for. It is easily shown that these poles are found at , where . (The sign distinguishes the poles resulting from the two terms of in (13).) They exist in the combined space spanning two complex planes, and so the residue must be formulated as an integral in accordance with the residue theorem for higher dimension complex functions. It is shown in Appendix B that the residue of the th pole is

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wavenumber and the poles of get very close to the saddle point. This case is particularly important to finding the complex dispersion relations of resonant arrays on surfaces, for which the propagation constant for traveling waves may be close to . A further regularization of the integral is therefore required. The resonant behavior resulting from the poles of may be explicitly extracted from the integrand, making it slowly varying, and then integrated in closed form. To simplify the notation, the integrand of may be written as . It is easily seen that the term is singular when . Near a pole singularity, may be well approximated by , i.e., the residue of the function divided by its singular component. To extract the resonant contribution of for a given value of the integer , one may subtract this term from the integral to regularize it and then separately integrate the subtracted terms in closed form

(14) where and . It may be further shown that only the poles at can contribute if , while the poles at can contribute only if . For most cases of interest, only one or two poles contribute to the integral. However, for very large (e.g., greater than 15) the spectral spacing between the poles becomes small enough that more may have to be considered. In general, the guided wave poles of the reflection coefficients are not crossed in the deformation to the SDP contours if the observation point is close to the array. If this does occur (e.g., for observers removed from the array), however, the contribution of that pole to the discrete spectrum (see Section V) must be excluded. The leaky wave poles of the reflection coefficients, in contrast, generally are crossed. These poles, which otherwise would not contribute to the discrete spectrum, must now be included, although in most cases their contributions are negligibly small. The integral formulation of (13) assumes that the observation coordinates and are small compared to , in which case the saddle point of the SDP integrations lie near and , as formulated in (12). As such, the presented integral representation is efficient for observation points relatively close to the array and surface (the separation should be smaller than ). For larger values of and , the Floquet mode representations of Section III.A become efficient. C. Extraction of Resonant Contributions of Poles The poles of both the reflection coefficient and the 1D spectral PGF still contribute significant variations to the integrand in (13) if they are located near the saddle point of the SDP integration. These variations can significantly slow down numerical integration. This is especially true in the “blind angle” regime, where the phase shift parameter approaches the free-space

(15) where erfc denotes the complementary error function, and we have used the fact that is even in to simplify the expression. The first term in (15) is the double integral that does not have variations resulting from the singularities at . The second term is the integral of the resonant parts of , and is represented as a 1D integral over that displays a Gaussian convergence. For most cases, the contribution of only one pair of poles must be extracted, though this procedure is easily extended to cases where multiple poles of are close to the saddle point. It can also occur that the guided wave poles of are sufficiently close to the origin in the plane to slow down the evaluation of , including both the double and single integrals in (15). This is generally the case for the TM reflected fields or for any “near-cutoff” guided mode. The resonant contribution of these poles may also be extracted to regularize the integrands. A complete analysis of the general case, in which poles of both and are close to the saddle point of the integral, is somewhat involved technically; a detailed treatment is given in Appendix C. As the electrical thickness of the multilayered medium increases, more poles of must be extracted to maintain a good convergence rate.

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V. DISCRETE SPECTRUM The discrete spectrum of guided modes must be included in the spectral expansion of the layered medium PGF. The propagation wavenumber of the th TM guided wave mode represents a pole in the TM reflection coefficient . Such poles may be found numerically using a pole or root-finding algorithm for complex valued functions. The discrete spectrum of the th mode is found as the residue of the integral in (10) evaluated in the plane at . The complete discrete spectrum of the component of is then found by summing this residue over all poles

(16) . The integral in (16) may where be expressed in terms of the scalar PGF for a linear array placed on the axis in 2D free space. The expression for the discrete spectrum then simplifies to

(17) may be computed rapidly by regularizing the specwhere tral integral in (16) [17], or by other methods [7], [18]. Note that is not evaluated at the free-space wavenumber , but rather the guided wavenumber component , indicating that it accounts for radiation into that guided wave mode. Although the expression (17) is accurate for the representation of (10), two important modifications are required for the accelerated integral representation of (13). First, in deforming the integration contours to the SDPs, the leaky wave poles are generally crossed and they must also be included in (17). On the other hand, if a guided wave mode is crossed, its contribution to the discrete spectrum must be excluded. Second, when a small number of sources near the unit cell of interest are excluded from the spectral integration, as described in Section IV.A, the same must be done for the discrete spectrum. That is, the computation of must be redefined so as not to include those source contributions. This may be done, for example, by replacing the 1D PGF in (16) by the second term of (11). VI. RIEMANN SHEETS OF THE GREEN’S FUNCTION In studying periodic structures, there is a frequent interest in identifying traveling wave modes and finding their complex propagation constants. For a given geometry, these modal propagation constants represent those values of the phase shift parameter that solve the relevant dispersion equation. (In integral equation formulations, they lead to a zero value for the determinant of the impedance matrix.) The dispersion relation is an implicit equation whose dependence on comes from

the Green’s function. When solving it in the complex plane, one must account for the branch cuts introduced by the Green’s function, as some leaky-wave solutions may appear on alternate Riemann sheets. It is therefore important to be able to compute the Green’s function on any Riemann sheet. The branch cuts result from the existence of Floquet modes. It is clear from the expression (3) that the free-space periodic Green’s function has an infinite number of square-root branch cuts in the plane, resulting from its dependence on the transverse Floquet mode wavenumbers , where . These branch cuts must also appear in the representation (4), as well as in the reflected fields Green’s function (5). The sign of the square root is typically chosen so that for all , corresponding to diffraction modes that are proper, and decay exponentially away from the array. In this case the Green’s function is said to be evaluated on the top Riemann sheet in the complex plane. To define a Green’s function on an alternate Riemann sheet, one may choose for a small number of Floquet modes. The branch cuts in the plane also exist in the spectral integral formulation of (or any representation of .) Mathematically, they result from the discontinuity of the integral in the plane when the integration contour crosses a pole of the 1D PGF (see Appendix B). Up to this point, this integral representation of has been presented for the top Riemann sheet. To define on the lower Riemann sheet with respect to the th Floquet mode, one must evaluate on the top Riemann sheet, then subtract the residue contribution of the th Floquet mode pole. That is, must be redefined as (18) where

is given in (14).

The discrete spectrum DS also contains square-root branch cuts, but of the form for the th guided wave mode. These branch cuts result from the 2D free-space periodic Green’s function in (17). The discrete spectrum may be defined on a lower Riemann sheet with respect to any of these branch cuts through the computation of , which may also be represented as a Floquet series [7]. VII. NUMERICAL IMPLEMENTATION This section discusses the numerical evaluation of the double SDP integrals of (13). The evaluation of the discrete spectrum via the 2D PGF, and of the contributions of the sources near the unit cell both have well-known numerical implementations and are not discussed here. The SDP integral may be evaluated using a quadrature rule with equally spaced nodes for both integration variables. One may choose a discrete set of nodes and (with and integers) in the ranges and , where and are the truncation values for each integration. These finite ranges represent the contribution zones of the integrals, beyond which the integral has negligible value. The computation of the integral is then reduced to a discrete double summation, where each term is simply the integrand evaluated at nodes , then

VAN ORDEN AND LOMAKIN: RAPIDLY CONVERGENT REPRESENTATIONS FOR PGFs OF A LINEAR ARRAY IN LAYERED MEDIA

multiplied by corresponding weights of (13) then becomes

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. The integration

(19) where and are number of nodes used in each integration. The error associated with this numerical integration depends on the number of nodes and used and the truncation values and . Assuming that the exponential decay dominates at the edge of the contributing zone, the truncation values may be taken as where is a dimensionless parameter, e.g., is generally sufficient for a single precision accuracy. The number of integration nodes and is then chosen to achieve the desired accuracy. In our implementation we choose , as the integral over is semi-infinite and requires half as many quadrature terms. It should be noted that in addition to the double SDP integral, a small number of single integrals must be evaluated to find the residue contributions and for the terms subtracted from to make it slowly varying (if required). These terms do not typically contribute significantly to the total computational cost (approximately 10% for the cases considered below), and can be evaluated using a similar quadrature rule, with summation terms. VIII. NUMERICAL RESULTS This section presents results showing the accuracy and convergence of the proposed method for finding . The case considered is an array of dipole sources placed directly on top of a dielectric slab in free space. Verification to high accuracy presents some complications, however, as it requires a suitable reference solution found using either a Floquet series (5) or a spatial summation of single source Green’s functions. Our method agrees well with both the Floquet series representation and spatial summations, assuming that those can be computed. We found that the Floquet series is only reasonably convergent for very thin slabs, as the integrals required for the Floquet summation are problematic to evaluate. Even for thin slabs, single precision accuracy is hard to achieve. For this reason we choose as a reference solution a spatial summation of single source Green’s functions, assuming a small amount of loss in the ambient medium. This loss is realized by making complex, with . Each term in the summation is evaluated, using Sommerfeld integration, to accuracy. The introduced loss permits truncating the sum to 3000 terms, ensuring a truncation error of or less for all presented cases. Fig. 2 shows the error of as a function of the number of quadrature nodes used to evaluate the SDP integral over . (The total number of summation terms in (19) is then .) It is shown for the observation point

Fig. 2. Error of the field component versus for array on top of a dielectric slab for 3 values of the slab thickness . The array has spacing , phase shift parameter , and the slab has permittivity . The fields are observed at .

TABLE I CPU TIMES FOR 200 EVALUATIONS OF THE DIAGONAL COMPONENTS OF AT , NOT INCLUDING THE CONTRIBUTION OF THE 5 SOURCES NEAR THE UNIT CELL

. Five sources in and near the unit cell are excluded from the integral and computed using the single source Green’s function (i.e., ) via the spectral method presented in [14]. The slab has relative permittivity , the array has spacing , and the dipoles are sequentially phase shifted with wavenumber . The error is shown for three different values of the slab thickness : , and , where is the wavelength in the slab medium. It is evident that the convergence is fast for all slab thicknesses. The error is small even for very small values of . For the slab thicknesses , and , the TM reflection coefficient has 1, 2, and 2 guided wave poles that must be included in the discrete spectrum, respectively, and 2, 3, and 3 poles whose resonant contribution must be subtracted to regularize the integrand for high accuracy solutions. Although these results are shown only for , the convergence rates of the other components of are very similar. Table I shows the computation times for evaluation of the 3 diagonal components of at 200 observation points (all taken to be the same as in Fig. 2). The contribution of the 5 sources in and near the unit cell is not included, as its evaluation time is a well-studied problem [19], and is not dominant compared to the evaluation time of the double integral. The CPU times are given for the choices of the integration parameters and (the latter being the number of nodes used to evaluate the single integrals required for pole extraction). The corresponding relative errors of the representation are given in Fig. 2. All computations were done on an Intel Xeon X5482 3.2 GHz processor.

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IX. EVALUATION OF 2D REFLECTED FIELDS PGF This section considers the 2D version of the problem, in which a periodic array of line sources resides near a surface. Because the fields have no dependence in the dimension, only the scalar PGF accounting for TE and TM reflections need be considered. Furthermore, there is only one dimension transverse to the periodicity axis, so the spectral representation involves only a single integral. The Floquet series representation of the free-space Green’s function is given by (20) Fig. 3. Field components observed on the array axis and slab surface for a lossless system, plotted over half the unit cell with the source contribution at , phase shift parameter the origin removed. The array has spacing , and the slab has permittivity and thickness .

with and . This may easily extended to the case of an array of line sources near a surface. Finding the reflected fields scalar Green’s function requires simply inserting the reflection coefficient into each series term, evaluated at the corresponding Floquet mode wavenumber component

(21)

Fig. 4. Error of the field component versus for a dense array with on top of a dielectric slab for 3 values of the parameter . The array , and the slab has permittivity has phase shift parameter and thickness . The fields are observed at .

As with the 3D case, however, this series converges very slowly when the observation point is close to the array and surface. An alternative is to use a transverse spectral expansion. This approach, presented in [17] to find the free-space PGF, may be extended to find the TE and TM reflected fields from a surface. The resulting integral becomes

(22) Fig. 3 shows the , and field components over half the unit cell for a lossless version of the system considered above, with phase shift parameter and slab thickness . Here the observation points are taken directly on both the surface and the array axis, and the source contribution at the origin is excluded to avoid the field divergence. Unlike , the and components include both TE and TM fields. Finally, Fig. 4 shows the convergence of the integral for a dense array with , in order to demonstrate low-frequency performance. The error of the PGF is shown at the observation point for three different values of . A larger number of sources near the unit cell must be excluded from the integration and evaluated separately for the integral to achieve the same convergence as shown in Fig. 2. Therefore, the performance of the presented method deteriorates for very small periodicities. It remains reasonably good, however, for periodicities from several wavelength down to values as small as , thus spanning the range used in most practical applications.

The discrete spectrum for each guided wave mode is found as the residue of the integral, evaluated at the mode wavenumber

(23) The integral in (22) is simply a plane wave expansion of the fields, with inserted to find the (TE, TM) reflection of each plane wave component. It may be regularized and evaluated using the exact same procedures outlined in Section IV. This approach is shown in [17] for the case where no surface is present, and detailed derivations are not provided here. The only modification that may be required is accounting for simple poles in the reflection coefficient that may reside near the saddle point of the integration. These poles correspond to guided and leaky wave modes in the surface medium, and their contribution may be extracted and integrated in closed form using the same procedure used for the poles of .

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APPENDIX A DYADIC COMPONENTS OF This section extends (10) to all the dyadic components of . Each of these integral expressions may be regularized and evaluated numerically as described in Sections IV–VIII

Fig. 5. Error of the field components versus for an array on top of a dielectric slab for 2 values of the slab thickness . The structure has the same is real and . parameters considered in Fig. 2, except

CPU TIMES AT

TABLE II 2000 EVALUATIONS OF THE TM REFLECTED FIELDS , NOT INCLUDING THE CONTRIBUTION OF 5 SOURCES NEAR THE UNIT CELL

FOR

To demonstrate the accuracy of this code, we use the Floquet representation (21) as a reference solution, as it can easily be computed to high accuracy for by taking many series terms. The integral (22), once regularized, may be evaluated using a simple quadrature rule with summation terms. Fig. 5 shows the convergence of and for the same configuration considered in Fig. 2, except the ambient medium is chosen to be lossless (i.e., real ) and . Table II shows CPU times for the TM reflected fields for 2000 evaluations at the observation point . The source contributions of the five sources in and closest to the unit cell are not included in these timings. (24)

X. SUMMARY Alternative dyadic PGF representations have been presented to find the electric fields resulting from an array of electric dipole sources near a planarly layered medium in two and three dimensions. They are based on a plane-wave expansion of the scalar periodic Green’s function, formulated as spectral integral and a discrete spectrum of guided wave modes. This representation is accurate for observation points close to and directly on the array axis and surface, including surfaces composed of gain and lossy media. It is furthermore accurate for complex phase shifts between sources and may be defined on alternate Riemann sheets with respect one or more diffraction modes. These properties make the spectral integral representation particularly well-suited to finding the complex dispersion relations of traveling wave modes supported by linear periodic arrays, as well as scattering by both near and far-field external sources. It has the potential to greatly accelerate the computational analysis and design of surface traveling wave structures and printed leaky-wave antennas.

APPENDIX B RESIDUES OF THE DOUBLE SPECTRAL INTEGRAL The simplest way to evaluate the residue of the integrals in (10) at the poles of is to change the integration variables into a polar form through the following transformation: (25) The integral for the

component in (10) then becomes

(26)

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where and . The important simplification here is that does not depend on , and has poles only in the plane. One may simply evaluate the residue of the integrand in the plane and then integrate this residue over the angular variable to recover the result given in (14). One problem that must be considered is how to determine if a pole is crossed by the double SDP integral (i.e., when a pole’s residue contribution should be included). In our formulation the integration in the plane is done first, but the integration path depends on through the coordinate transform in (12). This means that a pole may be crossed in the plane for some values of , but not others. This transition, for a given pole, represents a discontinuity of the integrand in the plane (i.e., the integrand of the second spectral integration). The th Floquet mode pole therefore has a corresponding pair of branch cuts in the plane whose path is defined by . Note that these branch cuts also appear in the Floquet series representation of (5). The branch cuts have branch points at . A pole’s residue contribution must be included if the SDP crosses this pair of branch cuts in the plane. It may be shown that only the poles at can contribute if , while only the poles at contribute if . It is interesting to note that the residue contribution of (26) may also be derived by finding the residue of the integral (with treated as a constant) and integrating it in the complex plane. The integration path must start at the branch point and go to the other branch point , where it crosses to the lower Riemann sheet and returns to . In the coordinate transform to the angular spectrum (25), the integrand does not explicitly contain these branch cuts, so the residue evaluation is more straightforward; the result, however, is the same. APPENDIX C EXTRACTING THE RESIDUE CONTRIBUTION OF POLES Suppose that in order to regularize the double integral in (13), the resonant part of multiple poles of both (found at ) and (denoted by ) must be extracted from the integrand, then integrated separately. One may begin this process with (15), in which only the singularities at are subtracted. Both terms in (15) contain integrals over that must be regularized with respect to the reflection coefficient. Following the same procedure, and extending it to the case of multiple poles of both terms, one obtains (27). This equation contains, in order, the nonresonant double integral, a single integral, in which the resonant terms of have been integrated, a regularized single integral over , and a sum of the integrated resonant components from the integral over

(27)

REFERENCES [1] R. E. Collin and F. J. Zucker, Antenna Theory: Part 2. New York: McGraw-Hill Education, 1969. [2] A. A. Olinear and R. C. Johnson, Leaky-Wave Antennas, Antenna Engineering Handbook, 3rd ed. New York: McGraw-Hill, 1993. [3] B. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [4] N. Engheta and R. Ziolkowski, Metamaterials: Physics and Engineering Explorations. Piscataway, NJ: Wiley-IEEE press, 2006. [5] A. F. Peterson et al., Computational Methods for Electromagnetics. New York: IEEE Press, 1998.

VAN ORDEN AND LOMAKIN: RAPIDLY CONVERGENT REPRESENTATIONS FOR PGFs OF A LINEAR ARRAY IN LAYERED MEDIA

[6] B. A. Munk and G. A. Burrell, “Plane-wave expansion for arrays of arbitrarily oriented piecewise linear elements and its application in determining the impedance of a single linear antenna in a lossy halfspace,” IEEE Trans. Antennas Propag., vol. 27, pp. 331–343, 1979. [7] A. W. Mathis and A. F. Peterson, “A comparison of acceleration procedures for the two-dimensional periodic Green’s function,” IEEE Trans. Antennas Propag., vol. 44, pp. 567–571, 1996. [8] G. Valerio et al., “Comparative analysis of acceleration techniques for 2-D and 3-D Green’s functions in periodic structures along one and two directions,” IEEE Trans. Antennas Propag., vol. 55, pp. 1630–1643, 2007. [9] H. Rogier, “New series expansions for the 3-D Green’s function of multilayered media with 1-D periodicity based on perfectly matched layers,” IEEE Trans. Microw. Theory Tech., vol. 55, pp. 1730–1738, 2007. [10] G. Valerio et al., “The array scanning method for the computation of 1D-periodic 3D Green’s functions in stratified media,” presented at the IEEE AP-S URSI, Toronto, 2010. [11] F. Capolino et al., “Efficient computation of the 2D Green’s function for 1D periodic layered structures using the Ewald method,” presented at the IEEE Antennas and Propagation Society Int. Symp., San Antonio, TX, 2002. [12] G. Valerio et al., “Efficient computation of mixed potential dyadic Green’s functions for a 1D periodic array of line sources in layered media,” presented at the Int. Conf. on Electromagnetics in Advanced Applications, Torino, Italy, 2009. [13] H. Rogier and D. De Zutter, “A fast converging series expansion for the 2-D periodic Green’s function based on perfectly matched layers,” IEEE Trans. Microw. Theory Tech., vol. 52, pp. 1199–1206, 2004. [14] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: IEEE Press, 1995. [15] M. I. Aksun and R. Mittra, “Derivation of closed-form Green’s functions for a general microstrip geometry,” IEEE Trans. Microw. Theory Tech., vol. 40, pp. 2055–2062, 1992. [16] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, 1973. [17] D. Van Orden and V. Lomakin, “Rapidly convergent representations for 2D and 3D Green’s functions for a linear periodic array of dipole sources,” IEEE Trans. Antennas Propag., vol. 57, pp. 1973–1984, Jul. 2009, 2009.

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[18] F. Capolino et al., “Efficient computation of the 2-D Green’s function for 1-D periodic structures using the Ewald method,” IEEE Trans. Antennas Propag., vol. 53, pp. 2977–2984, 2005. [19] M. I. Aksun, “A robust approach for the derivation of closed-form Green’s function,” IEEE Trans. Microw. Theory Tech., vol. 44, pp. 551–658, 1996.

Derek Van Orden (M’11) was born in San Francisco, California. He received the B.S. degree in applied physics from Rice University, Houston, Tx, in 2004 and the Ph.D. degree from the University of California, San Diego, in 2011. His research work is in computational and applied electromagnetics. His interests include periodic systems, surfaces structures, waveguiding structures and plasmonics.

Vitaliy Lomakin (SM’08) received the M.S. degree in electrical engineering from Kharkov National University, Ukraine, in 1996 and the Ph.D. degree in electrical engineering from Tel Aviv University, Israel, in 2003. From 1997 to 2002, he was a Teaching Assistant and Instructor in the Department of Electrical Engineering, Tel Aviv University. From 2002 to 2005, he was a Postdoctoral Associate and Visiting Assistant Professor in the Department of Electrical and Computer Engineering, University of Illinois at Urbana Champaign. He joined the Department of Electrical and Computer Engineering, University of California, San Diego, in 2005, where he currently holds the position of Associate Professor. His research interests include computational electromagnetics, computational micromagnetics/nanomagnetics, the analysis of microwave phenomena on structured surfaces, the analysis of optical phenomena in photonic nanostructures, the analysis of magnetization dynamics in magnetic nanostructures.

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A Novel Strategy for the Diagnosis of Arbitrary Geometries Large Arrays Aniello Buonanno and Michele D’Urso

Abstract—A general solution strategy for detecting faulty elements in phased arrays of arbitrary geometries is suggested. The proposed deterministic approach assumes as input data the amplitude and phase of near-field distributions and allows to determine the positions of the faulty elements. In particular, the method is founded on the well known Multiple Signal Classification (MUSIC) method, i.e., a spectral estimation technique. The proposed algorithm is also compared with a recently published method by the same authors, against experimental and numerical data. The results fully confirm the usefulness of the proposed technique, highlighting the advantages and the disadvantages of both methods. Index Terms—Antenna measurement, array diagnostics, inverse imaging, multiple signal classification (MUSIC).

I. INTRODUCTION AND MOTIVATIONS

T

HE identification of faulty elements in large (hundreds to thousands of elements) antenna arrays, e.g., radiotelescopes and radar, from complex field measurements is a problem of considerable practical and theoretical relevance [1]–[6]. The costs associated with maintaining continuous operation of such a sophisticated systems are rising as the functionality of the antennas and other equipment degrades as a result of age. With the ever-increasing sophistication of the antennas’ electronic subsystems, their life-cycle maintenance costs (related to per-unit testing and diagnosis cost of a faulty system and its recovery rate) could in the near future exceed the corresponding original capital investment. Frequently the fault detection and isolation tasks are performed by mission staff on a manual case-by-case basis. Most often, due to complexity of the system, diagnosis cannot be performed in real time, resulting in frequent loss of data acquisition. Indeed, the antenna diagnosis operation can be require multiple outages from a few minutes to several hours at times. The impact of such losses of data acquisition potentially could result in very difficult situations in critical operations (e.g., military radar). In order to know which element or elements are damaged, active antennas can include calibration systems. These systems

Manuscript received August 23, 2010; revised May 18, 2011; accepted July 20, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. The authors are with the Innovation Team, Intangible Capital Management Directorate, SELEX Sistemi Integrati S.p.A., I-80014 Giugliano di Napoli, Napoli, Italy. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173109

make an easy control of the system components, but it can fail if the calibration system is damaged too. Moreover, calibration systems can be also rejected because its inclusion means a critical increase in array volume, weight and costs. Due all this reasons, the necessity of development of an intelligent, comprehensive fault diagnosis unit becomes inevitable. Several deterministic and stochastic techniques have been developed [1]–[4] in the last years. Among the stochastic approaches, we point out the learning algorithms based on examples, such as neural networks [1], [2], and the genetic algorithms based approaches [3], [4]. These methods have the advantage to require small amount of samples of the radiated field and, in many cases, only amplitude data [2], but, due to the high size of search space, they can have poor performance. Note the diffused enthusiasm for physically inspired optimization techniques has induced to neglect the fact that all global optimization algorithms are limited in their performances by the computational cost required to get, within a given precision, the actual solution. This cost grows very rapidly with the number of unknowns [7], i.e., with the phased array antenna size. As a consequence, in large scale problems, due to the necessity of stopping the search after a given amount of flops, it is likely that only sub-optimal solutions will be generally achieved, which can be significantly worse than the actual optimal ones. Moreover, not only general global algorithms are computationally heavy: they are all essentially equivalent, as implied by the so called No Free Lunch Theorems [8]. These theorems state that a truly general-purpose universal optimization strategy does not exist [8]: on average the performances of any two optimization algorithms are the same across all possible optimization problems. Hence, for any algorithm, an elevated performance over one class of problems is exactly paid for in performance over another class. Now, for a given sufficiently general algorithm, neither it is practically possible to characterize the class of problems to which it is fitted, nor we can blindly refer to results obtained in a other area [8]. And so, the only way to devise an effective algorithm is to exploit the properties of the specific class of problems under consideration, thus possibly avoiding the use of global optimization schemes. In this paper we refer to deterministic methods. Among the deterministic methods, a simple and fast approach to estimate the array excitations from near-field measures is the Backward Transformation Method (BTM) [5], based on a proper exploitation of the Fast Fourier Transform (FFT) algorithm. Unfortunately the method can be only employed to planar arrays and measurement set-up. A different method has been proposed in

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[6] based on sources reconstruction method (SRM). It has been shown that the method in [6] is usable for arbitrary geometry source domain and field acquisition over arbitrary shape domains. In this contribution, we compare the method in [9], where a distributional formulation is exploited to linearize the problem and find the solution, with a novel method obtained re-interpreting the Multiple Signal Classification (MUSIC) published in [10] traditionally used to estimate the direction of arrival (DOA) of multiple plane waves. The main idea of this method is to assume that the incident signals and noise are uncorrelated. Thus it is possible to consider the subspace spanned by the noise orthogonal to the subspace spaced by the incident signals. We use the same information to solve the diagnostic problem at hand. In particular, differently to [9], where we determine the excitation coefficients of each radiating element and detect the failures by comparing the reconstructed ones with the nominal currents, herein we uptake the orthogonal proprieties of the signal and noise subspaces to estimate the faulty elements’ position. More in detail, by neglecting the mutual coupling between the radiating elements and by calculating the eigenvalues of the covariance matrix of the waveforms measured at the probe, we find the eigenvectors that belongs to the noise subspace. By calculating the spatial domain spectrum function, we estimate the position of each faulty radiating elements. As [9] and differently from traditional approaches [1], [6], the diagnostic tool herein proposed is free from eventual uncertainties on the measurement position due to not using the a priori knowledge of the antenna layout. Subspace methods in statistical signal processing [11], [12] are also used for detecting and locating targets from antenna array data. Moreover, in inverse scattering a comparison between the two different algorithms has been shown in [13]. The paper is organized as follows. In Section II, the MUSIC approaches are briefly recalled and applied to the problem at hand. In Section III the proposed method is experimentally compared to [9] for the case of a fully active phased array and then numerically tested for the case of a conformal array and spherical measurement system. Conclusions follow.

II. THE PROPOSED METHOD SCHEME The classical MUSIC algorithm [10] was proposed to improve the resolution of estimated direction of arrival of the incoherent waves incident on a linear array antenna system. Schmidt observed that the space of measured data can be decomposed into the direct sum , where the signal subspace is orthogonal to the noise subspace . In this section we apply the previous consideration to detect the faulty elements of a fully active phased array. Let and be the excitation coefficient and the electric-field radiation pattern of the -the radiating element, respectively (Fig. 1). A probe having effective height is placed in a known spatial point . We make no assumptions about the location of the antenna elements and, in particular, do not require them to lie in a plane or

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Fig. 1. Geometry of the problem showing the th radiating element position and the th measurement position of the probe.

be regularly spaced. The voltage at the probe output can be expressed as

(1) is the freewhere denotes the usual dot product, space wavenumber, and is the working wavelength. Moreover, and are the relative angles between the th measurement point and the th element position defined as

(2) with . Furthermore, we assume the system of targets is composed by identical radiating elements, so that . The aim of the problem at hand is to determine the locations of the faulty elements from a set of samples of amplitude and phase near-field radiation pattern, or values of voltages given by (1). To this aim we rewrite the measured voltage at probe as

(3) where is the number of correctly working radiating elements and is the number of faulty elements in the investigated phased array. Now, let us consider the matrix defined as (4)

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where represents the inner product,

and denotes the conjugation transpose. According to [10] we determine the signal subspace , i.e., the subspace spanned by the correctly working radiating elements, and the noise subspace . This subspaces can be revealed, for instance, via the singular value decomposition (SVD), which has the form where and are orthogonal matrices. In particular, as the matrix is an Hermitian one, it can be written and so . The columns of having nonzero eigenvalues span the signal subspace while the remaining columns span the null subspace . Note that the matrix only has one singular value different from zero, i.e., the signal subspace is only one-dimensional. This means the signal and noise subspace are simply distinguished; we a priori know the mono-dimensional property of the signal subspace, which is also the reason because we do not show the plot of the singular values. On the other hand, this is also a limitation in term of maximum number of detectable sources (disappearance of some of the sources may happen) or, equivalently, in term of achievable resolution. This is due to the fact that, increasing the number of sources the dimensionality of signal subspace does not increase (only one eigenvector). This means that the contribution of each sources decreases as it is shared by the same eigenvector representing the signal subspace. In other words, only a part of the vector representing the signal source is orthogonal to the noise subspace (i.e., the signal subspace, spanned by the “single” eigenvector having nonzero eigenvalue, is combination of all the signal sources). Nevertheless, for the considered application, this orthogonality property of the two defined subspaces, can be exploited to provide an implicit characterization of the faulty element locations. Indeed, observing that the faulty elements belong to the null subspace , we can detect this elements by means of the function (5) where

is a known function defined as (6)

with

(7) , , and the spatial domain containing all the radiating elements. The function defined in the (5) will peak (theoretically to infinity) at each correctly working radiating elements position. As the faulty elements belong to the null subspace this elements will reconstruct lower than the correctly working radiating elements. Clearly, due to the model describing the faulty elements, a radiating element having an excitation coefficient next to null will be reconstruct with intensity near to the faulty elements.

Fig. 2. The adopted experimental set-up and the considered phased array layout.

III. TESTING AGAINST EXPERIMENTAL AND NUMERICAL DATA As first example, the proposed diagnostic tool has been tested for the case of horn elements located as shown in Fig. 2. The data for the inversion have been collected using a traditional planar scanning placed at distance from the antenna under test aperture, and a fixed number of radiating elements have been forced off. The dimensions of the spatial domain and of the measurement plane are input data for the problem, see Fig. 2. The voltages (amplitude and phase) have been collected in points. The probe is a small horn antenna. All the measurement set-up is available at the Research and Development Department of Selex Sistemi Integrati. As first step, let us observe the desired and measured radiation patterns, see Fig. 3. The first one has been achieved by using a Taylor current distribution such to fulfill the design constraints. The measured one is obtained by standard near-field far-field transformation [14] starting from the measured near-field data. As it can be seen, according to the above hypothesis, the measured radiation pattern and the theoretical one are sensibly different. The overall pattern shapes and the sidelobes level are different, and a proper investigation aimed to detect possibly faulty elements becomes mandatory. To this aim, the measured near-field data have been first processed by using the method in [9]. Fig. 4 shows the amplitude distributions of the excitation coefficient. Then, the available data have been processed by adopting the diagnostic tool herein proposed, see Fig. 5. As it can be seen for this case (planar geometry and planar measurement system), the two methods obtain the position of faulty elements. Both the methods allow to identify that the main problem is an error of the elements of two semi-row of the phased array antenna under test. More in detail, the herein proposed method allows to achieve a better resolution than [9] discerning more clearly the faulty elements of two semi-row, see Fig. 5. It is even more evident by looking at the central element of the array under test that is not fed. Indeed, different to Fig. 4, this element can be clearly discerned in Fig. 5. Nevertheless, differently to the proposed method that “just” allows to localize the faulty elements, the tool in [9] has the capability of detecting array failure, as well as the possibility to evaluate the actual currents. We underline that, in the considered cases and in the proposed applications, the sources are not strongly correlated being the

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Fig. 5. Reconstructed faulty elements distributions by using the proposed method.

Fig. 3. A comparison between the theoretical radiation pattern (blue solid line) and the measured one (red dot-line) of the phased array layout in Fig. 2. (a) Azimuth cut; (b) elevation cut.

Fig. 4. Reconstructed amplitude distributions by using the method in [9].

mutual coupling between the radiating elements is negligible (the achieved experimental result is very interesting). Of course the correlation degrades the performance of the proposed algorithm (due to the mono-dimensional property of the signal subspace).

Moreover, both Figs. 4 and 5 diagnostic a mismatch along the -axis between the system reference centered on the AUT and the system reference centered on the measured domain proving the robustness against unknown measurement position errors. Note that according to the previous remarks the external radiating elements having the lowest excitation coefficient (Taylor current distribution) are reconstruct with a intensity level next (not equal) to the faulty ones (see Fig. 5). With regards to the computational complexity of the herein proposed algorithm note that the method employs the eigendecomposition of the matrix having elements that means a complexity order of . Note that also the method proposed in [9] requires the employment of the TSVD factorization. In particular, in [9] the matrix to factorize has elements where is the number of measures and is the number of considered pixels in the investigated domain. This means that, when the number of considered pixels is less than the number of measures, the method in [9] has a smaller computational cost. Note that the number of considered pixels can be chosen only according to the achievable resolution and the size of the investigation domain since the position and the number of radiating elements has been considered unknown. On the base of this consideration, the computational cost of the two methods can be considered similarly quite expensive but it does not represent, in both cases, a very relevant drawback considering the overall time for the array diagnosis. Different methods that are computationally less expensive can be considered in the literature. A good example is the Landweber iterative regularization method, that is characterized by a computational complexity of order , where is the number of iterations. However, differently from the TSVD based method where the regularization parameter can be easily chosen, the Landweber based algorithm has some regularization properties related to the number of iterations and it is not easy, in general, to estimate a priori the ill-conditioning of the system, especially for conformal arrays. As consequence, the choice of the best algorithm depends on the degree of confidence about the regularization parameter, i.e., the stopping rule. Also, the Landweber iterative regularization method requires the estimation of the relaxation parameter that affects the velocity of convergence of the algorithm.

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Fig. 7. Reconstructed amplitude distributions by using the method in [9]. The red circles show the position of the forced off radiating elements.

Fig. 6. The adopted measurement configuration (a) and the considered conformal phased array layout (b) for the numerical simulation. The red circles show the position of the forced off radiating elements.

In order to point out the effectiveness of the proposed approach for generic geometry arrays and measurement systems, we consider the case of a conformal array composed by Hertzian dipole located as shown in Fig. 6(b) (the large black dot). The data for the inversion have been simulated by using Computer Simulation Technology Microwave Studio (CST MWS) [15], a commercial full wave 3D electromagnetic (EM) simulator based on the Finite-Integration Technique. In particular, the spatial domain is a bounded surface domain of a cylinder described by equations , , and the measurement domain is a bounded surface domain of a sphere described by equations , , (see Fig. 6). The voltages (amplitude and phase) have been collected in points of the measurement domain . The red circled radiating elements in Fig. 6(b) have been forced off and the remaining ones are excited by a unitary current amplitude. The calculated data have been processed by adopting both [9] and the proposed diagnostic tool (see Figs. 7 and 8). As it can be seen, all the turned off elements are well localized for both the methods. Once more the result fully confirms the capability of the proposed method to obtain a better resolution than [9] and the usefulness for more complex geometry arrays and measurement systems.

Fig. 8. Reconstructed faulty elements distributions by using the proposed method. The red circles show the position of the forced off radiating elements.

IV. CONCLUSION A novel and effective deterministic method for detecting faulty elements in large phased arrays of generic geometry has been proposed and tested against experimental data measured at Selex Sistemi Integrati and numerical data obtained using CST MSW simulation tool. The proposed strategy assumes as input the amplitude and phase of the near-field distributions and allows to determine the locations and the number of faulty elementary antennas. By properly re-interpreting the classical MUSIC algorithm, the faulty radiating elements are localized by using their property to belong to the null subspace previously defined. The method has been compared to [9] showing that it allows to obtain a better resolution even if it cannot be used to the quantitative analysis of the actual currents. The achieved results fully confirm accuracy, usefulness and robustness against noise of the proposed technique. The main disadvantage is the computational effort required by the technique. However, the overall computational time required for the examples reported in this paper does not increase significantly the overall time required for the array diagnosis taking into account that the measurement time to collect near-field data is several hours. For future works, we are developing an extension to resolve this drawback based on a spatial smoothing procedure that allows to extend the dimension of the signal subspace improving the achievable performances also in the cases of correlated sources.

BUONANNO AND D’URSO: A NOVEL STRATEGY FOR THE DIAGNOSIS OF ARBITRARY GEOMETRIES LARGE ARRAYS

REFERENCES [1] A. Patnaik, B. Chowdhury, P. Pradhan, R. K. Mishra, and C. Christodolou, “An ANN application for fault finding in antenna arrays,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 775–777, Mar. 2007. [2] G. Castaldi, V. Pierro, and I. Pinto, “Efficient faulty elements diagnostics of large antenna arrays by discrete mean field neural nets,” PIER, Progress in Electromagnetics Research, vol. 25, pp. 53–76, 2000. [3] B. Yeo and Y. Lu, “Array failure correction with a genetic algorithm,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 823–828, May 1999. [4] J. A. Rodriguez, F. Ares, H. Palacios, and J. Vassal’lo, “Finding defective elements in planar arrays using genetic algorithms,” in Progress in Electromagnetics Research, PIER 29, J. A. Kong, Ed. Cambridge, MA: EMW Publishing, 2000, ch. 2, pp. 25–37. [5] J. J. Lee, E. M. Ferren, D. P. Woollen, and K. M. Lee, “Near-field probe used as a diagnostic tool to locate defective elements in an array antenna,” IEEE Trans. Antennas Propag., vol. 36, no. 6, pp. 884–889, Jun. 1988. [6] Y. Alvarez, F. Las-Heras, B. A. Dominguez-Casas, and C. Garcia, “Antenna diagnostics using arbitrary-geometry field acquisition domains,” IEEE Antennas Wireless Lett., vol. 8, pp. 375–378, May 2009. [7] A. S. Nemirovsky and D. B. Yudin, Problem Complexity and Method Efficiency in Optimization, ser. Interscience Series in Discrete Mathematics. New York: Wiley, 1983. [8] D. H. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Trans. Evol. Computat., vol. 1, pp. 67–82, 1997. [9] A. Buonanno, M. D’Urso, M. Cicolani, and S. Mosca, “Large phased arrays diagnostic via distributional approach,” PIER, Progress in Electromagnetics Research, vol. 92, pp. 153–166, 2009. [10] R. O. Schmidt, “Multiple emitter location and signal parameter extimation,” IEEE Trans. Antennas Propag., vol. AP-34, pp. 276–280, Mar. 1986. [11] C. Therrien, Discrete Random Signals and Statistical Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 1992, 18. [12] P. Stoica and R. Moses, Introduction to Spectral Analysis. Englewood Cliffs, NJ: Prentice Hall, 1997. [13] R. Solimene, A. Buonanno, and R. Pierri, “Comparison between two methods for small scatterer localization,” presented at the EUCAP 09 Conf., Berlin, Germany, Mar. 23–27, 2009, ISBN 978-3-8007-3152-7. [14] C. A. Balanis, Antenna Theory, 2nd ed. New York: Wiley, 1997. [15] CST Studio Suite 2008. [Online]. Available: http://www.cst.com

Aniello Buonanno received the Electronic Engineering Master degree (summa cum laude) and the Ph.D. degree from the Second University of Naples, Italy, in 2005 and 2008, respectively. He joined the Applied Electromagnetic Group at the Department of Information Engineering of Second University of Naples. He is currently with the Innovation Team of the SELEX Sistemi Integrati, an International Company working on Homeland Security and Homeland Defence Systems, as well as the development of radar for border control systems.

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He is the author and co-author of more than 40 papers, published on international scientific journals or in the proceedings on international conferences. His scientific interests are devoted to forward and inverse electromagnetic scattering methods for through wall imaging (TWI), ground penetrating radar (GPR), and life sign detection (LSD). Recently, he started to work on energy harvesting problems, integrated systems based on robotics and time domain antenna array design. Dr. Buonanno was awarded the EuWiT Young Engineering Prize in 2009.

Michele D’Urso was born in 1976. He received the Telecommunication Engineering Master degree (summa cum laude) and the Ph.D. degree from the University Federico II of Naples, Italy, in 2002 and 2006, respectively. After graduation, he joined the Applied Electromagnetic Group at the Department for Electronics and Telecommunication Engineering of the University Federico II in Naples, first as an Associate Researcher and then as a Ph.D. candidate, from 2003 to 2005. From September 2004 to January 2005, he was an intern in the Mathematics and Modelling Department of Schlumberger-Doll Research, Ridgefield, CT, under the supervision of Prof. T. Habashy and Dr. A. Abubakar. After 2006 he was an Associate Researcher at the University Federico II of Naples and then an Associate Professor, first at the University Mediterranea of Reggio Calabria and then at the University of Cassino, Italy. He is now the Director of a research team, the Innovation Team, working in SELEX Sistemi Integrati S.p.A, a Finmeccanica Company. His scientific interests are devoted to forward and inverse electromagnetic scattering methods, and to innovative strategies and efficient algorithms for array antenna synthesis problems, including time modulated and timed arrays. Recently, he started to work on energy harvesting problems, integrated systems based on robotics platforms. He is author and co-author of more than 100 papers, published on international scientific journals or in the proceedings on international conferences. Dr. D’Urso was the recipient of the G. Barzilai Award of the Italian Electromagnetic Society (SIEM) in 2004. He has also received the Best Young Research Paper Award at the European Wireless Technology Conference (EuWiT) in 2009. He is also the recipient of the (SELEX Sistemi Integrati) Premio Innovazione (Innovation Award) 2009 and 2010.

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Predicting Sparse Array Performance From Two-Element Interferometer Data James A. Nessel, Member, IEEE, and Roberto J. Acosta

Abstract—Widely distributed (sparse) ground-based antenna arrays are being considered for deep space communications applications with the development of the proposed Next Generation Deep Space Network. However, atmospheric-induced phase fluctuations can impose daunting restrictions on the performance of such an array, particularly during transmit and particularly at Ka-band frequencies, which have yet to be successfully resolved. In this paper, an analysis of the uncompensated performance of a sparse antenna array, in terms of its directivity and pattern degradation, is performed utilizing real data. The theoretical derivation for array directivity degradation is validated with interferometric measurements (for a 2-element array) recorded at Goldstone, CA, from May 2007—May 2008. With the validity of the model established, an arbitrary 27-element array geometry is defined at Goldstone, CA, to ascertain its theoretical performance in the presence of phase fluctuations based on the measured data. Therein, a procedure in which array directivity performance can be determined based on site-specific interferometric measurements is established. It is concluded that a combination of compact array geometry and atmospheric compensation is necessary to minimize array loss impact for deep space communications. Index Terms—Arrays, phase noise, propagation measurements, sparse array antennas.

I. INTRODUCTION

A

S the monolithic, large aperture (70-m) antennas of the Deep Space Network (DSN) are proving costly to operate/maintain and represent single points of failure, the concept of smaller aperture (34-m and below) ground-based antenna arrays for deep space communications applications is currently being considered by NASA as a means to offset these high maintenance costs, provide a graceful degradation of performance, and meet the stringent reflector surface accuracy requirements to realize higher frequency operation (Ka-band). However, the large separation distances necessary for aperture-based antenna arrays are subject to the increased influence of phase fluctuations induced by propagation through the atmosphere. This phenomenon is primarily a result of large amounts of inhomogeneous distributions of water vapor exposed to turbulent air flow

Manuscript received September 02, 2010; revised March 14, 2011; accepted June 04, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported in part by NASA’s Space Communications and Navigation (SCaN) Program under the Space Operations Mission Directorate (SOMD). The authors are with the National Aeronautics and Space Administration Glenn Research Center, Cleveland, OH 44135 USA (e-mail: [email protected]; roberto.j.acosta @nasa.gov). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173110

conditions in Earth’s upper atmosphere (troposphere), which directly leads to variations in the effective electrical path length (phase) of a received signal on spatiotemporal scales [1]–[3]. Such variations are seen as additional ‘phase noise’ and will inherently degrade the performance of antenna arrays. The radio science community has had to deal with this same issue for quite some time, but whereas radio science applications impose only receive-mode requirements (i.e., imaging) and observations are made on the order of minutes to hours (long integration times), communications applications require both transmit and receive capabilities, as well as real-time corrections at sub-second time scales. In the receive case, adaptive techniques have been utilized by the DSN since the 80’s to compensate for the atmosphere at frequencies up to X-band [4]. More recently, transmit arraying of a 7.15 GHz signal was successfully demonstrated in an experiment with the Mars Global Surveyor [5]. However, since atmospheric phase noise scales with frequency, at Ka-band (the frequencies of interest for future NASA DSN operations) the problem becomes much more severe and has yet to be successfully resolved, particularly in the uplink (transmit) scenario. In this paper, an analysis of the performance of a sparse antenna array, in terms of its directivity and array pattern degradation, is performed. The theoretical derivation for array directivity degradation is validated with real interferometric measurements recorded by a 2-element array at Goldstone, CA. Excellent agreement between theory and measurement is observed for the 2-element array case. With the validity of the model established, an arbitrary 27-element array geometry is defined at Goldstone, CA, to predict its performance based on the measured interferometric phase data. II. THEORY A. Array Directivity (With Deterministic Phase Difference) Consider a widely distributed array of reflector antennas arbitrarily spaced on a plane whose geometry is defined in Fig. 1, where is a unit vector in the direction of propagation, is the distance from the array origin to the th element, is the distance the signal travels from the array origin to the receiver, and is the distance the signal travels from the th element to the receiver. Since the results of our analysis are concerned with the evolution of primarily the main pencil beam (small off boresight) of large (with respect to the wavelength) apertures, from aperture field theory, the far field electric field of a paraboloidal reflector element, , can be accurately described by the Fourier transform of the electric field over the aperture plane [6]. If we assume, for analytical simplicity, that no blockage occurs and

U.S. Government work not protected by U.S. copyright.

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From the derivation for the peak directivity of the array, it is readily observed that for array elements coherently added in phase, a directivity equal to times the individual element directivity results, as expected for the case of large separation distance between elements. B. Array Directivity in Presence of Random Phase

Fig. 1. Geometry of a planar array of radiating elements located at arbitrary positions with arbitrary polarization. Inset: determination of excess path delay between array elements as a result of array geometry.

that the reflectors are uniformly illuminated by a linearly polarized feed with constant plane amplitude and phase excitation

The above analysis assumes a constant deterministic phase between elements, but if we now suppose that random phase fluctuations are present during signal transmission, such as those induced by water vapor in the atmosphere, the statistical distribution of the random process can be utilized to obtain a closedform solution. Let us assume (and later confirm this assumption) that the phase fluctuations induced by the atmosphere are normally distributed over the time scale of interest with mean zero and variance . The average peak directivity of an array, in terms of the statistical ensemble average of the random phase fluctuations between elements and , assuming independent distributions, can be determined, in closed form, as

(1) (4a) is the complex excitation coefficient of the rewhere flector feed, is the wave number, is the radius of the circular aperture, and the symmetry of the resultant field about is used. Translating this analysis to an array environment, an array of identical antennas will produce a far field electric field that will consist of the superposition of each individual element field plus a propagation delay, as determined by the geometry, relative to some specified origin (see inset of Fig. 1). This will result in an array far field, , which can be represented by the product of the individual element far field and an array factor [7]

(2) where, ; unit vector in direction of propagation; position vector of element relative to array origin;

(4b)

(4c) (4d)

, the classical Eulerian transformation where is employed in (4b), and the odd symmetry of the function is utilized in (4c). If we take the special case of independent and identically distributed random phase errors between antenna elements of , as it can be shown that the result in (4d) reduces to a generalization of the well known Ruze equation [8] where (5a) is derived from the fact that for and (by baseline) when

. From the far field array pattern, the radiation intensity, , and radiated power, , of an array can be determined, whose ratio defines the array directivity. If we assume the antenna elements are spaced many wavelengths apart, , then the peak and are pointed to zenith array directivity, , as a function of deterministic phase difference between elements and , given by , can be exactly described by

(5a) (5b) (5c) (5d)

(3a)

The average array loss can be described by the difference (in dB) of the ideal directivity and the actual directivity achieved in the presence of the phase fluctuations

(3b)

(6)

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III. VALIDATION

Fig. 2. Average synthetic array pattern based on a 27-element array with ele, spacing , for various ment aperture radius rms phase errors. Array geometry is as defined in Fig. 9(a).

For a two-element fies to

array, the above equation simpli-

The above theoretical derivation for array loss can be considered exact, so long as the original assumption that differential phase fluctuations induced by the atmosphere are zero-mean, normally distributed random variables is true. Therefore, before further analysis can proceed, we first validate this assumption by investigating the probability density function (PDF) of differential phase fluctuations, as measured by a two-element site test interferometer currently deployed in Goldstone, CA. The two-element interferometer developed by NASA Glenn Research Center utilizes a digital I and Q receiver which monitors an unmodulated beacon signal at 20.199 GHZ from a geostationary satellite, Anik F2, with a baseline separation of 256 m and an elevation angle of 48.5 deg. A localized 10 MHz GPSdisciplined rubidium oscillator provides the reference timing for all operations and data collection. A more detailed description of the system hardware and setup can be found in [9]. The signal is sampled at 3.64 MHz with an integration time of 144 ms and recorded every second. Since our measurements limit the resolution to which we can observe phase fluctuations to time scales greater than 1 second, we must make several assumptions as to the characteristics of these fluctuations at time scales comparable to a symbol period , the time scale of interest for communications applications. A. Statistics of Phase Fluctuations

(7a)

(7b) (7c) The preceding analysis determines a closed form solution to the average array loss factor in the direction of signal reception/transmission. Further insight into array performance may be gained by investigating how random phase fluctuations affect the evolution of the array directivity pattern, in a statistical sense. To do so, we repeat the previous derivation utilizing the spatial dependence of . Recognizing that the only random component of the formulation occurs in the differential phase term, , we obtain the following relationship

(8) A plot of the perturbed directivity pattern for various rms phase errors, as seen in Fig. 2, thus indicates an avg. pattern which possesses a noticeable broadening of the beamwidth of the main lobe. This is expected, as the perturbed pattern represents an average over all possible perturbed beam configurations based on a zero-mean Gaussian phase distribution.

A representative plot of the calibrated differential phase, as measured by our two-element interferometer, is shown in Fig. 3 for September 1, 2007 1. We analyze the statistics for a block of data from 02:00 to 03:00 GMT (where the atmosphere appears to be mildly turbulent) and 12:00 to 13:00 GMT (where the atmosphere appears to be calmer) at different time scales. Fig. 4 shows the PDF at time scales of 1 hr, 30 min, and 10 min for each of these times. From the plot, the statistics do appear to follow a zero-mean normal distribution as the analysis is extended to shorter and shorter time periods, but notice that the rms of the phase fluctuations as we move towards shorter time scales does possess some variance. Since the rms phase is the metric by which array degradation is measured, characterizing these changes at time scales comparable to a symbol period is tantamount to determining array performance for communications applications. From this analysis, we validate that the distribution of phase fluctuations induced by the atmosphere is indeed Gaussian for time scales larger than approximately 10 minutes. Due to the low sampling frequency, we assume that the trend observed in the statistics for this process continue to extend to shorter time scales; that is, at time scales comparable to a symbol period, the PDF of phase fluctuations remains a zero-mean, normally distributed process. To truly verify this assumption, the sampling rate of recorded phase data would need to be increased to sub-second intervals. 1Note: This day was arbitrarily chosen from a set of data blocks that possessed no erroneous data points. The statistics derived from this data is representative of all data collected, thus far.

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Fig. 3. Differential phase time series measured by the two-element site test interferometer at Goldstone, CA on Sep. 1, 2007.

Fig. 4. PDF’s of phase fluctuations for 1 hr (top left, 1), 30 mins (top right, 2), and 10 mins (bottom center, 6) at (a) 02:00–03:00 GMT and (b) 12:00–13:00 GMT on 9/1/07.

B. Two-Element Array Loss: Measurement Versus Predicted To compare the theoretically predicted array loss with the measured array loss for a two-element array, we assume that the differential phase statistics (mean, standard deviation, normal distribution) effectively describe the random process over the time scales of interest. In this way, we can directly compare the time-averaged directivity with the ensemble average directivity loss determined above. From the equation for peak directivity for a two-element array (9) The time-averaged directivity for a given time interval, can be described by

,

(10a) (10b) where

.

Returning to the data measured at Goldstone on September 1, 2007 (see Fig. 3), a plot of the instantaneous directivity for a 2-element array and the time-averaged directivity is shown in the upper and lower portions of Fig. 5, respectively. From the plot, it is observed that instantaneous phase errors induced by the atmosphere have the potential to completely nullify the signal power in the intended direction (anomalous refraction), if the integration time of the signal is on the order of 1 sec. However, since it is assumed that phase fluctuations are zero-mean normally distributed down to infinitesimally short time scales, the average directivity on the order of a symbol period is a more accurate determinator of array performance. From the lower portion of Fig. 5, the 10-min. time-averaged directivity is compared with the theoretically derived directivity for 2 elements, as predicted by (9). Over the course of the day, we observe very good agreement between measurement and theory. The plot of Fig. 6 shows the comparison between the predicted ensemble average array degradation, , for a given rms phase and the measured time-averaged directivity loss, , for the two-element array at Goldstone, CA, for the entire year. As 10 minutes is the smallest interval in which we possess enough data points to establish (verify) a normal distribution, 10 minute averages were used. From the plot,

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Fig. 5. Plot of instantaneous directivity time series of the 2-element array (upper) and the time-averaged directivity over 10-min. blocks compared to the theoretically derived directivity for the 2-element array (lower). Analysis utilizes measured phase data from Goldstone, CA, for September 1, 2007.

IV. PREDICTING EXPECTED ARRAY PERFORMANCE A. Phase Statistics at Different Baselines In order to utilize measured two-element interferometric data at a particular site to predict the performance of an -element array, some background into turbulence theory is required. The turbulent atmosphere consits of large scale structures where kinetic energy in the form of convection (i.e., solar radiation) or friction (i.e., rough boundaries) is constantly transfered down to the creation of smaller scale structures (eddies) of varying spatial scales [10]. In 1941, Andrey Kolmogorov developed a mathematical formalism to describe this turblence in a statistical manner, the structure function. A structure function describes the distribution of the variance of a particular parameter (e.g., temperature, phase, etc.) as a function of the spatial (or temporal) scale [11]. In this manner, a spatial phase structure function can be described by (11) Fig. 6. Measured versus theoretical array loss for varying rms phase as recorded from May 2007—May 2008 at Goldstone, CA.

we observe excellent agreement between the two curves, indicating the correctness of the theoretical derivation for array loss in the presence of atmospheric-induced phase fluctuations and the justification to utilize differential phase statistics to predict overall array performance for an abritrary geometry. Deviations from the theoretical curve are likely due to the lack of resolution to effectively determine a normal distribution over that time scale.

is the phase variance, is the phase at a point where de, is an arbitrary baseline separation distance, and notes the ensemble average of the enclosed quantity over all baselines. Thus, the spatial phase structure function simply describes how the variance in phase scales with distance. However, measuring the phase at the necessary number of baselines to obtain statistical relevancy requires many antennas at various separation distances and would be difficult to implement. Instead, if one assumes a frozen phase screen model (Taylor hypothesis), that is, that the statistics of the turbulence remain “frozen” for a significant amount of time and is advected across an array by the mean wind speed, one can relate temporal and

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Fig. 7. Temporal root phase structure function for the first 10-min block of data recorded on September 1, 2007 at Goldstone, CA.

spatial fluctuations over short time scales with a simple Eulerian transformation between baseline length and wind speed via (12) where is the velocity of the phase screen, and the factor of is merely a result of differencing two phases at two different times [12]. Therefore utilizing (12) in combination with the square root of the temporal phase structure function (13)

Fig. 8. CDF of rms phase for Goldstone, CA during first year of data collection for zenith pointing angle and 32 GHz frequency of operation.

which predicts an rms scaling factor of [14]. A more accurate representation of the scaling factor could be obtained if an interferometer element were added at a distance . This change in scaling factor is a direct result of the three-dimensional nature of the turbulence that is evident within the inner scale of the turbulence height (where ) versus the two-dimensional approximation of turbulence as the antennas are separated at appreciable fractions of the vertical extent of the turbulence layer (where ) [16].

provides the tool to scale the rms phase from one baseline to any other baseline. Fig. 7 shows the temporal root phase structure function for the first 10-min of the data set recorded September 1, 2007. From this plot, it is evident that a power law relationship exists for the phase difference at short time scales. Eventually, as the time scales approach the array crossing time of the phase screen ( sec for this particular 10-min time period), the rms phase saturates. Tatarski has shown that this scaling factor, as determined from the temporal structure function, can be used to scale the rms phase to other baselines (up to the height of the turbulent layer, ) via the relationship [13]

For varying elevation angles, it is assumed that the phase fluctuation statistics scales with the air mass through which the signal propagates [17], [18]. Therefore, the interferometric phase data is first scaled to zenith, from which it can be scaled to any other elevation angle through the equation

(14)

(15)

where is the slope of the temporal root phase structure func, tion at short time scales. From the slope in Fig. 7, which is close to the Kolmogorov theoretical value of 5/6 (0.8) for thick atmospheric layers. This method thus provides the basis in which two-element interferometer phase data statistics can be utilized to determine -element array performance. Over each block of time (in our case, 10-min), the slope of the root phase structure function can be determined and the rms phase over that same period of time can then be extrapolated to the other baselines, based on the geometry of the particular array. It should be kept in mind that this scaling factor is only applicable up to the thickness of the turbulent layer , which is specific to a particular site [14]. For Goldstone, CA, radiosonde measurements indicate [15]. For baseline distances greater than this layer thickness , our two-element interferometer has no spatial scaling information, so the Kolmogorov turbulence theory approximation must be used,

where is the measured rms phase at the fixed elevation angle, , and is the elevation angle at which the phase statistics are desired. Typically, for site comparisons, it is adequate to know the statistics corresponding to zenith , from which other elevation angles can be readily derived.

B. Phase Statistics at Different Elevation Angles

C. Phase Statistics at Different Frequencies As the refractive index of the atmosphere changes due to the turbulent field, path length fluctuations are imposed upon the propagating signal. These path length fluctuations will be independent of the frequency of operation, since the atmosphere is non-dispersive away from line centers [19]. Therefore, the phase fluctuation (in deg) associated with a given path length fluctuation, , is given by (16a)

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Fig. 9. (a) Model array geometries for Goldstone, CA, array loss calculation example with 250-m baseline geometry (o) and a 50-m baseline geometry (x), (b) resulting theoretical array loss versus rms phase curve for zenith pointing angle and 32 GHz frequency of operation, and array loss curves for (c) 250-m baseline geometry and (d) 50-m baseline geometry for various elevation angles as a function of zenith-determined rms phase.

and will scale with frequency

up to

as

(16b)

D. Procedure to Predict Array Performance To predict the directivity degradation of an array at a particular site in which phase data has been recorded, the following procedure is followed, where it is assumed that the interferometer baseline is less than the scale height of the turbulent layer. If this is not the case, then the theoretical scale values from Kolmogorov turbulence theory should be used where appropriate. For statistically valid results, phase fluctuations should be measured at a site for at least one year to ensure seasonal variations are taken into account.

STEP 1: Define a block size (period of time) over which sufficient data collection has occurred to ensure statistical validity (in our case of 1 Hz sampling, 10 minutes was validated as possessing enough samples to demonstrate zero-mean Gaussian phase noise). STEP 2: Calculate rms phase statistics, the temporal root phase structure function, and the slope of the structure function (at time scales less than the crossing time of the array) over each block. STEP 3: Scale the rms phase to the desired baseline length, elevation angle, and frequency of operation of the array via

(17) STEP 4: Calculate average array directivity from (4d).

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Fig. 10. (top) time series phase fluctuations, (middle) rms phase time series, and (bottom) calculated array loss for model arrays for July 25, 2007 measured data at Goldstone, CA, for zenith pointing angle and 32 GHz frequency of operation.

V. TYPICAL ARRAY PERFORMANCE AT GOLDSTONE, CA To determine the typical array loss at Goldstone, CA, we generate the cumulative distribution function (CDF) of the rms phase based on 1-yr data collected (May 2007—May 2008). To normalize our analysis, the data has been transformed to zenith and an operating frequency of 32 GHz. Recall the fixed interferometer baseline for the measured phase is 256 m. The resulting zenith rms phase CDF is shown in Fig. 8. From the CDF, we observe that 90% of the time (10% time exceedance), the rms phase for the two-element interferometer is better than 26.3 deg (at zenith). We extrapolate this data to elements for a particular array geometry by scaling the rms phase to different baselines, as described in the previous section. Note that the results of this analysis will be extremely geometry dependent [20]. As a simple example, let us consider an array geometry similar to the Very Large Array (VLA) in Socorro, NM, first, with an antenna spacing of 250 m between individual elements (blue circles in Fig. 9(a)), and one with a spacing of 50 m (red x’s in Fig. 9(a)). Since maximum directivity is only

a function of the number of elements (in widely-spaced arrays), these two geometries can be readily compared. In our analysis, we assume the theoretical Kolmogorov turbulence root phase and 1/3 structure function exponent of 5/6 [21]. We further assume that the average rms phase between antenna elements is similar for identical baseline separations, regardless of orientation or reference. Calculating the array loss curve based on the theoretical derivation for zenith pointing angle (Fig. 9(b)) and various other elevation angles (Fig. 9(c) and (d)), we observe that for the 250-m baseline array geometry in Goldstone, CA, we will need a margin of approximately 3.2 dB at zenith (13.9 dB at 10 elevation) to maintain 90% availability. This margin can be reduced, by reducing the baseline separation to the 50-m geometry, which only requires 0.6 dB at zenith (7.3 dB at 10 elevation). This is due entirely to the fact that small-scale fluctuations will contain much less energy than larger scale fluctuations, which would directly impact large baseline arrays. Thus, for communications applications, it will be desirable to maintain the most compact geometry possible to

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minimize array loss due to atmospheric phase fluctuations, as the furthest extent of the array will dominate this factor. To investigate the transient behavior of the model array, we can analyze the time-domain array performance for a particularly turbulent day at Goldstone, CA. Fig. 10 shows the phase fluctuations observed by the two-element interferometer on July 25, 2007, as well as the resulting rms phase and calculated array loss (for both the wide and compact array geometries described above) for zenith pointing angle and 32 GHz operational frequency. During extremely turbulent times (beginning of the GMT day), array degradation can exceed 10 dB with a mean array loss of 6.5 dB for the entire day (250-m baseline geometry). An approximate 4.5 dB improvement, on average, can be realized for the more compact array design (50-m baseline geometry). VI. CONCLUSIONS Herein we report on the theoretical performance of a sparse array whose signal degradation is primarily due to atmosphericinduced phase fluctuations. A procedure which utilizes interferometric phase measurements, combined with some requisite knowledge of the scale height of the turbulent layer, is established to predict array performance at a given site. The expected degradation in directivity of an element array in the presence of phase noise was derived theoretically and validated with measured data. Further, it is verified that the measured phase differential between two elements is indeed normally distributed (to the resolution limits defined by the experimental setup). Based on the theoretical analysis, it is observed that the directivity performance of an array in the presence of atmospheric-induced phase fluctuations is limited by the furthest extent of the array elements, and this geometry should remain as compact as possible to maximize array gain. That being said, optimal array performance may not necessarily coincide with maximum array gain, as the array signal-to-noise ratio would provide a better figure of merit for quantifying array performance for communications applications. There will inherently exist some trade-off in array spacing and optimal SNR, which is fundamentally determined by the physical location of additional noise sources within the array antenna pattern for the given geometry. Finally, the time series performance of an arbitrary array is shown for a particularly turbulent atmospheric day. For the geometries described herein, as well as for low elevation angles, there is still significant array losses observed and to prevent these losses, some form of compensation is necessary, particularly for uplink arraying to be viable.

[2] O. P. Lay, “The temporal power spectrum of atmospheric fluctuations due to water vapor,” Astron. Astrophys. Suppl. Ser., vol. 122, pp. 535–545, 1997. [3] O. P. Lay, “183 GHz radiometric phase correction for the Millimeter Array,” MMA Memo 209, 1998. [4] D. Rogstad, A. Mileant, and T. Pham, Antenna Arraying Techniques in the Deep Space Network. Hoboken, NJ: Wiley, 2003, pp. 99–109. [5] V. Vilnrotter, D. Lee, T. Cornish, R. Mukai, and L. Paal, “Uplink arraying experiment with the Mars Global Surveyor spacecraft,” IPN Prog. Rep. 42-166, pp. 1–14, Aug. 2006. [6] R. E. Collin, Antennas Radiowave Propagation. New York: McGraw-Hill, 1985, pp. 168–207. [7] C. A. Balanis, Antenna Theory Analysis and Design, 2nd ed. Hoboken, NJ: Wiley, 2002, pp. 309–314. [8] J. Ruze, “Antenna tolerance theory—A review,” Proc. IEEE, vol. 54, no. 4, pp. 633–640, Apr. 1966. [9] R. Acosta, B. Frantz, J. Nessel, and D. Morabito, “Goldstone site test interferometer,” in Proc. 13th Ka and Broadband Communications Conf., Turin, Italy, Sep. 24–26, 2007. [10] M. Bremer, “Ch. 11: Atmospheric fluctuations,” in Proc. 2nd IRAM Millimeter Interferometry Summer Sch., 2002, pp. 139–146. [11] A. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Proc. USSR Academy of Sciences, vol. 30, pp. 299–303, 1941. [12] G. I. Taylor, “The spectrum of turbulence,” Proc. Roy. Soc. London. Ser. A, Mathematical and Physical Sciences, vol. 164, no. 919, pp. 476–490, Feb. 1938. [13] V. I. Tatarski, Wave Propagation of the Turbulent Medium. New York: Dover, 1961. [14] C. Coulman, “Fundamental and applied aspects of astronomical seeing,” Ann. Rev. Astron. Astrophys., vol. 23, pp. 19–57, 1985. [15] R. Linfield, “Effect of Aperture Averaging Upon Tropospheric Delay Fluctuations Seen With a DSN Antenna,” 1996, TDA Prog. Rep. 42–124. [16] C. Ruf and S. Beus, “Retrieval of tropospheric water vapor scale height from horizontal turbulence structure,” IEEE Trans. Geosci. Remote. Sens., vol. 35, no. 2, pp. 203–211, Mar. 1997. [17] M. A. Holdaway, “Calculation of anomalous refraction on chajnantor,” MMA Memo #186, Sep. 1997. [18] B. Butler, “Another look at anomalous refraction on chajnantor,” MMA Memo #188, Oct. 1997. [19] S. Radford and M. A. Holdaway, “Atmospheric conditions at a site for submillimeter wavelength astronomy,” Proc. SPIE, vol. 3357, 1998. [20] L. D’Addario, “Combining Loss of A Transmitting Array Due to Phase Errors,” 2008, IPN Prog. Rep. 42–175. [21] C. Coulman, “Fundamental and applied aspects of astronomical seeing,” Ann. Rev. Astron. Astrophys., vol. 23, pp. 19–57, 1985. James A. Nessel (M’04) received the B.Sc. and M.Sc. degrees in electrical engineering from Arizona State University, Tempe, in 2002 and 2004, respectively. He is currently working toward the Ph.D. degree at the University of Akron. At ASU, he specialized in semiconductor device theory where his research involved the development of models for predicting the effects of gamma radiation on semiconductor microelectromechanical systems (MEMS) devices with Los Alamos National Laboratories. Since 2004, he has been employed as an Electronics Engineer with the Antennas and Optical Systems Branch, National Aeronautics and Space Administration Glenn Research Center, Cleveland, OH. His research interests include Ka-band propagation, microwave remote sensing, and active phase correction for transmit arraying of microwave signals. Mr. Nessel is a member of the American Geophysical Union and Secretary of the local Cleveland section of the IEEE AP-S/MTT/EDS Societies.

REFERENCES [1] V. I. Tatarski, Wave Propagation in a Turbulent Medium. McGraw-Hill, 1961.

New York:

Roberto J. Acosta, photograph and biography not available at the time of publication.

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Linear Aperiodic Array Synthesis Using an Improved Genetic Algorithm Ling Cen, Zhu Liang Yu, Member, IEEE, Wee Ser, and Wei Cen

Abstract—A novel algorithm on beam pattern synthesis for linear aperiodic arrays with arbitrary geometrical configuration is presented in this paper. Linear aperiodic arrays are attractive for their advantages on higher spatial resolution and lower sidelobe. However, the advantages are attained at the cost of solving a complex non-linear optimization problem. In this paper, we explain the Improved Genetic Algorithm (IGA) that simultaneously adjusts the weight coefficients and inter-sensor spacings of a linear aperiodic array in more details and extend the investigations to include the effects of mutual coupling and the sensitivity of the Peak Sidelobe Level (PSL) to steering angles. Numerical results show that the PSL of the synthesized beam pattern has been successfully lowered with the IGA when compared with other techniques published in the literature. In addition, the computational cost of our algorithm can be as low as 10% of that of a recently reported genetic algorithm based synthesis method. The excellent performance of IGA makes it a promising optimization algorithm where expensive cost functions are involved. Index Terms—Aperiodic arrays, beam pattern synthesis, Genetic Algorithms (GAs), linear arrays, Peak Sidelobe Level (PSL).

I. INTRODUCTION

U

NEQUALLY-SPACED arrays [1]–[3], also termed as aperiodic arrays have been studied for several decades. Compared with equally-spaced arrays, aperiodic arrays with optimally spaced sensors have the advantages that they are capable of achieving higher spatial resolution or lower sidelobe. Or we can use less sensors to meet similar pattern specifications by carefully designing the locations of array sensors. Over the past several decades, many analytical and numerical based array beam pattern synthesis techniques have been developed [4]–[19]. An example of the analytical techniques is reported in [14], [15], in which the inter-sensor spacings for a given array weight distribution are determined by performing a Legendre transformation on the array factor. Examples of numerical techniques include the use of linear or non-linear optimization

Manuscript received September 10, 2010; revised June 09, 2011; accepted July 16, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported in part by NSFC (60802068 and 61175114), Program for New Century Excellent Talents in University (NCET-10-0370), the Fundamental Research Funds for the Central Universities, SCUT (2012ZG0008), and SRF for ROCS, SEM. L. Cen is with the Institute for Infocomm Research (I2R), A*STAR, 138632 Singapore. Z. L. Yu is with the College of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, China (e-mail: [email protected]; [email protected]). W. Ser is with Nanyang Technological University, 639798 Singapore. W. Cen is with Elektrotechnik GmbH, 34063 Kassel, Germany. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173111

methods such as the simplex algorithm [9], simulated annealing algorithm [10], [11], differential evolution algorithm [12], and genetic algorithms [13], [17]–[19]. Although many studies have been published on array synthesis, further research is still needed for the problems described below. 1) In deriving the solution to beam pattern synthesis, many techniques proposed in the literature adjust either the weight coefficients alone [4]–[7] or the sensor positions alone [12], [13], [17]. For [8], [14], [15], both the weights and positions are adjusted but the adjustments are made separately. Intuitively however, it is more likely to achieve the truly minimum sidelobe level when both the weights and the positions are considered simultaneously. 2) Previous research efforts have focused mostly on symmetrical arrays [6]–[9], [12]–[15], [17]. The assumption of symmetrical arrays allows us to adjust only half of the total number of parameters, and hence, it reduces the design complexity. However, the constraint on the array being ‘symmetrical’ reduces the degree of freedom of the optimization process, leading to a sub-optimal solution eventually. 3) In order to reduce the design complexity, many published techniques confine the inter-sensor spacings to a finite set of candidate locations [7], [9]–[11], [13], [18]. However, this too, reduces the optimization degree of freedom. 4) Relatively fewer works have been reported on the synthesis of aperiodic arrays that use complex weight coefficients and assume an arbitrary main beam direction. Some papers have assumed the coefficients to be complex, but the techniques proposed assume uniform amplitude and adjust only the phase. Other papers adjust only the amplitude and position [8]–[11], [18] or only the phase and position [12]. Considering all the above four points, a Simulated Annealing (SA) based method was proposed in [20] to design asymmetric array by optimizing both the sensor positions and array complex weight coefficients. It does not simultaneously optimize all the parameters, but perturbs the weight coefficient and position of each sensor in turns. In addition, it searches the sensor positions over a grid space. Although the SA based method [20] has high performance in array pattern synthesis, it is possible that sparse arrays with continuously spaced sensors could have a high degree of freedoms in lowering the sidelobe level [14], [15], [17]. The challenge of determining optimum parameter values simultaneously stems from the non-linear and non-convex dependency of the array factor to the weights and the sensor positions [12]. The performance of the employed optimization scheme is an important factor in the success of a pattern synthesis method, in terms of solution quality, computational load, and stability.

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The intrinsic ability to cope with nonlinear problem makes the Genetic Algorithms (GAs) a suitable solution [17]. It has been employed in the pattern synthesis of aperiodic arrays for many years [13], [17], [18]. In [17], an iterative approach based on a Modified Genetic Algorithm (MGA) was proposed for the synthesis of linear aperiodic arrays. In [18], a GA was customized for adjusting the weight coefficients and the sensor positions simultaneously. However, as stochastic search techniques, the major disadvantages of the GAs are premature convergence and slow convergence speed, especially when they are employed to solve complicated problems with a large solution space and local optima [21]. As the optimization problem described in this paper has a cost function with real variables, the number of candidate solutions is infinite if the value precision is not constrained. This can not only increase the probability of premature convergence, but also lower computation speed. In order to overcome the four problems described before and the shortcoming of the GAs, an Improved Genetic Algorithm (IGA) with a self-supervised mutation method has been presented in [19]. This paper explains the IGA in more details. Specifically, the IGA is designed to achieve the following improvements compared to existing methods: 1) both the complex weight coefficients and the sensor positions are synthesized jointly; 2) the optimum solutions can be used for both symmetrical and asymmetrical arrays; 3) much lower computational cost; 4) a novel multi-section based chromosome arrangement that allows the optimization to handle a wide variety of constraints and evolution trends; 5) a novel crossover process for real variables; 6) a novel self-supervised (in place of the usual stochastic) mutation that makes the IGA more robust, statistically sound and faster in convergence. In this paper, an extensive study has been made on the performance of the IGA algorithm under different conditions. It is shown that our method can be applied in the presence of mutual coupling among the sensors. We also extend the investigations to include the effect of steering angles on sidelobe level with fixed mainbeam width. The remaining part of this paper is organized as follows. The array synthesis formulation for aperiodic arrays and the GA based synthesis method are briefly described in Section II. The IGA for the synthesis of aperiodic arrays is presented and discussed in Section III. The extension of the proposed method considering mutual coupling effect, is presented in Section IV. Section V describes the simulation study and shows the comparative performance of the presented technique. Concluding remarks are given in Section VI. II. METHOD ON ARRAY SYNTHESIS USING GENETIC ALGORITHM Assume an aperiodic and asymmetrical linear array with sensors. The array factor AF can be characterized as [2] (1)

where for for and , is the wavelength, is the angle of arrival of the incident wave measured with respect to the axis, is the distance between the first and the th sensors, is the inter-sensor spacing between the th and the th sensors, , and is the weight coefficient of the th sensor. Since is complex, it can be expressed as , where and are the amplitude and phase of respectively. Consequently, the array factor can be expressed as (2) The task of array synthesis is to design the parameters of the array so that it will produce a pattern close to the desired beam pattern. Since the array factor is an exponential or trigonometric function of sensor positions, the determination of the sensor positions is a nonlinear process with local minima. When applying the GA for array pattern synthesis, the sensor parameters are encoded and cascaded to form a chromosome which represents a potential solution. A specified number of chromosomes can be used to construct a population, which will then evolve through selection, breeding and genetic variation. With the help of such an evolutionary process, the parameters of the array can be synthesized. As the objective of optimization is to minimize the sidelobe level of the array pattern by adjusting the parameters of the array, subject to given design specifications and constraints, the fitness function can be defined with the evaluation of the Peak Sidelobe Level (PSL) as

(3) is the spanned angles within the sidelobe band, and is the range of the mainlobe. The function is then evaluated excluding the mainbeam. In (3), the PSL is measured in the unit of decibel. Here, we introduce the minus sign in order to make it a maximization problem. It is well known that array optimization should be organized along specific trade-off rule between sidelobe level and mainlobe width. To simplify this problem, only the minimization of sidelobe level is considered in the optimization. The mainlobe width is fixed to be within a given range according to the design specifications. where

III. IMPROVED GENETIC ALGORITHM (IGA) Over the past decades, the GA method has been widely applied to array synthesis [13], [17]–[19], [22]–[26]. Although the GA is able to solve non-convex pattern synthesis problems, it suffers from intensive computation and weak-guaranteed convergence, especially when the solution space is large [21]. In order to enhance the convergence performance of the GA, we proposed an Improved Genetic Algorithm (IGA) in [19], where a multi-section based real encoding scheme, a section-based

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crossover process, and a self-supervised mutation process were proposed. This section explains the IGA in more details.

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where is randomly generated within [0,1]. This expression ensures that and are confined to be within the upper and lower bounds.

A. Encoding Scheme An effective encoding scheme is important to the success of the GA optimization. In the IGA, instead of using binary coding representation (as is in most GA), the chromosomes are represented using floating-point numbers, which represent the parameter vectors of the array. The sensor weights and locations are concatenated into different variable-sections, e.g., weight-section and spacing-section (or amplitude-section, phase-section and spacing-section if the weights are complex numbers). A chromosome is formed by cascading all the sections. Arranging the chromosome in this multi-section way has the advantage that, it takes care of the possibility of different types of parameters that may have different constraints and evolutionary trends. A certain number of fitter chromosomes in a population are selected based on Roulette Wheel Selection strategy [21], so that highly fit chromosomes have better characteristic and more chances to be chosen and allowed to mate. B. Section-Based Crossover The selected chromosomes are treated as “parents” to produce new chromosomes called “children” by genetic operations. Crossover is a basic operation for yielding new chromosomes. In the IGA, the crossover is independently performed for each variable-section according to a crossover probability, . This allows independent control of evolution process for each type of array parameters. Three methods of crossover, i.e., uniform crossover, single-point crossover and multi-point crossover, are randomly applied in each generation during the evolutionary process. A control parameter, is randomly chosen from {0,1,2}. If , the uniform crossover is activated, in which the crossover is performed over the entire chromosome via a randomly generated mask. The mask is randomly generated with the same length as that of the chromosome and it consists of a bit-string of “0” and “1”. The children are generated according to the information contained in the mask. If 1 or 2, a single- or two-point crossover is performed. In the single-point crossover, one cut-point is randomly chosen from the parents and the parts located in the right of the cut-point are exchanged. Similarly, for the two-point crossover, two cut-points are selected and the parts between the two points are exchanged. The new cut-point genes, and , are obtained by taking a linear combination of the old cut-point genes, and , the upper and lower bounds, and , of the gene, and a randomly generated value, ,

(4)

C. Self-Supervised Mutation After the offspring are produced from the crossover process, they will then undergo a mutation process, which maintains the population diversity according to a mutation probability . One disadvantage of the stochastic mutation process is that the process may “miss” better solutions and waste much time on exploiting “bad” searching areas. In order to overcome this problem, a self-supervised mutation is proposed in the IGA. In contrast to conventional mutation methods, our mutation process can be applied to one gene for one or more times in each generation. Let us define each update of a gene during mutation as a Search. In the self-supervised scheme, the results obtained from the previous Searches are used to adjust the direction and the step size of the subsequent Search. At the beginning of the mutation process, a Search is performed on the gene in an arbitrary direction. If a better solution can be obtained in that direction, the mutation will continue with a new Search. The Searching step sizes are gradually decreased within a local area of the gene value. The Search in the same direction will stop when there is no significant solution improvement. This approach finds the best or near-best solution in one direction, and as such, it is capable of finding better solutions in “good” searching areas, and preventing the evolution from repeatedly exploiting “bad” areas. By this way, the IGA can be more efficient with less computational load over the GA. Let the gene to be mutated be denoted as , the new gene after one Search is calculated as if if

, ,

(5) where and are the lower and upper bounds of , respectively. The is a real number within the range of , which is randomly initialized at the beginning of the mutation. After each Search, is decreased by multiplying a factor to it as

(6) where is the new value of is the decreasing rate satisfying , and are the original and the new fitness values, respectively, is the average fitness value of the offspring from the crossover, and is the minimum acceptable improvement. The conditions and , are derived based on the following considerations. Firstly, it is found that, when is only slightly greater than , it is often hard to obtain a significant improvement from further Searches. It is therefore better to terminate the process at that point so as to save computation time. Secondly, the evaluation of additional individuals generated in each mutation Search

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increases the computational load. Based on the principle of nature selection, fitter individuals with higher fitness have a tendency to produce better quality offspring. In order to reduce computational cost, the Search and computational process are performed only on fitter individuals whose fitness values are larger than the average value. After updating and , a new Search will start with and . The mutation Search on is considered to be fully completed when the condition in (6) cannot be met. After the offspring is produced by crossover and mutation process, some of the new chromosomes will be chosen to replace the same number of chromosomes in the old population to form a new one. The replacement strategy used is the Generation-Replacement method incorporated with the Elitist Strategy [27]. The whole process from encoding and selection to crossover, mutation and replacement forms a generation of evolution. A new population is produced in each generation. Such process is iterated till it converges or the termination conditions are met. IV. EXTENSION OF PROPOSED METHOD TO INCLUDE MUTUAL COUPLING EFFECT Mutual coupling, which occurs in the array environment through radiation between sensors, can have significant effects on the sensor input impedances, the array gain, and the shape of the array pattern. The analysis and design of array patterns are complicated in the presence of mutual coupling. It is often ignored since it does not have explicit relationship with the array patterns. For the lack of elegant mathematics for calculating exact mutual coupling effect, it is also not considered in many array synthesis method. In the method proposed in the above sections, the mutual coupling effect is ignored because we adopt the constraint in [14], [15] for the inter-sensor spacing, , to make the minimum spacing not be smaller than , as , , so that the mean inter-sensor spacing in aperiodic array is relatively large with respect to equally-spaced arrays. This in turn alleviates the mutual coupling effect among the sensors. Considering practical applications, we extend the proposed method to work with mutual coupling effect. Let the array factor in (1) to be expressed in vector-form as (7) where denotes the transpose complex conjugate operation, and the vector of weighting coefficients, , and the nominal steering vector . The mutual coupling effects are different for each element depending on their positions in the array. Considering the minimum-scattering antennas, the mutual coupling effects can be evaluated via a coupling matrix [28]. With the existence of mutual coupling effects, the true array steering vector, , is then the multiplication of a mutual coupling matrix, , and the nominal steering vector [29], i.e., (8)

is a matrix function of sensor positions, and . The array factor in (7) is, then, modified as (9) The design procedure considering mutual coupling effect is summarized as below. Step 1. Initialization: Specify the initial parameter values of the IGA. A good starting point is important to the performance of stochastic based optimization [21]. In our implementation, the initial population is formed by using a uniform array with unit weights and an inter-sensor spacing of half a wavelength. In most of cases we examined, this provides a better starting point for the optimization process compared to random initialization. Step 2. Iteration: With the initial population, an iterative procedure of optimization is started. In the iteration, the array and , are updated with genetic operaparameters, i.e., tions. The is, then, obtained based on using the Method of Moment (MoM) [30]. The optimization aims to maximize the fitness function in (3) as

(10) The array factor is calculated according to (9). Step 3. Stopping Criteria: The iteration process in Step 2 is terminated if any one of the following three criteria is met: a. The design objective has been reached. b. The improvement of fitness during successive generations is smaller than an acceptable level. c. The maximum number of generations is reached. It should be noted that in some applications, one can choose any combinations of the above stopping criteria [e.g., (a) alone, or (a) and (b), or (a) and (c), etc.] in accordance to the design requirement. V. NUMERICAL EXAMPLES AND RESULTS ANALYSIS In order to illustrate the effectiveness of the proposed method, we compare the performance of IGA with that of MGA [17]. The beampatterns are evaluated as functions of , where is the steering direction. To get a fair comparison, we firstly use the same setting in [17]. An synthesis example considering all the possible is given in Section V-E with . In all the simulations, the main beams are confined to be within , and 17 and 37, respectively. In [17], two typical pencil-beam patterns were synthesized with the MGA method for a symmetrical aperture and using uniform weights, i.e., for all . Lower PSLs were achieved compared to those obtained from the analytical technique proposed in [14], [15]. Besides the design of the aperiodic array with a symmetrical aperture and uniform weights discussed in [19], an extensive study has been made on the performance of the IGA under different conditions in this section, where several configurations and weights constraints will be considered. The designs from the IGA will be compared with those reported in [17] in terms of the PSL, convergence speed and algorithm stability. In addition, the effects of steering

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angles and mutual coupling are also investigated. Without loss of generality, we set the parameters of (2) to

(11) In order to have same range of sensor positions used in [17], here we adopt the constraints of inter-sensor spacing between and . A very sparse array can be designed by modifying these constraints. Certainly, this does not require modification of the proposed algorithm.

Fig. 1. Resultant lowest PSLs obtained by the IGA in 10 runs dashed line shows the best-case lowest PSL for the MGA.

. The

A. Case I: Aperiodic Array With a Symmetrical Aperture and Uniform Weights In the IGA, the population size is set to 30. The initial population is formed by using a uniform array with an amplitude of 1 and an inter-sensor spacing of for each sensor. The parameters in the crossover and the mutation process are selected as , , , and , which is obtained according to the experience. The computational complexity is measured in terms of the number of fitness function evaluated, which is a common way of estimating the computational complexity of evolutionary algorithms [21]. Specifically, it involves the computation of the array factor for each where scans from 0 to . If 1024 points are sampled during the interval , we have to compute (2) 1024 times for each set of parameter values considered. The computational time required by the other processes involved in the algorithm is relatively short and can be ignored. In order to find the average performance of the IGA and gain an insight of its stability, the IGA is executed for 10 runs for each set of parameter values considered. The number of generations is 50 for each run. Fig. 1 shows the resultant lowest PSL in each individual run when the IGA is employed in the synthesis of a 17-sensor array. The best-case (i.e., lowest) lowest PSL for the IGA is 20.32 dB. In Fig. 1, the dashed line shows the best-case lowest PSL obtained by the MGA in [17]. For the MGA, the population size is 200 and the number of generations is 300. It can be seen from Fig. 4 that the best-case lowest PSL for the MGA is 0.5 dB higher than that of the IGA. Fig. 1 also shows that the lowest PSLs obtained by the IGA in 9 out of 10 runs are better than the best-case value obtained by the MGA. Simulation runs also show that the worst-case and average lowest PSLs for the MGA are also poorer than those from the IGA, repetitively. The sensor positions obtained by the IGA are shown in Fig. 2(a). The resultant beam patterns obtained by the IGA and the MGA are shown in Fig. 2(b). We can see from the enlarged view in Fig. 2(b) that the obtained 3 dB beamwith of the main lobe from the IGA is also narrower than that from the MGA. The above are reported for a 37-sensor array (i.e., ) and the corresponding results are shown in Figs. 3 and 4, respectively. As shown in these figures, the improvements made by the IGA are even clearer for a large number of sensors . The main comparative results shown in Figs. 1–4 are summarized in Table I. Besides the PSL values, the computational complexity is also compared, which is measured in terms of the number of

Fig. 2. (a) Sensor positions obtained using the IGA. (b) Resultant beam patterns . from the IGA and MGA

Fig. 3. Resultant lowest PSLs obtained by the IGA in 10 runs dashed line shows the best-case lowest PSL for the MGA.

. The

fitness functions evaluated. This is a common way of estimating the computational complexity of evolutionary algorithms [21] since the time taken for the other processes incurred in the computation is relatively insignificant and can be ignored [31]. It can be seen from the table that the IGA not only achieves a better PSL performance, but also requires a smaller population size (30 individuals) and involves a shorter process time (50 generations). Consequently, the average numbers of the fitness function evaluations required by the IGA are only about 10% or 17% , of that required by the MGA. This is a significant saving in the computational effort as compared to that needed by the MGA. B. Case II: Aperiodic Array With Asymmetrical Aperture and Uniform Weights In order to increase the degree of freedom of the optimization process, the 17-sensor array is now synthesized with an asym-

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Fig. 5. (a) Sensor positions obtained using the IGA. (b) Resultant beam pattern from the IGA. Fig. 4. (a) Sensor positions obtained using the IGA. (b) Resultant beam patterns . from the IGA and MGA.

TABLE I COMPARATIVE CONVERGENCE PERFORMANCE BETWEEN MGA [17] AND PROPOSED IGA. THE ARRAYS ARE DESIGNED WITH A SYMMETRICAL APERTURE AND UNIFORM WEIGHTS

Fig. 6. (a) Weights and sensor positions obtained using the IGA. (b) Resultant beam pattern from the IGA.

metrical aperture, where the search space is twice as large as that in the design Case I. The results are illustrated in Fig. 5. As Fig. 5 shows, the IGA is able to achieve a best-case lowest PSL of 20.6 dB which is about 0.3 dB lower than that for the symmetrical array (shown in Fig. 2). The number of the fitness function evaluations is now 9,207, which is higher than 6,074 required for the symmetrical array synthesis (due to the increase in the dimension of the search space). C. Case III: Aperiodic Array With Asymmetrical Aperture and Optimum Real Weights In this case, the weight coefficients and the sensor positions are determined simultaneously using the IGA. The weight coefficients are restricted to take on only real values here. The results are shown in Fig. 6. The best-case lowest PSL of 22.3 dB is obtained after 15,438 fitness function evaluations. Although the search space here is approximately 4 times larger than that in the design Case I, the number of fitness function evaluated is only about 25.7% of that required by the MGA (see Table I). The Current Taper Ratio (CTR) is the ratio between the maximum and minimum amplitudes of weight coefficients. The CRT is related to the effects of possible unforeseen occurrences regarding the sensors with the largest weights [11]. Generally, it is important to have smaller CRT when designing sensor weights. The amplitude constraint in (11) aims also to limit the CRT level. The design in this case has a low value of CRT equal to 2.73 dB.

Fig. 7. (a) Beam pattern of the array designed with steering direction of 60 . (b) Beam pattern of the array designed by considering all possible steering directions.

D. Case IV: Aperiodic Array With Asymmetrical Aperture and Optimum Complex Weights In this case, the complex weights and the inter-sensor spacings are determined simultaneously using the IGA. The mainbeam is steered to a direction of 60 measured with respect to the axis, with a same mainbeam width is retained. The results are plotted in Fig. 7(a). The best-case lowest PSL is now 16.7 dB with a CRT of 3.04 dB and the total number of the fitness function evaluated is 20,785. In this example, we use complex weights aiming to show the effectiveness of our method for simultaneously optimizing complex weights and positions of sensors. It should be noted that in

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TABLE II EFFECT OF STEERING ANGLE

TO

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PSL (dB)

some cases, e.g., aperiodic arrays with optimum placement of sensors, complex weights may not have significant advantage over real weights. In the above example, if the array is designed with real weights, the PSL can reach 16.42 dB. However, if there are some constraints on sensor placement, the advantage of using complex weights will be apparent. The reason of this phenomenon is that optimal placement of sensors also has the capability on signal phase adjustment as the complex weights do. E. Case V: Sensitivity of PSL to Steering Angle In the equally-spaced linear phased array, it has been shown that the mainlobe broadens with increasing steering angles from 0 to 90 for a given set of other parameters such as the sidelobe level [32]. This relationship between the directivity and the steering angle still holds in aperiodic arrays. On the other hand, if we keep the mainbeam width unchanged, the PSL will decrease when the steering angle increases from 0 to 90 , and increase when the angle increases from 90 to 180 . In order to show it, the method used for Case IV is applied to obtain the best-case lowest PSL for different mainbeam directions ranging from 0 to 180 . The PSL and the corresponding steering angle are listed in Table II to show the sensitivity of PSL to steering angles. It is clear that when the array steering direction is close to the endfire direction of array, the PSL will be very high because in such case, the effective array aperture is very small. It is interesting to synthesize an array by minimizing PSL considering all possible steering directions. We re-design the beampattern in Fig. 7(a) by considering , the resultant beampattern is shown in Fig. 7(b), where we can find that PSL is 11.72 dB which is higher than 16.7 dB in Fig. 7(a). This result coincides with the results in Table II because the minimum PSL achievable when steering direction close to endfire direction is around 12 dB. In practice, we only consider a specific steering direction, so the synthesized beampattern may have lower PSL as shown in Fig. 7(a). F. Case VI: Design in Presence of Mutual Coupling In the design case I–V, the mutual coupling effect is not considered in optimization as many other array pattern synthesis approaches do. Since mutual coupling can have adverse effect for a sensor array, we extend our method to work with the presence of mutual coupling in Section IV. The 17-sensor array is designed by limiting the maximum value of spacing to (i.e., , ). The mutual coupling effect is evaluated using a coupling matrix, which makes sense only in the case of minimum-scattering antennas [28]. As a good representative of minimum scattering antenna, the dipole antenna is then used in this example. All the dipole antennas have length and radius . These dipoles are fed with coaxial cable with impedance 50 . The direction of incoming signal is per-

Fig. 8. Resultant arrays designed using the IGA with and without considering mutual coupling effects in the design process (a) weights and sensor positions; (b) true beam pattern of the arrays.

pendicular to the dipole element. The weights and the sensor spacings are jointly optimized. The mutual coupling effect for each element depends on its position in the array. It is calculated by the MoM [30]. The array is synthesized based on the design procedure presented in Section IV. The weights and sensor positions obtained using our method are shown in Fig. 8(a), and the true beam pattern of the designed array is plotted in Fig. 8(b). If the array is synthesized without considering mutual coupling effect in the optimization process, as shown in Fig. 8 too, the true beam pattern of the designed array (taking mutual coupling effect into account at the level of pattern calculation) can be seriously distorted, as can be seen in Fig. 8(b) (dashed line), where the PSL increases to 9.75 dB. VI. CONCLUSION The objective of the proposed beam pattern synthesis method is to minimize the Peak Sidelobe Level (PSL) while maintaining a desired beam pattern. Extensive performance evaluation results show that the IGA is able to achieve lower PSL and much higher convergence rate compared to a newly reported GA based synthesis method [17]. Our design leads to savings on computational efforts of up to 90% compared to the results achieved in [17]. In addition, the stable performance of the IGA has been illustrated clearly from the statistic of multiple independent runs too. In nearly all of the runs, the IGA is able to find better designs than the best ones reported in [17]. We also extend our method in the presence of mutual coupling effect among sensors. To achieve a full understanding, the sensitivity of the PSL to steering angles is discussed in the paper. ACKNOWLEDGMENT The authors acknowledge all the anonymous reviewers for their constructive comments that helped to improve the quality of this paper. REFERENCES [1] D. King, R. Packard, and R. Thomas, “Unequally-spaced, broad-band antenna arrays,” IEEE Trans. Antennas Propagat., vol. 8, no. 4, pp. 380–384, Jul. 1960. [2] A. Ishimaru, “Theory of unequally-spaced arrays,” IEEE Trans. Antennas Propagat., vol. 10, no. 6, pp. 691–702, Nov. 1962. [3] A. Ishimaru and Y.-S. Chen, “Thinning and broadbanding antenna arrays by unequal spacings,” IEEE Trans. Antennas Propagat., vol. 13, no. 1, pp. 34–42, Jan. 1965.

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[4] C. Tseng and L. J. Griffiths, “A simple algorithm to achieve desired patterns for arbitrary arrays,” IEEE Trans. Signal Process., vol. 40, no. 11, pp. 2737–2746, Nov. 1992. [5] C. A. Olen and R. T. Compton, “A numerical pattern synthesis algorithm for arrays,” IEEE Trans. Antennas Propagat., vol. 38, no. 10, pp. 1666–1676, Oct. 1990. [6] L. Wu and A. Zielinski, “An iterative method for array pattern synthesis,” IEEE J. Oceanic Eng., vol. 18, no. 3, pp. 280–286, Jul. 1993. [7] S. Holm, B. Elgetun, and G. Dahl, “Properties of the beampattern of weight- and layout-optimized sparse arrays,” EEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 44, no. 5, pp. 983–991, Sept. 1997. [8] P. Jarske, T. Saramäki, S. K. Mitra, and Y. Neuvo, “On the properties and design of nonuniformly spaced linear arrays,” IEEE Trans. Acoust., Speech, Signal Process., vol. 36, no. 3, pp. 372–380, Mar. 1988. [9] R. M. Leahy and B. D. Jeffs, “On the design of maximally sparse beamforming arrays,” IEEE Trans. Antennas Propagat., vol. 39, no. 8, pp. 1178–1187, Aug. 1991. [10] V. Murino, A. Trucco, and C. S. Regazzoni, “Synthesis of unequally spaced arrays by simulated annealing,” IEEE Trans. Signal Process., vol. 44, no. 1, pp. 119–122, Jan. 1996. [11] A. Trucco and V. Murino, “Stochastic optimization of linear sparse arrays,” IEEE J. Oceanic Eng., vol. 24, no. 3, pp. 291–299, Jul. 1999. [12] D. G. Kurup, M. Himdi, and A. Rydberg, “Synthesis of uniform amplitude unequally spaced antenna arrays using the differential evolution algorithm,” IEEE Trans. Antennas Propagat., vol. 51, no. 9, pp. 2210–2217, Sep. 2003. [13] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propagat., vol. 42, no. 7, pp. 993–999, Jul. 1994. [14] B. P. Kumar and G. R. Branner, “Design of unequally spaced arrays for performance improvement,” IEEE Trans. Antennas Propagat., vol. 47, no. 3, pp. 511–523, Mar. 1999. [15] B. P. Kumar and G. R. Branner, “Generalized analytical technique for the synthesis of unequally spaced arrays with linear, planar, cylindrical or spherical geometry,” IEEE Trans. Antennas Propagat., vol. 53, no. 2, pp. 621–634, Feb. 2005. [16] R. K. Arora and N. C. V. Krishnamacharyulu, “Synthesis of unequally spaced arrays using dynamic programming,” IEEE Trans. Antennas Propagat., vol. 16, no. 5, pp. 593–595, Sept. 1968. [17] K. S. Chen, Z. S. He, and C. L. Han, “A modified real ga for the sparse linear array synthesis with multiple constraints,” IEEE Trans. Antennas Propagat., vol. 54, no. 7, pp. 2169–2173, Jul. 2006. [18] A. Lommi, A. Massa, E. Storti, and A. Trucco, “Sidelobe reduction in sparse linear arrays by genetic algorithms,” Microw. Opt. Technol. Lett., vol. 31, no. 3, pp. 194–196, Feb. 2002. [19] L. Cen, W. Ser, Z. L. Yu, and R. Susanto, “An improved genetic algorithm for aperiodic array synthesis,” in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing, Las Vegas, NV, Mar. 2008, pp. 2465–2468. [20] A. Trucco, “Synthesizing asymmetric beam patterns,” IEEE J. Oceanic Eng., vol. 25, no. 3, pp. 347–350, July 2000. [21] K. F. Man, K. S. Tang, and S. Kwong, Genetic Algorithms: Concepts and Designs. London, U.K.: Springer Verlag, 1999. [22] M. J. Buckley, “Linear array synthesis using a hybrid genetic algorithm,” in IEEE Antennas and Propagation Society Int. Symp., Baltimore, MD, Jul. 1996, vol. 1, pp. 584–587. [23] K. K. Yan and Y. L. Lu, “Sidelobe reduction in array-pattern synthesis using genetic algorithm,” IEEE Trans. Antennas Propagat., vol. 45, no. 7, pp. 1117–1122, Jul. 1997. [24] D. Marcano and F. Duran, “Synthesis of antenna arrays using genetic algorithms,” IEEE Antennas Propag. Mag., vol. 42, no. 3, pp. 12–20, Jun. 2000. [25] D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propagat., vol. 52, no. 3, pp. 771–779, Mar. 2004. [26] L. L. Wang and D. G. Fang, “Synthesis of nonuniformly spaced arrays using genetic algorithm,” in Proc. IEEE Asia-Pacific Conf. Environmental Electromagnetics, Nov. 2003, pp. 302–305. [27] K. S. Tang, K. F. Man, S. Kwong, and Q. H. He, “Genetic algorithms and their applications,” IEEE Signal Process. Mag., vol. 13, no. 6, pp. 22–37, Nov. 1996. [28] B. Clerckx, C. Craeye, D. Vanhoenacker, and C. Oestges, “Impact of antenna coupling on 2 2 mimo communications,” IEEE Trans. Veh. Technol., vol. 56, no. 3, pp. 1009–1018, May 2007. [29] I. J. Gupta and A. A. Ksienski, “Effect of mutual coupling on the performance of adaptive arrays,” IEEE Trans. Antennas Propag., vol. AP-31, no. 5, pp. 789–791, Sep. 1983.

[30] R. E. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968. [31] L. Cen, W. Ser, Z. L. Yu, S. Rahardja, and W. Cen, “Linear sparse array synthesis with minimum number of sensors,” IEEE Trans. Antennas Propagat., vol. 58, no. 3, pp. 720–726, Mar. 2010. [32] A. C. Clay, S. C. Wooh, L. Azar, and J. Y. Wang, “Experimental study of phased array beam steering characteristics,” J. Nandestructive Evaluation, vol. 18, no. 2, pp. 59–71, 1999. Ling Cen received the B.Eng. degree from the University of Science and Technology of China in 1997, the M.Eng. degree from the Chinese Academy of Sciences in 2001, and the Ph.D. degree in electrical and computer engineering from the National University of Singapore (NUS) in 2006. She was with the General Electronic Technology Institute, China, as a project engineer from 1997 to 1998. In 2005, she joined the Centre for Signal Processing, Nanyang Technological University (NTU), as a research associate, then became a Research Fellow. She is currently working as a scientist in the Institute for Infocomm Research (I2R), Singapore. Her research interests include digital signal processing, speech/singing synthesis, machine learning, and pattern recognition. Zhu Liang Yu (S’02–M’06) received the B.S.E.E. and M.S.E.E. degrees, both in electronic engineering, from the Nanjing University of Aeronautics and Astronautics, China, in 1995 and 1998, respectively, and the Ph.D. degree from Nanyang Technological University, Singapore, in 2006. He worked as a Software Engineer at the Shanghai Bell Company, Ltd., from 1998 to 2000. In 2000, he joined the Center for Signal Processing, Nanyang Technological University, as a Research Engineer, then became a Research Fellow. In 2008, he joined the College of Automation Science and Engineering, South China University of Technology, as an Associate Professor and was promoted to full Professor in 2010. His research interests include array signal processing, pattern classification and applications in brain signal processing. Wee Ser received the B.Sc. (Hon) and Ph.D. degrees, both in electrical and electronic engineering, from Loughborough University, U.K., in 1978 and 1982, respectively. He joined the Defence Science Organization (DSO), Singapore, in 1982, and became the Head of the Communications Research Division in 1993. In 1997, he joined Nanyang Technological University (NTU) as an Associate Professor and was since appointed Director of the Centre for Signal Processing at NTU. He has published about 90 papers in international journals and conferences. He holds five patents and has four pending patents. His research interests include microphone array and array signal processing in general, signal classification techniques, and channel estimation and equalization techniques. Dr. Ser was a recipient of the Colombo Plan scholarship and the PSC postgraduate scholarship. He was awarded the IEE Prize during his studies in the U.K. While in DSO, he was a recipient of the prestigious Defence Technology (Individual) Prize in 1991 and the DSO Excellent Award in 1992. He has served in several international and national advisory and technical committees and as reviewers to several international journals. Wei Cen received the B.Eng. degree and M.Eng. degree, both in electrical engineering, from the University of Science and Technology of China, China, and the Ph.D. degree from the Nueva Ecija University of Science and Technology, Philippines. Her research interests include effects of electromagnetic fields on biological systems, numerical methods, signal processing.

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Beamformer Design Methods for Radio Astronomical Phased Array Feeds Michael Elmer, Brian D. Jeffs, Karl F. Warnick, J. Richard Fisher, and Roger D. Norrod

Abstract—A major emphasis in current radio astronomy instrumentation research is the use of phased array feeds (PAF) to provide radio telescopes with larger fields of view. One of the challenges of PAF systems is the design of beamformers that provide sufficient sensitivity and known, stable beam pattern structure. High sensitivity has been achieved with the maximum sensitivity beamformer without regard to beam pattern shape. Deterministic beamformers provide the desired pattern shape control, but suffer from a significant reduction in sensitivity. We present a hybrid beamforming method, which balances the tradeoff between high sensitivity and precise beam pattern shape control. A comparison of each of these beamforming methods, using measured data, confirms the advantage of the hybrid approach. The pattern distortions introduced by modeled beamformers can be mitigated with a transformation step, but ultimately it is shown that PAF beamformer design is best done using measured calibrators. A PAF calibration vector quality metric based on minimum description length is also introduced.

Fig. 1. BYU 19-element PAF mounted on the NRAO 20 meter reflector antenna in Green Bank, WV.

Index Terms—Antenna array feeds, array signal processing, phased arrays, radio astronomy.

I. INTRODUCTION

T

YPICAL radio telescopes operate with a single feed horn antenna at the focal point of a large reflector dish. However, in recent years there has been considerable interest from the radio astronomy community in developing phased array feeds (PAF) of closely spaced antennas in the focal plane. Though no fully commissioned instrument currently uses a PAF, several projects and research groups, including ASTRON in the Netherlands, DRAO in Canada, CSIRO in Australia, NAIC Cornell and Arecibo, and a collaboration between Brigham Young University (BYU) and the National Radio Astronomy Observatory (NRAO) in the USA, are rapidly approaching deployment of science-ready instruments [2]–[10]. For example, the BYU-NRAO prototype array is shown in Fig. 1 mounted on the Green Bank 20 Meter Telescope at NRAO, West Virginia. The primary advantage of a PAF fed instrument is its ability to form multiple simultaneously steered beams covering a much greater field of view (FOV) than a single horn feed, and thus dramatically reducing sky survey times by between one and

Manuscript received February 24, 2011; manuscript revised July 07, 2011; accepted July 23, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. M. Elmer, B. D. Jeffs and K. F. Warnick are with the Department of Electrical and Computer Engineering, Brigham Young University, Provo, UT 84602 USA. J. R. Fisher is with the National Radio Astronomy Observatory, Charlottesville, VA 22903 USA. R. D. Norrod, retired, was with the National Radio Astronomy Observatory, Green Bank, WV 24944 USA. Digital Object Identifier 10.1109/TAP.2011.2173143

Fig. 2. Benefits of PAFs include the formation of multiple simultaneous beams from a single pointing, active control of the beam pattern shape, adaptation to changes in the noise environment, and cancelation of undesired signals.

two orders of magnitude. This is illustrated in Fig. 2, along with additional advantages of a PAF instrument. With this significant transition from traditional single-pixel radio telescopes to PAF systems, it is apparent that promised performance improvements can be achieved only with increased system complexity and after resolving a number of new technical challenges. Challenges include mutual coupling among the closely spaced array elements [11]–[13], hardware requirements for many (e.g., 40 to 200) identical analog and digital receiver chains and correlator/beamforming processing, and the need for

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new high performance beamformer design methodologies suitable for the PAF environment. The focus of this paper is a study of PAF beamformer design methods, which must balance the demand for high sensitivity with a known, stable beam pattern spatial structure. High sensitivity is required to detect the faint radio signals emitted by distant celestial bodies, which are typically tens of dB below the noise floor [14], while stability is needed for radiometry and dynamic range. This tradeoff occurs because the maximum sensitivity beamformer now used by all PAF development groups achieves its optimal SNR performance by adapting the beam pattern to the noise field and array response parameters, which causes some variation in the far-field beam pattern. It will be shown that the inherent tradeoff between these two parameters is a motivating force in PAF beamformer design. Phased antenna aperture arrays (i.e., bare arrays with no large reflector) have been used for 70 years in applications of wireless communication, radar, sonar, and remote sensing, providing benefits such as improved direction finding, spatial interference canceling, rapid beam steering, forming multiple simultaneous beams, gain optimization, etc. [15]–[18]. Recent work in the area of satellite communications has even included discussion on space platform orbital PAF-fed reflectors to provide adaptation to a changing radio environment [19], [20]. Though phased array antenna theory and design are relatively mature fields, the stringent demands of radio astronomical observation, including observing at very low signal to noise ratios (SNR) and the need for extreme pattern and gain stability, have until very recently kept these techniques from use in feed designs for the large dish instruments. However, several multiple horn array feed systems have been commissioned, including the NAIC ALFA array at Arecibo and the Australian (ATNF/CSIRO) Parkes 21 cm multibeam receiver [21]–[23]. Though not operated as a closely packed, electronically phased beamforming array, the fixed optics for the multiple separate feed horns packed together in the dish focal plane provide an increase in the number of pixels (beams on the sky) obtained for a single dish pointing. Each antenna in the array works independently to provide a sparse sampling of the field of view (FOV) and thus increase sky survey speed. In contrast, electronic beamforming capabilities of PAF technology offer further increases in survey speed while fully sampling the FOV with multiple simultaneous, perhaps overlapping beams. A variety of beamformers have been suggested for PAF use [24], [25], but the data-dependent max-SNR (max-sensitivity) beamformer [26], simultaneously introduced for astronomical PAF use by the ASTRON and BYU-NRAO teams, has been the only one successfully applied to create images of experimental PAF data [8], [27]–[32]. Other PAF beamforming work has been limited to simulation but has shown intriguing potential. This includes use of eigenbeams to reduce data transfer and storage requirements [33] and numerically optimized Gaussian beams steered without distortion while accounting for polarization effects [34]. The linearly constrained minimum variance beamformer (LCMV) is appealing because it has the ability to provide beam pattern constraints while minimizing the overall

noise response [17]. Conjugate field match beamforming has also been attempted, but has been found to be unsuitable due to the inability to control beam pattern shape when there is significant gain variation across the sensor array for far-field sources [25], [35]–[37]. Each of these design methodologies has drawbacks. The max-SNR beamformer does not guarantee pattern stability since it can be recomputed to optimize over variations in the noise field. The closely related LCMV approach is promising, though in some cases there are insufficient degrees of freedom in the beamformer for a typical PAF to constrain the side lobe pattern to be uniformly low while controlling the main beam shape. Deterministically derived beamformers offer complete control of the beam pattern shape but lack the necessary sensitivity for useful observations. Accurate array calibrations are required in any case, and this is a significant technical challenge. Desires for maximum sensitivity and complete control of the beam shape cannot be mutually satisfied. In this paper we compare options that are available to address the new challenges of practical PAF beamformer design. We show that while it is possible to design a deterministic beamformer in simulation, a transformation step is required before it can be satisfactorily applied to measured data. Additionally, we introduce a hybrid beamformer that balances the tradeoff between sensitivity and beam pattern shape control, providing a solution suitable for PAF operation. II. PRELIMINARIES A. Signal Model As depicted in Fig. 3, assuming narrowband operation of a element PAF, the complex basebanded data vector at time sample is given as (1) where is the normalized array response to a unit amplitude signal in the far field arriving from the direction of a point source signal of interest (SOI) , and is the array noise vector. Signals and are assumed to be zero mean random processes, statistically stationary across the samples obtained during the observation time. The array covariance matrix is defined as (2) where denotes expected value and superscript is complex conjugate transpose (Hermitian transpose). Assuming the SOI and noise are statistically independent we have (3) Making the simplifying assumptions that the SOI is a point source and noise is white then (4) where and are the power in the SOI and noise respectively, and is the identity matrix. Because of the non-isotropic

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use fixed man-made sources in the far field as calibration references. The only remaining option is to perform calibration on-reflector using the brightest available astronomical source. The calibration procedure can be summarized as follows. The radio telescope is steered, relative to the calibration source, in direction for which a response vector is desired. An on-source, signal-plus-noise covariance is obtained. The instrument is then steered several degrees in azimuth away from the source at the same elevation to avoid changing the spillover ground noise pattern, and an off-source, noise-only is obtained. The calibration vector is computed as Fig. 3. Block diagram for signal processing of a narrowband PAF.

(7)

distribution of spillover ground noise seen by the array, and mutual coupling between the antenna elements, PAF system noise is in fact correlated, but this equation is given to assist in understanding a later derivation. An estimate of for an -sample short-term integration (STI) window is calculated from observed data samples as

(5) The beamformer output is the weighted sum of the signals received by each array element and is computed as (6) where indexes one of main lobe beam steering angles and is the th beamformer complex weight vector. Narrowband beamformer operation is assumed in this notation. For the broadband case the signal is decomposed into many frequency channels and separate beamformers with distinct weights are computed as in (6) for each channel. B. Calibration A calibration vector of the array voltage response to a far-field point source is required in every direction that a beam is to be steered or where the pattern is constrained to a specified response value. Some details of our calibration procedure (reported in [8]) are repeated here since the information is crucial to understanding PAF beamformer design. On-reflector calibration is necessary for accurate response vector estimation. Even the most detailed numerical simulations cannot predict the real physical array response with sufficient accuracy to design beamformer weights, since they must account for signal interaction with the reflector as well as gain variations between channels. Off-reflector bare array measurements are likewise unsuitable. Antenna range calibration is unrealistic since radio telescopes are physically too large and array responses drift too much over time. Additionally, due to mechanical limitations, multipath, and thermal ground noise, a reflector dish cannot be steered to sufficiently low elevations to

where is the principal eigenvector determined by the generalized eigenvalue problem (8) A grid of response vectors is computed in the region surrounding a calibration source. Up to 1000 distinct pointings may be required depending on the desired number of simultaneous beams in the FOV and the number of pattern constraints to be incorporated in the beamformer. This can be a time-consuming process (e.g., 4 hours), but cannot be neglected because obtaining accurate calibration vectors is fundamental to PAF beamformer design. These may need to be updated periodically (e.g., every few days) to compensate electronic gain variations and structural perturbations in the system. The typical lifetime of a calibration set has not yet been determined, but preliminary analysis has revealed very little performance degradation after a 24—hour period. The quality of calibrations depends on SNR for the chosen bright calibrator source and integration time per pointing. SNR drops significantly for grid points outside the FOV imposed by the natural dish aperture response pattern. Thus, calibrations outside the first dish side lobe are unreliable and must be discarded. We have employed an algorithm based on the minimum description length principle (MDL) [18], [38], [39] to detect the presence of a sufficiently strong calibrator source in the received data. Eigenvalues of the noise-pre-whitened covariance are used to estimate , the number of dominant sources present with adequate SNR. Pre-whitening is required for MDL since PAF noise is correlated. The smallest eigenvalues of (4) will be clustered near , and the remaining eigenvalues correspond to source signals. Detection of a single dominant non-noise signal is an indication that the calibration vector is acceptable. Compute

(9)

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each given realistic array calibration data, and a performance analysis and comparison will be presented. A. Max-SNR Beamformer The optimum weight vector former is defined as [17], [18]

for the max-SNR beam(11)

and are computed as the In practice, estimates of on and off-source measurements, and , discussed in Section II-B, and the maximization in (11) gives the generalized eigenvalue problem (12) Fig. 4. Acceptable calibration vectors are determined by the MDL quality metric, which is an objective algorithm used to identify the existence of a single source. White spaces indicate a good calibrator and black spaces indicate a poor one. The source was Cassiopeia A.

where

are the eigenvalues of ordered as , and is from (5). Parameter , determines the degree of dominance (i.e., calibrator SNR) required. For , is the conventional MDL estimate of just detectable independent sources, while establishes a higher SNR threshold. If , then SNR is assumed insufficient for calibration at that grid point. Fig. 4 is a real data example of a measured 33 33 calibration grid using a 19 element PAF on the Green Bank 20 Meter Telescope. Positions of acceptable calibration vectors using are marked in white, showing that there are practical limits to the range of steering vectors that can be obtained with on-reflector calibration. C. Performance Metrics The following metrics are used to compare the beamforming methods presented in this paper: null-to-null beamwidth, peak side lobe level, and sensitivity. Sensitivity can be expressed as [25], [40] (10) is the effective receiving area of the PAF system for where the beamformer weight vector , is the equivalent noise temperature of the system at the output of the beamformer, is Boltzmann’s constant, and is the flux density of the source signal of interest (Jy). III. PAF BEAMFORMER DESIGN METHODS Of the several candidate beamformer methods mentioned in Section I, we will consider the two that most obviously represent the extremes in trading off high sensitivity with direct pattern control for stable on-sky responses: the max-SNR and numerically optimized approaches. A hybrid approach combining the positive aspects of each of these is also presented. In Sections IV and V we will study the practical implementation challenges for

whose solution is the max-SNR beamformer. A distinct weight vector is computed in this manner for each desired pointing direction of the multiple simultaneously formed beams. This beamformer is easily implemented since steering a beam in direction only requires a single pair of on and off-source measurements. When computed with previously acquired calibration data the beamformer is said to operate in “fixed-adaptive” mode, meaning that it is optimal for the calibration set but does not rely on current observation array covariance estimates. Fully adaptive operation requires frequent updates of throughout the observation time in order to cancel interfering signals or account for changes in the noise spatial structure such as spillover noise variation as a function of pointing elevation [8]. The max-SNR beamformer offers little direct control of the beam pattern shape, since it naturally responds to the noise covariance structure. Variations in the observation environment due to changes in spillover noise structure and electronic drift in the receiver system change the underlying array response vectors. This presents a dynamic range challenge since unpredictable beam pattern variations may introduce undesired signals from a nearby bright source. A natural extension of the max-SNR beamformer that provides constraints on the beam pattern structure is the LCMV beamformer, which will be discussed in Section IV-C. We will see that it has limitations similar to those of the max-SNR beamformer. B. Numerically Optimized Equiripple Beamformer Deterministic beamformers are designed to achieve a specified beam pattern structure and can be derived using a number of methods. This work utilizes an iterative optimization routine based on the minimax principle to construct an equiripple beamformer using calibration data (modeled or measured). The equiripple method was chosen for convenience and is simply a representative of a large class of deterministic beamformers. Complex beamformer weights are found by iteratively evaluating the far-field response at available calibration points throughout the beam pattern to achieve minimum gain equal ripple side lobes within the FOV while meeting designated main lobe shape constraints. Since this approach optimizes the beamformer weights only with respect to specified response pattern goals, the correlated spillover and mutually coupled

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array noise response is not considered, so true maximum sensitivity in not achievable. However, when strict control providing known pattern shapes across all formed beams is required, this approach may be desirable. A similar numerical optimization with respect to the dish illumination pattern could be used to control illumination spillover and thus directly reduce noise levels, but obtaining a sample calibration of points on the dish rather than in the on-sky farfield beam is difficult or impossible using available calibrator sources. Thus, only far-field pattern optimization has been considered in this analysis. A pattern magnitude response equality constraint in direction is specified as (13) where is the specified gain referenced to a unity gain main lobe peak. Additionally, the phase of the first element in is set to zero, eliminating an ambiguous degree of freedom. The peak response across all side lobes for a given weight vector is given by (14) indicates the norm (i.e., ) and the where columns of are the calibration vectors associated with the side lobe angles where the beam pattern is to be minimized. The constrained minimax side lobe response beamformer is found by applying a commercial numerical optimization code (such as fmincon in the MATLAB® Optimization Toolbox) to the objective function

(15) where is the number of point response constraints applied to the beam. Due to practical limitations of collecting very large grids of real data measured calibrations and the convenience of obtaining a wide, dense grid of modeled calibrators, this design approach appears best suited for simulation. However, it will be shown that beamformer weights designed from modeled calibrations, even with a detailed electromagnetic model, cannot be directly applied to PAF data without significant distortions, and that even after applying a corrective transformation, better performance is achieved with measured calibrators. In either case, while the resulting beamformer provides a closely controlled beam pattern shape, the associated sensitivity penalty makes this method undesirable for typical PAF use. C. Hybrid Beamformer We propose a hybrid beamformer that combines the benefits of both the max-SNR and numerically optimized equiripple beamformers, providing maximum sensitivity for a given amount of beam shape control. The beamformer is obtained through the numerical optimizer described above by solving

(16) where are the magnitude of the max-SNR beam response referenced to a unity gain main lobe peak and

(17) The user specified weighting parameter controls whether emphasis is placed on maximizing the array SNR [40], or on minimizing the equiripple side lobe levels. In order for and to produce the max-SNR and equiripple beamformers respectively, the equality pattern constraints of (15) were changed to -dependent inequality constraints. In (16), when the right-hand side inequality constraint is forced to zero and acts just like the constraint in (15). As the constraint region grows large enough around to include the max-SNR solution with some excess “elbow room” to allow the optimizer latitude in its search path. The hybrid beamformer can be designed using either modeled or measured calibration data, however, as mentioned above, it will be shown that measured calibrators are the best choice for this method. This hybrid beamformer will also be shown to provide both beam pattern shape control and sufficient sensitivity for PAF operation. D. Transformation Using modeled calibration vectors, deterministic beamformer design can be done relatively quickly, with constraints over an arbitrarily large field of view with a dense grid of points. However, such beamformer weights cannot be directly applied to real-world data because modeling inaccuracies lead to distortions in the final pattern. in situ calibration data is required. Fig. 5 verifies this conclusion. One half of both a modeled and measured equiripple beam pattern are shown side-by-side for comparison. The left half shows the modeled beam pattern resulting from an equiripple beamformer , designed using a very detailed electromagnetic model closely matching the real instrument (see the element pattern comparisons of Fig. 7). Response constraints of unity gain at the main lobe peak and a gain of dB at four additional points spaced 0.8 from boresight provide a null-to-null width of 1.6 (constraints placed on the side lobe or at the 3 dB point would be just as reasonable). Note that the main beam is of uniform radius and the first side lobe offers 27 dB of attenuation and is of constant height. The right half figure results when measured calibration vectors are used to calculate a beam response for . The result is noticeable and undesirable distortion in the main beam and throughout the side lobes. Such unacceptable results can be significantly improved by applying a transformation to map the modeled beamformer onto observed calibration data to correct for the inherent modeling inaccuracies. Even with a transformation, though, the simulated calibrators do not provide the best option for deterministic beamformer design. However, since the possibility of using modeled beamformers is reasonable and appealing, we present

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Fig. 6. BYU 19-element PAF: hexagonally spaced half-wave thickened on a ground plane. dipoles, separated by

Fig. 5. Effect of not performing a transformation before applying modeled beamformer weights to measured data. The left half is the modeled beam pattern. The right half is the measured beam pattern using the same modeled beamformer weights. There is noticeable distortion throughout the measured pattern.

the background and results of this approach for completeness, even though it is unlikely to be used in practice. The goal of a transformation is to find a mapping matrix such that real data beamformer produces a measured beam pattern closely matching the desired simulation designed beam pattern. A transformation of this kind would allow the advantages of designing modeled beamformers to be successfully exploited. Let be the beam pattern vector, where columns of are steering vectors associated with the angles where pattern matching is to be enforced. Simulated and measured beam patterns are given by and respectively, with consisting of measured calibration vectors, and those of simulation. The that minimize the squared difference between the beam patterns are computed as

(18) is the left matrix pseudoinverse where of . The weighted least squares version of (18), with diagonal weighting matrix , is written as

(19) This result provides more control over the transformation with improved matching at the desired points and a least squares fit over the remainder of the pattern. The main beam structure can be satisfactorily preserved by increasing the weighting of just a few points within the main beam. IV. RESULTS A. Experimental Setup Calibration data was collected in 2008 at the NRAO facility in Green Bank, WV. Experiments were conducted on a 20 meter

Fig. 7. Modeled and measured power patterns of array element 1 (center element). The close agreement between the modeled and measured patterns is an indication of the accuracy of the simulation model.

diameter reflector with a focal length over diameter (f/D) ratio of 0.43, using a 19-element PAF of thickened, half-wave dipoles spaced 0.6 wavelengths apart. The single-polarized, balun-fed array elements are positioned in two concentric hexagonal rings about a center element, mounted on a ground plane offset by , and designed to operate at a center frequency of 1600 MHz. A recent version of the array is shown in Fig. 6. Great effort has been made to design a simulation model that closely mirrors the true PAF and operating environment for the experimental array and 20 m dish [8]. Its accuracy is verified in Fig. 7, where modeled and measured individual element power patterns representing an azimuth slice through the center of the astronomical source Cassiopeia A (Cas A) are shown to be in close agreement. However, as shown in Fig. 5, this agreement is inadequate for the precision required for designing PAF beamformers in simulation. The calibration data set, measured with Cas A, is a 33 33 grid of 10-second pointings spaced 0.1 apart and ranging from to in both elevation and cross-elevation (i.e., the arc direction perpendicular to elevation) with respect to boresight. An off-source measurement was taken at each elevation, 8 from boresight, as an estimate of the noise field corresponding to all calibrators at that elevation. The MDL algorithm was used to identify useful calibrators (Fig. 4).

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Fig. 8. Measured far-field pattern of the max-SNR beamformer. The pattern exhibits high, uncontrolled side lobes.

B. Comparison of Beamformer Methods To compare performance of these beamformers we first look at the differences in the structure of each corresponding beam pattern. The max-SNR beam pattern is shown in Fig. 8 and demonstrates the concerns discussed previously: nonuniform side lobes with a peak only 13 dB below the main lobe. Nevertheless, the structure is optimal for obtaining maximum sensitivity in the calibration noise field. The modeled equiripple beamformer of Fig. 5 was constrained and transformed as described in Section III-D. The terms of the weighting matrix corresponding to the five optimization constraint points were set to a value of while the remaining diagonal terms were initialized to 1. This places a heavy penalty on deviations from the design equality constraint points. Fig. 9 compares the measured beam pattern after transformation to the original modeled pattern. In the measured right half image, the shape of the beam main lobe is slightly distorted, there is significant variation in the side lobe structure and an increase in the maximum side lobe level of 3–4 dB, Still, the shape much more closely resembles the desired pattern than when the modeled weights were applied without transformation. The need for a transformation can be entirely eliminated by applying the measured calibrators directly in the optimization of (15). In Fig. 10 the measured beam pattern of the right half image conserves the main and side lobe shapes of the modeled pattern. It still does not perfectly match the modeled image on the left (especially at the second null), but this approach does appear to provide the best overall option for deterministic design. This raises the question of whether there is a need for modeled beamformer design, which is fully addressed in Section IV-D. The hybrid beamformer, whose beam pattern with is shown in Fig. 11, offers characteristics of both the max-SNR and equiripple beamformers. The influence of the equiripple beamformer is seen in the uniformity across the pattern, while the increased width of the side lobe is more characteristic of the max-SNR beamformer. The 21 dB side lobes are several dB higher than those of the equiripple approach, but the tradeoff for increased side lobe levels is a desirable increase in sensitivity.

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Fig. 9. Effect of performing a weighted least squares transformation on the modeled beamformer weights before applying them to measured data. The left half of this figure is the modeled beam pattern. The right half is the measured pattern using the transformed modeled beamformer weights. The transformation causes some distortion to the pattern but makes modeled deterministic beamformers possible.

Fig. 10. Deterministic beamformers designed with measured calibration vectors closely resemble the modeled version. The left half of this figure is the modeled beam pattern. The right half is the measured beam pattern using the measured beamformer weights from a numerical optimizer. There are only slight differences noted between the two patterns.

A numerical comparison of the beamforming methods is given in Table I. Results of the hybrid beamformer with and 0.25 show that low side lobes can be achieved without fully sacrificing sensitivity. Certainly, based on the information in Table I, the hybrid beamformer is an admissible alternative for PAF operation. Fig. 12 shows results of the hybrid beamformer for all values of and the described set of constraints. As expected, as the value of approaches 1 the side lobes and the sensitivity both increase. The results of these figures can be used to determine the proper value of that should be used for designing a hybrid beamformer for a specific application. C. LCMV Beamformer With the ability to minimize output variance while meeting desired constraints, the LCMV beamformer appears to be a

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Fig. 11. Measured far-field pattern of the hybrid beamformer with . The circular shape of the main beam and the constant side lobe level are characteristics of the equiripple beamformer. The peak side lobe level is slightly greater than with the equiripple beamformer, but there is also a significant increase in sensitivity.

Fig. 12. The ability to suppress the beam pattern side lobes is greatest for the , and gradually decreases as the equiripple beamformer corresponding to hybrid beamformer approaches the max-SNR result. Similarly, the sensitivity of the hybrid beamformer is greatly improved as the beamformer becomes less deterministic.

TABLE I COMPARISON OF BEAMFORMER TECHNIQUES

strong candidate for PAF beamforming. The LCMV beamformer is computed as [17] (20) where the columns of are calibration vectors associated with constraint angles and is a vector of corresponding desired response values. This beamformer is easy to implement and only requires calibration vectors at the desired constraint points. However, due to the limited degrees of freedom available to the beamformer it is not possible to control the entire FOV and avoid undesired beam pattern structure. Introducing additional constraints to obtain more control uses degrees of freedom that are needed to minimize the variance, causing a decrease in sensitivity and distortions in the beam pattern. Example beam patterns using the LCMV beamformer are shown in Figs. 13 and 14. Fig. 13 was constructed with five constraint points matching those described in Section III-D. The constraints are met, but the remainder of the pattern is unpredictable. In order to mitigate the distortion four additional equally-spaced constraints were added to the first null. The result given in Fig. 14 compares well with that of the hybrid beamformer with , achieving a sensitivity of 2.86 , but due to the limited constraints in the LCMV, there is noticeable variation in the side lobe structure and the peak side lobe level of 15.24 dB is nearly 2 dB higher than in the hybrid result.

Fig. 13. Measured beam pattern from LCMV beamformer with a main beam constraint plus four others evenly spaced within the first null. The constraints are not adequate to provide the uniformly low side lobe structure given by the hybrid beamformer.

Fig. 14. Measured beam pattern from LCMV beamformer with a main beam constraint plus eight others evenly spaced within the first null. This pattern but has higher closely matches that of the hybrid beamformer with side lobes and decreased sensitivity.

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Fig. 15. The original measured calibration set used in this analysis was highly oversampled. There is only a slight difference between the measured beam patterns after full (left half) and sparse (right half) transformations of the modeled beamformer, but the calibration time difference is about 3 hours. The sparse transformation introduces distortions and a decrease in sensitivity.

D. Value of Modeled Beamformers The transformation process described in Section III-D requires a set of measured calibrators to best preserve the desired beam pattern structure, but this process can be bypassed if we use the measured calibrators directly to design the beamformer. Still, there are some potential benefits of modeled design that make it an appealing consideration. Modeled design would be useful if there was an advantage to (1) optimizing over a more dense grid of calibrators than is required for the transformation, or to (2) having calibration points on an increased span of angles. We explore these possibilities below. Using the MDL algorithm to remove poor calibrators from our highly oversampled 33 33 calibration grid, we are left with 817 pointings. Thinning this grid to just 44 points reduces calibration time by about 3 hours, but a modeled beamformer transformation based on this sparse calibration set results in beam pattern distortion as seen in Fig. 15. The left half pattern is the result of a full 817 point transformation and the right half image is that of the more sparse. A significant difference in the sensitivity is also obtained: 1.74 for the full transformation and 1.66 for the sparse. The pattern distortion and sensitivity loss may be tolerable for the given reduction in calibration measurement time, but the result is still less desirable than when the sparse calibration set is directly used itself to design the beamformer. The equiripple beamformer computed directly with the sparse set of measured calibrators still exhibits distortion, but offers much better sensitivity, achieving 1.78 . Because of the great dependence on the details of the transformation data set, any potential benefits of modeled dense calibration beamformer design are lost in the transformation process. Designing modeled beamformers over a larger angular region than can be covered with measured calibrators is only beneficial if the beam shape control in the extended region is not forfeited during the transformation procedure. The left half image of Fig. 16 was again obtained after a transformation with a full calibration set, while the right half image was transformed with

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Fig. 16. The eam pattern that extends beyond the calibration region cannot be controlled. A full transformation from the model gives the left half pattern and a reduced 19 19 transformation region (represented by the dash-lined box) gives the right half pattern. The modeled weights were computed based on a larger 33 33 grid of calibrators. The patterns are in relative agreement within the box, but not at all outside the box.

a reduced size grid of 19 19 points, bounded by the dashed line. The result is a loss of control of the side lobes outside the calibration region once a transformation is applied. Again we see that the benefits of modeled design are restricted by the need for, and the limits of the calibration set used in the transformation process. Based on these results, we conclude that it is impractical to design measured deterministic PAF beamformers using simulation models. Deterministic PAF beamformers are best designed using measured calibration data directly, avoiding pattern distortion and sensitivity loss that accompany a transformation from the model. E. Angular Limits of Pattern Control As noted in Section IV-D pattern shape control with deterministic beamformers is limited by the angular range of the calibration vectors used in the design process. This introduces concerns about the behavior of the beam pattern beyond the reach of good calibration vectors. The inherent dish aperture pattern, governed by the properties of the reflector dish, begins to dominate the combined array-dish pattern at some angle, after which we lose most control of the beam pattern shape. It is the region between the edge of the calibration set and the start of the dish aperture dominance that is of concern. Since we cannot measure good calibration vectors in this region of interest, we must draw conclusions through analysis of modeled results. We are interested in knowing whether the range over which good calibrators can be obtained extends to the angle at which the dish pattern begins to the dominate the PAF far-field pattern. If it does not, as seen in Fig. 16, there will be a region of the beam pattern that is uncontrollable. To determine if the measured calibration range is large enough to fill this gap, we have computed modeled equiripple beamformers using both a 33 33 grid (matching that obtained in practice) and a larger 101 101 grid of calibration points (same grid point density in each case). A comparison of the resulting beam patterns reveals no noticeable difference. Although in practice the calibration set

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is limited by the SNR of the calibration source, we conclude that it is adequate to provide control of all parts of the beam pattern that are not dominated by the dish aperture pattern. V. CONCLUSION Employing PAFs for radio astronomical observations introduces a need for beamformers that can provide both high sensitivity and beam pattern stability. The max-SNR and numerically optimized equiripple beamformers presented above represent the extreme cases: one offering the best sensitivity available and the other full beam pattern shape control. In order to satisfactorily manage the tradeoff between these conflicting goals we have introduced a hybrid beamforming method. The hybrid beamformer uses a numerical optimizer and the weighting parameter to provide both adequate sensitivity and the necessary pattern control. A comparison between the max-SNR, equiripple, and hybrid beamformers has shown the usefulness of the hybrid approach. Calibration of a PAF system must be completed on-reflector, using a bright astronomical source. This limits the angular range over which we can obtain accurate calibrators. To measure the accuracy of our calibration vectors we have used an algorithm based on the minimum description length principle. We have concluded that good calibrators can be obtained (i.e., a beam can be steered) in all directions within the limited FOV imposed by the reflector dish aperture pattern. Due to unpredictable PAF operating conditions which limit the accuracy of simulation models, beamformers designed using modeled calibration data introduce substantial distortions in the final measured beam pattern. A transformation step was introduced to mitigate these effects. Such a transformation requires a grid of measured calibration vectors, which can be used themselves to directly compute the desired beamformer. It has been shown that there is no benefit to designing beamformers with a model because the outcome of the required transformation is so highly dependent upon the limitations of the measured calibration data. The max-SNR and numerically optimized equiripple beamformers presented in this analysis are representative of any number of beamformers that could be used in PAF systems. They were used as simple examples in demonstrating solutions to general PAF beamforming challenges. Future work should focus on developing more complex beamformers that address the need for reduced system noise, more broadband systems, and greater control of the beam pattern details. Additional analysis on the persistence of calibration data is also of interest. REFERENCES [1] M. Elmer and B. D. Jeffs, “Beamformer design for radio astronomical phased array feeds,” in Proc ICASSP, Mar. 14–19, 2010, pp. 2790–2793. [2] B. Veidt and P. Dewdney, “A phased-array feed demonstrator for radio telescopes,” in Proc. URSI General Assembly, 2005. [3] W. A. van Cappellen, J. G. Bij de Vaate, M. V. Ivashina, L. Bakker, and T. Oosterloo, “Focal plane arrays evolve,” in Proc. URSI General Assembly, Chicago, IL, Aug. 2008. [4] R. P. Millenaar, T. Oosterloo, and M. Brentjens, “The radio observatory at ASTRON: News from the WSRT and LOFAR,” in Proc. URSI General Assembly, Chicago, IL, Aug. 2008. [5] J. R. Fisher, K. F. Warnick, B. D. Jeffs, G. Cortes-Medellin, R. D. Norrod, F. J. Lockman, J. M. Cordes, and R. Giovanelli, “Phased array feeds,” in Astro2010 Technology Development White Paper. Charlottesville, VA: National Radio Astronomy Observatory, 2009.

[6] S. G. Hay, J. D. O’Sullivan, J. S. Kot, C. Granet, A. Grancea, A. R. Forsyth, and D. H. Hayman, “Focal plane array development for ASKAP (Australian SKA Pathfinder),” in Proc. EuCAP, Nov. 2007, pp. 1–5. [7] S. G. Hay and J. D. O’Sullivan, “Analysis of common-mode effects in a dual-polarized planar connected-array antenna,” Radio Science, vol. 43, pp. RS6S04–RS6S04, 2008. [8] J. Landon, M. Elmer, J. Waldron, D. Jones, A. Stemmons, B. D. Jeffs, K. F. Warnick, J. R. Fisher, and R. D. Norrod, “Phased array feed calibration, beamforming, and imaging,” The Astronomical J., vol. 139, no. 3, pp. 1154–1167, 2010. [9] P. E. Dewdney, P. J. Hall, R. T. Schilizzi, and T. J. L. W. Lazio, “The square kilometre array,” Proc. IEEE, vol. 97, no. 8, pp. 1482–1496, Aug. 2009. [10] D. R. DeBoer, R. G. Gough, J. D. Bunton, T. J. Cornwell, R. J. Beresford, S. Johnston, I. J. Feain, A. E. Schinckel, C. A. Jackson, M. J. Kesteven, A. Chippendale, G. A. Hampson, J. D. O’Sullivan, S. G. Hay, C. E. Jacka, T. W. Sweetnam, M. C. Storey, L. Ball, and B. J. Boyle, “Australian SKA Pathfinder: A high-dynamic range wide-field of view survey telescope,” Proc. IEEE, vol. 97, no. 8, pp. 1507–1521, Aug. 2009. [11] K. F. Warnick and M. A. Jensen, “Effects of mutual coupling on interference mitigation with a focal plane array,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2490–2498, Aug. 2005. [12] M. V. Ivashina, M. Kehn, P.-S. Kildal, and R. Maaskant, “Control of reflection and mutual coupling losses in maximizing efficiency of dense focal plane arrays,” in Proc. EuCAP 2006, Nov. 2006, pp. 1–6. [13] M. N. M. Kehn, M. V. Ivashina, P.-S. Kildal, and R. Maaskant, “Definition of unifying decoupling efficiency of different array antennas-case study of dense focal plane array feed for parabolic reflector,” AEU—Int. J. Electron. Commun., 2009, to be published. [14] P. A. Fridman and W. A. Baan, “RFI mitigation methods in radio astronomy,” Astron. Astrophys., vol. 378, pp. 327–344, 2001. [15] J. E. Evans, D. F. Sun, and J. R. Johnson, Application of Advanced Signal Processing Techniques to Angle of Arrival Estimation in ATC Navigation and Surveillance Systems. Cambridge, MA: MIT Lincoln Laboratory, 1982. [16] B. Ottersten, “Array processing for wireless communications,” in Proc. 8th IEEE Signal Processing Workshop Statistical Signal Array Process. (Cat. No.96TB10004), Jun. 1996, pp. 466–473. [17] B. D. Van Veen and K. M. Buckley, “Beamforming: A versatile approach to spatial filtering,” IEEE Signal Process. Mag., vol. 5, no. 2, pp. 4–24, Apr. 1988. [18] H. L. V. Trees, Detection, Estimation, and Modulation Theory, Part IV, Optimum Array Processing. New York: Wiley, 2002. [19] M. N. M. Kehn and L. Shafai, “Characterization of dense focal plane array feeds for parabolic reflectors in achieving closely overlapping or widely separated multiple beams,” Radio Science, vol. 44, no. 3, pp. RS3014–RS3014, 2009. [20] K. J. Maalouf and E. Lier, “Theoretical and experimental study of interference in multibeam active phased array transmit antenna for satellite communications,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 587–592, Feb. 2004. [21] J. M. Cordes, P. C. C. Freire, D. R. Lorimer, F. Camilo, D. J. Champion, D. J. Nice, R. Ramachandran, J. W. T. Hessels, W. Vlemmings, J. van Leeuwen, S. M. Ransom, N. D. R. Bhat, Z. Arzoumanian, M. A. McLaughlin, V. M. Kaspi, L. Kasian, J. S. Deneva, B. Reid, S. Chatterjee, J. L. Han, D. C. Backer, I. H. Stairs, A. A. Deshpande, and C.-A. Faucher-Gigure, “Arecibo pulsar survey using ALFA I. Survey strategy and first discoveries,” Astrophys. J., vol. 637, no. 1, pp. 446–455, Jan. 2006. [22] L. Staveley-Smith, W. E. Wilson, T. S. Bird, M. W. Sinclair, and R. D. Ekers, “The Parkes 21 cm multibeam receiver,” in Proc. ASP Conf. Ser., Multi-Feed Syst. Radio Telesc., 1995, vol. 75, pp. 136–144. [23] L. Staveley-Smith, W. E. Wilson, T. S. Bird, M. J. Disney, R. D. Ekers, K. C. Freeman, R. F. Haynes, M. W. Sinclair, R. A. Vaile, R. L. Webster, and A. E. Wright, “The Parkes 21 cm multibeam receiver,” Publicat. Astronom. Soc. Australia, vol. 13, no. 3, pp. 243–248, Nov. 1996. [24] D. Hayman, R. Beresford, J. Bunton, C. Cantrall, T. Cornwell, A. Grancea, C. Granet, J. Joseph, M. Kesteven, J. O’Sullivan, J. Pathikulangara, T. Sweetnam, and M. Voronkov, “The NTD interferometer: A phased array feed test bed,” in Proc. Workshop Appl. Radio Sci., Queensland, Australia, Feb. 2008 [Online]. Available: http://www.ncrs.org.au/wars/wars2008/ Hayman%20et%20al%20WARS%202008.pdf [25] B. D. Jeffs, K. F. Warnick, J. Landon, J. Waldron, D. Jones, J. R. Fisher, and R. D. Norrod, “Signal processing for phased array feeds in radio astronomical telescopes,” IEEE J. Sel. Topics Signal Process., vol. 2, no. 5, pp. 635–646, Oct. 2008.

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[26] Y. T. Lo, S. W. Lee, and Q. H. Lee, “Optimization of directivity and signal-to-noise ratio of an arbitrary antenna array,” Proc. IEEE, vol. 54, no. 8, pp. 1033–1045, Aug. 1966. [27] T. Oosterloo, W. van Cappellen, and L. Bakker, “First results with APERTIF,” in Proc. CALIM2008, Perth, Australia, Apr. 2008 [Online]. Available: http://calim2008.atnf.csiro.au/twiki/pub/Main/WorkshopProgram/OosterlooCalim.pdf [28] B. D. Jeffs, K. F. Warnick, M. Elmer, J. Landon, J. Waldron, D. Jones, R. Fisher, and R. Norrod, “Calibration and optimal beamforming for a 19 element phased array feed,” in Proc. CALIM2008, Perth, Australia, Apr. 2008 [Online]. Available: http://calim2008.atnf.csiro.au/ twiki/pub/Main/WorkshopProgram/JeffsCalim.pdf [29] M. A. W. Verheijen, T. A. Oosterloo, W. A. van Cappellen, L. Bakker, M. V. Ivashina, and J. M. van der Hulst, Apertif, A Focal Plane Array for the WSRT, R. Minchin and E. Momjian, Eds. New York: AIP, 2008, vol. 1035, pp. 265–271. [30] K. F. Warnick, B. D. Jeffs, J. Landon, J. Waldron, D. Jones, J. R. Fisher, and R. Norrod, “Beamforming and Imaging With the BYU/NRAO Lband 19-Element Phased Array Feed,” in Proc. 13th Int. Symp. Antenna Technol. Appl. Electromagn. Canadian Radio Science Meeting, 2009, pp. 1–4. [31] W. A. van Cappellen, L. Bakker, and T. A. Oosterloo, “Experimental results of a 112 element phased array feed for the westerbork synthesis radio telescope,” in Proc. APSURSI, Jun. 2009, pp. 1–4. [32] M. V. Ivashina, O. A. Iupikov, R. Maaskant, W. A. van Cappellen, L. Bakker, and T. Oosterloo, “Off axis beam performance of focal plane arrays for the westerbork synthesis radio telescope—Initial results of a prototype system,” in Proc. APSURSI , Jun. 2009, pp. 1–4. [33] M. Voronkov and T. Cornwell, “On the calibration and imaging with eigenbeams,” in ATNF SKA Memo 12, Jan. 2007 [Online]. Available: http://www.atnf.csiro.au/projects/mira/newdocs/eigenbeams.pdf [34] T. Willis, “Simulations of synthesis telescope antennas equipped with focal plane arrays,” in Proc. CALIM2009, Socorro, NM, Mar. 2009 [Online]. Available: https://safe.nrao.edu/ wiki/pub/Software/CalIm09Program/agw_calim09.pdf [35] C. K. Hansen, “Beamforming Techniques and Interference Mitigation Using a Multiple Feed Array for Radio Astronomy,” M.S. thesis, Brigham Young University, Provo, UTAH, 2004. [36] K. F. Warnick, B. Woestenburg, L. Belostotski, and P. Russer, “Minimizing the noise penalty due to mutual coupling for a receiving array,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1634–1644, Jun. 2009. [37] J. S. Waldron, “Nineteen-Element Experimental Phased Array Feed Development and Analysis on Effects of Focal Plane Offset and Beam Steering on Sensitivity,” M.S. thesis, Brigham Young University, Provo, UTAH, 2008. [38] J. Rissanen, “Minimum description length,” Scholarpedia, vol. 3, no. 8, pp. 6727–6727, 2008. [39] M. Wax and T. Kailath, “Detection of signals by information theoretic criteria,” IEEE Trans. Acoust., Speech Signal Process., vol. 33, no. 2, pp. 387–392, Apr. 1985. [40] K. F. Warnick and B. D. Jeffs, “Efficiencies and system temperature for a beamforming array,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 565–568, 2008.

Michael Elmer received the B.S. degree (cum laude) in electrical and electronics engineering from California State University, Sacramento, in 2005. He is currently pursuing the Ph.D. degree from Brigham Young University, Provo, UT, in Electrical Engineering. Research interests include digital signal processing and RF design. His current research consists of beamforming methods and calibration procedures and techniques for phased array feeds.

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Brian D. Jeffs (M’90–SM’02) received the B.S. (magna cum laude) and M.S. degrees in electrical engineering from Brigham Young University, Provo, UT, in 1978 and 1982 respectively. He received the Ph.D. degree from the University of Southern California, Los Angeles, in 1989, also in electrical engineering. He currently holds the rank of Professor in the Department of Electrical and Computer Engineering at Brigham Young University, where he lectures in the areas of signals and systems, digital signal processing, probability theory, and stochastic processes. Current research activity includes array signal processing for radio astronomy and radio frequency interference mitigation. Previous employment includes Hughes Aircraft Company where he served as a sonar signal processing systems engineer in the antisubmarine warfare group. Projects there included algorithm development and system design for digital sonars in torpedo, surface ship towed array, and helicopter dipping array platforms. Dr. Jeffs was a Vice General Chair for IEEE ICASSP-2001 held in Salt Lake City, UT. He was a member of the executive organizing committee for the 1998 IEEE DSP Workshop, co organized the 2010 Workshop on Phased Array Antennas Systems for Radio Astronomy, and served several years as chair of the Utah Chapter of the IEEE Communications and Signal Processing Societies.

Karl F. Warnick (SM’04) received the B.S. degree (magna cum laude) (hons) and the Ph.D. degree from Brigham Young University (BYU), Provo, UT, in 1994 and 1997, respectively. From 1998 to 2000, he was a Postdoctoral Research Associate and Visiting Assistant Professor in the Center for Computational Electromagnetics at the University of Illinois at Urbana-Champaign. Since 2000, he has been a faculty member in the Department of Electrical and Computer Engineering at BYU, where he is currently a Professor. In 2005 and 2007, he was a Visiting Professor at the Technische Universität München, Germany. Dr. Warnick has published many scientific articles and conference papers on electromagnetic theory, numerical methods, remote sensing, antenna applications, phased arrays, biomedical devices, and inverse scattering, and is the author of the books Problem Solving in Electromagnetics, Microwave Circuits, and Antenna Design for Communications Engineering (Artech House, 2006) with Peter Russer, Numerical Analysis for Electromagnetic Integral Equations (Artech House, 2008), and Numerical Methods for Engineering: An Introduction Using MATLAB and Computational Electromagnetics Examples (Scitech, 2010). Dr. Warnick was a recipient of the National Science Foundation Graduate Research Fellowship, Outstanding Faculty Member award for Electrical and Computer Engineering (2005), and the BYU Young Scholar Award (2007). He has served the Antennas and Propagation Society as a member of the Education Committee and as a session chair and special session organizer for the International Symposium on Antennas and Propagation and other meetings affiliated with the Society. He is a frequent reviewer for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS. Dr. Warnick has been a member of the Technical Program Committee for the International Symposium on Antennas and Propagation for several years and served as Technical Program Co-Chair for the Symposium in 2007.

J. Richard Fisher received the B.S. degree in 1965 in physics from The Pennsylvania State University, University Park, PA, and the Ph.D. degree in 1972 in astronomy from the University of Maryland. He joined the NRAO scientific staff at Green Bank, WV immediately following graduate school and has held various positions such as Head of the Electronics Division, Site Director, and Project Manager of several instrumentation projects. In 1978 he took leave from the NRAO to spend 18 months at the Division of Radio Physics, CSIRO in Australia and 3 months at the Raman Research Institute, Bangalore, India. In February 2005 he moved to the Charlottesville, VA, offices of NRAO to pursue scientific and instrumentation research. His most recent position is Chief Technologist at NRAO. His research interests include cosmology, galaxy formation, antenna design, signal processing.

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Roger D. Norrod received the M.S.E.E. degree in 1979 from Tennessee Technological University, Cookeville. TN. During 1979–2011 he worked at the National Radio Astronomy Observatory, specializing in microwave receiver and systems design. He designed several cryogenic low-noise amplifiers and microwave receivers for radio astronomy, and supervised production of over twenty receivers for the VLBA project. During the 100-meter Green Bank Telescope project, he served as head of the

antenna optics group, then as the head of the Electronics group, and later as the manager of NRAO Systems development for the GBT, coordinating activities related to Electronics, Software, and Data Analysis. He served several years as the Electronics Division head at Green Bank, and as the lead engineer within the Microwave Group. He recently retired from the NRAO.

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Experimental Results for the Sensitivity of a Low Noise Aperture Array Tile for the SKA E. E. M. Woestenburg, Laurens Bakker, and Marianna V. Ivashina

Abstract—Aperture arrays have been studied extensively for application in the next generation of large radio telescopes for astronomy, requiring extremely low noise performance. Prototype array systems need to demonstrate the low noise potential of aperture array technology. This paper presents noise measurements for an Aperture Array tile of 144 dual-polarized tapered slot antenna (TSA) elements, originally built and characterized for use as a Phased Array Feed for application in an L-band radio astronomical receiving system. The system noise budget is given and the dependency of the measured noise temperatures on the beam steering is discussed. A comparison is made of the measurement results with simulations of the noise behavior using a system noise model. This model includes the effect of receiver noise coupling, resulting from a changing active reflection coefficient and array noise contribution as a function of beam steering. Measurement results clearly demonstrate the validity of the model and thus the concept of active reflection coefficient for the calculation of effective system noise temperatures. The presented array noise temperatures, with a best measured value of 45 K, are state-of-the-art for room temperature aperture arrays in the 1 GHz range and illustrate their low noise potential. Index Terms—Antenna array, low noise, noise coupling effects, radio astronomy.

I. INTRODUCTION

F

OR the Square Kilometre Array (SKA, [1]), the next generation of large radio telescopes with two orders of magnitude increase in sensitivity over existing telescopes, the radio astronomical community is considering the use of dense Aperture Arrays (AAs) for the SKA mid-frequency range from 400 MHz to 1400 MHz. Considerable effort has been put in the development of aperture arrays over the last ten years. Low frequency aperture arrays up to a few 100 MHz have been developed and are already in operation [2], [3] and several prototypes for higher frequencies up to 1.5 GHz have been built [4], [5], while development for dense aperture arrays over the frequency range from 100 MHz to 1500 MHz is continuing [6], [7]. The arrays for the SKA-mid frequency range will consist of a large number of 1 m tiles with approximately 100 flat antennas per tile with LNAs

Manuscript received February 16, 2011; revised May 30, 2011; accepted August 03, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. E. E. M. Woestenburg and L. Bakker are with the Netherlands Institute for Radio Astronomy (ASTRON), 7990 AA Dwingeloo, The Netherlands (e-mail: [email protected]; [email protected]). M. V. Ivashina was with ASTRON, Dwingeloo, The Netherlands. She is now with the Department of Earth and Space Sciences, Chalmers University of Technology, S-41296 Gothenburg, Sweden (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173140

and receivers, of which the signals will be combined to form a radio telescope with a (collecting) area of 500 m , as an alternative to reflector telescopes. Advantages of this AA-concept over conventional reflector antennas are the wide field of view, the possibility to avoid mechanical steering and maintenance (as beams will be formed and steered electronically) and the opportunity to observe with a large number of beams simultaneously. On the other hand, the arrays will have to operate at ambient temperature, because cooling cannot be realized at reasonable cost and complexity, due to the large number of antennas and LNAs. At the same time the array sensitivity is of utmost importance and is determined by the ratio of effective collecting area and system temperature Aeff/Tsys. The large collecting area of one square kilometer is the main reason for a greatly enhanced sensitivity, but requires that the system noise temperature will be at a competitive level with that of conventional radio telescopes. Emphasis in the development of AA-tiles for the concept demonstrator systems has thus far not been on achieving the lowest possible noise temperatures, which is a challenge because of the room temperature operation. Reported noise temperatures until now for aperture array tiles and systems in the 1 GHz frequency range are relatively high, around 170 K in [8] and approximately 100 K in [9], compared to the ultimately required 40 K maximum system noise temperature for the SKA near 1 GHz. Nevertheless, the results in [9] are similar to predicted values in [10] and a decreasing trend in noise temperature is obvious [11]. The focus for the AA-tiles and systems in [4] and [5] has been on proof of principle, large scale systems and adequate production techniques, limited costs and operability. The latter have been demonstrated in [9]. At the same time development of low noise AAs has been progressing at different groups within the SKA community. A similar development has been ongoing for Phased Array Feed (PAF) systems for reflector telescopes [12]–[15], while wide field imaging has been demonstrated with PAF prototype systems on such telescopes [15]–[17]. Simultaneous with the development and construction of AA- and PAFdemonstrator systems progress was made in the theoretical analysis and modeling of the noise properties of phased array systems with high sensitivity, in particular with respect to the effect of noise coupling between the antenna elements [18]–[22]. The understanding, resulting from this theoretical work, has favored the realization of prototype arrays with a factor 2, respectively 4 improvement in noise performance with respect to [9] and [8]. This paper discusses state-of-the-art results of noise measurements with a prototype AA-tile, which enable verification of the noise models.

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The AA results presented in this paper have profited from the development of a tile for a PAF system (APERTIF, APERture Tile In Focus, [17]), which will replace the existing 21 cm single pixel feeds of the Westerbork Synthesis Radio Telescope (WSRT), and should have competitive performance with the present 21 cm cryogenic receiver system. This development resulted in room temperature receivers and LNAs for an APERTIF prototype tile, giving a measured system temperature in the telescope below 68 K for an on-axis beam at 1.4 GHz [16], [17], using LNAs with close to 40 K noise temperature [23]. The construction of the PAF tile with individual antennas and receiver chains, allowed its use as an AA-tile, while the measured results as a PAF gave rise to great expectations with respect to the noise performance as an AA-tile. A description of the prototype tile as AA-tile, as well as the noise measurement set-up will be given in Section II. Measurement results of the array noise temperature will be presented in Section III, showing state-of-the-art noise performance for a room temperature AA-tile. Comparable performance has been recently shown for a smaller array in [24], but only measured data for single channel receiver noise temperatures are presented there. Results from our measurements are full array noise temperatures and will be presented for various beam configurations, both with analog and digital beam forming. In Section IV a noise temperature budget for the AA-tile will be presented and a comparison with simulated values is made. The latter are calculated using the equivalent system noise model in [22] and take into account the variation of system noise temperature as a function of scan angle [20], [21]. The results presented here have a lowest measured value of 45 K at 1200 MHz and illustrate the low noise potential of AAs for the SKA. II. MEASUREMENT METHOD The APERTIF tile used for the aperture array measurements consists of a total of 144 Vivaldi antenna elements and LNAs, configured in a two-dimensional array with 8 9 elements with 11 cm spacing, for each of the two linear polarizations [25]. The antenna array operates in the frequency range 1000–1750 MHz with the optimal signal-to-noise performance around 1.4 GHz at which the element spacing is close to the half wavelength. Details of the system for use as a PAF can be found in [16], [17] and [26]. For noise measurements as an aperture array, using the Y-factor hot/cold method [8], [15], the tile is placed horizontally on the ground as shown in Fig. 1. For the measurements two methods were followed, one using analog beam forming with 2 2 and 4 4 elements, the other using digital beam forming for various beam configurations. To assess the low noise potential of the tile, initial measurements were done with analog beam forming only. The output signals from the LNAs of a 2 2 and a 4 4 array in the centre of the tile were added with in-phase combiners, forming broadside beams. The analog output signals of the beam formers were fed to the input of an Agilent Noise Figure Meter 8970B and the noise temperature was determined with the Y-factor method, using the loads described next for the digital processing method. The first very promising measurement results urged the use of a more flexible system, with which beams in any direction could be formed. The digital beam forming and processing system of the APERTIF-pro-

Fig. 1. Prototype array situated for hot/cold measurements, with the hot load behind the tile. In the background some of the WSRT telescopes are visible.

totype (see [17] for a description of the digital processing hardware and architecture) was subsequently made available for offline data processing and calculation of the beam former output noise power (based on the measured spectral noise-wave correlation matrix of the array-receiver system according to the procedure in [26]). In order to compare to the previous measurements and verify the digital processing, the outputs of the analog beam formers were first connected to one channel of the digital processing system. For the digital processing method, a total of 49 individual antenna elements and LNAs (limited by the number of available receivers at the time of the measurements) are connected via 25 m long coaxial cables to a back-end. The electrical lengths of the cables have been made equal within 5 . All receiver channels, including the cables, were calibrated with respect to a reference channel. The differential phase stability of the receiver channels is 1 . The back-end electronics is located in a shielded cabin, with down converter modules and digital processing hardware [17]. Data are taken with the array facing the (cold) sky as a ‘cold’ load, after which a room temperature absorber panel is placed over the array for the measurement with the ‘hot’ load. This 1.2 m 1.2 m panel, shown behind the array in Fig. 1, is slightly oversized with respect to the outer dimensions of the tile (1 m 1 m), to avoid the edge elements seeing the cold sky beyond the edge of the absorber panel. Once placed over the array, the panel leaves approximately 0.3–0.5 wavelength (10 cm) space between the tips of the absorbers and the aperture plane of the antenna array. At this distance the absorbers do not influence the measured array S-parameters and provide a load with 30 dB return loss at normal incidence. Transmission through the absorbers, which have a depth of 18 cm and a 2.5 cm thick absorptive backing, is negligible at these low frequencies. The data processing takes into account correlations between individual receiver channels in calculation of the beam former output noise power. This is done through performing the eigenvalue decomposition of the measured spectral noise-wave correlation matrix of the array-receiver system and taking its dominant eigenvector. This vector holds the noise-wave amplitudes

WOESTENBURG et al.: EXPERIMENTAL RESULTS FOR THE SENSITIVITY OF A LOW NOISE APERTURE ARRAY TILE FOR THE SKA

at the receiver outputs that arise due to all internal and external noise sources and is a function of frequency. This vector is sometimes named the ‘calibration vector’ [28], as this procedure ensures that practical errors (such as electronic gain differences and non-equalization of the cables) do not impose limits on the accuracy of the noise temperature measurements and desired beam forming direction. Using off-line digital processing, beams with a combination of any of the 49 active elements can be formed and beams may be scanned in any direction by applying weights to the elements of this noise-wave vector (see (1) and (2) in [26]). In this way the array noise temperature as a function of frequency from 1.0 to 1.8 GHz has been determined for 2 2, 4 4, 5 5 and 7 7 element arrays, looking at broadside. Also the array noise temperatures as a function of scan angle for the 4 4 and 7 7 element arrays have been determined.

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Fig. 2. Comparison of results for analog and digital beam forming for 4- and 16-element arrays.

III. EXPERIMENTAL RESULTS For all measurements a cold sky load temperature of 4 K was assumed, slightly above the cosmic background noise of 2.7 K. The 4 K cold load temperature is considered to be a realistic value, as long as the beam is not directed at the horizon or directly pointed at the sun, which may be distinguished as an approximately 200 K source for the 49-element array in Fig. 5. The ambient temperature was taken as the hot load temperature, approximately 300 K during the measurements. The uncertainty in the measured ambient temperature is smaller than 1 K. Another source of error in the calculation of the noise temperature is the accuracy of the power measurements and the resulting Y-factor. The accuracy of the Y-factor measurement is estimated at 0.1 dB, which would result in a maximum error of 2 K for the measured Y-factors around 8 dB and the given load temperatures. The total results in a maximum absolute error in the noise measurement of approximately 5 K. The error between measurements at different frequency points, performed in the same measurement cycle, is much smaller and is estimated at 1 K maximum. It should be pointed out that the error in the hot load temperature is mitigated considerably by the large Y-factors measured. An error due to the measurement accuracy of the ambient temperature or the construction and size of the absorber panel would give a 1 K noise temperature error for a 6 K error in the hot load temperature, based on the measured Y-factors around 8 dB. A possible error due to the absorber panel would alter (lower) the hot load temperature, which could result in the presented noise temperatures being too pessimistic. In practice this effect appears to be negligible. A. Array Noise Temperatures at Broadside Fig. 2 shows a comparison of the measurement results for the analog and digital beam forming, for the broadside beams formed using 4 and 16 active elements in the centre of the array. The passive elements were connected to 75 loads during these initial measurements, which were performed mainly to verify the digital processing method. The measured array noise temperatures for the 4- and 16-element arrays are very similar, for both analog and digital beam formers, with some minor differences due to small differences between these beam formers and

Fig. 3. AA noise temperature for broadside beams with various numbers of active elements, compared to simulation results for the 49-element beam.

the shape of the beams in combination with environmental factors. The results are consistent with the data (not shown here) taken with the analog processing system described in the previous section and validate the results from the digital processing method within the 5 K error bars. Using the digital processing method four different array beams have been formed with 4, 16, 25, and 49 active elements. During the measurements producing the digital data, all antenna elements were connected to their LNA and subsequent receiver chain, but for the unused elements the weights were set to zero during the data processing. Complex weights for the other, active, elements were set to direct the beam to broadside or any other desired direction. Fig. 3 shows the measured noise temperatures for broadside beams with various numbers of active elements, compared to simulation results for the 49-element beam. The 25-element array shows the lowest noise on average as a function of frequency, close to that of the 16-element array, both being slightly better than the 49-element array. This may be explained by the particular noise coupling contribution for the 49-element array near broadside. At slightly different beam angles for the 49-element array this contribution is reduced and results in the same minimum value at, e.g., 1.4 GHz as for the 16- and 25-element arrays. The 4-element array has a broader beam with 9–12 dB directivity at 1–1.6 GHz as compared to the larger arrays which have

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Fig. 4. Noise temperatures as a function of scan angle for 16- and 49-element arrays at 1400 MHz, compared to simulation results for both arrays. The simulations include noise coupling contributions and the noise pick-up from the environment at the horizon. The detailed effects of the location and height of trees and buildings are not taken into account in these simulations.

directivity values higher than 15 dB at the lowest frequency, and hence suffers from considerable noise contributions from the environment, even when it is directed at broadside. The simulated noise temperatures for the 49-element array, as well shown on Fig. 3, compare well within the absolute measurement accuracy of 5 K to the measured values for that array, as well as those for the 16- and 25-element arrays, over most of the frequency range.

Fig. 5. Two-dimensional scan of Tsys at 1400 MHz for a 49-element array, showing the location of one of the WSRT telescopes (top right) and the sun.

TABLE I NOISE BUDGET IN [K] OF THE 49-ELEMENT ARRAY AS A FUNCTION OF SCAN ANGLE AT 1400 MHZ FOR THE NORTH-SOUTH DIRECTION

B. Array Noise Temperature as a Function of Scan Angle An important property of AAs is the varying noise coupling between the antenna elements as a function of scan angle, described by the active reflection coefficient [20]–[22]. Using the array and LNA properties a maximum of 13 K is calculated for the 49-element array at 45 scan angle, 25% of the system noise budget (see the simulation results in Fig. 4 and in Section IV). It is therefore interesting to see how the measured array noise temperature varies with scan angle and beam steering in general. Fig. 4 shows the variation in noise temperature at 1400 MHz for the 16- and 49-element arrays, for a maximum scan angle (from broadside) of 85 . Up to scan angles of 40 the noise temperatures only slowly vary due to the changing noise coupling, quite well in agreement with the simulations. For larger scan angles the broad beams introduce noise from the environment, which dominates over the calculated variation in noise coupling. Obviously the narrower 49-element beam allows further scanning without increase in noise than the 16-element beam. It is also shown that the narrower beam more effectively samples the environmental temperature near the horizon, giving a higher system noise temperature. C. Results for Two-Dimensional Scanning In Fig. 5 a plot is shown of the system noise temperature at 1400 MHz for two-dimensional scanning of the 49-element array. Over most of the scanning range the noise temperature is well below 80 K, with lowest values under 50 K. As expected a strong increase in noise temperature is visible near the horizon, with extended hot regions at the location of one of the WSRT telescopes (top right) and the sun (lower right).

IV. NOISE BUDGET AND COMPARISON WITH MODELING RESULTS Table I shows the noise budget (partly based on simulations) of the prototype AA-tile at 1400 MHz, with the noise coupling contribution for a few scan angles. The equivalent system representation of [22] was used and the simulations were performed with the numerical approach in [26], [27]. The simulations did not take into account the effect of the sky noise and contributions from obstacles (trees and telescopes/buildings) near the horizon. The array noise temperature was calculated as a sum of several noise contributions due to external and internal noise sources, using the equivalent system representation in [22]. The external noise contribution is the ground noise picked up due to antenna back radiation, which was computed from the simulated illumination pattern of the antenna array for the specified beam former weights. The internal noise contribution includes two components: • The thermal antenna noise due to the losses in the conductor and dielectric materials of TSAs and microstrip feeds. The conductor losses are computed through the evaluation of the antenna radiation efficiency using the methodology detailed in [29] and the dielectric losses are

WOESTENBURG et al.: EXPERIMENTAL RESULTS FOR THE SENSITIVITY OF A LOW NOISE APERTURE ARRAY TILE FOR THE SKA

computed based on the experimental evaluation of the feed loss [30]. • Multi-channel receiver noise which is calculated using CAESAR software that is an array system simulator, developed at ASTRON [27]. This noise component accounts for the antenna-LNA impedance noise mismatch effect and minimal noise of LNAs. This noise coupling component represents a combined effect of the noisy LNAs and active reflection coefficients of the array elements that are frequency and weighting dependent. It was computed from the noise-wave covariance matrix of the antenna-receiver system (see (1) and Fig. 1 in [26]) that was defined in the absence of the external noise sources and antenna thermal noise. The antenna noise in Table I comprises contributions due to the ground noise pick up (1 K), losses in the conductor and dielectric material of the antenna and the microstrip feed. is the measured LNA noise temperature in a 50 system, including the receiver second stage contribution. is the simulated noise coupling contribution. Adding these contributions leads to the simulated array noise temperature:

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Fig. 6. Two-dimensional scan of the simulated noise coupling contribution in [K] for the 49-element array at 1400 MHz.

(1) Table I compares the result of (1) with the measured array noise temperature , from which a ‘measured’ value for the noise coupling may be derived, according to the following formula: (2) The simulated noise coupling contributions at 1400 MHz for a full two-dimensional scan are presented in Fig. 6 for one polarization of the array, showing relatively low noise coupling over a large scan range. One-dimensional scans in the direction with the smallest change in noise coupling (east-west), as well as for the perpendicular direction are shown in Fig. 7. The noise coupling contribution remains below 10 K for scan angles up to 60 in the east-west direction and below 14 K over the full scan range. In the north-south direction a maximum value of 13 K for the noise coupling at 1400 MHz was calculated at 45 , increasing to 50 K for scanning near the horizon. This result underlines that the increase in measured system noise temperature at large scan angles in Fig. 5 is caused by noise pick up at the horizon. The sum of the simulated noise contributions and the LNA noise temperature in Table I is almost identical (within the measurement accuracy of 5 K) to the measured array noise temperature for small scan angles. This confirms the validity of the models, with the active reflection coefficient being the only variable as a function of scan angle in Figs. 6 and 7 and for small scan angles in Fig. 5. For scan angles larger than 40–50 the measured results in Figs. 4 and 5 are influenced by noise pick up from the environment. The presented results lead to the prediction that for larger arrays with a narrow beam (and relatively low side-lobes) the simulation results will more accurately predict the measurements at larger scan angles. This will be the subject of further study.

Fig. 7. Noise coupling contribution at 1400 MHz for a 49-element array, for two orthogonal scan directions.

V. CONCLUSION In this paper experimental noise temperature results for an AA-tile have been presented, which demonstrate the lowest array receiver noise temperature for an AA-tile to date, with a minimum measured value of 45 K in L-band. The properties as a function of frequency and the effects of scanning the beam on the noise temperature are shown and compared to simulation results, showing good agreement. Based on the measurement and simulation results a number of observations have been made, which lead to the following conclusions: — array noise temperatures below 50 K have been consistently measured, with a value of 45 K at 1200 MHz; — the use of the edge elements in the array may cause some increase in system noise temperature, but lead to lower noise temperature if a smaller part of the array is active; — for scan angles up to the increase in noise temperature due to noise coupling remains below 13 K; — a beam formed with more elements may be scanned to larger angles before the horizon introduces noise, due to the narrow beam;

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— scanning a smaller beam to the horizon results in a larger contribution from the horizon to the system noise, because the beam then ‘sees’ a larger part of a hotter environment; — verification that the variation in system noise temperature for large scan angles is caused solely by a change in the noise coupling contribution, can only be done with a larger array, i.e., with a narrow beam (with low side-lobes). The presented results illustrate the low-noise potential of AAs for application as future sensitive radio telescopes. REFERENCES [1] Square Kilometer Array (SKA). [Online]. Available: www.skatelescope.org [2] J. D. Bregman, G. H. Tan, W. Cazemier, and C. Craeye, “A wideband sparse fractal array antenna for low frequency radio astronomy,” in Proc. IEEE Antennas and Propagat. Symp., Salt Lake City, UT, 2000, pp. 166–169. [3] R. Bradley, D. Backer, A. Parsons, C. Parashare, and N. Gugliucci, “A precision array to probe the epoch of reionization,” presented at the American Astronomical Society 207th Meeting, Washington, DC, 2006. [4] J. G. Bij de Vaate and G. W. Kant, “The phased array approach to SKA, results of a demonstrator project,” in Proc. European Microwave Conf., Milan, Italy, 2002, pp. 993–996. [5] M. Ruiter and E. van der Wal, “EMBRACE, a 10000 element next generation aperture array telescope,” in Proc. European Microwave Conf., Rome, Italy, 2009, pp. 326–329. [6] E. de Lera Acedo, N. Razavi-Ghods, E. Garcia, P. Duffett-Smith, and P. Alexander, “Ultra-wideband aperture array element design for low frequency radio astronomy,” IEEE Trans. Antennas Propagat., vol. 59, no. 6, pp. 1808–1816, Jun. 2011. [7] Aperture Array Verification Program (AAVP). [Online]. Available: www.ska-aavp.eu [8] E. E. M. Woestenburg and K. F. Dijkstra, “Noise characterization of a phased array tile,” in Proc. European Microwave Conf., Munich, Germany, 2003, pp. 363–366. [9] G. W. Kant, E. van der Wal, M. Ruiter, and P. Benthem, S. A. Torchinsky, Ed., “EMBRACE system design and realisation,” in Proc. Wide Field Science and Technology for the SKA, Limelette, Belgium, 2009, ASTRON, ISBN 978-90-805434-5-4. [10] E. E. M. Woestenburg and J. C. Kuenen, “Low noise performance perspectives of wideband aperture phased arrays,” in The Square Kilometre Array: An Engineering Perspective. New York: Springer, 2005, pp. 89–99. [11] J. G. Bij de Vaate, L. Bakker, and R. Witvers, S. A. Torchinsky, Ed. , “Active antenna design and characterization,” in Proc. Wide Field Science and Technology for the SKA, Limelette, Belgium, 2009, ASTRON, ISBN 978-90-805434-5-4. [12] M. A. W. Verheijen, T. A. Oosterloo, W. A. van Cappellen, L. Bakker, M. V. Ivashina, and J. M. van der Hulst, “APERTIF, a focal plane array for the WSRT,” in Proc. Conf. ‘The Evolution of Galaxies through the Neutral Hydrogen Window’ (AIP), 2008, vol. 1035, astro-ph/0806. 0234. [13] B. Veidt and P. Dewdney, “A phased-array feed demonstrator for radio telescopes,” in Proc. URSI General Assembly, New Delhi, India, 2005. [14] S. G. Hay et al., “Focal plane array development for ASKAP (Australian SKA Pathfinder),” in Proc. 2nd European Conf. Antennas and Propagat., Edinburgh, U.K., 2007. [15] K. F. Warnick, B. D. Jeffs, J. Landon, J. Waldron, R. Fisher, and R. Norrod, “BYU/NRAO 19-element phased array feed modeling and experimental results,” in Proc. URSI General Assembly, Chicago, IL, 2007. [16] W. A. van Cappellen, L. Bakker, and T. Oosterloo, “Experimental results of a 112 element phased array feed for the Westerbork synthesis radio telescope,” in 2009 IEEE Int. Symp. Antennas and Propagation & USNC/URSI National Radio Science Meeting, Charleston, SC, 2009, pp. 1–4. [17] W. A. van Cappellen and L. Bakker, “APERTIF: Phased array feeds for the westerbork synthesis radio telescope,” presented at the IEEE Int. Symp. Phased Array Systems and Technology, Boston, MA, 2010. [18] J. P. Weem and Z. Popovié, “A method for determining noise coupling in a phased array antenna,” in IEEE MTT-S Int. Microwave Symposium Dig., 2001, vol. 1, pp. 271–274.

[19] C. Craeye, B. Parvais, and X. Dardenne, “MoM simulation of signal-tonoise patterns in infinite and finite receiving antenna arrays,” IEEE Trans. Antennas Propagat., vol. 52, pp. 3245–3256, 2004. [20] R. Maaskant and E. E. M. Woestenburg, “Applying the active impedance to achieve noise match in receiving array antennas,” presented at the IEEE Int. Symp. Antennas and Propagat., Honolulu, HI, 2007. [21] K. F. Warnick, B. Woestenburg, L. Belostotski, and P. Russer, “Minimizing the noise penalty due to mutual coupling for a receiving array,” IEEE Trans. Antennas Propagat., vol. 57, pp. 1634–1644, Jun. 2009. [22] M. Ivashina, R. Maaskant, and B. Woestenburg, “Equivalent system representation to model the beam sensitivity of receiving antenna arrays,” IEEE Antennas Wireless Propagat. Lett., vol. 7, no. 1, pp. 733–737, 2008. [23] J. G. Bij de Vaate, L. Bakker, E. Woestenburg, R. Witvers, G. Kant, and W. van Cappellen, “Low cost low noise phased-array feeding systems for SKA pathfinders,” presented at the 13th Int. Symp. Antenna Tech. and Applied Electromagnetics (ANTEM), Banff, Canada, 2009. [24] K. F. Warnick, D. Carter, T. Webb, J. Landon, M. Elmer, and B. D. Jeffs, “Design and characterization of an active impedance matched low noise phased array feed,” IEEE Trans. Antennas Propagat., vol. 59, no. 6, pp. 1876–1885, Jun. 2011. [25] M. Arts, M. Ivashina, O. Iupikov, L. Bakker, and R. van den Brink, “Design of a low-loss low-noise tapered slot phased array feed for reflector antennas,” presented at the 4th European Conf. Antennas and Propagation (EuCAP), Barcelona, Spain, 2010. [26] M. V. Ivashina, O. A. Iupikov, R. Maaskant, W. van Cappellen, and T. Oosterloo, “An optimal beamforming strategy for wide-field surveys with phased-array-fed reflector antennas,” IEEE Trans. Antennas Propagat., vol. 59, no. 6, pp. 1864–1875, Jun. 2011. [27] R. Maaskant and B. Yang, “A combined electromagnetic and microwave antenna system simulator for radio astronomy,” presented at the 1st European Conf. Antennas and Propagation (EuCAP), Nice, France, Nov. 2006. [28] B. D. Jeffs and K. F. Warnick, “Signal processing for phased array feeds in radio astronomical telescopes,” IEEE Trans. Signal Process., vol. 2, no. 5, pp. 635–646, Oct. 2008. [29] R. Maaskant, D. J. Bekers, M. J. Arts, W. A. van Cappellen, and M. V. Ivashina, “Evaluation of the radiation efficiency and the noise temperature of low-loss antennas,” IEEE Antennas Wireless Propagat. Lett., vol. 8, no. 12, pp. 1166–1170, Dec. 2009. [30] M. V. Ivashina, E. A. Redkina, and R. Maaskant, “An accurate model of a wide-band microstrip feed for slot antenna arrays,” in Proc. IEEE AP-S Int. Symp., Honolulu, HI, Jun. 2007, pp. 1953–1956.

E. E. M. Woestenburg received a degree in microwave engineering from the Technical University of Twente, Enschede, The Netherlands, in 1983. He has been involved in the design of low noise amplifiers and receiver systems, mainly for the Westerbork Synthesis Radio Telescope, since the start of his professional career at ASTRON, the Netherlands Institute for Radio Astronomy. He is presently head of the RF & Low Noise Systems group at ASTRON. His interests lie in the design of low noise amplifiers and the noise characterization of aperture arrays and phased array feed systems.

Laurens Bakker received the M.S. degree in electrical engineering from the University of Twente, The Netherlands, in 2001. From 2001 to 2006 he was a Researcher at the Eindhoven University of Technology, working on high speed optical data transmission technology and analog optical communication technology. Since 2006 he has been with The Netherlands Institute for Radio Astronomy (ASTRON). He is currently working as System Engineer for the APERTIF project—a phased array feed system that is being developed at ASTRON to replace the current horn feeds in the Westerbork Synthesis Radio Telescope (WSRT). His interests include communication systems, low noise systems, (RF) system design, digital signal processing and radio astronomy.

WOESTENBURG et al.: EXPERIMENTAL RESULTS FOR THE SENSITIVITY OF A LOW NOISE APERTURE ARRAY TILE FOR THE SKA

Marianna V. Ivashina received the Ph.D. degree in electrical engineering from the Sevastopol National Technical University (SNTU), Ukraine, in 2000. From 2001 to 2004 she was a Postdoctoral Researcher and from 2004 to 2010 an Antenna System Scientist at The Netherlands Institute for Radio Astronomy (ASTRON). During this period, she carried out research on an innovative Phased Array Feed (PAF) technology for a new-generation radio telescope, known as the Square Kilometer Array (SKA). The results of these early PAF projects have led to the definition of APERTIF, a PAF system that is being developed at ASTRON to replace the current horn feeds in the Westerbork Synthesis Radio Telescope (WSRT). Dr. Ivashina was involved in the development of APERTIF during 2008–2010 and acted as an external reviewer at the Preliminary Design Review

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of the Australian SKA Pathfinder (ASKAP) in 2009. In 2002, she also stayed as a Visiting Scientist with the European Space Agency (ESA), ESTEC, in the Netherlands, where she studied multiple-beam array feeds for the satellite telecommunication system Large Deployable Antenna (LDA). Dr. Ivashina received the URSI Young Scientists Award for GA URSI, Toronto, Canada (1999), APS/IEEE Travel Grant, Davos, Switzerland (2000), the 2nd Best Paper Award (‘Best team contribution’) at the ESA Antenna Workshop (2008) and the International Qualification Fellowship of the VINNOVA—Marie Curie Actions Program (2009) and The VR project grant of the Swedish Research Center (2010). She is currently a Senior Scientist at the Department of Earth and Space Sciences (Chalmers University of Technology). Her interests are wideband receiving arrays, antenna system modeling techniques, receiver noise characterization, signal processing for phased arrays, and radio astronomy.

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Direction Finding With Partly Calibrated Uniform Linear Arrays Bin Liao, Student Member, IEEE, and Shing Chow Chan, Member, IEEE

Abstract—A new method for direction finding with partly calibrated uniform linear arrays (ULAs) is presented. It is based on the conventional estimation of signal parameters via rotational invariance techniques (ESPRIT) by modeling the imperfections of the ULAs as gain and phase uncertainties. For a fully calibrated array, it reduces to the conventional ESPRIT algorithm. Moreover, the direction-of-arrivals (DOAs), unknown gains, and phases of the uncalibrated sensors can be estimated in closed form without performing a spectral search. Hence, it is computationally very attractive. The Cramér–Rao bounds (CRBs) of the partly calibrated ULAs are also given. Simulation results show that the root mean squared error (RMSE) performance of the proposed algorithm is better than the conventional methods when the number of uncalibrated sensors is large. It also achieves satisfactory performance even at low signal-to-noise ratios (SNRs). Index Terms—Direction-of-arrival (DOA), estimation of signal parameters via rotational invariance techniques (ESPRIT), partly calibrated arrays, uniform linear array (ULA).

I. INTRODUCTION

S

ENSOR array processing using antenna arrays has been successfully applied to many engineering fields including wireless communications and radar systems. In particular, the theoretical as well as applied aspects of direction finding have received great research interest during the last decades [1], [2]. Given an ideal antenna array without any uncertainties, direction-of-arrivals (DOAs) can be estimated with high accuracy using high- or super-resolution methods such as multiple signal classification (MUSIC) [3], root-MUSIC [4], estimation of signal parameters via rotational invariance techniques (ESPRIT) [5], and maximum likelihood (ML) algorithm [6]. However, antenna arrays in practice usually suffer from imperfections such as unknown or misspecified mutual coupling, imperfectly known sensor positions and orientations, gain, and phase imbalances [2]. It has been shown that conventional high- or super-resolution direction finding techniques are sensitive to array model errors, which will considerably degrade the performance of these techniques [2], [7]–[9]. A number of calibration methods have been proposed to deal with these problems, and the performances Manuscript received February 25, 2011; revised June 22, 2011; accepted July 28, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. The authors are with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173144

of conventional methods may be significantly improved by taking these antenna array uncertainties into account [10]–[16]. Theoretically, fully calibrated antenna arrays are preferred since high- or super-resolution direction finding techniques can be applied directly. Nevertheless, antenna arrays in some practical applications may be incompletely calibrated. Hence, the response of some sensor elements is poorly known or even unknown. This class of arrays is usually referred to as partly calibrated arrays, and a number of DOA estimation methods have been devoted to these arrays [17]–[24]. For instance, direction finding with partly calibrated arrays was addressed in [18] using the ML algorithm. This method requires the number of calibrated sensors to be larger than the number of signals. In [19], an algorithm for estimating the DOAs and the gains and phases of the uncalibrated sensors was proposed by minimizing a certain cost function. It has been shown that this method can achieve a satisfactory performance. However, due to the requirement of line searches and iterations, its complexity may be high, and the convergence to the global minimum cannot be guaranteed [19]. More recently, the approach in [20] extended the ML criterion used in fully calibrated arrays and employed a particle swarm optimization (PSO) algorithm to solve the problem of direction finding in partly calibrated arrays. Simulation results showed that it has a better performance than the approach in [19]. However, its complexity is high because the searching process is random in nature. In addition, the problem of DOA estimation using partly calibrated arrays containing multiple subarrays has been studied [21]–[24]. This is of great interest since in large subarray-based systems, it may be difficult to calibrate the whole array, though each subarray can be well calibrated [17]. A well-known class of methods is the rank-reduction (RARE) estimator proposed in [21]–[24]. The root-RARE algorithm in [21] and [22] is computationally efficient, but the subarrays are required to be linear identically oriented, and the interelement spacings should be integer multiples of a known shortest baseline. For more general cases where the geometries of subarrays are arbitrary, a spectral-RARE algorithm was proposed , subarray in [23] and [24]. However, the sensor number number , and source number have to satisfy the condition . Compared to the root-RARE algorithm, the complexity will be higher since an additional one-dimensional spectral search is needed. It is worth noting that although RARE algorithms are based on multiple subarrays, their applications to some common arrays such as uniform linear arrays (ULAs) are straightforward. However, there may exist some limitations when these methods are applied to ULAs, as we will show later in Section IV.

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In this paper, we consider the problem of direction finding with partly calibrated ULAs, which occurs in a number of practical applications [17]–[20]. A simple but efficient method based on the conventional ESPRIT algorithm is proposed. It is well known that the conventional ESPRIT algorithm generally requires the array to be fully calibrated and the subarrys be identically oriented. Unfortunately, as mentioned, the arrays available in practice may only be partly calibrated, and hence the ESPRIT algorithm is not directly applicable. In this study, the array manifold of the partly calibrated ULAs is modeled so that the conventional ESPRIT algorithm can be extended to this class of arrays by taking the imperfections into account. The proposed method does not require any spectral search, and the DOAs as well as the gains and phases can be jointly estimated in closed form. The rest of this paper is organized as follows. The models of ideal and partly calibrated ULAs are first introduced in Section II. The proposed method for DOA estimation using partly calibrated ULAs is presented in Section III. An analysis of the proposed method and the Cramér–Rao bounds (CRBs) of the partly calibrated ULAs are given in Section IV. Numerical examples are conducted in Section V to evaluate the performance of the proposed method. Finally, Section VI concludes the paper.

where is the signal covariance matrix, and denotes the statistic expectation. B. Partly Calibrated ULA Model We now consider the case where only part of the ULA is calibrated. Without loss of generality, it is assumed that the first sensors of the array are calibrated, whereas the last sensors are uncalibrated with uncertainties modeled as unknown, direction-independent gains and phases. Let and represent the array gain and phase vectors, respectively. Then, we have (5a) (5b) denotes an vector with all elements equal to where one, and and are the unknown sensor gains and phases of the uncalibrated sensors, respectively. Taking these unknown uncertainties into account, the steering vector of the partly calibrated ULAs can be written as (6) where

II. ARRAY MODELS

denotes the Schur–Hadamard product (7)

A. Ideal ULA Model isotropic To begin with, we consider an ideal ULA with sensors impinged by uncorrelated narrowband source signals, , from far field. The array output observed at the th snapshot consists of the outputs of the sensors and can be written as (1) is the steering vector corresponding to the where DOA of the th source, i.e., , and the array geometry, is the steering matrix (2) is the vector of the signal waveforms, and is the sensor noise vector that is commonly assumed to be additive white Gaussian noise (AWGN) vector with zero mean and covariance matrix , where and denote the noise variance and identity matrix, respectively. For the cases of ideal ULAs, the steering vector is given by

is an diagonal maand trix. Hence, the array covariance matrix becomes (8) where is the steering matrix of the partly calibrated ULA. The eigenvalue decomposition (EVD) of (8) can be written as (9) where is an diagonal matrix consisting of largest eigenvalues and is an diagonal matrix consisting of smallest eigenvalues. is the signal subspace matrix containing the eigenvectors with the largest eigenvalues, while is the noise subspace matrix containing the eigenvectors with the smallest eigenvalues. In cases of finite snapshots, the array covariance matrix and its EVD can be computed as , where is the total number of snapshots. The problem we are interested in is to estimate the DOAs as well as the unknown gains and phases from array observations.

(3) with , , and denoting the carrier wavelength, intersensor spacing, and DOA, respectively. From (1), the array covariance matrix of the array output is (4)

III. DOA ESTIMATION We now proceed to estimate the DOAs as well as the unknown gains and phases of the partly calibrated ULAs. Similar to the conventional ESPRIT algorithm, we divide the partly calibrated ULA into two overlapping subarrays. The first one

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comprises the first sensors, while the second one comprises the last sensors. The steering matrices of these two subarrays can be written as

are the eigenvectors of [5]. In order to show the relationship between , the DOAs, and the unknown gains and phases of the partly calibrated ULA, we let be the eigenvalues of . Hence, the DOAs can be estimated as

(10a) (10b) and denote the nominal steering matrices of the where subarrays, and they are equivalent to the first rows and last rows of , respectively. and denote the gain and phase vectors of these two subarrays, and

(19) where . Furthermore, given the vector the unknown gains and phases can be obtained as

(20)

(11a) (11b) It can also be noted that

and

satisfy (12)

. where Since and are still unknown, we propose to estimate them in the finite samples case according to (16) and (18) by solving the following optimization problem:

diagonal matrix of the phase delays of the where is an first and second subarrays for the sources, and it is given by (13) Since the signal subspace spans the same space as the steering matrix , i.e., , there exists an nonsingular matrix satisfying (14) consist Inspired by the conventional ESPRIT algorithm, let of the first rows of and represent the signal subspace of the first subarray, and consist of the last rows of and represent the signal subspace of the second subarray. Consequently, we have (15a)

in (18),

s. t.

(21)

where denotes the Frobenius norm. In order to solve this problem, we first minimize the objective function with respect to . This gives the least squares solution as follows: (22) Substitute this back to (21), and after some manipulation as shown in the Appendix, the problem in (21) can be finally reformulated as

s. t. where

(23)

is given by

(15b) , , and Since the matrices substitute (12) into (15) and get

(16) where the

matrix

is given by (17)

and being an

(24)

are nonsingular, one can

with vector as

and . Note that the formulation in (23) is derived based on the setting that the first sensors are calibrated. In fact, it can be applied to any partly calibrated ULAs with arbitrary placements of the calibrated sensors, provided that there is at least one pair of consecutive calibrated sensors. For instance, if the th and th sensors are calibrated, then the constraint in (23) should be replaced by . We now proceed to solve the optimization problem in (23) and estimate the DOAs and the unknown gains and phases using the Lagrange multiplier method. First, we note that the constraint in (23) can be represented as

(18) elements of are equal to one, Here, we note that the first i.e., , . It can be found in (17) that and are similar matrices. Therefore, the eigenvalues of must be equal to the diagonal elements of , and the columns of

(25) where

is an

matrix given by (26)

LIAO AND CHAN: DIRECTION FINDING WITH PARTLY CALIBRATED ULAs

Hence, the problem can be rewritten as

s. t.

(27)

To solve this problem using the Lagrange multiplier method, we form the Lagrangian function associated with (27) as follows

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to be arbitrary, and it can be directly applied to the partly calibrated ULAs by letting the first calibrated sensors be the first subarray and the other subarrays be of a single sensor. Therefore, the total number of subarrays is . According to the spectral-RARE estimator, the sensor number , subarray number , and source number must satisfy , i.e.,

(28)

(31)

where is the Lagrange multiplier. By setting the partial derivative of (28) with respect to to zero, one gets the first-order necessary condition for optimality as , which leads to the optimal solution

This indicates that the number of calibrated sensors in a ULA should be larger than the source number. For instance, the calibrated sensors in a ULA should be no less than four when the sources number is three. As a result, this method is not applicable when . However, in the proposed method, we only require the number of calibrated sensors to be no less than two. In fact, this is the basic property of a partly calibrated ULA because when there are no calibrated sensors in a ULA, the array should be an uncalibrated rather than partly calibrated one. A number of works have studied the problem of sensor array processing with uncalibrated arrays [10]–[16]. For instance, an ESPRIT-based technique has been proposed for spatial signature matrix, but not DOA estimation with uncalibrated ULAs, in [16]. Different from this method, in our proposed method, we aim to estimate the DOAs as well as unknown array gains and phase in closed form by taking advantage of the calibrated sensors. A special case of the proposed method occurs when the ULA is fully calibrated, i.e., . In this case, we have and . Consequently, the constrained problem in (21) is reduced to an unconstrained problem as follows:

(29) By substituting (29) back to the constraint (27), one can determine the Lagrange multiplier , and hence the final solution to (29) as follows: (30) , gains Consequently, the matrix , DOAs and phases can be estimated according to (19), (20), and (22). It should be noted that a sufficient condition for the existence of (30) is that is nonsingular. However, in the infinite samples case, is singular. One possible way to handle this problem is to employ diagonal loading as suggested in [17], [24], [26], and [27]. More precisely, a small multiple of the identity matrix is added to to form the diagonally loaded matrix . It is worth noting that in these robust algorithms, especially robust beamforming algorithms discussed in [26] and [27], the loading level is usually required to be optimally selected. Fortunately, in our case, we only require being nonsingular, and hence can be chosen as a small value. In fact, a large may degrade the accuracy of estimating as well as other unknown parameters. Moreover, it is found by extensive experiments that, in finite sample cases, the matrix is always nonsingular, and hence there is no need for diagonal loading in general. IV. COMPARISONS AND CRAMÉR–RAO BOUNDS A. Comparisons From the derivation in the previous section, it can be seen that the proposed method is similar to the conventional ESPRIT algorithm. Therefore, it is computationally efficient since the DOAs as well as the gains and phases can be estimated from (19) and (20) at the cost of an EVD, and no spectral search is required. Compared to ESPRIT, a more general case of partly calibrated ULAs is tackled. In [22], a root-RARE estimator was developed for partly calibrated subarray-based arrays with unknown vector translations. Since it generalizes the conventional root-MUSIC algorithm, it can be directly applied to fully calibrated ULAs. However, its extension to the case of partly calibrated ULAs is not straightforward. On the other hand, the spectral-RARE estimator derived in [24] allows the array geometry

(32) and its solution is given by

(33) Apparently, this is the solution of the conventional ESPRIT algorithm. In other words, the proposed method can be regarded as a generalized version of the conventional ESPRIT algorithm. It is interesting to note that another generalized version of the ESPRIT algorithm has been studied in [25]. Different from our proposed method, this method is proposed to deal with arrays where any sensor of the first subarray and the corresponding sensor of the second subarray are displaced by different displacement vectors. B. Cramér–Rao Bounds In this section, closed-form expressions for the CRBs of partly calibrated ULAs with zero mean and statistically independent Gaussian random vectors are given. The unknown vectors of DOAs , gains , and phases are given by , , and

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, respectively. The CRB for DOA estimation is given by [10], [19]

(34) where

denotes the real part of

and (35)

since by [19]

. The CRB for the gain estimation is given

(36) where is an

matrix with its if otherwise.

and th entry being (37)

Based on the derivations of CRBs for phase estimation of uncalibrated ULAs in [10] and gain estimation of partly calibrated ULAs in [19], the CRB for phase estimation of partly calibrated ULAs can be similarly derived and given by

Fig. 1. RMSE of DOA estimation versus SNR. The number of snapshots , and the number of calibrated sensors .

At first, the performance of the proposed method is evaluated at different SNRs. The determinant-based spectral-RARE algorithm [24] was also tested for comparison. Moreover, the results of MUSIC using the first five calibrated sensors and MUSIC using the whole array with known uncertainties were also obtained. A total of 200 Monte Carlo experiments are run at each SNR, and the number of snapshots in each experiment is . The following root mean squared error (RMSE) of DOA estimation is used as the performance measure:

(38) It should be noted that the unknown phases are modeled to be direction-dependent in [19] and direction-independent in this paper. Hence, the CRB for phase estimation given in (38) is different from that in [19]. V. SIMULATION RESULTS In order to evaluate the performance of the proposed method, computer simulation of a partly calibrated ULA with sensors separated by half a wavelength was performed. In all examples, the unknown gain and phase uncertainties are considered to be direction-independent and time-invariant. Three uncorrelated narrowband signals with identical power impinge on the array from the far field, and hence . The DOAs of them are assumed to be 10 , 10 , and 20 , respectively. The background noise is assumed to be AWGN. A. Example I In the first example, the first five sensor are assumed to be calibrated, i.e., , while the last five sensors are uncalibrated with unknown gain and phase uncertainties given by , , , , and .

where is the number of Monte Carlo experiments, and is the estimated DOA of the th signal in the th experiment. In all examples, we let . In Fig. 1, the RMSEs of the DOA estimates obtained by different methods versus SNR are compared, and the CRB is also displayed. Overall, it can be seen that, in the cases of partly calibrated ULAs, the proposed method can give better performance than the spectral-RARE algorithm and the MUSIC algorithm using the calibrated sensors. Moreover, it can be noted that the RMSEs of DOA estimated by all methods, except MUSIC with known uncertainties, cannot reach the CRB even at large SNRs. One possible explanation is that the performances of these methods are significantly dependent on the number of calibrated sensors. This will be shown in the last example, where we can see that the performances of these methods will be greatly improved with increasing number of calibrated sensors . For example, for a large , the RMSEs are close to the CRB even when the SNR is 5 dB. Fig. 2 shows the success probability of DOA estimation. Here, the success probability is defined as , where is the number of experiments in which all of the

LIAO AND CHAN: DIRECTION FINDING WITH PARTLY CALIBRATED ULAs

Fig. 2. Success probability of DOA estimation versus SNR. The number of , and the number of calibrated sensors . snapshots

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Fig. 4. RMSE of DOA estimation versus SNR. The number of snapshots , and the number of calibrated sensors and .

TABLE I ESTIMATED GAIN AND PHASE, BIAS MAGNITUDE, RMSE, AND CRB FOR THE FIRST UNCALIBRATED SENSOR AT DIFFERENT SNRS USING THE PROPOSED METHOD. TRUE VALUES ARE AND (rad)

Fig. 3. Bias magnitude of DOA estimation of the third signal versus SNR. The , and the number of calibrated sensors . number of snapshots

DOA estimate bias magnitudes are smaller than 0.5 , i.e., . Apparently, we notice that the proposed method can achieve the highest DOA estimation accuracy when the uncertainties are unknown. Even at low SNRs, the proposed method is able to successfully estimate all of the DOAs within the given bound with a high probability. This suggests that the proposed method is useful especially when the signals are seriously contaminated by noise. Fig. 3 illustrates the magnitude of DOA estimation bias of the third signal, which is defined as . We can find that even at a small SNR, the bias magnitude is rather small. When the SNR is larger than 0 dB, the estimation bias magnitude tends to be very small, whereas such a performance can only be obtained by the other methods with SNR larger than 10 dB.

In order to evaluate the performance of the proposed method in gain and phase estimation, the estimated gains and phases, bias magnitudes, and RMSEs are obtained by the proposed method with 200 experiments. The CRBs for gains and phases estimation are also calculated based on (36) and (38) for comparison. Table I shows the averaged gain and phase estimates, bias magnitude, RMSE, and CRB for the first uncalibrated sensor at different SNRs. Since the estimation results of the other four uncalibrated sensors obtained by the proposed method are similar to those of the first uncalibrated sensor, they are omitted for simplicity. B. Example II In this example, we will evaluate the effect of the number of calibrated sensors in a ULA on DOA estimation. First, we follow the settings in the previous example, but the calibrated sensors are assumed to be seven, i.e., . The unknown gains and phases of the uncalibrated sensors are identical to those in Example I, i.e., , , and . Fig. 4 shows the RMSEs of DOA estimation when . Moreover, the results of obtained in Example I are also displayed for comparison. It can be

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and phases. For a fully calibrated array, the proposed method reduces to the conventional ESPRIT algorithm. The DOAs and unknown gains and phases can be estimated in closed form without performing a spectral search. Thus, the proposed method is computationally attractive. The CRBs of the partly calibrated ULAs are also presented. Simulation results show that the proposed method outperforms the conventional methods especially when the number of uncalibrated sensors is large, and satisfactory performance can be achieved even at low SNRs. APPENDIX In this appendix, we briefly give the derivation of the problem in (23). Substituting (22) into (21), the objective function can be rewritten as Fig. 5. The RMSE of DOA estimation versus the number of calibrated sensors . The number of snapshots , the .

noted that the performances of the methods, especially the spectral-RARE algorithm and the MUSIC algorithm using calibrated sensors, are greatly improved by reducing the number of uncalibrated sensors. Next, we set 5 dB to and the number of snapshots to and evaluate the performance of the proposed method with different number of calibrated sensors. More precisely, the RMSE is calculated for . It should be noted that the gain and phase vector is chosen to be when there are calibrated sensors, where is defined as the following 1 8 vector:

Fig. 5 shows the RMSEs versus the number of calibrated sensors . We remark here that both spectral-RARE and MUSIC using calibrated sensors are not applicable when the number of calibrated sensors is less than four, i.e., , because these two algorithms require the number of calibrated sensors be larger than the number of sources in the case of ULA. From Fig. 5, we can see that the performance of each method can be improved by increasing the number of calibrated sensors. It is worth noting that when the number of calibrated sensors is 10, i.e., the ULA is fully calibrated without imperfections, the proposed method will reduce to the conventional ESPRIT algorithm, whereas both the spectral-RARE algorithm and MUSIC using calibrated sensors will reduce to the conventional MUSIC algorithm. This is the reason why the spectral-RARE algorithm achieves a better performance when . VI. CONCLUSION A new direction finding method for partly calibrated ULAs is presented. It extends the conventional ESPRIT algorithm by modeling imperfections of the ULAs as unknown gains

(39) where is an projection matrix. It is known that and for any matrix matrix , then (39) can be rewritten as

and

(40) where the property on the following identity [28]:

is utilized. Moreover, based

(41) where and are matrices, , and , the objective function (40) can be further simplified to

(42) Hence, the problem in (21) becomes

s. t.

(43)

which is identical to (23). ACKNOWLEDGMENT The authors would like to thank the reviewers and editors for their useful comments and suggestions. REFERENCES [1] H. Krim and M. Viberg, “Two decades of array signal processing research,” IEEE Signal Process. Mag., vol. 13, no. 4, pp. 67–95, Jul. 1996.

LIAO AND CHAN: DIRECTION FINDING WITH PARTLY CALIBRATED ULAs

[2] E. Tuncer and B. Freidlander, “Calibration in array processing,” in Classical and Modern Direction-of-Arrival Estimation. Amsterdam, The Netherlands: Elsevier, 2009. [3] R. O. Schmidt, “Multiple emitter location and signal parameter estimations,” IEEE Trans. Antennas Propag., vol. AP-34, no. 3, pp. 276–280, Mar. 1986. [4] A. J. Barabell, “Improving the resolution performance of eigenstructure-based direction-finding algorithms,” in Proc. ICASSP, Boston, MA, May 1983, pp. 336–339. [5] R. Roy and T. Kailath, “ESPRIT—Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal Process., vol. 37, no. 7, pp. 984–995, Jul. 1989. [6] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood, and Cramer Rao bound,” IEEE Trans. Acoust., Speech, Signal Process., vol. 37, no. 5, pp. 720–741, May 1989. [7] A. Swindlehurst and T. Kailath, “A performance analysis of subspacebased methods in the presence of model errors, part I: The MUSIC algorithm,” IEEE Trans. Signal Process., vol. 40, no. 7, pp. 1758–1773, Jul. 1992. [8] A. Ferreol, P. Larzabal, and M. Viberg, “On the asymptotic performance analysis of subspace DOA estimation in the presence of modeling errors: Case of MUSIC,” IEEE Trans. Signal Process., vol. 54, no. 3, pp. 907–920, Mar. 2006. [9] A. Ferreol, P. Larzabal, and M. Viberg, “On the resolution probability of MUSIC in presence of modeling errors,” IEEE Trans. Signal Process., vol. 56, no. 5, pp. 1945–1953, May 2008. [10] B. Friedlander and A. J. Weiss, “Direction finding in the presence of mutual coupling,” IEEE Trans. Antennas Propag., vol. 39, no. 3, pp. 273–284, Mar. 1991. [11] E. K. L. Hung, “A critical study of a self-calibration direction-finding method for arrays,” IEEE Trans. Signal Process., vol. 42, no. 2, pp. 471–474, Feb. 1994. [12] B. C. Ng and C. M. S. See, “Sensor-array calibration using a maximumlikelihood approach,” IEEE Trans. Antennas Propag., vol. 44, no. 6, pp. 827–835, Jun. 1996. [13] K. C. Ho and L. Yang, “On the use of a calibration emitter for source localization in the presence of sensor position uncertainty,” IEEE Trans. Signal Process., vol. 56, no. 12, pp. 5758–5772, Dec. 2008. [14] Z. Ye, J. Dai, X. Xu, and X. Wu, “DOA estimation for uniform linear array with mutual coupling,” IEEE Trans. Aerosp. Electron. Syst., vol. 45, no. 1, pp. 280–288, Jan. 2009. [15] X. Xu, Z. Ye, and Y. Zhang, “DOA estimation for mixed signals in the presence of mutual coupling,” IEEE Trans. Signal Process., vol. 57, no. 9, pp. 3523–3532, Sep. 2009. [16] D. Astély, A. L. Swindlehurst, and B. Ottersten, “Spatial signature estimation for uniform linear arrays with unknown receiver gain and phases,” IEEE Trans. Signal Process., vol. 47, no. 8, pp. 2128–2138, Aug. 1999. [17] S. Haykin and K. J. L. Ray, Handbook on Array Processing and Sensor Networks. Hoboken, NJ: Wiley, 2010. [18] P. Stocia, M. Viberg, K. M. Wong, and Q. Wu, “Maximum-likelihood bearing estimation with partly calibrated arrays in spatially correlated noise field,” IEEE Trans. Signal Process., vol. 44, no. 4, pp. 888–899, Apr. 1996. [19] A. J. Weiss and B. Friedlander, “DOA and steering vector estimation using a partially calibrated array,” IEEE Trans. Aerosp., Electron. Syst., vol. 32, no. 3, pp. 1047–1057, Jul. 1996. [20] M. Li and Y. Lu, “Source bearing and steering-vector estimation using partially calibrated arrays,” IEEE Trans. Aerosp., Electron. Syst., vol. 45, no. 4, pp. 1361–1372, Oct. 2009.

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[21] M. Pesavento, A. B. Gershman, and K. M. Wong, “Direction of arrival estimation in partly calibrated time-varying sensor arrays,” in Proc. ICASSP, Salt Lake City, UT, May 2001, pp. 3005–3008. [22] M. Pesavento, A. B. Gershman, and K. M. Wong, “Direction finding in partly-calibrated sensor arrays composed of multiple subarrays,” IEEE Trans. Signal Process., vol. 50, no. 9, pp. 2103–2115, Sep. 2002. [23] C. M. S. See and A. B. Gershman, “Subspace-based direction finding in partly calibrated arrays of arbitrary geometry,” in Proc. ICASSP, Orlando, FL, Apr. 2002, pp. 3013–3016. [24] C. M. S. See and A. B. Gershman, “Direction-of-arrival estimation in partly calibrated subarray-based sensor arrays,” IEEE Trans. Signal Process., vol. 52, no. 2, pp. 329–338, Feb. 2004. [25] F. Gao and A. B. Gershman, “A generalized ESPRIT approach to direction-of-arrival estimation,” IEEE Signal Process. Lett., vol. 12, no. 3, pp. 254–257, Mar. 2005. [26] Y. Hua, A. Gershman, and Q. Cheng, High-resolution and Robust Signal Processing. New York: Marcel Dekker, 2003. [27] J. Li and P. Stoica, Robust Adaptive Beamforming. Hoboken, NJ: Wiley, 2006. [28] J. R. Magnus and H. Neudecker, “Mathematical preliminaries,” in Matrix Differential Calculus with Applications in Statistics and Econometrics, 2nd ed. New York: Wiley, 1999.

Bin Liao (S’09) received the B.Eng. and M.Eng. degrees from Xidian University, Xi’an, China, in 2006 and 2009, respectively, and is currently pursuing the Ph.D. degree in electrical and electronic engineering at the University of Hong Kong, Hong Kong. His main research interests are array signal processing and adaptive filtering.

Shing Chow Chan (S’87–M’92) received the B.S. (Eng.) and Ph.D. degrees from the University of Hong Kong, Hong Kong, in 1986 and 1992, respectively. Since 1994, he has been with the Department of Electrical and Electronic Engineering, University of Hong Kong, and is now a Professor. He held visiting positions with Microsoft Corporation, Redmond, WA; Microsoft Research Asia, Beijing, China; University of Texas at Arlington; and Nanyang Technological University, Singapore. His research interests include fast transform algorithms, filter design and realization, multirate and array signal processing, communications and biomedical signal processing, and image-based rendering. Dr. Chan is currently a member of the Digital Signal Processing Technical Committee of the IEEE Circuits and Systems Society and an Associate Editor of the Journal of Signal Processing Systems. He was an Associate Editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS from 2008 to 2009, and Chairman of the IEEE Hong Kong Chapter of Signal Processing from 2000 to 2002.

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Calculation of MoM Interaction Integrals in Highly Conductive Media Joris Peeters, Ignace Bogaert, and Daniël De Zutter, Fellow, IEEE

Abstract—The construction of the impedance matrix in the method of moments requires the calculation of interaction integrals between the expansion functions, through the Green’s function and its derivatives. The singular behavior of the Green’s function poses considerable problems for an accurate numerical evaluation of these integrals, requiring techniques such as singularity extraction or cancellation. In this contribution we will show why these methods fail when the medium is highly conductive. A novel technique is proposed to handle these highly challenging integrals. The complexity of the new method is independent of the conductivity. Index Terms—Conductivity, electromagnetic shielding, integral equations, method of moments (MoM).

I. INTRODUCTION

T

HE method of moments (MoM) is one of the most powerful approaches for solving electromagnetic scattering problems in piecewise homogeneous media. Its main advantage compared to other techniques, such as the finite difference time domain (FDTD) method and the finite elements (FE) method, is that only the surface of the objects must be discretized. The disadvantage, however, is that the resulting system matrix is fully dense, describing the interaction between all expansion functions by integrals with the singular 3D Green’s function, given by , or its gradient as the kernel. Different techniques to calculate these integrals have been proposed in the past, focusing on regularizing the behavior. The two most prominent approaches are singularity extraction (SE) [1]–[4] and singularity cancellation (SC) [5]–[7]. However, both these techniques assume that the numerator of the Green’s function, i.e., , is a well-behaved function with a fairly small absolute value of the derivative. Indeed, for lossless media the wavenumber is real and for the usual discretization, the function is smooth. In that case, the wavenumber is given by , with . For a very good conductor (with conductivity ), we have that , with the skin depth [8]. Note that, in general, as the conductivity becomes larger, both the real part and imaginary part of grow, and are approximately Manuscript received June 24, 2010; revised July 17, 2011; accepted August 28, 2011. Date of publication October 21, 2011; date of current version February 03, 2012 J. Peeters was with the Department of Information Technology (INTEC), Ghent University, B-9000 Gent, Belgium. He is now with Computational Dynamics Ltd., London W6 7NL, U.K. (e-mail: [email protected]). I. Bogaert and D. De Zutter are with the Department of Information Technology (INTEC), Ghent University, B-9000 Gent, Belgium. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173105

equal to each other in magnitude. As a consequence, becomes a function that is both highly oscillatory and exponentially damped and can by no means be considered as a smooth function to be handled by the standard numerical quadratures. In fact, as will be shown later in this contribution, a very specialized approach, tuned to this damped behavior, is required in order to accurately evaluate the impedance integrals in highly conductive media. A similar topic has been treated in [9], but in a different manner that, to our knowledge, does not lead to a scalable solution (i.e. a calculation time that is independent of the conductivity , assuming the frequency does not vary). The outline of this paper is as follows. Section II introduces the MoM interaction integrals that occur when modelling a scattering problem at a body with complex and . Section III gives a short overview of the currently most widely used techniques for calculating the singular or near singular impedance integrals and also explains the reason for their breakdown when the interacting medium is highly conductive. Note that whenever SC is considered, only the Duffy transform is employed to illustrate the difficulties. Of course, a variety of other techniques exists within this field, that have not each been numerically tested by the authors. As will be shown further, though, the arguments against SC for conductive media are applicable to all these techniques. Section IV introduces our novel method for tackling the impedance integrals in these media and in Section V, this method is applied to a few challenging cases. Essentially, the full-wave treatment (as opposed to using a surface impedance) we propose is useful whenever the thickness of the conductor becomes of the order of or smaller than the skin depth. By means of numerical illustration and validation, Section VI applies our approach to the case of ‘tunnelling’ through a very thin conductive spherical shell. Additionally, some further fields of application are suggested that may benefit from this work. II. IMPEDANCE INTEGRALS IN THE MOM Discretization of the boundary integral equations (BIE) in the MoM leads to a dense linear system, the matrix elements of which describe the interaction, through the Green’s function, between the expansion functions. Scattering at objects with a permittivity and permeability (but neither infinitely lossy) requires the introduction of two equivalent surface current densities, electric and magnetic, which can be solved for as the solution of the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) [10] BIE. In this contribution, it will be assumed that the surface current densities are expanded into Rao-Wilton-Glisson (RWG) [11] functions (which we will ), defined on a mesh of flat triangles, although denote as the proposed techniques have a broader field of application (including an extension to a curvilinear mesh and the use of

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PEETERS et al.: CALCULATION OF MoM INTERACTION INTEGRALS IN HIGHLY CONDUCTIVE MEDIA

higher order basis functions). The resulting matrix elements require the calculation of the following integrals (as part of the and operators [2]) over the support of the test functions and the support of basis functions

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, can be integrated analytically. As such, for instance, can be rewritten as

(7) (1) (2) (3)

indicating the principal value of the integral. When with the supports of and overlap in at least a point, the above integrals have a non-continuous integrand, although they are integrable. For (the self patch case), becomes zero. In order to determine these integrals over a triangle, for each possible , it suffices to calculate the following integrals: (4) (5)

The second double integral is evaluated analytically and the first double integral, from which the singular part is extracted, is now regular. Note that, even though the first integrand is now continuous, it is not because the first derivative, in this example, displays a discontinuity at . Additional terms have to be extracted for continuity in the derivatives [2]. Singularity extraction can also be applied to the near-singular case, in order to smooth the integrand and thus increasing the efficiency. An essential assumption behind the philosophy of Singularity Extraction is that, by extracting the singular (or near-singular) static part, the remaining integral automatically becomes suitable for numerical quadrature. As we will see later, in the case of conductive media, this is not the case. A second technique, in competition with singularity extraction, is singularity cancellation. This method aims to regularize the integrand by a suitable change of coordinates. Considering again as an example, a simple yet effective transformation to polar coordinates in the inner integral would do the trick (8)

(6)

and , these integrals , To obtain the integrals , and are required, in addition to some others that are merely variations in terms of the presence or absence of or . The reason both and are included here, instead of just one of them, is to demonstrate in the examples that the presence of has no mentionable influence on the achievable accuracy. In short, if the three integrals above can be evaluated efficiently and accurately, this also guarantees accurate evaluation of all the integrals that are required in the impedance matrix. In the next section, we will briefly revisit the techniques of SE and SC, the workhorses behind most MoM implementations. III. CALCULATION OF IMPEDANCE INTEGRALS IN DIELECTRICS In order to obtain an accurate solution from the PMCHWT BIE, it is essential that the integrals described in the previous section are evaluated with a relatively high accuracy. When the expansion functions and are well-separated (i.e., their distance from each other is considerably larger than their size), the integrand is sufficiently smooth and a straightforward Gaussian quadrature rule allows for exponential convergence. More challenging are the cases when the supports overlap (singular) or are very close (near-singular). Both situations require a specialized approach that deals with the singular or near-singular behavior of the integrand. We will first elaborate on the concept of SE, which is based on the fact that interaction integrals with static kernels, for example

where we have assumed, in order to more clearly demonstrate the idea, that we are dealing with the self patch case . The Jacobian compensates the that appears in the Green’s function and as such regularizes the integrand to a function. An advantage of this approach is that it does not rely on the existence of analytical solutions for the static part. This allows for more flexibility in the expansion functions, paving the way for higher order solutions. IV. CALCULATION OF IMPEDANCE INTEGRALS CONDUCTIVE MEDIA

IN

In order to understand the difficulties that occur when calculating the impedance integrals in conductive media, it is instructive to look at the behavior of the Green’s function for various values of the conductivity , as shown in Fig. 1. The pulsation is chosen equal to 300 MHz. The distance is varied from 0 to , with the free space wavelength (with ). Note how even a relatively poor conductor (with ) dampens the Green’s function by more than five orders of magnitude over a distance of about . Copper, one of the most widespread conductors in industry, has , leading to a Green’s function that is extremely localized around the origin. This behavior explains why straightforward application of techniques such as SE or SC break down for high conductivity, because they neglect the highly oscillatory but at the same time exponentially damped character of . In addition, SE suffers from numerical cancellation issues between the different

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TABLE I (1/m) AND SKIN DEPTH (m) IN COPPER AS A FUNCTION OF THE FREQUENCY (Hz) AND FREE SPACE WAVE NUMBER

THE WAVE NUMBER

Fig. 1. The absolute value of the Green’s function

for a few values of .

extracted terms. A numerical comparison for conductive media between SE, SC (by means of the Duffy transform) and our novel approach will be given further in this paper. In order to introduce our new technique for treating these integrals in conductive media, the explanation will be based on . Further on it will also be shown how both and can be treated almost identically. So, in the remainder of this section, we will be looking at a way to efficiently evaluate the following integral: (9)

for arbitrary values of . In order to do this, a specialized approach is required for both the inner and outer integrals.

Fig. 2. The real part, imaginary part and absolute value of the function .

The behavior of this function is illustrated in Fig. 2 (with ). Beyond a certain electrical length, the numerator of the Green’s function drops to a fraction compared to its value in the origin. As such a certain cutoff value of can be determined, beyond which the remainder can be neglected, namely

A. Inner Integral First, we will take a look at evaluating the inner integral, namely (10)

where, although can be anywhere in space, the most challenging and practically interesting cases are when is very close to or even in it. The key to accurately integrate the strongly pulsed behavior is focusing the numerical quadrature points only in those regions where the Green’s function has a non-negligible value, based on a certain tolerance . The wave number in a good conductor approximately satisfies

(13) With this knowledge, the inner integral can now be evaluated to any desired tolerance independent of . As a first step, a similar transform as in the Singularity Cancellation method is employed, namely a Duffy transformation, see, e.g., [3]. As mentioned before, this allows for more flexibility in the integrand and will in fact allow us to treat the inner integrals of , and in an identical manner, despite the different kernel. With respect to a carefully selected the integral is transformed to polar coordinates (14)

(11) (see Table I for some numerical values using with copper as an example). This allows us to approximately express the numerator of the Green’s function in terms of only,

(12)

This point is found by first projecting into the plane of the triangle and calling this projection . If lies within , it is equal to . If lies outside the triangle, is that point on the edge of the triangle that lies closest to . This process of finding is illustrated in Fig. 3. Once is determined, is divided into one, two or three triangles (depending on the location of ), each having as one of their corners. This is illustrated in Fig. 4. The total integral is

PEETERS et al.: CALCULATION OF MoM INTERACTION INTEGRALS IN HIGHLY CONDUCTIVE MEDIA

Fig. 3. The point is found as the point on the triangle (or its edge) that is closest to . This is illustrated for three different possibilities of .

Fig. 4. The division into subtriangles for three different cases. The location of is indicated by the small circle. Left: lies in the triangle, which is sublies on the edge of the triangle, which is divided into three parts. Middle: lies on the corner of the triangle and no subdivided into two parts. Right: subdivision is needed.

Fig. 5. The truncation of the radial integration domain from to keep all quadrature points within a distance from .

to

expressed as the sum of the integrals over these subtriangles, a similar approach as, e.g., [7] and [12]. The integration over one subtriangle can be rewritten as (15) Let us first look at the radial integration for a subtriangle (16) is dependent on the triThe endpoint of the integration angle shape, on the angular coordinate and also on the value of , which might truncate the integration domain. The latter occurs when the endpoint is further away from than the distance . In that case, the integration is carried out from to , such that the new endpoint is away from . This is illustrated in Fig. 5. Clearly, this cutoff does not compromise the accuracy, due to the rapid decay of the Green’s function.

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Regarding the shape of the integrand, it must be noted that, while the polar coordinate transform is capable of cancelling out a singularity, leads to a singularity. In addition, due to the conductive behavior, the function will in any case have its most dominant behavior around , although the limitation of radial distance to largely solves this problem. One interesting approach to tackle integrands such as with possible endpoint singularities is the double exponential (DE) transform [13], essentially mapping a [ 1, 1] region on a region that can be handled with a trapezoidal rule to exponential accuracy. This allows for the desired flexibility in terms of kernel and expansion functions. The radial integrand now becomes (suppressing dependencies of and assuming to indicate the integration endpoint, whether or not truncated to )

(17) in which and with the so-called double exponential transform given by . To our knowledge, the DE transform was first employed for the calculation of impedance integrals in [12], which also contains a large amount of background on the technique. The essential difference with regard to conductive integrals is the use of in this work. An alternative for DE is using Gauss quadrature. Even though it cannot handle the singular behavior of , it performs better for those integrals that are regularized by the Duffy transform (i.e. achieves roughly one or two orders of magnitude additional precision for the same number of quadrature points). So, in the case of the self-patch, when is zero, it would lead to a more efficient solution. However, in any case different from the self-patch we would need the K-operator in addition to the T-operator, so our recommendation is to use the DE transform to calculate the different radial integrals simultaneously, which reduces the number of evaluations of the Green’s function, whilst still achieving any practically desired tolerance. So, in the remainder of this article we will use the DE transform (like in our own MoM implementation for these integrals), but the reader should be aware that Gauss quadrature can be a decent alternative in some cases, but unfortunately fails in others. As an example, integral (17) is evaluated for the following data: , , and . The results are given in Table II for a few choices of the parameters. The use of the truncation distance essentially imposes a maximal absolute error on the integral. If the interaction distance is well beyond the skin depth , this may lead to a large relative error (because the value of the integral is very small). However, in the MoM scheme, it is pointless to evaluate these integrals to higher precision because they barely contribute. Essentially, the more distant an interaction, the less accurate its evaluation needs to be. That is exactly what the use of accomplishes. Note that in all numerical experiments, both here and in the next sections, the values of are obtained through

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TABLE II AND ABSOLUTE ERROR FOR THE THE RELATIVE ERROR NUMERICAL EVALUATION OF THE RADIAL INTEGRAL (17). THE NUMBER OF QUADRATURE POINTS FOR THE DOUBLE EXPONENTIAL FORMULA IS

TABLE III AND ABSOLUTE ERROR FOR THE RELATIVE ERROR NUMERICAL EVALUATION OF THE ANGULAR INTEGRAL (18)

THE

TABLE IV FOR EVALUATION OF THE RELATIVE ERRORS AS A FUNCTION OF THE INNER INTEGRAL USING SINGULARITY EXTRACTION (WITH 15 TERMS), SINGULARITY CANCELLATION (WITH 17 QUADRATURE POINTS BOTH FOR THE RADIAL PART AND THE ANGULAR PART) AND OUR NOVEL APPROACH (USING THE SAME AMOUNT OF QUADRATURE POINTS AS SC AND FOR A TOLERANCE OF )

Fig. 6. The integration domain when is smaller than the dimensions of the becomes larger, the integration domain becomes triangle. In contrast, when the entire triangle.

comparison with a numerical result using a much higher amount of quadrature points, which is used as the reference result. Having control over the radial integral, it is now used as the integrand for the angular quadrature (18) Regarding the choice of quadrature rule and number of sample points needed to evaluate (18), it is important to notice that, when is small compared to the dimensions of the triangle, the integrand is actually not strongly dependent on because in that case only a limited portion of the triangle has to be integrated over. This is illustrated in Fig. 6. As such, in those cases, as little as one integration point is usually sufficient. When the conductivity is high, these cases will occur quite often and it is worth detecting them. If the complete triangle plays a role, then a Gaussian quadrature rule is employed. For most practical purposes, 8 sample points in turn out to be sufficient. In order to illustrate the obtainable accuracy and the fact that the complexity is independent of the conductivity, we consider the following example. The triangle is defined by the vertices (0, 0, 0), (1, 0, 0) and (0, 1, 0), the first of which is chosen to be . Table III shows the accuracy of the angular integral for a few locations of the observer point , a few values of and different numbers of sample points for the angular integration. The radial integration was performed with sufficient accuracy so as not to influence the results.

It is clear that the previously described methods allow for efficient and accurate evaluation of the inner integral (14). Application of the DE technique and the introduction of makes the calculation time and accuracy independent of the conductivity. The inner integral will now serve as the integrand of the outer integral, over triangle . However, before moving on to the outer integral, the numerical accuracy and efficiency of our treatment of the inner integral will be compared with that of Singularity Extraction and Singularity Cancellation. The latter comes in many shapes, but here we will simply employ our previously discussed technique, but setting , which reduces to a typical Duffy transform. Of course, other cancellation approaches may lead to different results, but this example merely serves as an illustration of the problems that will occur in, to our belief, all of them. For singularity extraction, we use a formulation without a “regular remainder,” i.e., we extract as many analytical terms as is necessary for a sufficiently accurate result (in this case 15 terms) if there was no conductivity. For the numerical example, we take a source triangle that has vertices at (0, 0, 0), (1, 0, 0) and (0, 1, 0), with an observer point above its center of mass and we use a wavenumber of the form . The results are shown in Table IV. As can be observed, the SE method becomes numerically unstable and diverges, due to the many terms that suffer from numerical cancellation. If only one or two terms are extracted (instead of 15) and the remainder is integrated numerically, a similar problem as with the SC technique will appear, which can not keep up with the increasingly rapid variation of the Green’s function and loses accuracy. For the inner integral, the computational cost of the novel approach is approximately equal to that of the considered singularity cancellation approach, because the same amount of quadrature points is considered and any preprocessing is negligible compared to the evaluations of the Green’s function. The novel approach is less accurate at low

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TABLE V NORMALIZED TIME FOR THE EVALUATION OF THE INNER INTEGRAL, COMPARING AN ADAPTIVE IMPLEMENTATION OF SC WITH THE NOVEL METHOD

Fig. 7. The situation of two triangles that only partially overlap when projected onto each other, with the dashed line indicating the projection of the top triangle on the plane of the bottom triangle.

losses, due to the cutting procedure, but manages to stay within the chosen tolerance for the cases of high conductivity, whereas other methods fail in this region. In order to demonstrate the non-scalable behavior of the SC technique, the same experiment as before is repeated, but now using adaptive refinement for SC until the tolerance is achieved. The results, using our own implementation of both methods, are demonstrated in Table V, displaying the normalized time for SC and the novel method. In contrast with the previous numerical example, the total number of quadrature points (and hence the computational time) required for SC increases as increases. This is because the SC approach requires increasingly deeper local refinement near the observation point, in order to achieve the desired accuracy. The novel approach is essentially independent of the conductivity because it identifies the critical region during preprocessing. SE was omitted here due to its inherent instability.

Fig. 8. The domain of (lying in the yz-plane) is reduced to the darkly shaded area (the polygon ABCD), which is the intersection between and the (infinite) (dashed thin line in the xy-plane) and its volume described by the plane of . upward projection (dashed thick line) over a distance

B. Outer Integral The outer integral is given by (19) and the others have a similar form and can also be treated in a completely identical manner as will be described in this subsection. However, for the sake of the argument, the approach will be focusing on . For the inner integral, the key to efficient evaluation was a focusing of quadrature points in the regions where the integrand is non-negligible (through the choice of and the use of the DE transform). A similar objective lies behind the philosophy of the proposed method to evaluate the outer integral. As an example, and to illustrate the difficulties, Fig. 7 shows two triangles and that, when projected onto each other, overlap only partially. When the conductivity is high, the parts on that are not very close to (basically within the range as previously determined) will hardly contribute to the outer integral. If the two triangles are parallel and right above each other, the integrand will in fact hardly change at all. The only regions on where the outer integrand is not smooth are those that are very close to an edge of , because in that case the inner integrand and hence the result of the inner integration changes rapidly. In order to accurately evaluate the integral, these latter regions will require special care. Our novel approach is designed to determine those parts of that contribute to the outer integral and to focus the quadrature points in those regions where the integrand changes rapidly. As a first step, the integration region on is reduced by eliminating those parts that are too far from the plane of to give

any contribution. This is obtained by calculating the intersection (if any) between and the region between two planes, one at a distance below and parallel to and a similar plane above . Depending on the configuration, this leads to a single polygon with three, four or five edges. If there is no intersection, is too far away from and the entire interaction integral, in view of the previously chosen tolerance , can be considered zero. This process is illustrated in Fig. 8. The next step attempts to further reduce the integration domain and also identifies those regions where a rapid change of the integrand can be expected. This in turn leads to a subdivision of the integration domain in judiciously chosen subtriangles, such that in the end quadrature points are distributed in such a way that the overall integration precision is guaranteed. In order to achieve this, is first projected onto the plane of . This projection is subsequently extended (in the plane of ) with polygons, covering a distance of at least from the original projection. The reduced integration domain for the outer integral is then determined as the intersection between and ’s projection including its extensions. This process is illustrated in Fig. 9. In this particular example, the plane of triangle is parallel to that of (the geometry is shown in Fig. 7). The solid black line in Fig. 9(a) represents the projection of in the plane of . We now first extend this projection over a distance to the outside. This extension is also shown in Fig. 9(a) (the dashed lines). From this it follows that the integration over can be restricted to the darkly shaded area (denoted ). Referring to the reasoning put forward w.r.t. the integration over in (18), it is clear that the integrand will

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Fig. 10. (a) A detail of the absolute value of the inner integrand (in dB), in the region indicated by the white dashed lines in Fig. 9(b). In that example, is equal to 0.05 and was chosen to achieve a accuracy. Here we see the exponential behavior of the integrand near the edges (indicated by the black beyond a distance . The outward extension of lines), dropping below the projection is indicated by the white dashdot line. (b) Similar to (a), but now showing the absolute value of the inner integrand (in dB) minus its value at (0.4, 0.4, 0). The white dashdot lines indicate the inward extension.

Fig. 9. (a) Triangle (lightly and darkly shaded area) is reduced to the region (solid line) of triangle and forming its intersection with the projection (dashed lines). The darkly its extensions to the outside over a distance shaded area is the resulting domain for the outer integration. (b) In addition to must also be extended inwards. This the outward extension, the projection does not change the total integration domain, but it influences the division into polygons. The white dashed lines indicate the region that is shown in detail in Fig. 10(c) Schematic representation of the total integration domain (identical to the darkly shaded area in (a) and (b)), subdivided into the polygons over which the individual integrations takes place.

not vary uniformly over . In order to guarantee the overall integration precision, the boundary of the projected triangle is now also extended to the inside as depicted in Fig. 9(b), finally leading to the subdivision of in elementary integration polygons (6 in this particular example), as shown in Fig. 9(c). The numerical integration over these polygons now leads to an overall positioning of the sample points accounting for the exponential variation of the integrand imposed by the Green’s func-

tions (see Fig. 10 for a detail of the behavior of this integrand), as such making sure that the precision obtained for the inner integration (10) does not get compromised when performing the outer integration (19). To further illustrate the principles put forward by means of the example of Fig. 9, we again turn to the example shown in Fig. 8. In this special case, the projection of on reduces to the line in Fig. 8, as the planes of and are perpendicular. Applying the procedure followed in the example of Fig. 9 now simply amounts to the reduction of the outer integration domain to the polynomial . The combination of the first and second steps guarantees that the integrand in each polygon is non-negligible and that each possible steep variation is covered by one polygon. In a final step the actual integration needs to be carried out over these domains. The easiest approach, which delivers accurate results, is to divide each polygon into triangles and then consider all these triangles separately. Numerical quadrature over a triangle is already present in most implementations, reducing the amount of programming required. A possible alternative to this approach

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TABLE VI THE RELATIVE ERRORS (

AND ) FOR THE NUMERICAL EVALUATION OF THE IMPEDANCE INTEGRALS AND NUMBER OF QUADRATURE POINTS USED FOR THE OUTER INTEGRATION (19) IS GIVEN BY

would be adaptive outer integration on the entire outer domain, which was in fact the first method attempted by the authors. However, the computational cost quickly becomes prohibitive as it goes up rapidly with the conductivity. In Table V it was shown how an adaptive approach does not scale for the inner integral, due to the increasingly deeper refinement around the observation point. For the outer integral, the situation is even worse because, in general, refinement will be required along lines instead of points. An adaptive approach, for both outer and inner integral, can easily take more than a thousand times longer than our novel method. The novel approach requires some geometrical preprocessing, but this is negligible in comparison to the evaluation of the Green’s function. In the current implementation, every separate outer subtriangle is treated with the same amount of quadrature points. Further gains in efficiency may possibly be obtained by selecting this number more carefully for each subtriangle or even through adaptive quadrature per subtriangle. However, that is beyond the scope of the current work. V. PERFORMANCE This section will evaluate the performance and accuracy for calculating the impedance integrals for a few of the most interesting and challenging cases. The techniques described in the previous sections will be applied to each of the integrals , and . Three particular geometrical situations will be considered that are of particular importance to potential applications. These are the so-called self patch (when two triangles overlap), the orthogonal neighbor patch (when they touch in a line and have orthogonal planes) and the case of two parallel triangles that are close to each other. Note that the self patch for is always zero and consequently that the self patch for does not need to be calculated. Note that the accuracy of all results has been obtained through self-convergence (using the same method but with higher precision and, consequently, more quadrature points). For the low conductivity cases, our technique for the inner integral has been compared with SE and SC (see also Table IV), which verifies the implementation. The evaluation of the outer integral has been compared with an adaptive technique (progressive refinement of the integration region into more triangles), which is incredibly slow for higher conductivity but does, eventually, confirm our results. For every result, we generated a reference value that is at least two orders of magnitude more precise, in terms of all parameters (number of quadrature

IN THE CASE OF A SELF PATCH.

THE

points for the radial, angular and outer integral, as well as the tolerance for ). A. Self Patch The first example under consideration is that of the interaction between two identical triangles, which is the cornerstone of the impedance matrix. The triangle is defined by the vertices (0, 0, 0), (1, 0, 0) and (0, 1, 0). The material through which they interact is chosen to be copper and the self patch integral is studied at different frequencies. The challenging situations are those for which is small (or, equivalently, is large), which happens in the limits of high conductivity and high frequency. The results are shown in Table VI. Note that the self patch contribution to the K-operator is always zero [10], hence the omission of for this example. The results show that our approach is stable for small and can also achieve a desired tolerance, for the frequency ranging over many orders of magnitude. Further numerical tests show that our approach is stable for at least as small as , indicating the inherent robustness of our approach. Actually, the critical parameter in determining the behavior of the integrand is , with the typical size of the mesh elements (so for the self patch example). Taking a closer look at realistic values of , two frequency ranges need to be treated. In the case of high frequencies, will be of the order of (with the wavelength in the background medium), while in the low frequency regime, is determined by the geometry and can be considered independent of the frequency. In the high frequency regime, with , we have that . As the frequency increases, the skin depth decreases as , but the discretization of the triangles as , eventually leading to a situation where our special approach is no longer required as the dimensions of the triangles become even smaller than . However, for copper, for (or ), so in practice any high frequency simulation for the microwave and millimeter wave range involving copper (or other good conductors) requires the techniques we previously described. At low frequencies, we have and the parameter depends both on the frequency and the geometry. An important aspect that has not yet been discussed before are the conditions under which it is allowed to use triangles that are considerably larger than the skin depth . Roughly said, this is valid when the curvature of the geometry is sufficiently small

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TABLE VII FOR THE NUMERICAL EVALUATION OF THE IN THE CASE OF AN ORTHOGONAL NEIGHBOR PATCH

TABLE VIII AND FOR THE NUMERICAL EVALUATION THE RELATIVE ERRORS IN THE CASE OF NEAR-SINGULAR PARALLEL OF THE IMPEDANCE INTEGRAL TRIANGLES

in comparison with . As a result, near sharp corners of a conducting object it will still be necessary to refine the mesh in order to accurately catch the electromagnetic behavior. However, this can be done in a localized manner, without affecting the mesh of those parts that are smooth. B. Neighbor Patch Whilst the self patch is critical for the contribution due to the operator of the PMCHWT formulation [10], the associated impedance integrals discretizing the operator are zero. The most common neighbor patches, namely those where the two triangles lie in the same plane, also result in a zero contribution [10]. As such, here we will consider the case of two orthogonal triangles that touch in one line (as shown in Fig. 8), which, incidentally, is also of considerable practical importance. is again defined by the vertices (0, 0, 0), (1, 0, 0) and (0, 1, 0), while has (0, 0, 0), (1, 0, 0) and (0, 0, 1) as its corners. The remaining logarithmic edge singularity in the outer integral is a well-known issue [10], but due to the focusing of our quadrature points in a small region near the common edge, relatively good and stable results can be obtained by simply applying a brute force Gaussian integration. The results, shown in Table VII, demonstrate that it is possible to obtain an accuracy that is more than enough for most applications. If a still better accuracy is required, certain approaches could be followed (e.g., [10]) to get rid of the remaining edge singularity, but that is beyond the scope of this paper. C. Thin Plate Triangles In a practical application, many of the impedance integrals (for interaction through a conductive medium) will be negligible, simply because the triangles are too distant and the kernel is highly lossy. In many cases, only the self patch, neighbor patches and point patches (when two triangles touch in exactly one point) contribute (the so-called singular integrals). However, one exception is that of very thin plates, with a thickness of the order of the skin depth or smaller. In that case, the interaction between the two walls through the conductive medium has an important contribution and needs to be accounted for. An important aspect regarding accuracy is that these integrals do not require the same accuracy as the self patch contribution because, due to the lossy nature of the medium, they are perturbations of the diagonal. If the self patch is known to accuracy and the distance of the wall leads to a drop in interaction strength, then only approximately relative accuracy is required for the interactions through the wall. Any additional

Fig. 11. The geometry for the numerical example.

accuracy would get numerically lost in the uncertainty on the self patch. Our approach automatically takes this into account through the value of . So, two types of relative errors will be given in the results, namely and , with the evaluation of the self patch integral corresponding to . To make it more challenging, we will consider triangles that, while parallel (as is the case for thin walls), do not have a completely overlapping support. This creates some difficulties for the outer integral, solved by our approach. is defined by the vertices (0, 0, 0), (1, 0, 0) and (0, 1, 0) and by (1, 1, d), (0, 1, ) and (1, 0, ), where is the thickness of the plate (and the distance between the triangles). The results will again focus on the accuracy of . The results are shown in Table VIII. The cases where are the result of being smaller than (meaning that the integral will be evaluated to zero). However, as shown by , this is within our desired tolerance. VI. NUMERICAL EXAMPLE To illustrate the previously developed techniques, we will consider the practical case of very thin, conductive walls, which was in fact the original motivation for this work. If the wall thickness is of the order of the skin depth or smaller, the “tunneling effect” cannot be neglected and a full-wave solution is required. In order to allow verification of the numerical result, a configuration will be chosen that allows comparison with an analytical solution. Fig. 11 displays this geometry (not to scale), which consists of a hollow conductive sphere with radius and thickness . The parameters are chosen as follows: , , (copper). The incoming plane wave

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Fig. 12. A comparison (between simulation and analytical result) of the electric field after scattering at a very thin conductive shell.

has a frequency of (so for the background medium) and is linearly polarized with and . The skin depth of copper at this frequency is . The surfaces of each sphere are discretized in 584 triangles, leading to a total of 3504 unknowns. The impedance integrals were calculated with a tolerance of . The results are displayed in Fig. 12, comparing the total simulated field with the analytical result obtained from the Mie series. The results are plotted along the dashed line shown in Fig. 11 (which is the x-axis). The error is represented as , which is a measure for the distance in the complex plane. As such, it compares the complex field values, taking both amplitude and phase into account. The distance between the data and the error can be interpreted as the relative accuracy of the result. This is better than 1%, except close to the walls. This is due to geometrical meshing error (flat triangles are used to model a curved surface). Similar results are very difficult to obtain with a method that discretizes the volume instead of the boundaries. In order to catch this behavior it is, however, necessary to accurately evaluate the impedance integrals. An identical simulation, but using traditional Singularity Cancellation (without ) instead of our method, failed to converge. In a second simulation, using the same geometry as shown in Fig. 11, we evaluate the shield penetration (SP) for these enclosures for various values of . The SP, in this case, is defined as (20) The results are shown in Fig. 13 for ranging from 0.1 to 10, with the error defined in the same way as for Fig. 12. Clearly the simulations agree very well with the analytical solution throughout the entire domain. Of course, the accurate modeling of the tunneling effect through a conductor is not the only application of this work. A full-wave treatment (as opposed to using, e.g., a surface impedance approximation) is necessary whenever the thickness becomes of the order of the skindepth or when the inside becomes important for other reasons, e.g. in the study of the effect of corners or of impurities within the conductor. Additionally,

Fig. 13. The shield penetration as a function of , calculated both analytically and numerically.

it provides a smooth extension of the full-wave approach for dielectrics to conductors, without requiring a sudden transition to surface impedances, possibly leading to more reliable results in the transition zone. It may also serve as a reference against which different high conductivity approaches can be evaluated. A deeper investigation of all these applications will be the subject of future work. VII. CONCLUSION In this paper, the accurate and scalable evaluation of impedance integrals in a conductive medium has been treated. An error-controllable approach was proposed that is stable for the high conductivity limit, evaluating both the inner and outer integral with care. The main novelty is with regard to the use of a cutoff distance—at various places in the algorithm–to more efficiently focus numerical effort. The performance of the approach was shown through a few challenging case studies (self patch, neighbor patch and near singular case) and the example of very thin conductive shells. Finally, some suggestions for application of this technique were listed. REFERENCES [1] D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag., vol. 32, no. 3, pp. 276–281, Mar. 1984. [2] P. Ylä-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix RWG functions,” IEEE Trans. Antennas elements with RWG and Propag., vol. 51, no. 8, pp. 1837–1846, Aug. 2003. [3] R. Graglia, “On the numerical integration of the linear shape functions times the 3D-Green’s function or its gradient on a plane triangle,” IEEE Trans. Antennas Propag., vol. 41, no. 10, pp. 1448–1455, Oct. 1993. [4] T. Eibert and V. Hansen, “On the calculation of potential integrals for linear source distributions on triangular domains,” IEEE Trans. Antennas Propag., vol. 43, no. 12, pp. 1499–1502, 1993. [5] M. Khayat and D. Wilton, “Numerical evaluation of singular and nearsingular potential integrals,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3180–3190, Oct. 2005. [6] L. Rossi and P. Cullen, “On the fully numerical evaluation of the linearshape function times the 3D Green’s function on a plane triangle,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 4, pp. 398–402, Apr. 1999. [7] R. Graglia and G. Lombardi, “Machine precision evaluation of singular and near singular integrals by use of Gauss quadrature formulas for rational functiuns,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 981–998, Apr. 2008.

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[8] J. Van Bladel, Electromagnetic Fields, ser. IEEE Press Series on Electromagnetic Wave Theory. Hoboken, NJ: Wiley, 2007, 978-0-47126388-3. [9] S. Chakraborty and V. Jandhyala, “Evaluation of Green’s function integrals in conducting media,” IEEE Trans. Antennas Propag., vol. 52, no. 12, pp. 3357–3363, 2004. [10] P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Analysis of surface integral equations in electromagnetic scattering and radiation problem,” Engrg. Analy. Bound. Elements, vol. 32, no. 3, pp. 196–209, Mar. 2008. [11] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, no. 3, pp. 409–418, May 1982. [12] A. Polimeridis and J. Mosig, “Evaluation of weakly singular integrals via generalized Cartesian product rules based on the double exponential formula,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 1980–1988, June 2010. [13] M. Mori, “Discovery of the double exponential transformation and its developments,” RIMS, Kyoto Univ., vol. 41, pp. 897–935, 2005.

Joris Peeters was born in Antwerp, Belgium, in 1983. He received the M.Sc. degree in physical engineering and the Ph.D. degree in applied physics from the University of Ghent, Belgium, in 2006 and 2010, respectively. In 2006, he joined the Electromagnetics Group, Department of Information Technology, University of Ghent, to do further research. He then focused on efficient techniques within the context of boundary integral equations, with much attention to complex real-life problems in the field of computational electromagnetism. After a brief period working as a Postdoctoral Researcher at the University of Ghent, he joined Computational Dynamics Ltd., London, U.K., in March 2011, to work within the extended field of computational continuum mechanics.

Ignace Bogaert received the M.S. degree in physical engineering and the Ph.D. degree in applied physics from Ghent University, Ghent, Belgium, in 2004 and 2008, respectively. In 2004, he joined the Electromagnetics Group, Department of Information Technology (INTEC), Ghent University. Currently, his research is supported by a postdoctoral grant from the Research Foundation-Flanders (FWO-Vlaanderen). His research interests include boundary integral equations for the modeling of various physical systems, with the emphasis on robustness, efficiency and accuracy.

Daniël De Zutter (F’00) was born in 1953. He received the M.Sc. degree in electrical engineering and the Ph.D. degree, and he completed a thesis leading to a degree equivalent to the French Aggrégation or the German Habilitation from Ghent University, Ghent, Belgium, in 1976, 1981, and 1984, respectively. Between 2004 and 2008, he served as the Dean of the Faculty of Engineering, Ghent University, where he is now a Full Professor of electromagnetics and the Head of the Department of Information Technology. His research focusses on all aspects of circuit and electromagnetic modelling of high-speed and highfrequency interconnections and packaging, on electromagnetic compatibility (EMC) and numerical solutions of Maxwell’s equations. As an author or coauthor he has contributed to more than 180 international journal papers (cited in the Web of Science) and 200 papers in conference proceedings. Prof. De Zutter was elected to the grade of Fellow of the IEEE in 2000. He was an Associate Editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES.

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Electromagnetic Scattering From General Bi-Isotropic Objects Using Time-Domain Integral Equations Combined With PMCHWT Formulations Ze-Hai Wu, Student Member, IEEE, Edward Kai-Ning Yung, Fellow, IEEE, Dao-Xiang Wang, Member, IEEE, and Jian Bao, Student Member, IEEE

Abstract—Electromagnetic scattering by general bi-isotropic objects is calculated by using the time-domain integral equations which are incorporated with the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulations. By introducing a pair of equivalent electric and magnetic sources, electromagnetic fields inside a homogeneous bi-isotropic region can be represented by these sources over its boundary after applying fields splitting method. A series of coupled surface integral equations are obtained after imposing boundary conditions. These equations are then solved numerically using the method of moment (MoM) which involves separate spatial and temporal testing procedures. The Rao-Wilton-Glisson (RWG) functions are used as the spatial expansion and testing functions, and the weighted Laguerre functions are derived as the temporal basis and testing functions. Numerical results such as transient currents, far scattered fields, and normalized radar cross sections are presented and compared with analytical results as well as MoM-based frequency-domain analysis, and good agreements are observed. Index Terms—Bi-isotropic medium, method of moments (MoM), scattering, time domain integral equations.

I. INTRODUCTION

B

I-ISOTROPIC (BI) medium is a special class of complex materials that can produce both electric and magnetic polarizations when excited by either an electric or magnetic source [1]. Among all, Chiral and Tellegen materials represent two subclasses of BI medium [2], [3]. Chiral medium is optically active, which means that the polarization plane of an electromagnetic (EM) wave is rotated when propagating through it. Investigations show that Chiral medium possesses the property of reciprocity, while Tellegen medium is nonreciprocal. Many efforts have been made in the fabrication of these BI materials [4]–[6] because of their great potential in the millimeter-wave and microwave applications such as antenna radomes [7], chiro-microstrip antennas [8], modes convertors [9], and polarization rotators [10]. Such material brings about new challenges to the EM

Manuscript received December 21, 2009; revised December 06, 2010; accepted August 08, 2011. Date of publication October 20, 2011; date of current version February 03, 2012. This work was supported in part by the Research Grants Council of Hong Kong Special Administrative Region (SAR), China, under Grant CityU 124308. Z.-H. Wu was with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR. He is now with Argus Technologies, Guangzhou Development District, Guangzhou, China. (e-mail: [email protected]). E. K.-N. Yung, D.-X. Wang, and J. Bao are with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2173098

theory since its constitutive relationships enforce an additional coupling between electric and magnetic fields. Hence, considerable attention has been paid to developing accurate numerical methods to solve the EM propagation, scattering and radiation problems associated with the BI medium [11]–[21]. This paper is concerned with the EM scattering by general BI objects. Although an analytical method has been previously proposed for BI cylinders by Monzon where a contour integral technique is combined with the dyadic Green’s function [11], [12], it can only calculate two-dimensional BI bodies at normal incidence. Kluskens considered the scattering of a chiral cylinder with arbitrary cross section [13]. Integral equations are formulated for it, along with its solution in method of moment (MoM). Later this method was extended to three-dimensional chiral scatterer by Worasawate [15] and chiral revolution by Yuceer [16]. Wang applied MoM to solve the surface integral equations that were incorporated with the Poggio-MillerChang-Harrington-Wu-Tsai (PMCHWT) formulations for the scattering by general BI objects [17] and BI coated conductors [18]. Although there have been many frequency-domain techniques reported for the scattering of BI media, very little work has been done in the time domain. Most of the time-domain schemes available for BI media focus on the finite difference methods, such as finite-different time-domain (FDTD) [19], conformal FDTD [20], BI-FDTD [21], and so on. The examples available are restricted to chiral spheres whose solutions can be analytically calculated by using the modal expansion theory. Therefore, the applicability of the FDTD method still needs verification for general BI objects. The time-domain integral equation (TDIE) solver is commonly used for analyzing complex EM scattering phenomenon [22]–[26]. Although the FDTD method has been the dominant tool for time-domain simulations, the TDIE approach is preferable in some applications especially for analysis of transient scattering by large-size bodies. The reason is that the TDIE method solves fewer unknowns by using surfaces discretization and requires no artificial absorbing boundary condition (ABC). The most popular method to solve a TDIE is the marching-on in time (MOT) scheme [23], [24]. However, many researchers have pointed out that the MOT method may suffer from latetime instabilities in the form of high frequency oscillation. Recently, the marching-on in degree (MOD) method [25], [26] using a set of scaled Laguerre polynomials as the temporal expansion and testing functions is proposed for the TDIE, and stable results can be obtained even for late time. To the best of

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our knowledge, this TDIE solver has not been used to deal with the scattering by BI media. For the first time, this work presents the application of the MOD-based TDIE method for three-dimensional homogeneous BI objects with arbitrary shape. In this paper, pair of new sources and two integro-differential operators are first defined and, later, they are introduced to formulate the far scattered fields by homogeneous dielectric objects in time domain. Then the method is extended for constructing scattered fields inside and outside the BI medium. A field splitting scheme [1], also known as Bohren Decomposition [27], is employed to simplify the expression of the EM fields inside the bodies. It turns out that the fields in the BI media can be decomposed into two uncoupled wavefields, which individually satisfy Maxwell’s equations. In this sense, the BI media can be replaced by two isotropic dielectrics each of which is characterized by its own isotropic parameters, and the fields for BI media can be easily obtained as the summation of two wavefields. In order to achieve stable solutions, PMCHWT formulations [28], [29] are used to construct the surface integral equations. After enforcing boundary conditions, a series of coupled integral equations are established, and they are solved numerically by the MoM involving separate spatial and temporal testing procedures. The Rao-Wilton-Glisson (RWG) functions are used as the spatial expansion and testing functions, and the weighted Laguerre functions are used as the temporal expansion and testing functions. The use of the Laguerre functions completely removes the time variable from computation, and the matrix equation is solved recursively using a MOD procedure. To validate the accuracy of the proposed TDIE method, the scattering of BI objects is analyzed, and the transient currents, far scattered fields and bistatic radar cross-sections are presented and compared. II. THEORY AND INTEGRAL EQUATION FORMULATIONS

(6) is the wave impedance in the medium surrounding where the scatterer, and and are the scattered fields outside the dielectric body. The integro-differential operators and in (5) and (6) are defined as

(7)

(8) where represents the distance between the observation point and the source point , is the retarded time, is the velocity of the propagation of EM wave in space, and denotes the surface with the singularity at removed from the surface . It can be seen that here we introduce a pair of new sources and instead of using the equivalent electrical current and magnetic current to construct the far scattered fields. In this case, the time derivative of the electric and magnetic vector potentials can be easily handled because there is no time-integral term existed. B. Integral Equations in BI Medium The expression of the electric and magnetic fields inside the BI region is relatively complex because of the introduction of the bi-isotropic constitutive relations, namely (9)

A. Equivalent Sources for Homogeneous Dielectric Bodies Here, we consider a homogenous dielectric body with permittivity of and permeability of which is embedded in an infinite homogenous medium with permittivity of and permeability of . A pair of new sources and on the surface of the dielectric body is defined by (1) (2) and are the equivalent electric and magwhere netic surface currents. According to the equation of continuity, the electric charge density and magnetic charge density can be written as (3) (4) With the use of equivalent principle [30], the scattered fields and can be formulated in terms of the equivalent sources on the surface by (5)

(10) and are Tellegen and Pasteur parameters, respecwhere tively, and and are the permittivity and permeability of the BI medium. To represent the fields in the BI region, a field splitting scheme is applied [1]. Both the electric and magnetic fields and in the homogeneous BI medium are divided into the right- and left-circularly polarized wavefields. The right-polarized fields are denoted by “ ” subscript, while the left-polarized components are denoted by “–” subscript. Therefore, we can write (11) (12) and are independent and The wavefields uncoupled in the homogeneous BI medium. They are related to respective medium characterized by , and , which are defined by (13) (14)

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(15) and . Since two wavefields where are independently governed by Maxwell’s equations, and can be expressed by

in which the equivalent electric and magnetic sources are represented by using the RWG functions [32]. The equivalent sources and are expanded by (24)

(16)

(25)

(17) where the integro-differential operators as

and

are defined

where and are the time-domain coefficients to be determined, is the number of the inner edges, and represents the RWG function. The coefficients for the temporal expansion functions and which are assumed to be causal response functions for , can be expanded as

(18)

(26) (27)

(19) As can be seen, the expressions of the scattered wavefields and in the media induced by and are similar to those of free space except that the material parameters are different. Here the relations of and can be obtained from Maxwell’s equations, (20) (21) To determine the unknown sources and , the boundary condition needs to be enforced on the surface of the BI scatterer. That means the total tangential fields should be continuous across the surface of the BI object. Hence, a set of coupled field integral equations can be obtained as

where

and are the unknown coefficients, and is the temporal basis function. is the Laguerre function of order [33] with a scaling factor . Through the Galerkin’s method, we take both spatial testing with and temporal testing with to the integro-differential operators and , respectively. With reference to [26], is the maximum order of the Laguerre functions which is the time-bandwidth product of the incident waveform. When computing the integrals, the distance between two triangles is assumed to be constant, hence (28) where , and can be either or –, and is the distance between the center point of triangles and With these assumptions, we obtain the following equations

.

(22)

(23)

(29)

and are the incident electric and magnetic fields where respectively, and the subscript “tan” represents the tangential components. III. NUMERICAL SOLUTION PROCEDURE A. Basis Functions and Testing Scheme MoM [31] is adopted to solve (22) and (23). For the implementation of MoM, the surface of an arbitrarily shaped BI object is meshed by using a number of planar triangular patches,

(30)

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represents a spatial testing which

is defined by multiplying the function and integrating in the triangle pairs , and represents a temporal testing which is done by multiplying the function and integrating from zero to infinity. With reference to [25], the temporal integral can be simplified as

(39)

(31) where the spatial integrals given by

,

,

,

and

are (40) (32) (33) (34) (35) (41) (36)

where is a unit vector along the direction . It is noted that the time and space variables are separated in the computation, and the time variable is replaced by the order degree of the Laguerre functions. We apply both the spatial and temporal testing procedures to (22) and (23). The matrix below is obtained after some mathematical manipulations,

and the elements on the right hand are (42) (43) where

and

are given by (44) (45)

(37) where the elements on the left hand side of the matrix are

(38)

where the definitions for , , , and are given in appendix (A1)–(A4). For the above equations, the calculation of the system matrix elements , , , and is very time consuming because the double integrals , , and require much computation. In the cases of , all the integrals are regular and they can be calculated numerically [34]. When the source and observation edges are in the same triangles or , analytical formulas [35] are used to remove the singularity of the spatial integrals. In addition to that, the integral is not needed for the computation during the whole solution because of automatic cancellation of the PMCHWT formulations. We can see from (37) to (45) that, to obtain the coefficients and , the matrix (37) should be solved recursively on the order of the degree of Laguerre function. Particularly, in the first step when , the parameters , , , and are all equal to zero. As a result in such case, only system , , , and are needed matrix elements where the LU decomposition can be stored for further use. In the following -th step, we only have to compute , , , on the right side of the matrix, which are the sums of and

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Fig. 1. The 3-D Bi-isotropic sphere and coordinates illustrations.

the previous solved coefficients and , as shown in appendix (A1)–(A4). Since all the spatial integrals will be needed in the last step , we multiply all the spatial integral with and store these values generated from the first step. This can save much computation cost. The complexities of this method are for the filling procedure and for the iteration procedure. B. Far Scattered Fields and Bistatic Radar Cross Sections After solving the matrix (37) in a marching-on in degree manner, the transient electric and magnetic current coefficients are expressed by using (1) and (2). Once the equivalent currents on the scatterer are determined, the far scattered fields can be computed [25]. After computing the total scattered field at a point , it can be rewritten by using the sphere coordinates as (46) With the use of the fast Fourier transform (FFT), the total scattered and incident fields and at a frequency band can be obtained. Using the scattering amplitude matrix, the and the cross-polarco-polarized bistatic radar cross section are defined [18]. ized bistatic cross section

Fig. 2. Transient currents at the point (0.0096, 0.0022, 0.0005) on a 0.02 m diameter BI sphere.(a) real and imaginary part, (b) magnitude.

IV. NUMERICAL RESULTS AND DISCUSSIONS In this section, the numerical results for the Bi-isotropic scatterer placed in free space will be presented to validate the previously described TDIE scheme. In this part, and are the speed of wave propagation and the wave impedance of free space, respectively. The scatterers are illuminated by a Gaussian plane wave, in which the electric and magnetic fields are given by (47) (48) where , is the unit vector in the direction of wave propagation, is the pulse width, and is the time delay. In this work, the field is incident from and with and , as shown in Fig. 1. A. Sphere Scattering In the first example, we consider a BI sphere of radius 0.01 m centered at the origin. The exact solutions obtained using the matrix Ricatti equations [14] are also presented for comparison.

Fig. 3. Normalized forward scattered fields of the BI sphere .The sphere has a , , , and radius of 0.01 m, and other parameters are .

The sphere has a relative permittivity of and a relative permeability of , and it is meshed with 616 triangular patches and 924 unknowns. The Gaussian pulse of and is used in this numerical computation, and it

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Fig. 4. Forward co- and cross-polarized bistatic echo widths of the BI sphere as a function of frequency. The sphere has a radius of 0.01 m, and other parameters , , , and . are

Fig. 5. Co-polarized bistatic echo widths of the BI sphere as a function of the evaluation angle. The BI sphere has a radius of 0.01 m, and other parameters , , , and . are

has a spectrum of 9 GHz. (The unit “m”denotes a light meter, and one light meter is the amount of time taken by EM wave to travel 1 m.) The scattering geometry of the sphere is shown in Fig. 1, where the forward direction is defined by and . In the TDIE computation, we set , which is sufficient to get accurate results. The value can be different for BI spheres with different , and it will be illustrated in the convergence test. The exact solution was computed at 30 discrete frequencies between 0 and 9 GHz. The scattering of a BI sphere with constitutive parameters and is first computed by using the proposed TDIE solver. In this computation, the maximum temporal order is set to be 120. Fig. 2 displays the transient response of the electric and magnetic currents at the point (0.0096, 0.0022, 0.0005) of the sphere. The real and imaginary parts of the current are plotted in Fig. 2(a), and it is obviously seen that the computed value of currents does not increase, even at very late

Fig. 6. Cross-polarized bistatic echo widths of the BI sphere as a function of the evaluation angle. The BI sphere has a radius of 0.01 m, and other parameters , , , and . are

Fig. 7. Forward co- and cross-polarized bistatic echo widths of the BI sphere for different values of . The BI sphere has a radius of 0.01 m, and other pa, , , and . rameters are

time. The magnitude of the currents are shown in Fig. 2(b), and a gradual decrease is observed in the magnitude of both the electric and magnetic currents. Fig. 3 displays the normalized forward scattered electric fields, in which both the - and -component are stable. The computed forward co- and cross-polarized bistatic radar cross-sections are also shown in Fig. 4 in the frequency band from 0 to 9 GHz, and they are in good agreement with the exact results. The co- and cross-polarized bistatic echo widths of the BI sphere as a function of evaluation angle are shown at the frequencies of 4 GHz and 6 GHz in Figs. 5 and 6, respectively, and they agree very well with the exact results. It is found that the number of the meshed triangles and the maximum temporal order both significantly influence the accuracy of the numerical result. Since the convergence test of the triangle number has already been discussed in [15], this part will only show the effect of the parameter on convergence. At first, the values of used for a BI sphere with parameters described in the last part are 80, 100, and 120. Fig. 7 shows the results of the convergence test in the forward scattering direction.

WU et al.: ELECTROMAGNETIC SCATTERING FROM GENERAL BI OBJECTS USING TDIEs COMBINED WITH PMCHWT FORMULATIONS

Fig. 8. Forward co- and cross-polarized bistatic echo widths of the BI sphere for different values of . The BI sphere has a radius of 0.01 m, and other pa, , , and . rameters are

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Fig. 11. Co-polarized bistatic echo widths of the BI cylinder as a function of the evaluation angle. The cylinder has a radius of 0.02 m with a height of 0.04 , , , and . m. Other parameters are

Fig. 9. Forward co- and cross-polarized bistatic echo widths of the BI sphere for different values of . The BI sphere has a radius of 0.01 m, and other pa, , , and . rameters are Fig. 12. Cross-polarized bistatic echo widths of the BI cylinder as a function of the evaluation angle. The cylinder has a radius of 0.02 m with a height of , , , and . 0.04 m. Other parameters are

Fig. 10. The 3-D Bi-isotropic cylinder with the definition of coordinates.

It is observed that the accuracy increases for a higher . It has been found that the TDIE results converge to those given by the exact solutions when . The second convergence test is performed when the Pasteur parameter of the BI sphere is increased to 0.5 with the Tellegen parameter remaining unchanged. The maximum temporal orders are 100, 120, and 130 in this computation. The computed forward co- and cross-polarized bistatic radar cross-sections of the BI sphere are shown

in Fig. 8. It is noted that the TDIE solutions converge to the exact results when , larger than in the first test. The third convergence test is done when the Tellegen parameter of the BI sphere is increased to 0.8 with a Pasteur parameter . The maximum temporal orders are chosen to be 80, 100, and 120. Fig. 9 displays the co- and cross-polarized bistatic radar cross-sections of the BI sphere. The TDIE solutions converge to the exact results when , which is the same as the first convergence test. From these convergence tests, it can be concluded that the maximum temporal order should be increased to get satisfactory results when increasing Pasteur parameter , while it is not affected by changing the Tellegen parameter only. B. Cylinder Scattering The second example is the scattering from a finite length BI cylinder, which is computed using the proposed TDIE tech-

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(A1)

(A2)

(A3)

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(A4)

nique, and the results are compared with those of the frequencydomain integral equation (FDIE) analysis based on the MoM solution [17]. The cylinder has a radius of 0.02 m with a height of 0.04 m, and other parameters are chosen to be , , , and . Fig. 10 displays the meshed cylinder, and it has 610 triangular patches with 915 unknowns. The parameters of the incident Gaussian pulse remain unchanged and is used in the TDIE computation. Figs. 11 and 12 show the co- and cross-polarized bistatic radar cross-sections of the BI cylinder at the frequencies of 2.0 GHz, and 4.0 GHz, respectively. The results of the proposed time-domain method agree well with the frequency-domain data for the co-polarized radar cross-section. Slight difference is observed for the cross-polarized radar cross-section near degree at 4.0 GHz. A perfect match is not observed at some angles, and the reason is that the comparison is made between two different numerical techniques. V. EXTENSION TO THE DISPERSIVE CASE Owing to the fact that the constitutive parameters are non-dispersive, the equations abovementioned are set up for high idealized model. It is not very difficult for us to extend the proposed method for frequency dependent materials. The TDIE method based on the MOD procedure is one of the recursive convolution techniques that allow linear dispersion to be incorporated like FDTD formulation [19], [21]. Taking the dispersive chiral media for example, the electric and magnetic fields are decomposed into the wavefields and the scattering problem is treated as the sum of two problems in associated isotropic media. After using the fields splitting scheming, the (13)–(15) will become (49) (50) (51)

where the frequency variation of the chirality term is expressed by the Condon model [1]. The fields inside the chiral media can be constructed using the integro-differential operators and . Different from nondispersive formulations (18) and (19), these two operators will have an additional time-integral term. After applying boundary condition, the coupled integral equations can be obtained. For the testing procedure, the temporal integral (31) will become a little complicated because of the additional time-integral term. This integral and the whole solution are considered as the further work in the near future. VI. CONCLUSION In this paper, a TDIE solver using the MoM technique is applied to calculate the scattering problem of the general BI medium. Two pairs of new sources with integro-differential operators have been introduced to construct a series of coupled surface integral equations which are combined with the PMCHWT formulation. The equations are solved with Galerkin’s method that involves separate spatial and temporal testing procedures. The RWG functions are used as the spatial expansion and testing functions. Also the weighted Laguerre polynomials are used as the temporal expansion and testing functions. Numerical results including the transient currents, far scattered fields, and co- and cross-polarized bistatic radar cross-sections are given. Comparisons with the analytical results are made, and good agreements are observed. Also, the proposed TDIE method is verified by the FDIE approach based on the MoM solution. APPENDIX In this Appendix, for the marching-on in degree method to solve the matrix (37), the detailed expression of the elements , , , and in (42) and (43) are given by (A1)–(A3) on the previous page and (A4) at the top of this page.

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REFERENCES [1] I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media. Boston, MA: Artech House, 1994. [2] K. F. Lindman, Ann. Phys., vol. 63, pp. 621–644, 1920. [3] B. D. H. Tellegen, “The gyrator: A new electric network element,” Phillips Res. Rep., vol. 3, p. 81, 1948. [4] V. V. Varadan, R. Ro, and V. K. Varadan, “Measurement of the electromagnetic properties of chiral composite materials in the 8–40 GHz range,” Radio Sci., vol. 29, no. 1, pp. 9–22, 1994. [5] A. J. Bahr and K. R. Clausing, “An approximate model for artificial chiral material,” IEEE Trans. Antennas Propag., vol. 42, no. 12, pp. 1592–1599, Dec. 1994. [6] S. A. Tretyakov, S. I. Maslovski, I. S. Nefedov, A. J. Viitanen, P. A. Belov, and A. Sanmartin, “Artificial Tellegen particle,” Electromagn., vol. 23, no. 8, pp. 665–680, 2003. [7] S. A. Tretyakov and A. A. Sochava, “Proposed composite material for nonreflecting shields and antenna radomes,” Electron. Lett., vol. 29, pp. 1048–1049, Jun. 1993. [8] N. Engheta and P. Pelet, “Reduction of surface waves in chirostrip antennas,” Electron. Lett., vol. 27, pp. 5–7, Jan. 1991. [9] P. Pelet and N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag., vol. 38, pp. 90–98, Jan. 1990. [10] I. V. Lindell, S. A. Tretyakov, and M. I. Oksanen, “Conductor-backed Tellegen slab as twist polarizer,” Electron. Lett., vol. 28, pp. 281–282, 1992. [11] J. C. Monzon, “Radiation and scattering in homogeneous general biisotropic regions,” IEEE Trans. Antennas Propag., vol. 38, no. 2, pp. 227–235, Feb. 1990. [12] J. C. Monzon, “Scattering by a biisotropic body,” IEEE Trans. Antennas Propag., vol. 43, no. 11, pp. 1288–1296, Nov. 1995. [13] M. S. Kluskens and E. H. Newman, “Scattering from a chiral cylinder of arbitrary cross section,” IEEE Trans. Antennas Propag., vol. 38, no. 9, pp. 1448–1455, Sep. 1990. [14] D. L. Jaggard and J. C. Liu, “The matrix Riccati equation for scattering from stratified chiral spheres,” IEEE Trans. Antennas Propag., vol. 47, no. 7, pp. 1201–1207, Jul. 1999. [15] D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three dimensional homogeneous chiral body,” IEEE Trans. Antennas Propag., vol. 51, no. 5, pp. 1077–1084, May 2003. [16] M. Yuceer, J. R. Mautz, and E. Arvas, “Method of moments solution for the radar cross section of a chiral body of revolution,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1163–1167, Mar. 2005. [17] D. X. Wang, E. K. N. Yung, R. S. Chen, and P. Y. Lau, “Scattering characteristics of general bi-isotropic objects using surface integral equations,” Radio Sci., vol. 41, no. 2, Apr. 2006. [18] D. X. Wang, P. Y. Lau, E. K. N. Yung, and R. S. Chen, “Scattering by conducting bodies coated with bi-isotropic materials,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2313–2319, Aug. 2007. [19] V. Demir, A. Z. Elsherbeni, and E. Arvas, “FDTD formulation for dispersive chiral media using the Z transform method,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3374–3384, Oct. 2005. [20] H. X. Zheng, X. Q. Sheng, and E. K. N. Yung, “Computation of scattering from conducting bodies coated with chiral material using conformal FDTD,” J. Electromagn. Waves Applicat., vol. 18, no. 11, pp. 1471–1484, 2004. [21] A. Akyurtlu and D. H. Werner, “BI-FDTD: A novel finite-difference time-domain formulation for modeling wave propagation in bi-isotropic media,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 416–425, Feb. 2004. [22] S. M. Rao, Time Domain Electromagnetic. New York: Academic, 1999. [23] B. P. Ryne and P. D. Smith, “Stability of time marching algorithms for the electric field integral equation,” J. Electromagn. Waves Applicat., vol. 4, pp. 1181–1205, 1990. [24] P. J. Davies, “On the stability of time-marching schemes for the general surface electric-field integral equation,” IEEE Trans. Antennas Propag., vol. 44, pp. 1467–1473, Nov. 1996. [25] B. H. Jung, T. K. Sarkar, Y. S. Chung, S. P. Magdalena, Z. Ji, S. Jang, and K. Kim, “Transient electromagnetic scattering from dielectric objects using the electric field integral equation with Laguerre polynomials as temporal basis functions,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2329–2339, Sep. 2004.

[26] Y. S. Chung, T. K. Sarkar, B. H. Jung, and Z. Ji, “Solution of time domain electric field integral equation using the Laguerre polynomials,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2319–2328, Sep. 2004. [27] C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett., vol. 29, pp. 458–462, Dec. 1974. [28] J. R. Mautz and R. F. Harrington, “Electromagnetic scattering from a homogeneous body of revolution,” AEU, vol. 33, pp. 71–80, Feb. 1979. [29] Y. Chu, W. C. Chew, S. Chen, and J. Zhao, “Generalized PMCHWT formulation for low frequency multi-region problems,” in Proc. IEEE Int. Symp. Antennas Propagation Society, Piscataway, NJ, 2002, pp. 664–667. [30] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [31] R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968. [32] S. M. Rao, “Electromagnetic Scattering and Radiation of Arbitrarily Shaped Surfaces by Triangular Patch Modeling,” Ph.D. dissertation, Univ. Mississippi, , 1980. [33] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products. New York: Academic, 1980. [34] D. A. Dunavant, “High degree efficient symmetrical Gaussian quadrature rules for the triangle,” Int. J. Numer. Methods Engrg., vol. 21, pp. 1129–1148, 1985. [35] P. Arcioni, M. Bressan, and L. Perregrini, “On the evaluation of the double surfaces integral arising in the application of the boundary integral method to 3-D problems,” IEEE. Trans. Microwave Theory Tech., vol. 45, no. 3, pp. 436–439, Mar. 1997.

Ze-Hai Wu (S’07) was born in Jingzhou, China. He received the B.S. degree in information engineering and the M.S. degree in communication and information systems from South China University of Technology (SCUT), Guangzhou, China, in 2002 and 2005, respectively, and the Ph.D. degree from City University of Hong Kong, Hong Kong, in 2010. He is currently a R&D Engineer at Argus Technologies (China) Ltd., Guangzhou, China. His current research interests include novel base station antenna, and time-domain electromagnetic computational methods.

Edward Kai-Ning Yung (M’85–SM’85–F’12) was born in Hong Kong. He received the B.S., M.S., and Ph.D. degrees from the University of Mississippi, University, in 1972, 1974, and 1977, respectively. He worked briefly in the Electromagnetic Laboratory, University of Illinois at Urbana-Champaign. He returned to Hong Kong in 1978 and began his teacher career at the Hong Kong Polytechnic. He joined the newly established City University of Hong Kong in 1984 and was instrumental in setting up a new department. He was promoted to Full Professor in 1989, and in 1994, he was awarded one of the first two personal chairs in the University. He is the Founding Director of the Wireless Communications Research Center, formerly known as Telecommunications Research Center. Despite his heavy administrative load, He remains active in research in microwave devices and antenna designs for wireless communications. He is the principle investigator of many projects worth tens of million Hong Kong dollars. He is the author of over 450 papers, including 270 in referred journals. He is also active in applied research, consultancy, and other technology transfers. Prof. Yung was the recipient of many awards in applied research, including the Grand Prize in the Texas Instrument Design Championship, and the Silver Medal in the Chinese International Invention Exposition. He is a Fellow of the Chinese Institution of Electronics, the Institute of Electrical Engineers, and the Hong Kong Institution of Engineers. He is also a member of the Electromagnetics Academy. He is listed in the Who’s Who in the World and Who’s Who in the Science and Engineering in the World.

WU et al.: ELECTROMAGNETIC SCATTERING FROM GENERAL BI OBJECTS USING TDIEs COMBINED WITH PMCHWT FORMULATIONS

Dao-Xiang Wang (M’08) was born in Nanjing, China. He received the M.S. degree from Nanjing University of Science and Technology (NJUST) in 2004 and the Ph.D. degree from City University of Hong Kong, Hong Kong, in 2007. Since 2007, he has been a Research Fellow at City University of Hong Kong. His research interests include computational electromagnetics, electromagnetic scattering and propagation in complex media, and signal integrity.

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Jian Bao (S’11) was born in Zhejiang, China. He received the B.Sc. degree in electronic engineering from Zhejiang University, Hangzhou, China, in 2005 and the M.Phil. degree in electronic engineering from City University of Hong Kong, Hong Kong, in 2007, where he is currently working towards the Ph.D. degree. His research interests include numerical method in electromagnetic, antennas, electromagnetic scattering and propagation in complex media, and fast and efficient algorithms.

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Efficient Surface Integral Equation Using Hierarchical Vector Bases for Complex EM Scattering Problems Li Ping Zha, Yun Qin Hu, and Ting Su

Abstract—A set of hierarchical divergence-conforming vector basis functions based on curved triangular patches is presented for method of moment (MoM) solutions of surface integral equations in this paper. The higher order method of combined-field integral equation (CFIE) solves perfect electric conductor electromagnetic scattering problems, the higher order electric-magnetic current combined-field integral equation (JMCFIE) formulations solves dielectric objects, and their combination is able to efficiently analyzes the scattering of hybrid PEC-dielectric objects. The expressions of the divergence- conforming hierarchical basis functions up to order 3.5 are reported in this paper. The multilevel fast multipole algorithm (MLFMA) is then employed to reduce the memory requirements and computational complexity. Numerical experiments indicate that the proposed hierarchical vector basis functions can provide well-conditioned linear system for iterative solution. Index Terms—Complex electromagnetic scattering, hierarchical vector basis functions, surface integral equation.

I. INTRODUCTION

F

REQUENCY domain surface integral-equation (SIE) formulations is a powerful tool in electromagnetic simulation, since they require fewer unknowns than volumetric methods. In terms of the geometrical modeling and current discretization, traditional methods are low-order techniques and the structure is modeled by surface geometrical elements that are electrically very small and the currents within the elements are approximated by low-order basis functions [1]. When the size of problem is very large, the plane triangles used to discrete the surface will produce a large number of unknowns, but cannot provide enough flexibility and efficiency in modeling curved structures. Further more, the accuracy of solution while using the low-order bases is improved slowly with increases in the number of unknowns. Higher order numerical methods resolve such problems. Hierarchical higher order basis functions enable to use different orders of current/field approximation in different elements, and also a -adaption scheme [2] comparing to interpolatory higher order Manuscript received January 28, 2011; revised April 27, 2011; accepted June 16, 2011. Date of publication September 15, 2011; date of current version February 03, 2012. This work was supported in part by the National Natural Science Foundation of China under Grants 60701003, 60701005, and 60825102 and in part by the Major State Basic Research Development Program of China (973 Program) under Grant 2009CB320201. The authors are with the Department of Communication Engineering, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2167932

vector bases. However, the hierarchical basis functions suffer from a weakness that as the order increases, the system matrix becomes ill-conditioned, which worsens the convergence rate of the iterative solver. Recently hierarchical vector bases with good orthogonality have received intense attention [3]–[10]. The hierarchical vector bases fall into two categories: hierarchical vector basis functions based on triangular elements and those based on quadrangular elements. Hierarchical Legendre basis functions [5] for curved quadrilaterals use a near-orthogonal expansion of the surface current and lead to a low conditioned MoM matrix. Comparing to quadrangular patches, the triangular mesh can provide more accurate and effective geometrical discretization for arbitrary surfaces, especially those contains cuspate structure. Webb [3] use orthogonal polynomials based on curved triangular elements and Whitney bases as the lowest order vector functions to construct hierarchical vector bases in 2-D finite element method (FEM) computations. In [8], a nearly orthogonal set of hierarchical vector basis functions based on flat triangular patches were used in MoM-SIE formulations. But if higher order basis functions for currents are used on flat patches, many small patches may be required for the geometrical precision of the model, and then higher order basis functions actually reduce to low-order functions (on small patches) [1]. An efficient method of SIE, Poggio-Miller-Chang-Harringt-on-Wu-Tsai (PMCHWT) [11]–[13], is employed to solve the electromagnetic scattering problems for homogeneous dielectric objects. However the iteration convergence of PMCHWT formulation is found to be slow [14] because the magnetic surface current isn’t well tested. For this reason, a new combined-field integral equation proposed by Pasi Ylä-Oijala and Matti Taskinen [15] is used. This electric-magnetic current combined-field integral equation (JMCFIE) can lead to a matrix equation with a high convergence rate. In this work, the higher order formulation of JMCFIE for dielectric electromagnetic analysis is presented. This paper is organized as follows. In Section II, the higher order hierarchical basis functions based on curved triangular elements used in MoM are derived from the curl-conforming hierarchical vector bases proposed by Graglia et al. [9], [10]. The expressions of the divergence-conforming hierarchical basis functions up to order 3.5 are given in detail. An efficient combined-field SIE formulation of higher order method for complex EM scattering problems is presented in Section III. In Section IV, the multilevel fast multipole method (MLFMM) is employed to reduce the memory requirements and computational complexity, and numerical results are also given to show

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TABLE I EDGE-BASED AND FACE-BASED HIERARCHICAL POLYNOMIALS UP TO THE THIRD ORDER ASSOCIATED WITH EDGE 1

Fig. 1. Position vector of arbitrary point determined by normalized face coordinates on curved parametric triangular patch.

the improved performance of the proposed techniques in terms of iterative convergence rate and computational time. Without any further orthogonalization of the vector functions, the hierarchical vector basis functions in [16] are used to compare with the hierarchical bases described in Section II.

the order of 3.5 associated with edge 1 can be expressed in simplex coordinates as

II. HIERARCHICAL DIVERGENCE-CONFORMING VECTOR BASIS FUNCTIONS The class of basis functions defined on curved parametric triangles proposed here is a set of divergence-conforming hierarchical-type vector basis functions for modeling surface currents. The lowest order of divergence-conforming bases as well as higher order functions for triangular elements can be obtained by forming the cross product of the element normal with the curl-conforming functions given in [10]. In order to simplify the derivation of their divergence type, we rewrite the lowest-order (order of 0.5) divergence-conforming basis functions on a triangle element. They can be expressed in normalized area cooras dinates

(2) The first subscript of denotes the edge number of the triangle element, and the second subscript labels the order number of edge-based hierarchical polynomials associated with that edge. The remaining two subsets of edge-based functions associated with edge 2 and edge 3, can be obtained by rotating the simplex in (2), and then multiplying by the corcoordinates responding lowest-order basis in (1). The face-based functions associated with edge 1 can be expressed as

(3)

(1) where is the position vector of the point determined by normalized face coordinates on curved parametric triangular patch, as shown in Fig. 1. is the element Jacobian. Then we extract parts of the new set of orthogonal scalar polynomials proposed by Graglia et al. [9], [10] for triangular patches. As shown in Fig. 1, the element-edges are labeled and . Without by using the three parent variables , loss of generality, we consider basis functions associated with edge 1. The higher order bases are derived from the generating polynomials multiplied by the lowest-order bases. The edgebased and face-based hierarchical polynomials up to the third , are given in Table I, order associated with edge indicates the Legendre polynomial of order , where . So the edge-based basis functions up to and

The remaining two subsets of face-based functions associated with edge 2 and edge 3 can be obtained by rotating the simplex in (3), and then multiplying by the corcoordinates responding lowest-order basis in (1). According to [17], a 2-D triangle element can have only two independent tangent vectors, therefore one of the three subsets of face-based functions should be discarded. Finally, the divergence of the hierarchical vector basis functions (4) is obtained by [17] (5)

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Accordingly, problem 2 has no incident wave in the internal equivalent problem. The continual boundary condition is also used for the tangential component of the total field. We formulate the traditional and on the dielectric surface, and the and on the conductor surface, respectively, as (8) (9) Fig. 2. Arbitrarily shaped PEC object coated by homogeneous dielectric.

(10) (11)

III. EFFICIENT COMBINED-FIELD SIE FORMULATION HO METHOD

OF

The efficient combined-field surface integral equation for complex electromagnetic analysis presented here especially specifies the combination of the combined-field integral equation (CFIE) and the electric-magnetic combined-field surface integral equation (JMCFIE), and its formulations based on the generalized Huygens’ Principle [15]. As shown in Fig. 2, the homogeneous dielectric body is characterized by , and the conductor body is coated by that dielectric. Consider a general purpose scattering problem of a time harmonic plane wave , incident on the surface of the coated object. Using the equivalence principle, the problem can be solved by considering two simple equivalent problems, an external equivalent problem and an internal equivalent problem. The external equivalent problem corresponds to the scattering field induced by the equivalent surface electric current and magnetic current residing on the surface of the dielectric body in free space, and the internal equivalent problem corresponds to the scattering field induced by the equivalent surface currents residing on the surface of the dielectric body and the conductor body in the homogeneous domain characterized by , respectively. Now let’s take the external equivalent problem as problem 1, and the internal equivalent problem as problem 2. In problem 1, there is a time harmonic plane wave incident on the surface of the dielectric body. Using the continual boundary condition for the tangential component of the total electric field and total magnetic field on dielectric surface, we formulate the traditional and for the dielectric surface, as

is the outward unit normal on the dielectric surface or conductor surface. Integral operators , with the superscript and represent the dielectric surface and the conductor surface, respectively, while the subscript 1 and 2 represent the external problem and internal problem, respectively. The integral operators and are defined as (12) (13) , and is the boundary where surface of the considered domain. Then by operating with to (6)–(11), and combining them as follows, the new combined-field surface integral equations (JMCFIE-CFIE) for this conductor-dielectric combined problem are obtained as in (14), shown at the bottom of the page, where denotes the wave impedance in free space, and . The coefficients satisfy , are in successful procedures, the coefficients are equal to . When only a metallic object exists in our discussion, formulation (14) reduces to the traditional CFIE. Additionally, as the coefficients are set to and , the PMCHWT-EFIE formulations are obtained. Both unknown currents and can be discretized by the higher order hierarchical vector basis functions proposed in Section II, (15) (16)

(6) (7)

where and are the unknown expansion coefficients. The total number of unknowns is . The equations in (14)

(14)

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TABLE II SEVERAL PARAMETERS AFFECT THE SOLUTION PRECISION FOR THE CALCULATIONS OF BI-RCS OF A PEC SPHERE WITH A RADIUS OF 5 , USING CFIE WITH FINE MESH

are all testing by the same HO basis functions using Galerkin’s testing method. Direct numerical Gauss quadrature is applied to nonsingular integrals. When the source point coincide with or is close to the observation point , the integrals of Green’s functions will be singular or near-singular. A singularity subtraction method [21]–[23] has been used for the evaluation of singular and near-singular potential integrals in this paper. In the singularity subtraction scheme, the singular and near-singular integrals are decomposed into two parts: a nonsingular integral that can be numerically calculated by Gaussian quadrature and singular terms that can be analytically evaluated. For the higher order basis functions on curved triangles, the evaluation of the singular terms becomes even more challenging. In our implementation, for the calculation of inner integrals, after the singular terms has been extracted, a tangent plane triangle at the projection point of is substituted for the source triangle, where is an integral point on the observation triangle, and then the singular terms can be calculated analytically on this tangent plane triangle by use of the method in [21]. Equations (14) lead to a well condition HO impedance matrix which can be easily solved by an iteration algorithm. The good convergence property will be discussed in Section IV.

IV. NUMERICAL RESULTS In this section, numerical results demonstrate the efficiency of the proposed method. In all considered cases, the multilevel fast multipole algorithm (MLFMA) is employed to reduce the memory requirement and accelerate the iterative solution. The cube size on the finest layer is adjusted properly for large patch when the higher order hierarchical vector basis functions are directly applied in MLFMA. The inner-outer flexible generalized minimal residual (FGMRES) algorithm was used for the iterative solution of the system matrix. In the FGMRES algorithm, the inner and outer restart numbers are both taken to be 10, and the stop precision for the inner and outer iteration is

1.E-2 and 1.E-3 respectively. All computations, unless otherwise mentioned, were carried out on a computer with 1.87 GHz CPU and 1.96 GB RAM. The first example considers the bistatic radar cross section (Bi-RCS) of a perfectly electrically conducting (PEC) sphere of radius to test the accuracy of the proposed hierarchical basis functions. Root mean square (RMS) error was used as a measuring tool for the comparison of accuracies of the solutions and is defined as (17) denotes the calculated radar cross section (RCS) where and denotes the reference Mie solution, both measured in decibels, and is the number of sampling points, which are the angles of observation [19]. In Table II, various parameters affect the solution precision and computational efficiency for fixed order basis functions are presented. As discussed in [20], the FMM box size is chosen to be a little bit larger than the average patch size in all numerical calculations. As shown in Table II, for different mesh sizes giving rise to different scales of unknowns, we obtain various RMS errors for a fixed order of HO bases. Results indicate that the smaller mesh size produced lower RMS error, but more memory and computation time are needed. When the mesh sizes reach a maximum for each order basis functions (soln. no. 3, 6, 10 and 12), the RMS error is about 0.5 dB, and then the accuracy should be questioned. The gauss integral points on the patch also affect the solution precision (see soln. no. 8 and 9). By using more integral points on the element patch, one can obtain more accurate results, but the matrix filling time is greatly increasing. It is suitable that the number of integral points is slightly greater than the number of degrees of freedom (DOF) for a triangle element. As the second example, we consider the electromagnetic scattering of Tomahawk missile model. That simulation model with a length of and a radius of is given in Fig. 3. The CFIE-

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Fig. 3. The simulation model of Tomahawk missile.

Fig. 6. (a) Number of iterations with respect to the number of unknowns for the dielectrically coated warhead example. (b) Computation time with respect to the number of unknowns for the dielectrically coated warhead example. Fig. 4. Bistatic RCS of -polarization of that missile at 1.2 GHz, for the hierarchical basis functions of order 2.5 and RWG basis functions.

Fig. 5. Bistatic RCS for -polarization of a dielectrically coated warhead at 3 GHz, with the RWG basis functions, and the hierarchical basis functions of order 3.5 based JMCFIE-CFIE formulation, respectively.

MLFMM method with higher order basis functions is also employed to solve this problem and the hierarchical basis functions of order 2.5 are employed. As Fig. 4 shows, the Bi-RCS is computed for a -directed and - polarized incident plane wave at 1.2 GHz. The HO method produces 15372 electric current unknowns, and only requires 26 outer iterative steps and 134 s for FGMRES, which includes the both inner and outer iteration CPU times. The numerical results agree well with RWG basis functions’. Another example is a simple dielectrically coated warhead in free space, as shown in Fig. 5. A PEC warhead was coated by a homogeneous dielectric , with the thickness of

coating 0.05 m, and a maximum size of 1.08 m along the direction. This scatterer is discretized with 6008 curvilinear triangular patches for order 1.5 hierarchical basis functions, 2378 for order 2.5, and 1266 for order 3.5, giving rise to 46390, 38598, and 35712 HO unknowns respectively. The incident angles of plane wave are , at 3 GHz. The Bi-RCS at the scattered angle for - polarization was shown in Fig. 5. The HO results of order 3.5 is compared with the LO method. JMCFIE-CFIE was used for these calculations. In the LO method, this example is discretized using 698 RWG basis functions [18] per wavelength, producing 222168 unknowns. Note that the result for RWG basis functions was carried out on a computer with 2.83 GHz CPU and 8 GB RAM. It can be found that there is an excellent agreement among them. The use of higher order techniques greatly reduces the number of unknowns for a given problem [1]. Fig. 6 shows the plot of the convergence behavior of the hierarchical basis function based on the FGMRES method for the JMCFIE-CFIE formulation of the warhead example in Bi-RCS computation. We compare the order of 2.5 hierarchical basis function’s result with that order of 2.5 in [16]. Increasing the frequency of the incident wave from 600 MHz to 3 GHz, while adjusting the real mesh size for each frequency point, results in 4452 HO unknowns for 900 MHz, 9219 for 1.4 GHz, 17199 for 2 GHz, 38598 for 3 GHz, and 97419 for 5 GHz. The outer iterative steps and total (inner and outer) iteration CPU times with respect to the number of unknowns are given in the plots. It can be observed that the iteration convergence behavior of HO JMCFIE-CFIE technique has been improved largely; the outer iterative steps and iteration time for the hierarchical bases of order 2.5 proposed in this paper is less than the hierarchical basis function of order 2.5 in [16] by at least a factor of 7.

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V. CONCLUSION This paper presents a set of hierarchical divergence- conforming vector bases based on curved triangular element for surface integral equations solved using the MLFMA with a reduced computational complexity. HO methods need less memory requirements without compromising the accuracy of geometry modeling in comparison with LO methods. Since the combined-field surface integral equation (JMCFIE-CFIE) formulation can lead to a well tested equation system, these proposed hierarchical vector bases can also provide a wellconditioned system matrix for iterative solution which is based on the inner-outer flexible generalized minimal residual (FGMRES) algorithm. The numerical results of the proposed higher order basis functions have been compared with other set of hierarchical divergence-conforming vector bases which are also based on curved triangular patches. It can be found that the iteration convergence behavior of the proposed HO technique has been improved largely. REFERENCES [1] B. M. Notaros, “Higher order frequency-domain computational electromagnetics,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2251–2276, Aug. 2008. [2] J. Wang and J. P. Webb, “Hierarchical vector boundary elements and p-adaption for 3-D electromagnetic scattering,” IEEE Trans. Antennas Propag, vol. 47, no. 8, pp. 1244–1253, Aug. 1997. [3] C. Carrie and J. P. Webb, “Hierarchal triangular edge elements using orthogonal polynomials,” in Proc. IEEE Trans. Antennas Propag. Soc. Int. Symp., Montreal, Canada, Jul. 1997, vol. 2, pp. 1310–1313. [4] D. A. White, “Orthogonal vector basis functions for time domain finite element solution of the vector wave equation,” IEEE Trans. Magn., vol. 35, no. 3, pp. 1458–1461, May 1999. [5] E. Jorgensen, J. L. Volakis, P. Meincke, and O. Breinbjerg, “Higher order hierarchical Legendre basis functions for electromagnetic modeling,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 2985–2995, Nov. 2004. [6] M. Djordjevic and B. M. Notaros, “Double higher order method of moments for surface integral equation modeling of metallic and dielectric antennas and scatterers,” IEEE Trans. Antennas Propag., vol. 52, no. 8, pp. 2118–2129, Aug. 2004. [7] D. S. Sumic and B. M. Kolundzija, “Efficient iterative solution of surface integral equations based on maximally orthogonalized higher order basis functions,” in IEEE Trans. Antennas Propag. Soc. Int. Symp., Montreal, Canada, Jul. 2005, vol. 4A, pp. 288–291. [8] Ismatullah and T. F. Eibert, “Surface integral equation solutions by hierarchical vector basis functions and spherical harmonics based multilevel fast multipole method,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 2084–2093, Jul. 2009. [9] R. D. Graglia, A. F. Peterson, and F. P. Andriulli, “Hierarchical polynomials and vector elements for finite methods,” in Proc. ICEAA, Torino, Italy, Sep. 2009, vol. 1, 1, pp. 1086–1089. [10] R. D. Graglia, A. F. Peterson, and F. P. Andriulli, “Curl-conforming hierarchical vector bases for triangles and tetrahedral,” IEEE Trans. Antennas Propag, vol. 59, no. 3, pp. 950–959, Mar. 2011. [11] K. Umashankar, A. T. Ove, and S. M. Rao, “Electromagnetic scattering by arbitrarily shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag, vol. 34, pp. 758–766, 1986. [12] J. Q. He, T. J. Yu, N. Geng, and L. Carin, “Method of moments analysis of electromagnetic scattering from a general three-dimensional dielectric target embedded in a multilayered medium,” Radio Sci., vol. 35, pp. 305–313, Mar.-Apr. 2000. [13] J. Q. He, T. J. Yu, N. Geng, and L. Carin, “Multilevel fast multipole algorithm for three-dimensional dielectric targets in the vicinity of a lossy half space,” Microw. Opt, Tech Lett., vol. 29, no. 2, pp. 100–104, Apr. 2001. [14] P. Yla-Oijala, M. Taskinen, and S. Jarvenpaa, “Analysis of surface integral equations in electromagnetic scattering and radiation problems,” Eng. Analy. Boundary Elements, vol. 32, pp. 196–209, 2008.

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[15] P. Yla-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag, vol. 53, no. 3, pp. 1168–1173, Mar. 2005. [16] D. Z. Ding, R. S. Chen, Z. H. Fan, and P. L. Rui, “A novel hierarchical two-level spectral preconditioning technique for electromagnetic wave scattering,” IEEE Trans. Antennas Propag, vol. 56, no. 4, pp. 1122–1132, Apr. 2008. [17] R. D. Graglia, D. R. Wilton, and A. F. Peterson, “Higher order interpolatory vector bases for computational electromagnetics,” IEEE Trans. Antennas Propag, vol. 45, no. 3, pp. 329–342, Mar. 1997. [18] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag, vol. 30, no. 5, pp. 409–418, May 1982. [19] K. Gang, J. M. Song, C. C. Weng, K. C. Donepudi, and J. M. Jin, “A novel grid-robust higher order vector basis function for the method of moments,” IEEE Trans. Antennas Propag, vol. 49, pp. 908–915, Jun. 2001. [20] Z. P. Nie, W. Ma, Y. Ren, Y. Zhao, J. Hu, and Z. Zhao, “A wideband electromagnetic scattering analysis using MLFMA with higher order hierarchical vector basis functions,” IEEE Trans. Antennas Propag, vol. 57, no. 10, pp. 3169–3178, Oct. 2009. [21] S. Jarvenpaa, M. Taskinen, and P. Yla-Oijala, “Singularity subtraction technique for high-order polynomial vector basis functions on planar triangles,” IEEE Trans. Antennas Propag, vol. 54, no. 1, pp. 42–49, Jan. 2006. [22] P. Yla-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix functions,” IEEE Trans. Antennas elements with RWG and Propag., vol. 51, no. 8, pp. 1837–1846, Aug. 2003. [23] R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green’s function or its gradient on a plane triangle,” IEEE Trans. Antennas Propag, vol. 41, no. 10, pp. 1448–1455, Oct. 1993. Li Ping Zha was born in Anhui Province, China, in 1987. She received the B.S. degree in electronic information engineering from Anhui University of Architecture, China, in 2008, and is currently working toward the Ph.D. degree at Nanjing University of Science and Technology (NJUST), Nanjing, China. Her current research interests include computational electromagnetics, electromagnetic modeling of scattering problems, wave scattering and propagation from random media, and numerical techniques for electrically large objects.

Yun Qin Hu was born in Jiangsu Province, China, in 1982. She received the B.S. degree in electronic information engineering from Jiangsu University, China, in 2005, and is currently working toward the Ph.D. degree at Nanjing University of Science and Technology (NJUST), Nanjing, China. Her current research interests include various aspects of computational electromagnetics with focus on precondition- ing and fast solution of frequency domain integral equation, electromagnetic modeling of microwave integrated circuits and microstrip antennas.

Ting Su was born in Anhui Province, China, in 1985. She received the B.S. degree in communication engineering from Nanjing University of Science and Technology (NJUST), China, in 2006, where she is currently working toward the Ph.D. degree. Her current research interests include computational electromagnetics, antennas and electromagnetic scattering and propagation, electromagnetic modeling of microwave integrated circuits.

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Accelerated FDTD Analysis of Antennas Loaded by Electric Circuits Yuta Watanabe, Member, IEEE, and Hajime Igarashi, Member, IEEE

Abstract—A fast FDTD method for the analysis of antennas loaded by nonlinear electric circuits is introduced. In the present analysis, the modified nodal analysis (MNA) method is coupled with the FDTD method. The time-periodic explicit error correction (TP-EEC) method is applied to the MNA method for accelerated computation of the transient processes. The present method is applied to analysis of simplified models of an RFID tag composed of a nonlinear electric circuit and line antenna. It is shown that the present method can effectively shorten the computational time by accelerating the transient processes. Index Terms—Electromagnetic waves, FDTD method, modified nodal analysis, RFID tag, TP-EEC method.

I. INTRODUCTION

F

ELD COMPUTATION METHODS, such as the finite -difference time-domain (FDTD) method [1], [2] and the method of moment [3], have widely been used for analysis of high frequency electronic devices. In recent years, these methods have been applied to coupling analysis of high-frequency electromagnetic fields and electric circuits for the design of high-frequency electronic devices and analysis of electromagnetic compatibility (EMC) problems [4]. In the coupling analysis, circuit simulation involving nonlinearity requires time domain computations. For this reason, the coupling analysis of electromagnetic fields and nonlinear electric circuits usually requires high computational cost. When the time constant of the circuit is much longer than the time period of electromagnetic waves, this problem becomes quite severe because the number of time steps must be considerably large. It would be possible to reduce the computational cost if one could effectively shorten the time constant, that is, accelerate the computation of the transient processes of the circuit. The time-periodic explicit error correction (TP-EEC) method [5], [6], which accelerates the transient processes of time-periodic systems, has been introduced for reduction of computational costs. The TP-EEC method is based on the assumption that the unknown variables are temporally periodic in the steady state and slowly converging components without periodicity can be separated from them. The slowly converging components are then determined by solving small-scale correction Manuscript received April 07, 2011; revised June 30, 2011; accepted August 03, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. The authors are with the Graduate School of Information Science and Technology, Hokkaido University, Sapporo, 060-0814, Japan (e-mail: ywata@em-si. eng.hokudai.ac.jp; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173148

equations to correct the transient solutions. The TP-EEC method is an extension of the EEC method [7], which gives a general framework of the error correction based on the decomposition of unknowns to fast and slowly converging components. It has been shown that the EEC has a common theoretical basis with the deflation methods [8], which have been applied to linear systems of poor convergence [9], [10]. The TP-EEC method has been applied to finite element (FE) analysis of motors and coupling FE analysis of circuit and eddy current fields [6], in the latter of which inductance is computed in the FE analysis by taking magnetic saturation into account. In this paper, we will discuss the effectiveness of the TP-EEC method when applied to antenna analysis, where coupling between electromagnetic waves and a nonlinear circuit is considered. In particular, we consider here transient analysis of dipole antennas loaded by a nonlinear circuit, which are simplified models of the UHF-band RFID tag. In the design optimization of antennas for RFID tags, the coupled problem between the electromagnetic waves and the circuit must be repeatedly solved [11]. Hence the reduction in the computational cost for the coupling analysis is of fundamental importance. Moreover, in this paper, we will introduce a theoretical basis of the TP-EEC method for explanation of the reason why it is effective for acceleration of the transient processes. In this work, the FDTD method and modified nodal analysis (MNA) [12], [13] are employed for the coupling analysis of a high frequency electromagnetic field and a nonlinear circuit. This paper will be organized as follows: in Section II, the coupled method with the FDTD method and MNA will be formulated. Moreover a computational procedure of the present method will be described. In Section III, the TP-EEC method will be formulated, and effect of the TP-EEC method will be discussed, while in Section IV, numerical results will be shown to verify the present method.

II. HYBRIDIZATION OF FDTD METHOD AND MNA The hybridization of the FDTD method and MNA will be described in the following. The Maxwell equations (1a) (1b) are considered in this paper where the conduction current density is determined from the voltage-current characteristics of the nonlinear circuit. In the FDTD process, (1a) and (1b) are

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Fig. 2. Equivalent circuit for FDTD method of antenna loaded by nonlinear circuit.

Fig. 1. Line antenna loaded by nonlinear circuit.

approximated by the central differences in time and space and explicitly solved in turns as follows [2]:

(2a) (2b) Let us consider the line antenna loaded by a nonlinear circuit, shown in Fig. 1, parallel to the z-axis, where , and are the cell size of the FDTD method. The spatial size of the nonlinear circuit is assumed to be sufficiently smaller than that of the FDTD cell. By integrating (1a) on the surface of the FDTD cell, we obtain (3) where

is the voltage imposed to the circuit, is the capacitance of the FDTD cell, is the current flowing into the nonlinear circuit and the is total current given by (4) The equivalent circuit governed by (3) is shown in Fig. 2 [12], [13]. This circuit is composed of a parallel circuit of the current source computed from (4), the capacitance and the nonlinear circuit. The equivalent circuit shown in Fig. 2 is analyzed by MNA in this work. Modified nodal analysis determines the voltages between nodes in the electrical circuit according to Kirchhoff’s Current Law. A system of the nonlinear circuit equations of the form (5) is solved, where is composed of the unknown nodal voltages, current source driven by the antenna, nonlinear function which includes the effects of the active devices such as diodes and transistors, and C is the capacitance matrix. Note here that (5) includes (3).

Fig. 3. Flow diagram.

Fig. 3 shows the flow diagram of the coupling analysis of the FDTD method and MNA. In the FDTD process, the magnetic field is computed, where denotes the time step, and is computed from by (4). Then, is substituted to the right-hand side of (5), which is solved by MNA for including . The resultant electric field is substituted to (2-b), which is solved by the FDTD method for . These computations of , , and are repeated until a steady solution is obtained.

III. ACCELERATION OF CONVERGENCE IN CIRCUIT ANALYSIS This section describes the TP-EEC method which accelerates the transient processes of the circuit.

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A. Discretized Circuit Equation Discretization of (5) with the finite difference leads to a system of nonlinear equations of the form

where and represent the correction vector and matrix composed of the slowly converging components whose explicit forms will be described below. The constant represents the degree of the correction. The approximated solution given by (12) is substituted to (11) to obtain the residual vector given by

(6) (13)

where

(7a)

We expand (13) around for linearization, and then employ the Galerkin approximation that in (13) is orthogonal to the column vectors in W to obtain the correction equation given by

(7b) (7c) and . When equals to 1, for example, this scheme corresponds to backward Euler method. In this work, (6) is solved with the Newton-Raphson method at each time step.

B. TP-EEC Method When the time constant of the system governed by (6) is much longer than , high computational cost is required to obtain the steady state solutions which do not vary in time. To accelerate the convergence to the steady state, the TP-EEC method is applied to (6). It is assumed that is periodic, which satisfies . The solution to (6) is then expected to have periodicity in the steady state. On the other hand, in the transient state, periodicity would approximately hold, that is

(14) and

where

.. . .. . .. .

..

(15)

(16a) (16b) The correction matrix of the 0th order , in which the slowly converging components are assumed to be temporally constant, is given by (17)

(8) To apply the TP-EEC method to (6), we introduce the vectors defined for each period as

.

where is the unit matrix. By substituting (17) to the left-hand side of (14), one has (18)

.. .

(9)

.. .

(10)

Equation (6) for one period is now expressed in the form

Then it is assumed that (11) can be solved with sufficiently high accuracy except at where imperfect periodicity would result in nonnegligible residuals. Under this assumption, the residual can be expressed in the form

.. .

(19)

Substitution of (18) and (19) to (14) yields (11)

To accelerate convergence to the steady state, is decomposed into fast and slowly converging components as follows: (12)

(20) By solving (20) for , is corrected from (12). Convergence to the steady state can further be accelerated by use of the correction of the 1st order in which

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the slowly converging components are assumed to be temporally constant and linear. The corresponding correction matrix is given by

(21a) where (21b) By substituting (21) to (14), one obtains

Fig. 4. Analysis model.

(22a) (22b)

which is a projection matrix satisfying seen from (26) that . The error in the form

. It can easily be can be decomposed

(27)

(22c) where

(28)

(22d)

(22e) The unknown is then corrected with the vectors determined by solving (22) as follows:

and

(23) Although increase in the degree of the correction, e. g. from 0th to 1st, is expected to give better convergence, it also results in increase in the unknowns in the correction equation.

Therein, the errors and represent the slowly and fast converging errors, and they satisfy A-orthogonal relation . It can be seen from (25) to (28) that the slowly converging error is eliminated as by the TP-EEC method. On the other hand, has no effects from the correction because of the property . Thus this error component is reduced by the iterative solution of (11) where its convergence is expected to be fast by definition. It is known that the multigrid method, which effectively eliminates the slowly converging components with spatially smooth profiles by mapping them to coarser meshes, is also based on the above mentioned decomposition and selective elimination [14]. More detailed discussions on the TP-EEC method are can be found in [15].

C. Effect of TP-EEC Method The mathematical property of the TP-EEC method has been discussed for the scalar linear diffusion equation [5]. We give here more general discussion on this method. By substituting and (14) to (12), we obtain (24) The steady solution to (11) is here expressed by . Then, the error is modified after the error correction in the form (25) where

and P is defined by (26)

IV. NUMERICAL RESULT A. CR Diode Series Circuit The half-wave dipole antenna loaded by the nonlinear circuit shown in Fig. 4 is analyzed by hybridization of the FDTD method and MNA to test acceleration of convergence to the steady state by applying the TP-EEC method to MNA. The nodal voltages of the nonlinear circuit are obtained from MNA. The half-wave dipole antenna is assumed to be illuminated by the plane wave. The amplitude and frequency of incident wave is assumed to be 20 V/m and 1 GHz. The size of the FDTD cell, , is set to 3mm. The perfect matched layer is employed to enforce the free space conditions on the domain boundary. The half-wave dipole antenna is parallel to -axis and set to perfect conductor .

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Fig. 5. CR diode serial circuit.

Fig. 7. Time evolution error of node voltage.

Fig. 6. Time evolution of

Fig. 8. Cockcroft-Walton circuit.

.

The nonlinear circuit is composed of a capacitance, diode and resistance as shown in Fig. 5. The capacitance is set to 10pF and the resistance is set to . The diode circuit is assumed to obey V-I characteristics given by

.

(29)

where is the voltage at diode. For negative voltages, the diode current is nearly zero. The explicit form of the circuit (5) for the circuit shown in Fig. 5 is given in the Appendix. In this analysis, is set to 1 in MNA and is set to 200. Fig. 6 shows the time evolution in the voltage . It can be seen in Fig. 6 that the convergence to the steady state is clearly accelerated by the present method. Fig. 7 shows the absolute error between the steady and transient solutions. The numbers of time steps required to satisfy for no correction, for 0th correction, and for 1st order correction, are approximately 205000 steps (1025 ns), 2400 steps (12 ns) and 1600 steps (8 ns), respectively. This means that the TP-EEC method of the 1st order correction provides convergence to the steady state 128.1 times faster than that for noncorrected computation. Moreover, the three solutions in the steady state are found to be in good agreement.

Fig. 9. Time evolution of output voltage

.

B. Cockcroft-Walton Circuit The TP-EEC method is now applied to the half-wave dipole antenna loaded by the CW circuit shown in Fig. 8. The explicit form of the circuit (5) for the circuit shown in Fig. 8 is given in the Appendix. The parameters of the FDTD method and MNA are the same as those used in Section IV-A. Fig. 9 shows the time evolution of the output voltage of the CW circuit. It can be seen in Fig. 9 that the convergence to the steady state is clearly accelerated by the TP-EEC method.

Fig. 10. Time evolution error of node voltage.

Fig. 10 shows the absolute error between the steady and transient solutions. The number of time steps required to satisfy for no correction, for 0th order correction, and for 1st order correction, are approximately 8880 steps (44 ns), 3545

WATANABE AND IGARASHI: ACCELERATED FDTD ANALYSIS OF ANTENNAS LOADED BY ELECTRIC CIRCUITS

steps (18 ns) and 1800 steps (9 ns), respectively. That means that the TP-EEC method of the 1st order correction provides convergence to the steady state 4.9 times faster than that for noncorrected computation. Moreover, it is observed that the three solutions in the steady state are in good agreement. It can be seen in Figs. 7 and 10 that the effects in acceleration by TP-EEC method depend on circuits. V. CONCLUSION In this paper, it has been shown that convergence to the steady state of a nonlinear circuit driven by antenna voltage, which was analyzed by FDTD and MNA, is effectively accelerated by using the present method. The theoretical reason why the present method can improve convergence to the steady state has been discussed. To test the present method, it has been applied to analysis of a half-wave dipole antenna loaded by nonlinear circuits including diodes. It has been numerically shown that the TP-EEC method effectively accelerates convergence to the steady state. APPENDIX This appendix describes the nodal equations for the circuit discussed in this paper. The nodal equations for the CR diode series circuit shown in Fig. 5 are given by (A1) (A2) (A3) where is the input current obtained by the FDTD computation and is the capacitance for the FDTD cell. The nodal equations for the CW circuit shown in Fig. 8 are given by (A4) (A5)

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[3] R. F. Harrington, Field computation by moment methods. Hoboken, NJ: Wiley, 1993. [4] C. R. Paul, Introduction to Electromagnetic Compatibility. Hoboken, NJ: Wiley, 2006. [5] Y. Takahashi, T. Tokumasu, M. Fujita, S. Wakao, T. Iwashita, and M. Kanazawa, “Improvement of convergence characteristic in nonlinear transient eddy-current analyses using the error correction of time integration based on the time-periodic FEM and the EEC method,” IEEJ Trans. PE, vol. 129, no. 6, pp. 791–798, 2009. [6] Y. Takahashi, T. Tokumasu, A. Kameari, H. Kaimori, M. Fujita, T. Iwashita, and S. Wakao, “Convergence acceleration of time-periodic electromagnetic field analysis by the singularity decomposition-explicit error correction method,” IEEE Trans. Magn., vol. 46, no. 8, pp. 2947–2950, Aug. 2010. [7] T. Iwashita, T. Mifune, and M. Shimasaki, “Similarities between implicit correction multigrid method and A-phi formulation in electromagnetic field analysis,” IEEE Trans. Magn., vol. 44, no. 6, pp. 946–949, Jun. 2008. [8] H. Igarashi and K. Watanabe, “Deflation techniques for computational electromagnetism: Theoretical considerations,” IEEE Trans. Magn., vol. 47, no. 5, pp. 1438–1441, May 2011. [9] C. Vuik, A. Segal, and J. A. Meijerink, “An efficient preconditioned CG method for the solution of a class of layered problems with extreme contrasts in the coefficients,” J. Comp. Phys., vol. 152, pp. 385–403, 1999. [10] Y. Saad, M. Yeung, J. Erhel, and F. Guyomarch, “A deflated version of the conjugate gradient algorithm,” SIAM J. Sci. Comput., vol. 21, no. 5, pp. 1909–1926, 2000. [11] Y. Watanabe, K. Watanabe, and H. Igarashi, “Optimization of meander line antenna considering coupling between non-linear circuit and electromagnetic waves for UHF-band RFID,” IEEE Trans. Magn., vol. 47, no. 5, pp. 1506–1509, May 2011. [12] V. A. Thomas, M. E. Jones, M. Piket-May, A. Taflove, and E. Harrigan, “The use of SPICE lumped circuits as sub-grid models for FDTD analysis,” IEEE Micr. Guid. Wave Lett., vol. 4, no. 5, pp. 141–143, 1994. [13] M. J. Piket-May, A. Taflove, and J. Baron, “FDTD modeling of digital signal propagation in 3-D circuits with passive and active loads,” IEEE Trans. Microw. Theory Tech., vol. 42, pp. 1514–1523, Aug. 1994. [14] D. Braess and W. Hackbusch, “A new convergence proof for the multigrid method including the V-cycle,” SIAM J. Numer. Anal., vol. 20, no. 5, pp. 967–975, 1983. [15] H. Igarashi, Y. Watanabe, Y. Itoh, and K. Watanabe, “Why error correction methods realize fast computations,” IEEE Trans. Magn., vol. 48, no. 2, Feb. 2012. Yuta Watanabe (M’10) received the B.S. and M.I. degrees in engineering from Akita National College of Technology and Hokkaido University, Sapporo, Japan, in 2008 and 2010, respectively, and is currently pursuing the Ph.D. degree at the same institution. His primary research interest is in the area of computational electromagnetics and design optimization.

(A6) (A7) (A8) REFERENCES [1] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. AP-14, pp. 302–307, 1966. [2] A. Taflove, Computational Electrodynamics, The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, 1998.

Hajime Igarashi (M’95) received the B.E. and M.E. degrees in electrical engineering from Hokkaido University, Sapporo, Japan, in 1982 and 1984, respectively, and the Ph.D. degree in engineering from Hokkaido University in 1992. He has been a professor at the Graduate School of Information Science and Technology, Hokkaido University, since 2004. He has worked as a research engineer at Canon Co. Ltd., [during] from 1984 to 1989. From 1989 to 1999, he was a research associate with the Faculty of Engineering, Hokkaido University. He was a guest researcher at Berlin Technical University, Germany, under support from the Humboldt Foundation from 1995–1997. He was an associate professor from 1999 to 2004 at Kagawa University, Japan, and Hokkaido University. His research area is computational electromagnetism, design optimization and RFID technologies. He has authored and coauthored more than 90 peer-reviewed journal papers.

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An Angle-Dependent Impedance Boundary Condition for the Split-Step Parabolic Equation Method Chad R. Sprouse, Member, IEEE, and Ra’id S. Awadallah, Member, IEEE

Abstract—A formulation of the impedance boundary condition (IBC) is derived, which can be used in split-step parabolic wave equation solvers to accurately represent an interface with arbitrary dielectric properties. The approach relies on ensuring that the plane-wave decomposition of the field satisfies the appropriate IBC for each spectral component. Numerical experiments illustrating the robustness of the approach for low-contrast interfaces and angles-of-incidence near the Brewster angle are presented. Index Terms—Electromagnetic propagation, Maxwell equations, microwave propagation.

I. INTRODUCTION

P

ARABOLIC equation solvers [1], [2] provide a powerful modeling capability for propagation of electromagnetic waves over long distances in complex environments. They are used extensively for predicting radar coverage in ducting enviroments over rough surfaces. Two primary classes of solvers are used in conjuction with the parabolic wave equation. The first is a finite-difference approach that relies on fine discretization of the spatial domain for accurate representation of the field propagation. The second is the split-step approach [3], [4] that implements propagation in the Fourier domain and utilizes a series of “phase-screens” to account for refraction effects. One of the primary advantages of the split-step method with respect to the finite-difference approach is its insensitivity to discretization size, i.e., while finite-difference methods may require discretization in both altitude and range on the order of a tenth of a wavelength or finer, the split-step method needs only half-wavelength discretization in altitude while range steps can be large compared to the wavelength. In practice, the discretization for the split-step method is driven by the variation in the environment rather than a requirement to resolve the fast variation of the fields. A consequence of the use of Fourier techniques, however, is that the split-step method requires that the dielectric properties of the domain be slowly varying in both altitude and range. By contrast, the finite-difference method is able to handle sharp interfaces between regions of different dielectric properties natManuscript received July 29, 2010; revised June 27, 2011; accepted August 22, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. The authors are with the Milton S. Eisenhower Research Center, The Johns Hopkins University Applied Physics Laboratory, Laurel, MD 20723 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173107

urally by solving for the fields both above and below the interface and enforcing continuity of the tangential fields at the interface. In the split-step method, propagation over such an interface must be treated as propagation in a half-space along with an appropriate boundary condition at the interface. In this paper, we present a new method for treating the interface using an impedance boundary condition (IBC). The investigation leading to this result was motivated by the problem of propagation over lunar terrain that has near-zero conductivity and low permittivity . Previous implementations of the IBC did not handle such low-constrast interfaces accurately, and so a new formulation was required. The resulting “consistent-field” IBC formulation is presented in Section II with numerical experiments for the lunar terrain in Section III. II. FORMULATION A. Split-Step Method In this paper, we consider a cylindrical coordinate system with azimuthal symmetry in both the fields and environment. The Helmholtz equation for this geometry is (1) where is the auxiliary field which is equal to for horizontal polarization (H-pol) and for vertical polarization (V-pol). Here and throughout this paper, we assume and omit a harmonic time variation of for all fields. In the far field, the term proportional to may be neglected. Factorizing the resulting far-field wave equation into forward and backward components gives (2) where (3) is negligible and conAssuming that the commutator sidering only outgoing waves, we have the parabolic equation (4) This equation has the formal solution (5) The well-known (wide-angle) split-step method computes the field taking into account the refractivity by making the

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approximation (see [4], for example; this approximation was introduced and discussed in [5]) (6) with approximated as

from (3). With this approximation, (5) is (7)

where the symmetric splitting of the “refractive” portion of the propagator is used and is allowed to be a slowly varying function of both range and altitude. The “diffractive” portion of the propagator, , is handled in the Fourier -space conjugate to the -coordinate. In -space, the diffractive propagator is thus , where . If, in addition to the slow variation assumption required for the above derivation, we have , then this propagator is accurate for angles from to . In particular, it is exact for free-space propagation. The split-step procedure for advancing the field thus becomes

(8)

Fig. 1. Plane wave incident on the interface between two homogeneous halfin this figure. spaces. Note that with the sign convention used here,

Elimination of the subsurface fields from the boundary condition requires that (12) Considering fields of the form (10), the incident and reflected fields are given by (13) (14) respectively in the upper domain (see Fig. 1), while the field in the lower domain is given by

represents the Fourier transform with respect to the where -coordinate.

(15)

B. Impedance Boundary Condition Propagation above a discontinuous dielectric interface using the split-step method must be incorporated in the diffractive propagator due to the slow-variation requirement for . This interface is incorporated by considering only fields in the half-space bounded below by the interface and using an impedance boundary condition to include the effects of the presence of the interface. While the full boundary condition at the interface requires continuity of both tangential electric field and magnetic field , at the boundary, the half-space problem admits only a single boundary condition. Thus, a single impedance condition on the propagated field given by (9) is used. This condition is valid for the case where the field is a plane wave of the form (10) with (we take for waves having a component that propagates in the negative -direction). For V-pol, the propagated field is and from Ampére’s Law, we have . Denoting the fields below the surface with primes, a linear combination of the boundary conditions on the tangential and fields is (11)

where given by

with

. Hence,

is

(16) Substituting (16) into (11) gives

(17) The H-pol case is similar, providing the proper definition of the coefficient in (9) as V-pol H-pol

(18)

where the branch cut of the square-root is taken to be along the negative real axis such that . The boundary condition for the field in the upper domain then gives us the condition (19) for the field amplitudes. The traditional IBC formulation [6] assumes a unit incident field amplitude, , such that , where is the reflection coefficient (20)

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It should be noted that is associated with the direction of propagation of the incident field, which explains the choice of sign in the definition of . With this formulation, the field in terms of the field is given by

where

and

, while for

(31)

(21) where (22) Here and in the remainder of this paper, we neglect any surface wave contributions and focus solely on the spatial waves. For the case of lossless dielectrics (real ) considered in Section III, there is no surface wave component, and the results presented are exact. Equation (22) can be written in terms of the regular Fourier transforms of the even and odd extensions of as (23) where (24) Equation (21) then becomes (25) . This is the where “mixed-transform” approach. The singularity at in the integrand may be handled by taking the Cauchy Principal Value (CPV) of the integral. As will be shown in Section III, this formulation exhibits artifacts associated with this singularity. Alternative handling of the singular integrand does not resolve the issue. For example, adding and subtracting the singular part of the integrand to construct a regular integrand that can be numerically integrated and analytically integrating the singular part gives (26) with

(27) where

is the residue at

, and (28)

, we set

This method exhibits numerical artifacts similar to those encountered with the simpler CPV approach. Previous efforts [7], [8] have sought to mitigate these artifacts through the use of a variety of numerical techniques, but rely on an angle-independent impedance assumption, , such as [9]. One such approach defines an auxiliary field (32) and solves for using sine transforms (the boundary condition for is simply ). The field is then computed as the solution to the forced differential equation (32). If has any components proportional to the homogeneous solution to this equation, , the solution for will diverge due to resonance. The alternative central/forward/backward difference approach of [8] effectively shifts the homogeneous solution away from any component of resulting in a well-behaved solution. However, this approach has the undesirable side-effect of changing the reflection amplitude for incidence angles away from grazing. An example of this effect is shown in Section III. In contrast, this paperrevisits (19) to derive a formulation that avoids the singular behavior of (25). C. Consistent-Field IBC Formulation For a general field, (19) must be satisfied for the plane-wave decomposition of the field , i.e., (33) This formulation focuses on ensuring that the propagated field is consistent with the presence of the interface. In particular, and , where is the solution to . For H-pol, there is no such solution, while for V-pol, there is a solution, , which corresponds to boundary reflections of waves incident on the surface at the Brewster angle. As the boundary condition is entirely accounted for through modification of the diffractive part of (8), without loss of generality, we consider the case with , such that (34)

Here, we have used the relation

For

(29)

must satisfy (33). By decomposing into the where sum of functions and , which are respectively even and odd in , and substituting into (33), it is shown that

(30)

(35)

, the residue is given by

SPROUSE AND AWADALLAH: ANGLE-DEPENDENT IBC FOR SPLIT-STEP PARABOLIC EQUATION METHOD

where . Here, we have used the fact that the Fourier transform preserves even/odd symmetry, i.e., (36) Thus, the problem may be solved completely in terms of an odd function of . The intial field is specified for , from which we must determine such that

(37) To this end, define

so that with known and Taking the Fourier transform, we have

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erywhere above the interface. Thus, any field component that is upwardly propagating must have been reflected from the surface at some point. For , since , this implies that this component must be zero. Realistically, however, the source is the aperture field of an antenna located at a finite range. The antenna may produce components in the direction , and these must be accurately captured if this method is to be useful in practice. We note that since and hence components for which will not interact with the surface during propagation. Thus, it is a good approximation to propagate these components in free space. To accomplish this, the incident field can be partitioned into two portions, , propagated in the presence of the surface and in free space, respectively. One such splitting is given by

(38)

(44)

(39)

(45)

unknown.

where is a unitless parameter that determines the splitting, with larger including more components in the free-space field. The field is propagated in the presence of the surface by propagating the antisymmetric field given by (35), while the field is propagated in free space using (8). The total field at the next range step is then given by

(40) To solve for , we will consider the discrete Fourier transform such that (40) becomes

(46) A detailed outline of this algorithm is included in the Appendix. (41) III. NUMERICAL EXPERIMENTS where for version of (33) gives

. Substitution into the discrete

(42) which may be solved in the least-squares sense as an overdetermined linear system. This procedure is computationally intensive, but would only need to be performed for the initial field (e.g., at ). In practice, for initial fields that are nonzero only away from the interface, it is a good approximation to simply set for . It is this latter approach that is used in the results presented in Section III. In theory, the field could simply be computed from (35). For this to be valid, however, it must be true that (43) is the solution to . For a genwhere, as above, eral antenna pattern, this condition will not be satisfied. This issue arises from the fact that (33) was derived for a plane-wave decomposition of the fields in which the components exist ev-

Current methods perform well for high-contrast interfaces and/or shallow grazing angles, so the experiments presented here will focus on scenarios in which these methods break down. As the singularity in (25) occurs at the negative of the Brewster angle, a worst-case scenario is expected for excitations near this angle. Therefore, a low-contrast interface at separating regions of free space (above) and (below) is used. The simulation is initiated with a Gaussian source field [2]

(47) with m ( GHz), m, , and . The value of was chosen such that the beam is incident on the interface near the Brewster angle . Fig. 2 shows the power distribution of the field for this scenario as computed by the mixed-transform method. Severe numerical artifacts dominate the result with variations in the implementation details determining the particular form of the artifacts. Treating the singularity as in (26) yields a similar result. Mitigation techniques such as those described in [8] remove the singular behavior by effectively modifying the surface properties near the Brewster angle.

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Fig. 2. Gaussian beam with 1 half-beamwidth and elevation angle inbetween regions of free space and cident on the interface at computed using the mixed-transform approach. Numerical artifacts dominate the result.

Fig. 4. Gaussian beam with 1 half-beamwidth and elevation angle inbetween regions of free space and cident on the interface at computed using the consistent-field formulation. Vertical sam. There is a clear null in the reflected power exactly at the Brewster pling is as expected. angle

Fig. 5. Histogram of the difference between the results shown in Fig. 4 and the plotted as a function finite-difference result using a spatial resolution of of power.

Fig. 3. Gaussian beam with 1 half-beamwidth and elevation angle inbetween regions of free space and cident on the interface at computed using finite-differences in both upper and lower . The null in half-spaces. Spatial sampling in both height and range is the reflected field at the Brewster angle is clearly indicated.

The finite-difference solution to the above scenario is shown in Fig. 3. This method solves for the fields in both upper and lower half-spaces and treats the boundary condition at the interface by enforcing continuity of the tangential fields. Hence, there is no impedance boundary condition assumption so that this solution serves as a useful “ground-truth” for comparison with the split-step solutions. Unlike the split-step method, however, the finite-difference method is sensitive to the resolution used. A very fine sampling of in both range and altitude was used to ensure an accurate baseline for comparisons.

Fig. 4 shows the result computed using the consistent-field IBC formulation. The numerical artifacts seen in Fig. 2 are absent, and the expected null in the reflected field along the Brewster angle is accurately predicted. Fig. 5 shows a bivariate histogram of the difference between the consistent-field split-step result and the finite-difference result as a function of power over the domain shown in Fig. 4. This plot shows the degree of discrepancy between the two results as well as the power levels at which the differences occur. Large differences at low power levels are less significant than at higher power levels. Agreement between the two results is extremely good with generally less than 1 dB of difference and less than 0.5 dB for power levels greater than 70 dB. To test the convergence properties of both the split-step and finite-difference methods, the spatial sampling was increased and the simulations rerun. Increasing the altitude sampling of the split-step method from to did not affect the

SPROUSE AND AWADALLAH: ANGLE-DEPENDENT IBC FOR SPLIT-STEP PARABOLIC EQUATION METHOD

Fig. 6. Histogram of the difference between the results shown in Figs. 4 and 3 as a function of power.

result. Sampling in range was not varied as the split-step method is exact with respect to range for a free-space upper medium as it is simply a process of serial application of Huygens’ Principle. Doubling the sampling (in both range and altitude) of the finitedifference method from to reduced the error as shown in Fig. 6. Thus, these plots represent a measure of the accuracy of the finite-difference method with the split-step result representing “ground-truth.” An example from the lunar propagation scenario which motivated this investigation is shown in Fig. 7. It shows a comparison of the consistent-field approach to the method of [8] for a wide-angle antenna pattern. The antenna pattern used is the tapered cardioid defined by (48) where

is the elevation angle and

is the taper function otherwise.

(49)

This was one of two proposed antenna patterns for the communications system on a lunar lander intended for the International Lunar Network mission currently under study by NASA. The simulation was again performed at a frequency of 2.25 GHz with the antenna located 1.5 m above the interface . The consistent-field result uses (46) with and provides much better agreement with the finite-difference result than the method of [8], which was developed for high-conductivity surfaces and does not correctly capture the Brewster-angle effect (manifested here as a damping of the interference pattern due to the vanishing of the reflected field at the Brewster angle). IV. CONCLUSION A formulation of the impedance boundary condition has been developed, which eliminates divergences seen in previous approaches while preserving the correct behavior of reflections from the surface. The technique is based on an application of

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Fig. 7. Tapered cardioid antenna pattern propagated in free space (dashed between regions of free space curve) and over an interface at and computed using the finite-difference method (solid black curve), an implementation of the method of [8] (fine dashed curve, blue online), and (46) (dotted curve, red online).

the IBC for a single plane wave to the spectral decomposition of a general field. Experiments show excellent agreement with full-space solutions. It should be noted that the approach described here is an alternate method for representing the impedance boundary condition within the split-step formalism. Although the results presented in this paper are for the case of free space above the surface, atmospheric refractivity can be incorporated via (8) in the usual way. Future work will investigate the possibility of including surface-wave effects and combining this approach with techniques for incorporating surface roughness such as the linear shift map [10].

APPENDIX CONSISTENT-FIELD SPLIT-STEP ALGORITHM This appendix provides an outline of the split-step algorithm using the consistent-field impedance boundary condition incorporating the field-splitting (46). Let the domain encompass ranges from to and altitudes from to with , where the surface is located at . We discretize this domain in range and altitude with steps and . We image the domain in altitude so that we have for . The algorithm to propagate the auxilliary field to the next range step consists of the following steps. 1) Multiply the field above the surface by the first half of the refractivity correction for the range step (50) for . 2) Apply a -space windowing function to simulate an unbounded upper half-space by forcing the field to zero at the boundary (51)

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for . Many choices of filter will serve this purpose; we have used a tapered cosine window

10) Transform the field back to -space

(64)

(52) 11) Apply the second half of the refractivity correction

3) Transform the field to -space

(65) (53)

for

. REFERENCES

. for 4) Split the field into the two portions needed to propagate to the next range step and compute the -field (54) (55) 5) Transform both fields back to -space (56) (57) 6) Enforce antisymmetry on the -field to simulate the effect of the surface and taper the -field to zero below the surface (58) (59) for

where (60)

for the results for some positive (we used presented above). 7) Transform the fields to -space and propagate to the next range step (61) (62) 8) Recombine the fields at the new range (63) 9) Apply a -space window analogous to Step 2.

[1] M. F. Levy, Parabolic Equation Methods for Electromagnetic Wave Propagation. London, U.K.: IEE, 2000. [2] F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, Computational Ocean Acoustics. New York: AIP Press, 1994. [3] R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev., vol. 15, p. 423, 1973. [4] D. J. Thomson and N. R. Chapman, “A wide-angle split-step algorithm for the parabolic equation,” J. Acoust. Soc. Amer., vol. 74, no. 6, pp. 1848–1854, Dec. 1983. [5] M. D. Feit and J. A. Fleck, Jr., “Light propagation in graded-index fibers,” Appl. Opt., vol. 17, no. 24, pp. 3990–3998, Dec. 1978. [6] R. Janaswamy, “Radio wave propagation over a nonconstant immittance plane,” Radio Sci., vol. 36, no. 3, pp. 387–405, May–Jun. 2001. [7] G. D. Dockery and J. R. Kuttler, “An improved impedance-boundary algorithm for Fourier split-step solutions of the parabolic wave equation,” IEEE Trans. Antennas Propag., vol. 44, no. 12, pp. 1592–1599, Dec. 1996. [8] J. R. Kuttler and R. Janaswamy, “Improved Fourier transform methods for solving the parabolic wave equation,” Radio Sci., vol. 37, no. 2, p. 1021, Mar. 2002. [9] J. R. Kuttler and G. D. Dockery, “Theoretical description of the parabolic approximation/Fourier split-step method of representing electromagnetic propagation in the troposphere,” Radio Sci., vol. 26, no. 2, pp. 381–393, Mar.–Apr. 1991. [10] D. J. Donohue and J. R. Kuttler, “Propagation modeling over terrain using the parabolic wave equation,” IEEE Trans. Antennas Propag., vol. 48, no. 2, pp. 260–277, Feb. 2000. Chad R. Sprouse (M’09) received the B. S. degrees in mathematics and physics from Washington State University, Pullman, in 1998, the M. S. degree in computer science from The Johns Hopkins University, Baltimore, MD, in 2002, and is currently pursuing the Ph.D. degree in electrical engineering at the Virginia Polytechnic Institute and State University, Blacksburg. He is currently a Senior Staff Member with The Johns Hopkins University Applied Physics Laboratory, Laurel, MD, where he has been since 2000. From 1998 to 2000, he was with Computer Sciences Corporation, where he was a Flight Dynamics Analyst with the NASA Goddard Space Flight Center, Greenbelt, MD. His research interests include electromagnetic and acoustic propagation and scattering as well as hydrodynamics and fluid-structure interaction modeling.

Ra’id S. Awadallah (S’97–M’98) was born in Jerusalem, Israel, in 1966. He received the B.S.E.E. and M.S.E.E. degrees from the Jordan University of Science and Technology, Irbid, Jordan, in 1988 and 1991, respectively, and the Ph.D. degree in electrical engineering from the Virginia Polytechnic Institute and State University, Blacksburg, in 1998. From 1991 to 1993, he worked as a Lecturer with the Department of Electronics, Jerusalem University College of Sciences, Abu-Dis/Jerusalem, Israel. In 1998, he joined The Johns Hopkins University Applied Physics Laboratory, Laurel, MD, as a Research Associate, where he is presently Principal Professional Staff Member of the Milton S. Eisenhower Research Center. His research interests include tropospheric propagation, electromagnetic scattering from randomly rough surfaces, radar cross section of complex targets, and applied electromagnetism. Dr. Awadallah is a Member of Commission F of the International Scientific Radio Union (URSI).

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A Nested Multi-Scaling Inexact-Newton Iterative Approach for Microwave Imaging Giacomo Oliveri, Member, IEEE, Leonardo Lizzi, Member, IEEE, Matteo Pastorino, Fellow, IEEE, and Andrea Massa, Member, IEEE

Abstract—A microwave imaging technique based on the integration of the Inexact-Newton method within a multi-scaling strategy is proposed in the framework of the contrast field formulation of the electromagnetic inverse scattering. The inversion problem is solved by means of a nested procedure that considers three different logical levels: (a) an outer multi-focusing loop aimed at implementing a synthetic zoom for focusing the scatterer support within the investigation domain; (b) a local linearization of the original full-nonlinear inverse scattering function; and (c) a truncated Landweber inner loop devoted to regularize the arising ill-posed linear problem. Thanks to the features of the integrated approach, a reliable inversion technique able to suitably face the non-linearity and the ill-posedness/ill-conditioning issues of the imaging problem is designed. A numerical validation dealing with different objects, measurement setups, and noise conditions is carried out to assess the features and the potentialities as well as the limitations of the proposed strategy. Comparisons with bare approaches and other multi-resolution formulations are presented, as well. Index Terms—Electromagnetic inverse scattering, microwave imaging, Inexact Newton method, iterative multiscaling approach.

I. INTRODUCTION

I

N RECENT years, microwave imaging systems have been widely studied and developed because of their applications to non-invasive and non-destructive testing problems [1]–[4] arising in subsurface prospecting [5], biomedical imaging [6]–[11], and material characterization [12]. In this framework, several interesting results from both the algorithmic [13]–[18] and instrumentation viewpoint [19]–[21] have been reported in the scientific literature. However, a limited number of microwave imaging systems are at present used in real-world applications due to the limitations of present-day instruments and techniques still confined at the laboratory experimentations. As far as inversion procedures are concerned, further work is needed for developing reliable, stable, and efficient inversion algorithms usable in practical applications. This is mainly due to the theoretical difficulties that arise when solving inManuscript received December 30, 2010; revised May 12, 2011; accepted August 09, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. G. Oliveri, L. Lizzi, and A. Massa are with the ELEDIA Research Group@DISI, University of Trento, Povo 38123, Trento, Italy (e-mail: [email protected]; [email protected]; andrea.massa@ing. unitn.it). M. Pastorino is with the Department of Biophysical and Electronic Engineering, University of Genova, 16145 Genova, Italy (e-mail: pastorino@dibe. unige.it). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173131

verse problems, namely the ill-posedness and the non-linearity [22]. In order to properly address these issues, several techniques have been proposed in the last few years. On the one hand, the use of global optimization techniques [23]–[31] as well as of alternative formulations (e.g., Rytov and Born apformulation [14], [32]–[34], qualitaproximations [22], tive methods [35]–[41]) has been proposed in order to mitigate the presence of local-minima caused by the non-linearity of the inverse scattering problem at hand [42]. Moreover, since the occurrence of local minima is related to the degree of nonlinearity, the data information content, and the number of unknowns [42], information retrieval techniques able to suitably allocate the unknowns within the domain of interest have been proposed, as well. More specifically, multi-resolution strategies [43]–[47] proved to be very effective in better exploiting and enhancing the information collectable from the measurements, thus yielding accurate reconstructions in different conditions [48]–[52] with a high computational efficiency. On the other hand, efficient regularization techniques have been introduced to mitigate the ill-posedness/ill-conditioning that causes the non-uniqueness and the numerical instabilities of the solution. Indirect regularization techniques, based on the introduction of suitable multiplicative or additive terms to the error cost function, have been applied to both deterministic [14] and to stochastic [15] methodologies. Direct regularization approaches have been recently proposed, as well. In this latter framework, the exploitation of the Inexact Newton method has been investigated and its regularization features have been assessed dealing with both numerical [53] and experimental data [54]. This paper is aimed at presenting a new microwave imaging technique based on the integration of an efficient multi-focusing strategy, namely the iterative multi-scaling approach (IMSA) [44], [46], with the Inexact-Newton solution method (INM) [53], [55] to effectively tackle both the non-linearity and the ill-posedness/ill-conditioning of microwave imaging problems by exploiting the best properties of the two strategies and mutually overcoming their limitations. More in detail, a nested approach is developed by considering three different procedural steps: (a) an outer multi-scaling loop aimed at iteratively locating the regions-of-interest (RoIs) where the scatterers are supposed to be located within the investigation domain and adaptively refining the inversion grid within those areas, (b) a linearization loop that locally approximates the nonlinear inversion problem with a linear one to address the non-linearity issues, and (c) a truncated Landweber loop devoted to find the regularized solution of the approximated linear

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inverse problem. Such a choice is motivated by the following considerations: • the IMSA approach proved to be a suitable countermeasure against local minima problems thanks to the effective exploitation of the available information, thus enabling the use of local search strategies for solving the arising inverse scattering problems [44]; • the INM, a numerically efficient local search approach, has shown reliable and stable regularization features despite few regularization parameters to be set [53], [55]; • the integration of the two approaches is very easy thanks to the modularity of the multi-scaling scheme [46]. The expected outcome is a set of guidelines/indications/ranges for the interested user to identify the working conditions/scenarios when the IMSA-IN performs in an optimal fashion (i.e., the optimal trade-off between reconstruction accuracy and computational costs also in view of the envisaged application) in comparison with state-of-the-art methodologies, as well. The outline of the paper is as follows. Section II briefly summarizes the mathematical model of the scattering problems of interest. In Section III, the inversion strategy combining the IMSA with the INM is described. A numerical validation is then presented (Section IV) to analyze features, potentialities, and limitations of the proposed IMSA-IN method with reference to cylindrical configurations under transverse magnetic (TM) illumination conditions. Finally, some conclusions are drawn (Section V). II. PROBLEM FORMULATION Let us consider a cylindrical scatterer of arbitrary bounded cross section embedded in a homogeneous lossless non-magnetic background with permittivity . The unknown scatterer is successively probed by known incident transverse-magnetic (TM) monochromatic waves whose time-dependence is assumed and omitted hereinafter. The material properties of the nonmagnetic scatterer are invariant along the symmetry axis and they are described by the object function [44]

where

is the measurement curve outside is the two-dimensional free space Green’s function [56]. The objective of the reconstruction procedure is that of finding the unknown distributions of and in starting from the knowledge of within and in a set of measurement points in ( ). Equations (2) and (3), rewritten in a more compact notation as follows: (4) mathematically describe the relationships between the unknown vector and the data vector being the nonlinear scattering operator:

(5) The inversion of the nonlinear operator (5) for determining the unknown term yields to an ill-posed problem [22]. Therefore, a regularized solution of (4) is looked for by applying the multi-resolution iterative linearization scheme detailed in the following section. III. IMSA-IN INVERSION PROCEDURE To numerically address the inverse-scattering problem at hand, (4) is firstly discretized according to the Richmond’s procedure [60]. At each th step of the IMSA ( being the step index), the RoI is parsquare cells centered at ( titioned into ), being the number of degrees of freedom of the scattered field [58], to obtain the following algebraic nonlinear equation: (6) where

(1) being the position vector and and are the relative dielectric permittivity and electric conductivity, respectively. Under these hypotheses, the scattered, , total, , and incident, , fields for each illumination comply with the following integral equations [56]:

(2)

(3)

being the discretized version of applied to . Thanks to the suitable choice of the ratio between measurement data and unknowns (i.e., ) and the reduced occurrence of local minima [42], numerically efficient local search algorithms can be profitably used as solution tools for (6) at each th step. The INM is here adopted to benefit from its strong regularization capabilities, numerical efficiency, accuracy, and robustness [53]. Such an approach stems from the classical Newton technique, which is, in its simplest scalar implementation, a local root-finding algorithm. At each iteration ( being the INM iteration index) the INM performs the following two phases [53]: • Linearization—Truncation at the first term of the Taylor expansion of to determine its linear approxi; mation

OLIVERI et al.: A NESTED MULTI-SCALING INEXACT-NEWTON ITERATIVE APPROACH FOR MICROWAVE IMAGING

• Update—Computation of the new guess solution , where satisfies the following relationship . As for the linearization, let us consider that the Fréchet derivative of (6) at , is defined as follows:1

where is a variation in , then the first term of the Taylor expansion of turns out to be . in correAccordingly, the linear approximation of spondence with results

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3) Update. Update

. If , then , and terminate, else Goto 2. The INM is then iterated by successively repeating the “linearization” and the “update” phases until steps are completed ). Once the INM loop has been terminated, (i.e., a new IMSA step takes place and the “filtering” and “clustering” processes are performed for updating [44]. Successively, a higher spatial resolution is adopted only within the RoI and the inversion of (6) is carried out on finer and finer ( discretization grids by updating ). The multi-step process is iterated until suitable termination conditions hold true [44] and is finally assumed as the retrieved solution. IV. NUMERICAL RESULTS

(7) being and . To update the th trial solution (Outer INM is comloop—Fig. 1), the increment puted according to the classical Newton scheme by solving the linear equation or, in an equivalent fashion exploiting (7), by the following:

This section is aimed at numerically assessing the potentialities and the limitations of the IMSA-IN method in terms of accuracy, computational complexity, and robustness when dealing with various scatterers, measurement setups, and noise conditions. Besides the pictorial representations of the retrieved dielectric distributions, the accuracy of the reconstructions is quantitatively evaluated by computing the reconstruction errors as in [53], [44] and defined as follows:

(8) Unfortunately, (8) is ill-posed [53] and an approximated (i.e., inexact) solution of (8) has to be found through regularization. Towards this end, a finite number of Landweber iterations is applied2 [57]. More in detail, a regularized solution of the leastsquare counterpart of (8), that is

(9) where is the number of cells of the whole investigation domain or belonging to the scatterer support or to the background region . Moreover, and denote the retrieved and the actual contrast, respectively. A. Homogeneous Square Cylinder

being the adjoint operator of [57], is computed by performing the following truncated Landweber loop (Inner INM loop—Fig. 1): 1) lnitialize. Let . Set the total number of Landweber iterations, ; 2) Computation. Evaluate the step value

for a given

;

1The Fréchet derivative can be quite easily computed by exploiting the bi-linearity of the integral operators involved in (2) and (3), as detailed in [53]. 2Several other approaches could be used such as the Tichonov method, the , the method, or the truncated factorization [53]. truncated However, the truncated Landweber method has been shown to be more suitable for this application in terms of numerical efficiency and simplicity of the setup of the regularization parameter [53].

The first set of experiments deals with a single lossless homogeneous scatterer [ —Fig. 2(a)] off-centered in a square investigation domain of size and illuminated by a set of TM plane waves coming from . For each view, the synthetically-generated scattered field is collected at measurement points uniformly-spaced on a circle in radius. It is worthwhile to notice that the values of and have been chosen following the guidelines in [58] to collect from the scattered field all the information available on the scenario under test. The IMSA-IN and BARE-IN reconstructions have been carried out by setting [53], and [59]. Moreover, the maximum number of multi-scaling steps has been fixed to [44]. The retrieved profiles in Fig. 2 indicate that the IMSA-IN [Fig. 2(b)] performs better than the bare INM (BARE-IN) [Fig. 2(d)] as pointed out by the smaller value of the “external” error ( versus ). As regards the evolution of the estimated distribution at different IMSA steps [Fig. 3(a)–(d)], the reconstruction accuracy improves step-by-step as confirmed by the behavior of the error figures in Fig. 3(e) (Noiseless case). More in detail, monotonically decreases of about one order in magnitude ( versus ) as

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Fig. 2. Off-Centered Square Cylinder Reconstruction ( , Noiseless Data)—Actual contrast distribution (a). Reconstructed profile with (b) BARE-IN, (c) BARE-CG, (d) IMSA-IN, and (e) IMSA-CG.

Fig. 1. Flowchart of the IMSA-IN method.

well as the internal error ( versus ). On the other hand, the external error does not exhibit a monotone decrement versus because of the free-space initialization

Fig. 3. Off-Centered Square Cylinder Reconstruction ( , Noiseless Data)—IMSA-IN recon, (b) , (c) , and (d) . structions at intermediate steps: (a) (e) Behavior of the error indexes versus .

sults have been yielded by setting

. These reto its minimum value

OLIVERI et al.: A NESTED MULTI-SCALING INEXACT-NEWTON ITERATIVE APPROACH FOR MICROWAVE IMAGING

Fig. 4. Off-Centered Square Cylinder Reconstruction ( , Noiseless Data)—Reconstructed profiles with (a), (c), (e) the IMSA-IN and (b), (d), (f) the IMSA-CG for different values of : (a), (b) , (c), (d) , and (e), (f) .

for collecting all the available information [58]. However, the condition is usually assumed when applying the INM [53]. Therefore, further experiments have been developed by choosing to fairly compare the standard INM and its enhanced multiscaling version. Towards this end, the same scatterer has been reconstructed in successive simulations with 60, 240, 360 [Fig. 4(a), (c), (e)]. As it can be observed, the reconstruction with the IMSA-IN significantly enhances with [Fig. 2(d)— versus Fig. 4(e)— ] as also confirmed by the corresponding error figures in Fig. 6. As an example, increasing above yields a total error reduction of around one order of magnitude [ —Fig. 6(a)]. Such a behavior is not due to the IMSA procedure, but it is rather related to the intrinsic nature of the INM method. As a matter of fact and unlike the IMSA-IN [Fig. 4(a), (c), (e)], the inversions with a state-of-the-art multi-resolution conjugate gradient approach (IMSA-CG) [44] highlight that is already sufficient to achieve a low reconstruction error [Fig. 2(e)— ] and the improvements are not significant for larger values [Fig. 4(b), (d), (f)]. This is quantitatively assessed by the plots of the error indexes of the IMSA-CG in Fig. 6 where, as a representative example, the total error turns out to be almost constant and equal to whatever . On the other hand, it should be noticed that although when , the IMSA-IN performances result equivalent or

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Fig. 5. Off-Centered Square Cylinder Reconstruction ( , Noiseless Data)—Reconstructed profiles with (a), (c), (e) the BARE-IN and (b), (d), (f) the BARE-CG for different values of : (a), (b) , (c), (d) , and (e), (f) .

slightly better than that of the IMSA-CG when more scattered field samples are processed [ versus being —Fig. 6(a)]. An explanation of the dependence of the IMSA-IN accuracy on is that overconstraining the inversion (i.e., using more than measurement points) helps the reduction of the approximation error introduced by the linearization of the INM Outer loop. It is worth remarking that such a consideration and the above remarks on the relationship between inversions and hold true also when dealing with BARE approaches (Fig. 5). Indeed, while the BARE-CG reconstruction for [Fig. 5(f)] does not significantly improve that with [Fig. 2(c)], the BARE-IN performs more and more accurately increasing [Fig. 5(a), (c), (e)] as confirmed by the error indexes computed for the BARE inversions in Fig. 6. Moreover and as expected, the total reconstruction error for the IMSA-based implementations (whether based on CG or IN) is always below that obtained with the BARE methods (Fig. 6) with an improvement of about one order in magnitude for every choice of [e.g., versus for the IN formulation—Fig. 6(a)]. As for the computational issues, Fig. 7 shows the plot of the total inversion time3 for the IMSA-IN method versus in comparison with that needed by the IMSA-CG approach and 3On

a standard laptop with 2.16 GHz CPU clock and 2 GB of RAM.

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Fig. 7. Off-Centered Square Cylinder Reconstruction ( , Noiseless Data)—Behavior of the inversion time as a function of .

Fig. 6. Off-Centered Square Cylinder Reconstruction ( , Noiseless Data)—Plots of (a) versus . and (c)

, (b)

,

their BARE counterparts. As it can be noticed, the IN-based inversions are more computationally-efficient than the CG-based ones (e.g., [s] versus for —Fig. 7) because of the less complex nature of the linear problem to be solved compared to

the fully-nonlinear one of the CG formulation. More specifically, although the CPU time of both approaches depends on the same manner on (as well as on the discretization grid), such a result is related to the faster convergence of the INM approach (i.e., smaller number of iterations to reach a threshold of the cost function) as well as the lower computational complexity of each INM step, which does not require any univariate search unlike the CG [44]. Moreover, it must be also noticed that the IMSA further enhances the numerical efficiency of the IN method, despite the multi-step processing, since significantly smaller-dimension problems are actually solved at each step of the zooming procedure. Quantitatively, only [s] are required by the IMSA-IN method when with a reduction of about 54% with respect to the standard INM ( [s]—Fig. 7). Because of the favorable trade-off between inversion accuracy and computational burden, the setting will be considered hereinafter as reference setup for the IMSA-IN procedure. To provide further insights on the IMSA-IN, the next numerical experiment is aimed at assessing its robustness against noisy data. Towards this end, the field data scattered from the dielectric profile in Fig. 2(a) have been corrupted with different signal-to-noise ratio (SNR) levels of an additive zero mean complex Gaussian noise. The enhanced robustness of the IMSA-IN with respect to the INM can be well recognized by comparing the retrieved object profiles when SNR 30 dB [Fig. 8(a) versus (b)], SNR 20 dB [Fig. 8(c) versus (d)], and SNR 10 dB [Fig. 8(e) versus (f)]. The IMSA extension allows one satisfactory inversions for both high and moderate SNRs [Fig. 8(b)–(d)] as well as (unlike its BARE counterpart) an acceptable dielectric estimate when processing highly-corrupted data [Fig. 8(e) versus Fig. 8(f)]. These considerations are quantitatively confirmed by the values of the error indexes in Fig. 9(a). As it can be noticed, the total reconstruction error is on average halved mainly thanks to the “cleaning” effect outside the support of the scatterer enabled by the zooming

OLIVERI et al.: A NESTED MULTI-SCALING INEXACT-NEWTON ITERATIVE APPROACH FOR MICROWAVE IMAGING

Fig. 8. Off-Centered Square Cylinder Reconstruction ( , Noisy Data)—Dielectric profiles retrieved by means of (a), (c), (e) the BARE-IN and (b), (d), (f) the IMSA-IN [dB], (c), (d) [dB], for different SNR values: (a), (b) [dB]. and (e), (f)

process. As for the comparison between IN-based methodologies [Fig. 9(a)] and CG-based ones [Fig. 9(b)], previous outcomes are still confirmed in the presence of noise. B. Homogeneous “Two-Lines” Cylinder In order to verify whether the previous outcomes on the IMSA-IN still hold true also when dealing with more complex dielectric shapes and different values, the homogeneous “two-lines” cylinder in Fig. 10(a) has been chosen as another benchmark geometry. Such an object is assumed to be located in a square domain of size and the scattered fields due to differently-directed TM illuminations have been collected at measurement samples on a circle of radius . For comparison purposes, let us analyze the dielectric distributions retrieved by the IMSA-IN and the INM when

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Fig. 9. Off-Centered Square Cylinder Reconstruction ( , Noisy Data)—Error indexes versus SNRs for (a) INM-based and (b) CG-based methodologies.

SNR 30 [dB] and . As expected, the advantages of the multi-step implementation are still confirmed [Fig. 10(b) versus (c)]. Such enhancements are even more evident when reconstructing stronger scatterers as in second and third rows of Fig. 11. As a matter of fact, while the IMSA-IN inversions turn out to be always acceptable for larger and larger contrasts [Fig. 11(d)— ; Fig. 11(f)— ], the performances of the bare INM significantly worsen when , until the retrieved profile fully differs from the actual one as in Fig. 11(e) concerned with the case . For completeness, the inversions for a weaker distribution are also reported [Fig. 11(a)–(b)] to give the reader the full picture on the effectiveness of the IMSA-enhanced implementation versus . The result is that the IMSA-IN fully improves the accuracy of the bare INM version as pointed out by the error plots in Fig. 12. Although the error indexes increase for both INM and IMSA-IN as the value grows, it turns out that whatever the actual contrast and, on average,

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Fig. 11. “Two-Line” Cylinder Reconstruction ( [dB])—Dielectric profiles reconstructed with (a), (c), (e) the BARE-IN and , (c), (d) , and (e), (f) (b), (d), (f) the IMSA-IN when (a), (b) .

Fig. 10. ‘Two-Line’ Cylinder Reconstruction ( [dB])—Actual contrast (a). Dielectric profiles reconstructed with (b) the BARE-IN and (c) the IMSA-IN.

,

and

. It is also worth observing that these outcomes are not only limited to lossless scatterers as indicated by the results in Fig. 13 concerned with the same profile of Fig. 10, but with .

Fig. 12. “Two-Line” Cylinder Reconstruction ( 30 [dB])—Error indexes versus .

SNR

C. Non-Homogeneous Cylinder In order to assess the performances of the IMSA-IN in the presence of non-homogeneous scatterers, the set of experiments of this section considers the reference profile in Fig. 14(a) characterized by and for the inner and outer square profiles, respectively. The geometry and measurement

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Fig. 15. Non-Homogeneous Cylinder Reconstruction ( [dB])—Dielectric profiles estimated with (a) the BARE-IN, (b) the BARE-CG, (c) the IMSA-IN, and (d) the IMSA-CG.

Fig. 13. Lossy “Two-Line” Cylinder Reconstruction ( [dB], )—Actual (a), (b) and retrieved profiles by (c), (d) BARE-IN and (e), (f) IMSA-IN: (a), (c), (e) real and (b), (d), (f) imaginary part of the contrast functions.

Fig. 14. Non-Homogeneous Cylinder Reconstruction ( [dB])—Actual contrast distribution (a). Dielectric profiles estimated with (b) the BARE-IN, (c) the BARE-CG, (d) the IMSA-IN, and (e) the IMSA-CG.

setup of Section IV-B have been maintained, while the scattering data have been blurred with a SNR 30 dB noise level.

Because of the undoubted and already highlighted advantages of the IMSA-IN over the standard INM, the following analysis will be mainly focused in comparing the IMSA-IN with the IMSA-CG, while the results for the BARE-IN will be only reported for completeness. By comparing the inversions results from the IMSA-IN [14(d)] and the IMSA-CG [14(e)], it turns out that the reconstruction error of the INM-based technique is lower than that of the full nonlinear CG-based approach ( versus —Table I) despite the non-trivial contrast. Such an indication, in conjunction with the higher numerical efficiency of the IMSA-IN compared to the fully-nonlinear CG formulation ( [s] versus [s]—Table I), could yield one to conclude that the IMSA-INM approach is the best strategy. However, it is worth to note that those results have been obtained by considering an “over-sampling” of the measurement domain that unavoidably leads to a more complex/expensive measurement setup in terms of both acquisition time and measurement precision (e.g., receiver positioning) with respect to the case in which measurement points were employed. In order to further (besides the analysis carried out in Section IV-A) investigate on the IMSA-IN performance reduction when a smaller number of measures are available and to give more argumentations on the choice between IMSA-IN and IMSA-CG, the same numerical example has been solved processing measurements for each view. As it can be observed (Fig. 15), the reconstruction from the IMSA-IN is not accurate and the shape of the inner cylinder is completely lost even though the scatterer support is still correctly located [Fig. 15(c)]. On the opposite, the IMSA-CG provides a low reconstruction error ( versus —Table I) also in these working conditions [Fig. 15(d)] at the cost of a longer inversion time ( [s] versus [s]—Table I).

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TABLE I NON-HOMOGENEOUS CYLINDER RECONSTRUCTION (

SNR

30 [dB])—ERROR AND COMPUTATIONAL INDEXES

Fig. 17. “Three-Hollow” Cylinder Reconstruction ( [dB])—Actual contrast profile (a). Dielectric profile retrieved from (b) the BARE-IN, (c) the BARE-CG, (d) the IMSA-IN, and (e) the IMSA-CG.

Fig. 16. “Three-Hollow” Cylinder Reconstruction ( [dB])—Actual contrast profile (a). Dielectric profile retrieved from (b) the BARE-IN, (c) the BARE-CG, (d) the IMSA-IN, and (e) the IMSA-CG.

The results from the last two experiments indicate that (a) when a large number of measurements is available , the IMSA-IN approach is more efficient and accurate than the IMSA-CG, while (b) the fully non-linear CG-based approach overcomes the IN-based method in terms of accuracy when . Moreover, it is again worth remarking that the reconstructions with the “bare” methods are unsatisfactory when using the INM formulation [Figs. 14(b)–15(a)] and very rough when exploiting the nonlinear CG [Figs. 14(c)–15(b)] whatever the choice of . D. “Three-Hollows” Cylinder The last test case is aimed at further assessing whether the above “guidelines” still apply for more complex geometries exhibiting higher-resolution details. Towards this end, the dielectric scatterer in Fig. 16(a) is imaged starting from the data collected with the same measurement setups described in

Section IV-B and blurred with an additive noise with SNR 20 dB. From the pictorial representations of the retrieved dielectric profiles (Figs. 16–17) and the values of the corresponding errors (Table II), the IMSA-IN and IMSA-CG methods guarantee a comparable accuracy when [Fig. 16(d) versus 16(e)] ( versus —Table II), while, as expected, the full nonlinear approach is more effective when [Fig. 17(c) versus (d)]. Moreover, the following considerations come out: (a) the total inversion time needed by CG-based approaches is generally two-orders in magnitude longer than that of IN-based retrievals (e.g., [s] versus [s] when —Table II); (b) the multi-zooming scheme significantly reduces of both IN-based (e.g., [s] versus [s] for —Table II) and CG-derived methods (e.g.: [s] versus [s] for —Table II). V. CONCLUSIONS AND REMARKS A new strategy for microwave imaging has been presented. The proposed reconstruction method is based on the integration of a direct regularization method (INM) in a multi-scaling loop (IMSA) to mitigate the non-linearity and the ill-conditioning of the inverse problems at hand. The IMSA-IN approach has been validated through simulations involving scatterers with simple

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TABLE II “THREE-HOLLOW” CYLINDER RECONSTRUCTION (

as well as complex cross sections and different dielectric profiles under various noisy conditions. The numerical analysis has pointed out the following key issues: • the inversion performances of the IMSA-IN are always better than those of the standard INM whatever the measurement setup, the scatterer profile, and the blurring conditions (Section IV); • the IMSA-IN method is significantly more computationally efficient than the IMSA-CG as well as of the bare INM (e.g., Fig. 7); • the IMSA-IN provides smaller or comparable reconstruction errors than the IMSA-CG technique with reduced computational costs when redundant measurements are possible . Otherwise , the IMSA-IN performances degrade compared to the IMSA-CG ones (Sections IV-B–IV-D) even though they still remain superior to those of the bare counterpart (INM). Future works will be aimed at extending the proposed approach to deal with multiple separable scatterers [61] and fullinverse problems [62]. Of course, further validations of the method potentials against experimental data will be matter of future analysis.

REFERENCES [1] G. C. Giakos, M. Pastorino, F. Russo, S. Chiwdhury, N. Shah, and W. Davros, “Noninvasive imaging for the new century,” IEEE Instrum. Meas. Mag., vol. 2, pp. 32–35, Jun. 1999. [2] J. Ch. Bolomey, “Microwave imaging techniques for NDT and NDE,” in Proc. Training Workshop on Advanced Microwave NDT/NDE Techniques, Supelec/CNRS, Paris, France, Sep. 7–9, 1999. [3] R. Zoughi, Microwave Nondestructive Testing and Evaluation. Amsterdam, The Netherlands: Kluwer Academic, 2000. [4] D. Lesselier and J. Bowler, “Special Issue on Electromagnetic and ultrasonic nondestructive evaluation,” Inverse Problems, vol. 18, no. 6, Dec. 2002. [5] C.-C. Chen, J. T. Johnson, M. Sato, and A. G. Yarovoy, “Special Issue on Subsurface sensing using ground-penetrating radar,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 8, Aug. 2007. [6] Z. Q. Zhang and Q. H. Liu, “Three-dimensional nonlinear image reconstruction for microwave biomedical imaging,” IEEE Trans. Biomed. Eng., vol. 51, no. 3, pp. 544–548, Mar. 2004. [7] S. Caorsi, A. Massa, M. Pastorino, and A. Rosani, “Microwave medical imaging: Potentialities and limitations of a stochastic optimization technique,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1909–1916, Aug. 2004. [8] M. El-Shenawee and E. Miller, “Spherical harmonics microwave algorithm for shape and location reconstruction of breast cancer tumors,” IEEE Trans. Med. Imag., vol. 25, pp. 1258–1271, Oct. 2006.

SNR

20 [dB])—ERROR AND COMPUTATIONAL INDEXES

[9] T. Rubk, P. M. Meaney, P. Meincke, and K. D. Paulsen, “Nonlinear microwave imaging for breast-cancer screening using Gauss-Newton’s method and the CGLS inversion algorithm,” IEEE Trans. Antennas Propagat., vol. 55, pp. 2320–2331, Aug. 2007. [10] H. Zhou, T. Takenaka, J. Johnson, and T. Tanaka, “Breast imaging model using microwaves and a time domain three dimensional reconstruction method,” PIER, vol. 93, pp. 57–70, 2009. [11] S. Caorsi, A. Massa, and M. Pastorino, “Numerical assessment concerning a focused microwave diagnostic method for medical applications,” IEEE Trans. Antennas Propag., vol. 48, no. 11, pp. 1815–1830, Nov. 2000. [12] S. Kharkovsky and R. Zoughi, “Microwave and millimeter wave nondestructive testing and evaluation—Overview and recent advances,” IEEE Instrum. Meas. Mag., vol. 10, pp. 26–38, Apr. 2007. [13] H. Harada, D. J. N. Wall, T. Takenaka, and T. Tanaka, “Conjugate gradient method applied to inverse scattering problems,” IEEE Trans. Antennas Propag., vol. 43, pp. 784–792, Aug. 1995. [14] P. M. van den Berg and A. Abubakar, “Contrast source inversion method: State of the art,” PIER, vol. 34, pp. 189–218, 2001. [15] P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, “Evolutionary optimization as applied to inverse scattering problems,” Inverse Problems, vol. 25, no. 12, pp. 1–41, Dec. 2009. [16] M. Pastorino, “Stochastic optimization methods applied to microwave imaging: A review,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 538–548, Mar. 2007. [17] E. Bermani, A. Boni, S. Caorsi, and A. Massa, “An innovative real-time technique for buried object detection,” IEEE Trans. Geosci. Remote Sens., vol. 41, no. 4, pp. 927–931, Apr. 2003. [18] A. Massa, A. Boni, and M. Donelli, “A classification approach based on SVM for electromagnetic sub-surface sensing,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 9, pp. 2084–2093, Sep. 2005. [19] P. M. Meaney, M. W. Fanning, D. Li, S. P. Poplack, and K. D. Paulsen, “A clinical prototype for active microwave imaging of the breast,” IEEE Trans. Microw. Theory Tech., vol. 48, pp. 1841–1853, Nov. 2000. [20] J.-C. Bolomey and F. E. Gardiol, Engineering Applications of the Modulated Scatterer Technique. Boston, MA: Artech House, 2001. [21] T. Henriksson, N. Joachimowicz, C. Conessa, and J.-C. Bolomey, “Quantitative microwave imaging for breast cancer detection using a planar 2.45 GHz system,” IEEE Trans. Instrum. Meas., vol. 59, no. 10, pp. 2691–2699, Oct. 2010. [22] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Berlin, Heidelberg: Springer-Verlag, 1998. [23] M. Pastorino, A. Massa, and S. Caorsi, “A microwave inverse scattering technique for image reconstruction based on a genetic algorithm,” IEEE Trans. Instrum. Meas., vol. 49, no. 3, pp. 573–578, Jun. 2000. [24] S. Caorsi, A. Massa, and M. Pastorino, “A computational technique based on a real-coded genetic algorithm for microwave imaging purposes,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 4, pp. 1697–1708, Jul. 2000. [25] S. Caorsi, A. Massa, M. Pastorino, and A. Randazzo, “Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the Memetic algorithm,” IEEE Trans. Geosci. Remote Sens., vol. 41, no. 12, pp. 2745–2753, Dec. 2003. [26] A. Massa, M. Pastorino, and A. Randazzo, “Reconstruction of twodimensional buried objects by a hybrid differential evolution method,” Inverse Problems, vol. 20, no. 6, pp. 135–150, Dec. 2004. [27] A. Qing, “Dynamic differential evolution strategy and applications in electromagnetic inverse scattering problems,” IEEE Trans. Geosci. Remote Sens., vol. 44, pp. 116–125, 2006.

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[50] D. Franceschini, M. Donelli, G. Franceschini, and A. Massa, “Iterative image reconstruction of two-dimensional scatterers illuminated by TE waves,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1484–1494, Apr. 2006. [51] G. Franceschini, M. Donelli, R. Azaro, and A. Massa, “Inversion of phaseless total field data using a two-step strategy based on the iterative multi-scaling approach,” IEEE Trans. Geosci. Remote Sens., vol. 44, no. 12, pp. 3527–3539, Dec. 2006. [52] D. Franceschini, M. Donelli, R. Azaro, and A. Massa, “Dealing with multi-frequency scattering data through the IMSA,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2412–2417, Aug. 2007. [53] G. Bozza, C. Estatico, M. Pastorino, and A. Randazzo, “An inexact Newton method for microwave reconstruction of strong scatterers,” IEEE Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 61–64, Dec. 2006. [54] C. Estatico, G. Bozza, A. Massa, M. Pastorino, and A. Randazzo, “A two-step iterative Inexact-Newton method for electromagnetic imaging of dielectric structures from real data,” Inverse Problems, vol. 43, pp. S81–S94, Dec. 2005. [55] G. Bozza, C. Estatico, A. Massa, M. Pastorino, and A. Randazzo, “Short-range image-based method for the inspection of strong scatterers using microwaves,” IEEE Trans. Instrum. Meas., vol. 56, no. 4, pp. 1181–1188, Aug. 2007. [56] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, May 1989. [57] L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math., vol. 73, no. 3, pp. 615–624, Jul. 1951. [58] O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag., vol. 37, pp. 918–926, Jul. 1989. [59] G. Bozza and M. Pastorino, “An Inexact Newton-based approach to microwave imaging within the contrast source formulation,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1122–1132, Apr. 2009. [60] J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross shape,” IEEE Trans. Antennas Propag., vol. 13, no. 3, pp. 334–341, May 1965. [61] S. Caorsi, M. Donelli, and A. Massa, “Detection, location, and imaging of multiple scatterers by means of the iterative multiscaling method,” IEEE Trans. Microw. Theory Tech., vol. 52, pp. 1217–1228, Apr. 2004. [62] G. Franceschini, D. Franceschini, and A. Massa, “Full-vectorial three-dimensional microwave imaging through the iterative multi-scaling strategy—A preliminary assessment,” IEEE Geosci. Remote Sens. Lett., vol. 2, no. 4, pp. 428–432, Oct. 2005.

Giacomo Oliveri (M’09) received the B.S. and M.S. degrees in telecommunications engineering and the Ph.D. degree in space sciences and engineering from the University of Genoa, Italy, in 2003, 2005, and 2009, respectively. Since 2008, he has been a member of the ELEDIA Research Center at the University of Trento, Italy. His research work is mainly focused on cognitive radio systems, electromagnetic direct and inverse problems, and antenna array design and synthesis.

Leonardo Lizzi (M’10) received the B.S. and M.S. degrees in telecommunications engineering and the Ph.D. degree in information and communication technology from the University of Trento, Italy, in 2004, 2007, and 2011, respectively. His research work mainly focuses on multi-band and ultra-wideband (UWB) antenna design and synthesis. He is now a member of the LEAT (Laboratoire d’Electronique, Antennes et Telecommunications), University of Nice–Sophia Antipolis, CNRS UMR 6071, Valbonne, France.

OLIVERI et al.: A NESTED MULTI-SCALING INEXACT-NEWTON ITERATIVE APPROACH FOR MICROWAVE IMAGING

Matteo Pastorino (M’90–SM’96–F’09) received the Laurea degree in electronic engineering and the Ph.D. degree in electronic engineering and computer science, both from the University of Genoa, Genoa, Italy, in 1987 and 1992, respectively. At present, he is the Director of the Applied Electromagnetics Group, Department of Biophysical and Electronic Engineering, University of Genoa, where he is a Professor of electromagnetic fields. He teaches the university courses of “Electromagnetic Fields,” “Remote Sensing and Electromagnetic Diagnostics,” and “Antennas and Remote Sensing.” His main research interests are in the field of microwave and millimeter wave imaging, direct and inverse scattering problems, industrial and medical applications, smart antennas, and analytical and numerical methods in electromagnetism. He is a coauthor of more than 300 papers in international journals and proceedings of conferences. Prof. Pastorino is an Associate Editor of the IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENTS.

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Andrea Massa (M’03) received the Laurea degree in electronic engineering from the University of Genoa, Genoa, Italy, in 1992 and the Ph.D. degree in electronics and computer science from the same university in 1996. From 1997 to 1999, he was an Assistant Professor of electromagnetic fields in the Department of Biophysical and Electronic Engineering, University of Genoa, teaching the university course of “Electromagnetic Fields 1”. From 2001 to 2004, he was an Associate Professor at the University of Trento. Since 2005, he has been a Full Professor of electromagnetic fields at the University of Trento, where he currently teaches electromagnetic fields, inverse scattering techniques, antennas and wireless communications, and optimization techniques. At present, he is the director of the ELEDIALab at the University of Trento and Deputy Dean of the Faculty of Engineering. He is a member of the IEEE Society, of the PIERS Technical Committee, of the Inter-University Research Center for Interactions Between Electromagnetic Fields and Biological Systems (ICEmB) and Italian representative in the general assembly of the European Microwave Association (EuMA). His research work since 1992 has been principally on electromagnetic direct and inverse scattering, microwave imaging, optimization techniques, wave propagation in presence of nonlinear media, wireless communications and applications of electromagnetic fields to telecommunications, medicine and biology. Prof. Massa is an Associate Editor of IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

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Fast and Shadow Region 3-Dimensional Imaging Algorithm With Range Derivative of Doubly Scattered Signals for UWB Radars Shouhei Kidera, Member, IEEE, and Tetsuo Kirimoto, Senior Member, IEEE

Abstract—Ultra-wideband (UWB) radar with its high range resolution and applicability to optically harsh environments, offer great promise for near field sensing systems. It is particularly suitable for robotic or security sensors that must identify a target in low visibility. Some recently developed radar imaging algorithms proactively employ multiple scattered components, which can enhance an imaging range compared to synthesizing a single scattered component. We have already proposed the synthetic aperture radar (SAR) method considering a double scattered, which successfully expanded a reconstructible range of radar imagery with no a priori knowledge of target or surroundings. However, it requires a multiple integration of the received signals, requiring the fifth times integration in the 3-D case. Thus, this method requires an intensive computation and its spatial resolution is insufficient for clear boundary extraction such as edges or specular surfaces. As a substantial solution, this paper proposes a novel shadow region imaging algorithm based on a range derivative of double scattered signals. This new method accomplishes high-speed imaging, including a shadow region without any integration process, and enhances the accuracy with respect to clear boundary extraction. Results from numerical simulations verify that the proposed method remarkably decreases the computation amount compared to that for the conventional method, especially for the 3-D problem, enhancing the visible range of radar imagery. Index Terms—Fast and shadow region imaging, multiple scattered wave, range derivative of double scattered signal, range points migration, ultra-wideband (UWB) radar.

I. INTRODUCTION

U

ltra-wideband (UWB) pulse radar with high range resolution fulfills its potential for near-field sensing techniques. A robotic sensor is one of the most promising applications of UWB radar, able to identify a human body even in optically blurry visibilities, such as dark smog in disaster areas or high-density gas in resource exploration scenes. It is also

Manuscript received December 09, 2010; revised May 31, 2011; accepted August 29, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported in part by the Grant-in-Aid for Scientific Research (B) (Grant 22360161), and the Grant-in-Aid for Young Scientists (Start-up) (Grant 21860036), promoted by the Japan Society for the Promotion of Science (JSPS). The authors are with the Graduate School of Informatics and Engineering, University of Electro-Communications, Tokyo 182-8585, Japan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173128

in demand for non-contact measurement in manufacturing reflector antennas or aircraft bodies requiring high-precision surfaces. Furthermore, it has a potential for accurate surface extraction of the human breast for detecting breast cancer, where the surface reflection from a breast often causes severe interference [1], [2]. While various radar imagery algorithms have been developed based on the aperture synthesis [3], the time reversal approach [4], [5], the range migration [6], [7] or genetic algorithm (GA)-based solutions for domain integral equations [8], they are not suitable for the above applications because it is, in general, difficult to achieve both low computation cost and high spatial resolution. To conquer the problem in the conventional techniques, we have already proposed a number of radar imaging algorithms, which accomplish real-time and high-resolution surface extraction beyond a pulse width [9], [10]. Although these algorithms have been applied to surface imaging, such as breast cancer detection [2], through-the-wall imaging [11], or human activity recognition [12], they are actually applicable only to simple shapes such as convex objects. As a high-speed and accurate 3-D imaging method feasible for complex-shaped targets, the range points migration (RPM) algorithm has been established [13]. This algorithm directly estimates an accurate direction of arrival (DOA) with the global characteristic of observed range points, avoiding the difficulty of connecting them. Although RPM is based on a simple idea, it offers accurate and super-resolution surface extraction by incorporating a frequency domain interferometer [14]. However, the above methods including [13] and [14] have the unresolvable problem that aperture size strictly constrains the imaging range of a target surface. In many cases, a major part of a target shape, such as a side of the target, falls into a shadow region, that is not reconstructed since only single scattered components are used for imaging. To resolve this difficulty and enhance imaging range, the SAR algorithm considering a double scattered path has been developed [15]. Although this method shows that shadow region imaging is possible by positively using double scattered signals without preliminary observations or target models, which are required in other algorithms [16], [17], the method requires multiple integrations of the received signals. This incurs a large computation cost, especially for obtaining a full 3-D image. Moreover, the spatial resolution of SAR is often insufficient to identify target shapes particularly for edges or wedges owing to a range resolution limited by frequency bandwidth of UWB pulse, even if a large aperture size, i.e., high azimuthal resolution, is obtained.

0018-926X/$26.00 © 2011 IEEE

KIDERA AND KIRIMOTO: FAST AND SHADOW REGION 3-D IMAGING ALGORITHM

As an essential solution for these problems, this paper proposes a novel imaging algorithm based on the range derivative of doubly scattered signals, where an initial image obtained by RPM is used to the best effect. This method is based on an original proposition that each DOA of the double scattered points is strictly derived from the derivative of range points both in the 2-D and 3-D cases. This proposition enables us to directly estimate a target boundary corresponding to the doubly scattered centers without any integration procedures. The results of numerical simulations, investigating various target shapes and computational complexities, show that the proposed method accomplishes high-speed target boundary extraction in situations, which produce a shadow using existing techniques. II. 2-D PROBLEM A. System Model Fig. 1 shows a system model for the 2-D model. It assumes a mono-static radar with an omnidirectional antenna scanning along the -axis. A static target with an arbitrary shape is assumed, the spatial gradient of conductivity or permittivity on its boundary is expressed with Dirac’s delta function [9], a so-called clear boundary. This assumption is generally acceptable for most indoor UWB sensors, for which omnidirectional radiation can be achieved by a small micro-strip antenna, as in [14], and the surroundings of sensors should be artificial objects such as furniture or walls with clear boundaries in terms of the center wavelength of a general UWB pulse. Due to the static object assumption, the scanning velocity is not relevant here. The propagation speed c of the radio wave is assumed to be a known constant. A transverse electric (TE) mode wave and cylindrical wave propagation is considered. A mono-cycle pulse is used as the transmitting current. The space in which the target and antenna are located is expressed by the . The parameters are normalized by , which parameters is assumed for is the central wavelength of the pulse. simplicity. is defined as the electric field received at at time . is calculated antenna location as by applying the Wiener filter to (1) where .

is the signal in the frequency domain of is defined as (2)

where , and is the reference signal in the frequency domain, which is the complex conjugate of that is a constant for dimension consisof the transmitted signal. tency. This filter is an optimal mean square error (MSE) linear is now converted to filter for additive noises. using the valuable conversion where c is the speed of the radio wave. B. Conventional Imaging Algorithms Two methods are introduced as the conventional imaging algorithms for comparison with the proposed method. One

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Fig. 1. System model in the 2-D problem.

is RPM, which achieves accurate high-speed target imaging even with a complex-shaped boundary, and employs a single scattered wave [13], [14]. The second is SAR extended for double scattered waves [15]. 1) RPM: As one of the most promising imaging algorithms applicable to various target shapes, the RPM algorithm has been established. First, this method extracts the group of range points , which satisfy the local maxima of . Basias exists cally, RPM assumes that a target boundary point and radius , following from the on a circle with center assumption of an omnidirectional antenna and cylindrical propagation of a TE mode wave, and employs an accurate DOA ( in Fig. 1) estimation by making use of the global characteristics is calculated as of the observed range map. The optimum (3) where and is the number of the range points. denotes the angle from the axis to the intersection point of the circles, with parameters and . The target boundary for each range point is expressed as and . This algorithm ignores range points connection, and produces accurate target points, even if an extremely complicated range distribution is given. It also has the significant and range point advantage that each target point satisfies a one-to-one correspondence, which takes a substantial role in the proposed method described in the following section. The performance example of RPM is presented here, where the received electric field is calculated by the finite-difference are time-domain (FDTD) method. The range points which are beyond extracted from the local peaks of the preliminary determined threshold [13]. An example of this method for the target shape shown in Fig. 1 is presented. Fig. 2 shows the output of the Wiener filter, and the extracted range . The received signals are calculated at 401 points as . A noiseless environment is locations for assumed. The positive local maxima are regarded as the range points originated from the single scattered waves and, on the contrary, the negative ones are regarded as the range points originated from the double scattered components, because they have, in general, an anti-phase relationship. Fig. 3 shows the and estimated target points obtained by RPM.

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method calculates the image using double scattered waves as

(4) where space,

denotes the region of real is the output of the Wiener filter, and

. The minus sign in (4) creates a positive image focused by double scattered waves that have an antiphase relationship from a single scattered one. Here, the is defined as the original SAR image as initial image Fig. 2. Output of Wiener filter s(X; Z ) for the multiple targets and range points as (X; Z ) and (X; Z ).

(5) Equation (4), simple expression of the aperture synthesis of the received signals by considering a double scattered path, can be regarded as a coherent integration scheme because denotes the amplitude and its positive outputs offer the target boundary. Any extension of the SAR algorithm, such as omega-k migration [18] range-Doppler, can be used in creating . The final image is defined as (6)

Fig. 3. Estimated image with RPM for the multiple targets.

Fig. 4. Estimated image with the SAR I (r ) using double scattered signals for the multiple targets.

are empirically determined by considering robustness and accuracy for imaging, as detailed in [13]. This figure indicates that the target points accurately express the front side of the target boundary, but the rectangular side of the boundary mostly falls into a shadow region. This is because each antenna receives a distinguishable echo from the target boundary, which is perpendicular to the direction of the line of sight from each antenna location. This is an inherent problem in all algorithms that use only a single scattered wave for target reconstruction. 2) SAR With Double Scattered Signals: SAR employing the double scattered signal has been developed to enhance the imaging range, including a shadow [15]. Here, the same system and signal models are used as in Section II-A. In general, a double scattered wave propagates with a different path from that of a single scattered one. It, therefore, often provides independent information as to the two scattering points. This

where is the Heaviside function. The performance evaluation of this method is shown as folfor the previous lows. Fig. 4 shows the estimated image is normalized target case, using the same data as in Fig. 2. by its maximum value. Fig. 4 shows that the part of the side region of the rectangular target can be reproduced, and that the visible ranges of the circle and rectangular boundaries are remarkably expanded. The reason is that double scattered waves are effectively focused on the part of the target side in (4). It also claims that this method does not require target modeling or a priori information of the surroundings. However, it requires a triple integration for imaging and its calculation time goes up to around 60 s for Intel Pentium D 2.8-GHz processor. In the 3-D case, such a large calculation burden becomes more severe for a robotic sensor because it basically requires a fifth times integration for each image frame. Moreover, some false images occur around the target boundary, due to the range sidelobe of filter responses or other components like triple scattering ones; it also offers a blurry boundary, where its spatial resolution is strictly limited by half of a pulse width. C. Proposed Method To overcome the problems described for conventional methods, this paper proposes an accurate high-speed imaging algorithm for the shadow region. This method employs target points, which are preliminarily created by RPM, and directly reconstructs the target points corresponding to the double scattered signals, where each derivative of the range points is employed. 1) Principle of Proposed Method: This subsection describes a basic theory of the proposed method, indicating the relationship between the range points and the doubly scattered centers. Here, two target points originating from the doubly scat-

KIDERA AND KIRIMOTO: FAST AND SHADOW REGION 3-D IMAGING ALGORITHM

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tering are defined as and , respectively. As previously described, a double scattered signal has an anti-phase relationship to a single scattered one [15]; the negaare mostly regarded as double scattered tive peaks of is defined as the range point of a double echoes. Then, scattered wave, which is extracted from the local minimum of . denotes an antenna location. In this case, the following proposition holds. Proposition 1: When exists on , the next relationship holds: (7) where and are defined, hold. This method assumes that a double and scattered path satisfies the law of reflection on each scattering boundary; that is regarded as a phase stationary condition. The proof of this proposition is described in Appendix A. It , which is used for the is naturally derived as actual procedure of the proposed method, as described in the holds, regarded as single scattered following section. If case, (7) is equivalent to inverse boundary scattered transform is expressed as (IBST) in [9]. Here, (8) Once a first scattering point

is determined,

is given as (9)

In addition, if the normal vector as reflection derives as

on

is given, the law of

(10) where (11) holds with is described in Appendix B. condition as

. The derivation of (10) obviously satisfies the following

(12) where holds. Fig. 5 shows the relationship among the scattered points and the antenna location . 2) Incorporation With RPM: A substantial idea of the proposed method is that it makes uses of the preliminary estimated with target points by RPM as the first scattering location its normal vector . As previously described in Section II-B1, RPM directly converts the range points to the target points, satisfying a one-to-one correspondence. Here we define each target and point and range point with RPM as , where is the

Fig. 5. Relationship among the double scattered points p ; p , and the antenna location p .

total number of target points by RPM. In addition, each normal on is given by vector

(13) This relationship is derived from the assumption that each antenna receives a strong echo from the target boundary, which is perpendicular to a direction for a line of sight [13]. Equation (13) indicates that the inclination of the target boundary is directly estimated without using derivative operations; it is applicable even for a non-differentiable point like an edge. In addition, target points obtained by RPM on edges are reconstructed from different antenna locations, because an edge diffraction wave can be received in a wider observation range. Such points have different normal vectors, which are directly related to , and contribute to the search for a secondary scattering center first diffracted from an edge. from a set This algorithm determines an optimal of target points obtained by RPM, which is defined as . Here, the parameter vector is also defined as (14) (15) (16) , and is deterwhere mined in (11), similarly. To select the optimal from , two conditions for are introduced as follows. First, using (8), is defined as

(17) and are calculated in (10) and (9), respecwhere tively. Second, considering an another condition in (12), is defined as (18)

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Step 2) Range points are extracted as from the output of a Wiener filter, according to the condition given as (23)

Fig. 6. Relationship between p and p as two candidates for p .

where . Fig. 6 shows as and the relationship between two candidates for . Then, the proposed method determines the optimum for each as candidate

where is the total number of range points of the double scattered signals, and is empirically determined. is calculated by the Step 3) For each difference approximation with Gaussian weighting as

(24) (19) The optimum second scattering point

is determined as (20)

where is defined as , when the evaluation value in the right term in (19) becomes minimum. This optimization scheme is is the actual target point, based on the assumption that if it must satisfy both (8) and (12). Note that this method does not employ an integration of the scattered signals but directly determines the double scattering points using the derivative of the range points. Furthermore, a false image reduction scheme is introduced. is regarded as a false image, if In the postprocessing, the following condition is satisfied: (21) where is an empirically determined threshold, and is defined as

is empirically determined and dewhere , notes the number of range points of . The which satisfy parameter is created in (15) is stored into a set . as Step 4) For each from is created in (16). and are determined in Then, (19) and (20), respectively. does not satisfy the condition in (21), it Step 5) If is added to the set of the target points . becomes empty. Step 6) Steps 4) to 5) are iterated until Step 7) Obtain the final set of target points as . Step 3) avoids the fatal sensitivity caused by the derivative op. Although this erations by taking an appropriate value for method needs an optimization procedure for , it requires no integration process, directly locating the accurate target points. D. Performance Evaluation Using Numerical Simulation

(22) where and . increases when the following two conditions are satisfied. First, a large number of the first Fresnel zones determined by target points with RPM, . regarded as , include the double scattering point as exists close to the antenna location compared Second, . We assume that to the observed range of single scattering both situations are inadmissible for the actual target points. The similar scheme for a false image reduction is described in [15]. 3) Procedure of the Proposed Algorithm: The actual procedure of the proposed method is summarized as follows. is obtained by RPM. Step 1) A set of target points

This section presents numerical examples performed by the proposed method for two target cases. Fig. 7 illustrates the target points reproduced by RPM and the proposed methods, where the true range points for single and double scattered signals are given by a geometrical optics approximation [15]. Here, and are set. This figure indicates that the proposed method accurately creates the target points around the side of the rectangular boundary. This verifies that if the actual range points are given, our method obtains extremely accurate boundary extraction, including shadow regions. As a realistic case, Fig. 8 presents the estimated target points with RPM and the proposed method, where the same data as in Fig. 2 are used. This figure exemplifies that this method produces many accurate target points around the rectangular side. Here, it should be noted that the RPM reconstructs the rectangular edge points as in Fig. 3, which are converted from on the part of the hyperbolic the multiple range points and in Fig. 2. curve for which

KIDERA AND KIRIMOTO: FAST AND SHADOW REGION 3-D IMAGING ALGORITHM

Fig. 7. Estimated image with RPM and the proposed method for the multiple objects, when true range points are given.

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Fig. 9. Estimated image with RPM and the proposed method for the concave target, when true range points are given.

Fig. 10. Output of Wiener filter s(X; Z ) for the concave target and range points as (X; Z ) and (X; Z ). Fig. 8. Estimated image with RPM and the proposed method for the multiple objects, when range points are extracted from s(X; Z ).

Those target points, which can be reconstructed from the different antenna locations, have multiple normal vectors as related to the second scattering directions as described in the proposed method. Consequently, even in the case that the includes first scattering points exist on an edge, i.e., an edge diffraction, this method selects the optimum first scatin (19) and sequentially determines for each tering point . Also, while possibly includes no double scattered range points accidentally extracted from range sidelobes of single scattered signals, or other scattering components, this figure validates the fact that the false image reduction postprocessing successfully eliminates these false points. Besides, it has a great advantage in computation time, which requires less than 0.4 s for Intel Pentium D 2.8-GHz processor with 800 MB effective memory. As previously mentioned in Section II-B2, the conventional method costs around 60 s, and it is distinctly improved as to the computation burden. However, some fluctuations of the estimated points occur around the rectangular side, regardless of a noiseless situation. This is because the method employs the range derivative in (9), which tends to enhance small errors caused by the scattered waveform deformations or other interference effects. For another target case, a deep-set concave boundary is investigated. Fig. 9 shows the target points estimated by RPM and the proposed method, respectively, where the true range points are given. It is confirmed that our method successfully enhances the imaging range around the side of the concave boundary, using the double scattered range points. Fig. 10 shows the output of a

Fig. 11. Estimated image with RPM and the proposed method for the concave targets, when range points are extracted from s(X; Z ).

Wiener filter, which is calculated by FDTD, and each extracted and . Fig. 11 presents the estirange point as mated image created by the proposed method. It verifies that, in the practical case, it can produce accurate target points around the side of the concave boundary, which are not seen for RPM. The calculation time is also around 0.4 s with the same processor previously described, which is definitively improved from that required by the conventional method around 60 s. Here, the quantitative analysis is introduced by defined as (25) where and express the locations of the true and estiis the total number of . mated target points, respectively. Fig. 12 plots the number of estimated points for each value of

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Fig. 12. Number of target points for each  , in the case of Figs. 8 and 11.

Fig. 13. Estimated image with RPM and the proposed method for the multiple dB. targets at SN

= 30

in both cases as Figs. 8 and 11. It verifies that the number of accurate target points with the proposed method significantly , simultaneously enhancing the imaging increases around for the mulrange. The mean value of as is tiple objects, and this result quantitatively proves the effectiveness of the proposed method as to accurate imaging. In the case , because of the concave boundary, becomes there are many false image points over the actual boundary as in Fig. 11 for both RPM and the proposed method. This inaccuracy is mainly contributed by the conventional RPM, not by the proposed method. The reason for this is that more than a triple scattering effect produces an unnecessary image. Furthermore, an example in noisy situation is investigated, whereby white Gaussian noise is added to each received signal . Fig. 13 shows the estimated points obtained by the as proposed method, where the mean S/N is 30 dB. S/N is defined as the ratio of peak instantaneous signal power to the average noise power after applying the matched filter with the transmitted waveform. Although the accuracy of the estimated target points deteriorates due to the false range points extracted from noisy components, the whole image can offer a significant target boundary including the side of the rectangular boundary. Also, it should be noted that the range fluctuations caused by noise are effectively suppressed by the Gaussian weighted difference . approach in (24) using an appropriate Next, the relationship between and S/N is investigated for . The figure shows multiple objects as in Fig. 14 for each that the proposed method obtains a sufficient accuracy less than



Fig. 14. Relationship between  and S/N for each  jects.

as to the multiple ob-

over SN dB using any value of . While this method requires an apparently high S/N to hold the accuracy, the definition of S/N used in this paper overestimates the practical S/N because it considers not only a frequency localization of the received pulse but also a temporal one. Indeed, the actual UWB radar system can achieve this level of S/N. because we assume a near field measurement, where each receiver obtains an intensive echo from objects even under the spectrum mask of the UWB signal [19], and random noises in received signals can be considerably suppressed using coherent averaging. Moreover, to consider the applicability of the method to a realistic scenario, the sensitivity of antenna location to inaccuracy is investigated. This is mainly caused by mechanical errors in the scanning system. Figs. 15 and 16 show the output of Wiener filter with extracted range points and the estimated dB, image with RPM and the proposed method at SN respectively. The spatial errors with the Gaussian distribution in calculating the are added to each antenna location received signals. The standard deviation of the errors is set to , which corresponds to 1-mm accuracy for the antenna positioning, in the case of 100-mm center wavelength of the UWB pulse. This can be obtained by the real scanning systems used in [14]. The figure shows that the proposed method still creates an accurate image including the shadow region; denotes in this case. The first reason for this is that RPM has a significant tolerance to random noise or system errors because it employs the global characteristic of the range points distribution; there is only a slight image degradation as shown in the figure. The second reason is that the proposed method employs the smoothing scheme in calculating the range derivative in (24), and the false image reduction as postprocessing, in order to avoid image distortions from system errors or receiver noise and enhance the credibility of the obtained images. Finally, the limitation of this method is discussed as follows. The proposed method, which does have some limitations, has two main requirements: a scanning orbit and a clear boundary assumption for objects. First, while this paper assumes linear antenna scanning, it can be extended to arbitrary curvilinear scanning by modifying (7). However, the scanning line must be differentiable because the method employs derivative operation along it. Second, although the method assumes a clear boundary for objects, in the case of a human body, this assumption is

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Fig. 17. System model in 3-D problem. Fig. 15. Output of Wiener filter and extracted range points for the multiple targets, where random errors are given to antenna locations ( 0).

X;

Fig. 18. Relationship among

Fig. 16. Estimated image with RPM and the proposed method for the multiple targets where random errors are given to antenna locations ( 0).

X;

hardly acceptable, and then, both the conventional and the proposed methods offer blurry images expressing statistical scattering centers. On the other hand, with respect to a UWB waveform, any waveform holding the same equivalent bandwidth with lower a range sidelobe is applicable to this method, for example, a pulse, chirping signal, or spread spectrum (M-sequence) waveforms. This is because the proposed method uses only the significant ranges, which can be extracted from the local maxima or minima of the output of Wiener filter using any possible transmitted waveform with enough S/N level. Consequently, the performance limitation depends mainly on the bandwidth of the transmitted pulse and the S/N level. In addition, while this paper assumes mono-static observation, it is easily extended to a bi-static model, required for a realistic radar constitution, employing the same approach in the proposed method. This is our future task.

p ;p

and

p

in 3-D model.

as the output of the Wiener filter with the transmitted waveform. As similar to the 2-D case, The two sets of range points are and extracted from the local maxima and minima as , respectively. B. Proposed Method for the 3-D Problem This section describes the 3-D model of the proposed method. . The Here, an antenna location is redefined as first scattering point as and the second one as are also redefined as

(26) where and hold. Fig. 18 and in the 3-D model. shows the relationship among Similar to the 2-D model, the following proposition holds. and exist on each range Proposition 2: If point , the next formulations hold:

III. 3-D PROBLEM A. System Model Fig. 17 shows a system model for the 3-D problem. The target model, antenna, and transmitted signal are the same as those assumed for the 2-D problem. The antenna is scanned on the . It assumes a linear polarization in the direction of plane, the -axis. A spherical wave propagation is assumed. R-space . We assume for is expressed by the parameter simplicity. is defined as the received electric field at . is defined the antenna location

(27)

The derivation of this proposition is described in Appendix C. Once is determined, and can be calculated as

(28)

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(29) where denotes an imaginary unit. In this extension, target points redefined as corresponding to the range points as obtained by is also RPM are employed [13]. Each normal vector calculated as (30) Fig. 19. Estimated 3-D image with RPM and the proposed method for the targets as in Fig. 17, when true range points are given.

And then, the parameter vector defined in (14) is constituted by redefining , and upin (11). dating Using the above parameters, the target points in the 3-D model are basically calculated according to the procedure of the proposed method in the 2-D case, as in Section II-C3. Some modifications for the procedure are described below. As in Step is extracted from the output 2), the range point , where the following condition of a Wiener filter satisfies: (31) Also, in Step 3),

Fig. 20. Extracted range points of single and double scattered waves for the targets in Fig. 17.

is calculated as

(32)

where , and points which satisfy .

denotes the number of the range around Fig. 21. Estimated 3-D image with RPM and the proposed method for the targets as in Fig. 17, when range points are extracted from s(X; Y; Z ).

C. Performance Evaluation Using Numerical Simulation This section presents two examples of the proposed method with different target cases, using a numerical simulation. The , where the mono-static radar is scanned for number of locations on each axis is 101. Here, and are set. First, the target boundary is assumed as in Fig. 17. Fig. 19 illustrates the estimated target points obtained by RPM and the proposed method, where the true range points are given by the geometrical optics approximation, similar to 2-D case. This figure verifies that the imaging points express a quite accurate target boundary including the side of the cylindrical objects. This is because the double scattered wave propagates along the side of the toric and cylindrical boundaries. Next, Fig. 20 shows the range points as and extracted from the output of a Wiener filter, which is calculated by FDTD. Figs. 21 and 22 depict the es, timated 3-D image and its cross-section at respectively. These figures show that the obtained image of the proposed method, in this case, accurately creates the side of the cylindrical boundary, which cannot be reconstructed by RPM.

There are some divergent images around the target side, which are basically caused by the errors of range derivatives. This method creates the target points, not the intensified SAR image, which contributes to the identification of the edge or wedge region. Note that, the proposed method requires only 10 s for a full 3-D image after creating the target points with RPM. This amount is prominently reduced from that of the conventional method based on the fifth times integral for imaging after SAR s. processing [15], requiring around For another example, a concave target is shown in Fig. 23. Fig. 24 offers the estimated 3-D boundary performed by the proposed method, where the true range points are given. This verifies that the proposed method accomplishes an accurate target imaging including the side of the concave boundary. Fig. 25 shows the extracted range points from the output of a Wiener filter. Figs. 26 and 27 present the 3-D target image and its cross , respectively, where the received section at data is calculated by FDTD. This figure also proves that the proposed method creates an accurate image around the deep side of

KIDERA AND KIRIMOTO: FAST AND SHADOW REGION 3-D IMAGING ALGORITHM

Fig. 22. Cross-section image of Fig. 21 for

00 1   0 1. :

x

:

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Fig. 26. Estimated 3-D image with RPM and the proposed method for the targets in Fig. 23, when range points are extracted from s(X; Y ; Z ).

Fig. 27. Cross-section image of Fig. 26 for

00 1   0 1. :

x

:

Fig. 23. True concave target boundary.

Fig. 24. Estimated 3-D image with RPM and the proposed method for the targets in Fig. 23, when true range points are given.

Fig. 28. Number of target points for each  in the case of Figs. 21 and 26.

Fig. 25. Extracted range points of single (black) and double (red) scattered waves for targets in Fig. 23.

the concave boundary, which is focused by the double scattered signal. Furthermore, a quantitative analysis for these examples is presented as follows. Fig. 28 shows the number of target points for each accuracy , that is defined as in (25) for the target cases as in Figs. 21 and 26. This figure shows that the proposed and avoids method increases accurate target points around the accuracy distortion for both target cases. The mean value of

this error index as is for the toric and cylindrical tarfor the deep-set concave targets, respectively. gets, and This quantitatively shows that the proposed method enhances the imaging range even in the 3-D model, accomplishing much faster image processing. In addition, an example of a noisy situation is presented. . White Gaussian noise is added to the received signal Fig. 29 shows the estimated target boundaries of the RPM and the proposed method, with S/N around 30 dB. This figure shows that, while the proposed method suffers from image fluctuations caused by random noise, it still offers a significant image expansion around the side of the torus boundary. in this case. Finally, the computational complexities of the algorithms are compared. The conventional SAR based method requires , where and around denote the sampling numbers for the antenna location , and , respectively, and and the spatial coordinates gives the Landau notation. This is because the conventional

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Fig. 29. Estimated 3-D image with RPM and the proposed method for the tardB. gets in Fig. 17 at SN

= 30

wedge region and helps to classify the target structure. Numerical simulations in the 2-D and 3-D models, including multiple objects and concave shaped objects showed that the proposed method substantially extended the imaging range with extremely high speed, even if the model errors and random noises are added to the received data. This exemplifies a significant applicability to the realistic radar sensing scenario. Particularly for times improvement comthe 3-D problem, it is a more than pared with that of the conventional SAR-based method as to the computational complexity. Consequently, this method can significantly contribute to the design of real-time sensors, as found in robots or security systems.

TABLE I CALCULATION TIME (FOR INTEL PENTIUM D 2.8 GHz PROCESSOR) AND COMPUTER COMPLEXITY OF EACH ALGORITHM

APPENDIX A DERIVATION OF (7) is divided into three terms as (33)

SAR-based method should employ a quintuple integration for each image voxel as in [15]. On the contrary, the proposed , since it requires only a method requires around searching operation to the first scattering points obtained by for each range point . Table I RPM shows a comparison for computational times using an Intel Pentium D 2.8-GHz processor with 800-MB memory, and the computational complexity for each method. In this case, and , where . This table shows that the computation each voxel size is times that of required for the proposed method is reduced to the conventional method. Moreover, even if the fast processing of the SAR like omega-k migration [18] were to be adopted to double scattering aperture synthesis, it would have an essential problem that the computation complexity severely depends, in principle, on the voxel size or imaging range. On the contrary, the proposed method, based on range points migration, is quite different from SAR, and it does not need to determine the voxel size or imaging region, owing to the mapping from the observed range points to the target boundary points. The computation required depends only on the number of observed range points, which is on the order of the square of the antenna scanning samples.

is expressed as

Here,

(34) where hold. Each partial derivative of

is given as (35) (36) (37)

where (38) (39) (40)

IV. CONCLUSION This paper proposed a novel imaging algorithm for expanding the imaging range, which efficiently utilizes the range derivative of double scattered waves. The proposed method elicits some inherent characteristics in the RPM method and achieves direct shadow imaging without using any integration process. This method has an outstanding advantage that it accomplishes extremely high-speed imaging by specifying a clear boundary extraction, simultaneously extending the visible region without a priori knowledge of target or surroundings. It has the additional advantage that the target boundary can be expressed as a group of target points, which enables the identification of an edge or

(41) (42) hold. In this case, we assume that the reflection path of the double scattered path satisfies the law of reflection and the following relationships hold: (43) (44)

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(53) (54) (55) (56) and hold, where the similar relationships hold as

Fig. 30. Relationship among e ; e ; e @p p =@X; and @p p =@X .

are used. Then, (57)

Fig. 30 shows the relationship among those parameters. Substituting these equations to (35)–(37), (7) is obtained.

(58) Substituting them to (49)–(51), (27) as to Using the same approach, (27) for

APPENDIX B DERIVATION OF (10) Using expressed as

and

defined in Section II-C,

is (45)

Obviously, the next equation holds from the definition of

: (46)

Squaring the both side of (46) and rearranging as to , (10) is obtained. This relationship obviously holds in 3-D problem.

APPENDIX C DERIVATION OF (27) We introduce the partial derivatives of tions along and axis as

for the two direc-

(47) (48) .

where

and hold. In the similar to the 2-D case in (35)–(37), the following equation holds: as to (49) (50) (51) where

is used, and (52)

is derived. is obtained.

REFERENCES [1] T. C. Williams, J. M. Sill, and E. C. Fear, “Breast surface estimation for radar-based breast imaging system,” IEEE Trans. Biomed. Eng., vol. 55, no. 6, pp. 1678–1686, Jun. 2008. [2] D. W. Winters, J. D. Shea, E. L. Madsen, G. R. Frank, B. D. Van Veen, and S. C. Hagness, “Estimating the breast surface using UWB microwave monostatic backscatterer measurements,” IEEE Trans. Biomed. Eng., vol. 55, no. 1, pp. 247–256, Jan. 2008. [3] D. L. Mensa, G. Heidbreder, and G. Wade, “Aperture synthesis by object rotation in coherent imaging,” IEEE Trans. Nucl. Sci., vol. NS-27, no. 2, pp. 989–998, Apr. 1980. [4] D. Liu, G. Kang, L. Li, Y. Chen, S. Vasudevan, W. Joines, Q. H. Liu, J. Krolik, and L. Carin, “Electromagnetic time-reversal imaging of a target in a cluttered environment,” IEEE Trans. Antenna Propag., vol. 53, no. 9, pp. 3058–3066, Sep. 2005. [5] D. Liu, J. Krolik, and L. Carin, “Electromagnetic target detection in uncertain media: Time-reversal and minimum-variance algorithms,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 4, pp. 934–944, Apr. 2007. [6] J. Song, Q. H. Liu, P. Torrione, and L. Collins, “Two-dimensional and three dimensional NUFFT migration method for landmine detection using ground-penetrating radar,” IEEE Trans. Geosci. Remote Sens., vol. 44, no. 6, pp. 1462–1469, Jun. 2006. [7] F. Soldovieri, A. Brancaccio, G. Prisco, G. Leone, and R. Pieri, “A Kirchhoff-based shape reconstruction algorithm for the multimonostatic configuration: The realistic case of buried pipes,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 10, pp. 3031–3038, Oct. 2008. [8] A. Massa, D. Franceschini, G. Franceschini, M. Pastorino, M. Raffetto, and M. Donelli, “Parallel GA-based approach for microwave imaging applications,” IEEE Trans. Antenna Propag., vol. 53, no. 10, pp. 3118–3127, Oct. 2005. [9] T. Sakamoto and T. Sato, “A target shape estimation algorithm for pulse radar systems based on boundary scattering transform,” IEICE Trans. Commun., vol. E87-B, no. 5, pp. 1357–1365, 2004. [10] S. Kidera, T. Sakamoto, and T. Sato, “High-resolution and real-time UWB radar imaging algorithm with direct waveform compensations,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 11, pp. 3503–3513, Nov. 2008. [11] S. Hantscher, A. Reisenzahn, and C. G. Diskus, “Through-wall imaging with a 3-D UWB SAR algorithm,” IEEE Signal Process. Lett., vol. 15, no. 2, pp. 269–272, Feb. 2008. [12] H. Wang, H. Xue, and M. Ge, “Application and practice of system integration approach in intelligent human motion recognition,” in Proc. Int. Conf. Comput. Sci. Software Eng., 2008, vol. 1. [13] S. Kidera, T. Sakamoto, and T. Sato, “Accurate UWB radar 3-D imaging algorithm for complex boundary without range points connections,” IEEE Trans. Geosci. Remote Sens., vol. 48, no. 4, pp. 1993–2004, Apr. 2010. [14] S. Kidera, T. Sakamoto, and T. Sato, “Super-resolution UWB radar imaging algorithm based on extended capon with reference signal optimization,” IEEE Trans. Antenna Propag., vol. 59, no. 5, pp. 1606–1615, May 2011.

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[15] S. Kidera, T. Sakamoto, and T. Sato, “Experimental study of shadow region imaging algorithm with multiple scattered waves for UWB radars,” in Proc. PIERS’09, Aug. 2009, vol. 5, no. 4, pp. 393–396. [16] J. M. F. Moura and Y. Jin, “Detection by time reversal: Single antenna,” IEEE Trans. Signal Process., vol. 55, no. 1, pp. 187–201, Jan. 2007. [17] G. Shi and A. Nehorai, “Cramer–Rao bound analysis on multiple scattering in multistatic point-scatterer estimation,” IEEE Trans. Signal Process., vol. 55, no. 6, pp. 2840–2850, Jun. 2007. [18] X. Xu, E. L. Miller, and C. M. Rappaport, “Minimum entropy regularization in frequency-wavenumber migration to localize subsurface objects,” IEEE Trans. Geosci. Remote Sens., vol. 41, no. 8, pp. 1804–1812, Aug. 2003. [19] Federal Communications Commission (FCC), Office of Engineering and Technology (OET) Bulletin No. 65, Supplement C, Aug. 1997, p. 35. Shouhei Kidera (M’11) received the B.E. degree in electrical and electronic engineering from Kyoto University, Kyoto, Japan, in 2003 and M.I. and Ph.D. degrees in informatics from Kyoto University in 2005 and 2007, respectively. He is an Assistant Professor in the Graduate School of Informatics and Engineering, University of Electro-Communications, Tokyo, Japan. His current research interest is in advanced signal processing for the near field radar, UWB radar. Prof. Kidera is a member of the Institute of Electronics, Information, and Communication Engineers of Japan (IEICE) and the Institute of Electrical Engineering of Japan (IEEJ).

Tetsuo Kirimoto (M’91–SM’97) received the B.S., M.S., and Ph.D. degrees in communication engineering from Osaka University, Osaka, Japan, in 1976, 1978, and 1995, respectively. From 1978 to 2003, he was with Mitsubishi Electric Corp. studying radar signal processing. From 1982 to 1983, he was a Visiting Scientist at the Remote Sensing Laboratory, University of Kansas. From 2003 to 2007, he was with the University of Kitakyushu as a Professor. Since 2007, he has been with the University of Electro-Communications, Tokyo, Japan, where he is a Professor at the Graduate School of Informatics and Engineering. His current study interests include digital signal processing and its application to various sensor systems. Prof. Kirimoto is a member of the Institute of Electronics, Information, and Communication Engineers (IEICE) and the Society of Instrument and Control Engineering (SICE) of Japan.

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High-Resolution ISAR Imaging by Exploiting Sparse Apertures Lei Zhang, Zhi-Jun Qiao, Member, IEEE, Meng-Dao Xing, Member, IEEE, Jian-Lian Sheng, Rui Guo, and Zheng Bao, Senior Member, IEEE

Abstract—Compressive sensing (CS) theory indicates that the optimal reconstruction of an unknown sparse signal can be achieved from limited noisy measurements by solving a sparsity-driven optimization problem. For inverse synthetic aperture radar (ISAR) imagery, the scattering field of the target is usually composed of only a limited number of strong scattering centers, representing strong spatial sparsity. This paper derives a new autofocus algorithm to exploit the sparse apertures (SAs) data for ISAR imagery. A sparsity-driven optimization based on Bayesian compressive sensing (BCS) is developed. In addition, we also propose an approach to determine the sparsity coefficient in the optimization by using constant-false-alarm-rate (CFAR) detection. Solving the sparsity-driven optimization with a modified Quasi-Newton algorithm, the phase error is corrected by combining a two-step phase correction approach, and well-focused image with effective noise suppression is obtained from SA data. Real data experiments show the validity of the proposed method. Index Terms—Bayesian compressive sensing (BCS), compressive sensing (CS), inverse synthetic aperture radar (ISAR), sparse aperture (SA).

I. INTRODUCTION

D

UE TO the superiorities over other remote sensing tools, such as high probability of target identification, robust performance under all-weather circumstances, and very long operating distance, inverse synthetic aperture radar (ISAR) is widely applied in many civilian and military fields [1], [2]. To realize these applications, the two-dimensional (2-D) high resolution is usually required to characterize target features in detail. In general, high down-range resolution depends on the system bandwidth. To mitigate this dependence, stepped frequency waveforms (SFWs) [3] are employed. High cross-range resolution is obtained by exploiting the multiple diversities of radar-viewing angles to the target, and then Doppler analysis can resolve scattering centers into different Doppler bins. The

Manuscript received January 03, 2011; revised June 16, 2011; accepted July 20, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported by the “973” Program of China under Grant 2010CB731903. The work of Z.-J. Qiao was supported by the U. S. Army Research Office under Grant No. W911NF-08-1-0511 and the Texas Norman Hackerman Advanced Research Program under Grant No. 003599-0001-2009. L. Zhang, M.-D. Xing, J.-L. Sheng, R. Guo, and Z. Bao are with the National Key Laboratory of Radar Signal Processing, Xidian University, Xi’an 710071, China (e-mail: [email protected]). Z.-J. Qiao is with the Department of Mathematics, University of Texas–Pan American, Edinburg, TX 78539-2999 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173130

cross-range resolution depends on both the available CPI and intrinsic motion characteristics of the target. As is known, achieving high cross-range resolution usually requires a long CPI. However, in the situation of multitargets, long observation for a single target imaging is no longer acceptable in a modern radar system, attributed to its multiple functions, such as searching, locating, and tracking multiple targets simultaneously. Since targets may locate in different channels and beams as well as with different velocity vectors, radar system has to switch among different line-of-sight (LOS) angles to capture them. As a result, observation interval is assigned evenly to each target, resulting in sparse apertures (SAs) and gaps in the collected data. SAs would be also introduced in synthetic aperture radar (SAR) imaging with multiply angular diversities [4], [5], where a target is illuminated by several sensors from different angles independently and each sensor collects only a small angular region composing a sparse aperture. In SA-ISAR imaging, if the motion error is eliminated, a simple way to achieve image would be to apply Fourier transform with the missing data set to zero, bringing serious grating lobes in the image. To reduce the discontinuous aperture effects on ISAR imagery, many novel approaches are ready to use. These approaches can be sorted into three groups: 1) CLEAN techniques [6]–[8] treat image formation from SA data as a deconvolution procedure. They estimate and subtract the main lobes of the strong scattering centers iteratively until reach a convergence. CLEAN techniques are usually efficient but sensitive to noise. 2) A number of modern spectral estimate approaches can cope with SA data effectively. They estimate the complex-valued amplitude and position of strong scatterers from gapped data based on interpolation of the missing data under certain constraints. The gapped-data amplitude and phase estimation (GAPES) [4], [9] and its extensions [10], [11] are representative approaches of this group. They can handle quite general SA patterns and perform well under some noisy circumstances. 3) Interpolation and extrapolation algorithms can be also solve the data missing problem in some situations. By fitting the available data into linear predication models, the missed data can be interpolated or extrapolated from the observed data. These methods also apply some modern spectral estimation techniques to obtain the coefficients of the prediction model. See some detailed approaches in [12]–[16]. The conventional approaches usually perform well in coping with SA data in some situations. However, they are more or less sensitive to additive noise and usually take nominal model error into consideration, especially evitable phase errors

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induced by undesired target motion. It should be emphasized that due to the discontinuities between subapertures, current approaches [17]–[21] are not satisfactory to be directly applied for phase adjustment. As the complex phase errors and noise are inevitable, conventional methods for the SA imaging encounter inherent limitations in real applications. Therefore, the error correction should be accounted in dealing with SA imaging to a large extent. Another significant factor, the signal-to-noise ratio (SNR) gain of the formatted image, also plays an important role in SA imagery. In ISAR imaging, radar signal is usually contaminated by strong noise, which is usually overcome by the coherent accumulation processing, providing a high SNR in the final image. It is well known that the SNR gain is proportional to the amount of signal accumulated. In SA-ISAR imaging, due to a large portion of data missed, the negative effect of noise would degrade the performance of current approaches to generate image with high SNR. Notably, the negative effects from both motion error and strong noise should be accounted seriously in SA-ISAR imaging. The ISAR image demonstrates the distribution of strong scattering centers of the target’s scattering field in the range-Doppler (RD) plane. The dominant scattering centers take only a fraction of the whole bins in the plane. In this sense, ISAR image represents strong spatial sparsity in the RD domain. Exploiting such sparsity is meaningful to achieve improved performance, such as super-resolution [22], feature enhancement [23], SAR imaging with model error correction [24], and simplicity of data acquirement [25]. More importantly, recent developing theory of the compressive sensing (CS) tells us that an unknown sparse signal is able to be exactly recovered from a very limited number of measurements with high probability by exploiting the sparsity of signal. This is implemented by solving a -norm optimization problem [26]–[28]. In other words, the ill-posed problem, recovering high-dimensional signal from low-dimensional observations, could be solved by exploiting sparsity of the objective signal. Following this idea, in this paper, we propose a novel algorithm for SA-ISAR imaging. In this algorithm, we focus on image formation from SA data and correction of the phase errors induced by translational motion. The SA-ISAR imaging and model error correction are converted into a problem of solving a sparsity-driven optimization problem corresponding to the maximum a posteriori (MAP) estimate in Bayesian compressive sensing (BCS) [29]. The sparsity-driven optimization is based on the assumption that the additive noise is subject to a zero-mean Gaussian distribution with unknown variance and the signal components corresponding to the dominant scattering centers follow a Laplace distribution with coefficient independently. In the sparsity-driven optimization, the sparsity coefficient is directly related to and . In order to precisely estimate the statistic parameters from SA data, we utilize the constant-false-alarm-rate (CFAR) detector to discriminate signal from noise in the subaperture images approximately. Using the pure noise and target components, both and can be obtained via maximum likelihood (ML). In the SA-ISAR imaging algorithm, the phase adjustment is indispensably required as model error correction, and conventional CS solvers

Fig. 1. SA geometry.

are not directly available herein. Therefore, we apply a modified Quasi-Newton algorithm for image formation jointed with phase adjustment, which is implemented in an iterative manner. A two-step phase adjustment is developed for coarse correction of motion error, which can reduce the phase error in a small level. In order to improve the efficiency of the solver, fast Fourier transform and conjugate gradient algorithm can be applied in its implementation. Real data experiments show that the sparsity-driven algorithm is capable of overcoming the grating lobes and yielding ISAR image with high SNR, even when the observations are very limited. This paper is organized as follows. In Section II, we introduce the SA-ISAR imaging algorithm and the statistic estimation of and . In Section III, the Quasi-Newton solver is presented in detail, together with a two-step phase adjustment for acceleration of the solver. In Section IV, we present results of real data experiments to validate the proposed method, and we give some conclusions in last section. II. SPARSITY-DRIVEN OPTIMIZATION FOR SA-ISAR IMAGING A. Signal Model for SA-ISAR Imaging Considering that an ISAR system observes multiple targets simultaneously, radar illumination has to switch from one target to another evenly, resulting in sparse apertures for each target. At first, conventional range compression and range alignment are applied to the SA data with some current approaches [30]–[33], which are identical to those in the conventional ISAR imagery. The range-compressed and aligned signal is denoted by . Without loss of generality, we assume that there are subapertures for a target consisting of a long sparse aperture. Fig. 1 shows the geometry of the sparse aperture. The full aperture should contain pulses with index from 0 to , and each pulse composes range bins. Suppose that the th subaperture consists of pulses (whose index is from to ). The range-compressed data set corresponding to the th subaperture is given by (1), shown at the bottom of the next page. Then, the SA data matrix is .. . .. .

(2)

We note that the SA data set has pulses. Clearly, in an ideal ISAR data collection, the returned signals can be regarded as a measuring patch of the two-dimensional Fourier transform of the target scattering field corresponding to some aspect angles. Due to the maneuver, the phase errors from complex motion of the target should be

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accounted in signal modeling. Then, the echoed signals with phase error are rewritten in the following form: (3) where is an matrix and denotes the 2-D ISAR image, whose pixel values are corresponding to scattering center amplitudes. is the additive noise matrix with the same size as . stands for a partial Fourier matrix in size , whose construction is corresponding to the SA structure. It is given by

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B. SA-ISAR Imaging via Exploiting Sparsity Generally, the components of are approximated as a zeromean complex Gaussian noise, namely, its imaginary and real parts (denoted by and , respectively) independently follow Gaussian distributions with unknown variance . As a result, its probability density function is given by

.. .

(8) (4)

.. .

The

notation for a matrix denotes . Therefore, we have the Gaussian likelihood model of the observation, which is

where

.. .

.. .

..

.. .

.

(5) is the partial Fourier matrix in accordance with the th subaperture. In our SA-ISAR imaging, the structure of the sparse aperture of a certain target is assumed to be obtained. In other words, we can construct the exact partial Fourier matrix of in advance. is an matrix and represents the phase errors from pulse to pulse (6)

(9) ISAR imagery demonstrates the distribution and amplitudes of limited dominant centers of the target, which usually represents strong sparsity. According to Bayesian compressive sensing [29], the sparsity can be formulized by placing a sparseness-promoting prior on . Herein, this sparseness prior is represented by the Laplace density function. (10) . Then, SA-ISAR imwhere agery is shifted into a classical problem to estimate from noisy observation . For this purpose, the MAP estimator is used, which is given by

where

(11) .. .

Using the Bayes rule, one gets (7)

.. .

(12) Clearly, (11) is also equivalent to

and denotes the phase error vector corresponding to the th subaperture.

.. .

.. .

(13)

.. .

.. .

(1)

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Substituting (9) and (10) into (13), the MAP estimator becomes

(14) where is the sparsity coefficient, which is directly related to the unknown statistic of noise and target signal. The optimization problem consists of two different terms: The -norm preserves the data fidelity of the solution, and the -norm imposes it to be sparse. Clearly, based on the assumption of Gaussian and Laplace distributions, the MAP estimator of SA-ISAR imagery is corresponding to an -norm regularization optimization problem, which is often called basis pursuit denoising (BPDN) [34]. BPDN is often concerned as a well-suited estimator of sparse signal from limited measurements. It also can overcome the noise interference effectively. Clearly, different from conventional SA imaging algorithms, the MAP estimator aims at reconstructing denoised image with full resolution. Compared to the point-enhanced algorithm [23], the sparsity-weighting coefficient in the optimization (14) has an analytic expression in mathematics: It is directly associated with the statistics of noise and target in a Bayesian sense. However, besides the unknown phase error, a significant problem of solving (14) for SA imaging lies in the determination of . Only when a precise value of the sparsity coefficient is given can we reconstruct an optimal ISAR image from limited noisy SA data. Nevertheless, if is set too large, weak scatterers together with noise will be rejected in the reconstruction of image, and only the dominant scattering centers are preserved. If is overly small, then a significant part of noise elements may be left in the image, degrading the image quality. Derived from the MAP estimation, the sparsity coefficient is deterministic if we have the prior information about and . In Section II-C, we propose an approach to estimate them from subaperture images. C. Estimation of

and

The estimation of and from SA data can be used as prior information. Estimation of the noise variance is available since Gaussian noise usually distributes evenly, and there exists a large number of cells containing only noise in the RD plane. Given enough noise samples by those pure noise bins, we can estimate with high accuracy. Meanwhile, of the Laplace distribution placed on can be estimated from the signal bins. Herein, the estimation of the statistical parameters of noise and ISAR image contains the following two steps. Step 1) SA images are generated by conventional ISAR imaging procedure. Noise variance is estimated by using pure noise samples. Step 2) Scattering centers of target are determined by CFAR detector in the SA images. Meanwhile, the noise bins are set to be zero, and maximum likelihood estimation of is performed by using the denoised SA images. In Step 1, we first perform conventional imaging processing to each SA data, including translational motion removal and

azimuth compression. As conventional range alignment and phase adjustment are suitable to correction of translational motion within the subaperture data, each self-organized subaperture is ready to generate a low-resolution image with the range-Doppler algorithm. In the SA image, the target is placed round the zero Doppler as the Doppler shift is removed in the translational motion compensation, which indicates that the cells corresponding to high Doppler frequency contain noise only. Herein, we use these bins as noise samples. As we assume the noise follows zero-mean Gaussian distribution, the ML estimator of is the variance of all real and imaginary parts of noise samples (defined as ). Because there are SA images, one usually has enough noise samples, and trends to the exact value. In Step 2, the first is the detection process to separate scattering centers from noise in the SA images. Due to high SNR gained from the 2-D coherent integration, strong scattering centers are distinctive from noise in the SA images. Herein, discriminating target bins from noise in the SA images corresponds to a problem of distributed target detection under the background of Gaussian noise. Utilizing the noise samples from Step 1, a CFAR detector for target scatters detection is straightforward. Then, we can use these noise samples to develop a CFAR detector for strong scattering centers. For an extensive study of CFAR detector, [35] and [36] can be consulted. Then, bins with amplitudes larger than the CFAR threshold are determined as target components, and the rest are regarded as noise and set to be zero. After the above denoising processing, the th SA image is defined as . The ML estimator of is found by maximizing

(15) It is equivalent to maximizing with respect to produces

. Differentiating

(16) Setting (24) equal to zero yields the ML estimator of the th SA image (17)

Clearly, the ML estimator of is the the reciprocal of the mean of all pixel values. Finally, we average the estimates of all SA images to obtain the estimation of , which gives (18) and , the sparsity coefficient is obtained easily, By using which is given by . For clarity, we give conceptual flowchart by using the Yak-42 data in Fig. 2.

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This problem is a 2-D optimization. To make it is easy to solve. We first convert it to one-dimensional (1-D) problems. The optimization (20) can be rewritten as

(21) Fig. 2. Statistic estimation from SA image.

Generally, as we use the target samples from subaperture images to estimate the statistics of the full aperture image, some estimation error is inevitable. However, in real applications, we find performs well in different situations, although it usually trends to be excessively large. A drawback of the estimator for lies in the need of a considerable amount of pulses in a subaperture, which may limit its usage in some special cases, such as very short SA and random sampled SA patterns. In these cases, the high SNR gain by the 2-D coherent integration may not be achievable, and thus signal components are submerged by the strong noise, which will be studied in our future work. III. MODIFIED QUASI-NEWTON SOLVER IMAGING

FOR

SA-ISAR

and denote the SA signal where of the th range cell and the th column of the high resolution 2-D image, respectively. Due to the independence between the range cells, solving the 2-D optimization (20) is equivalent to figure out the following 1-D optimization for all range cells separately. For the reconstruction of the th column of , we have the following optimization:

(22) Hence, the conjugate gradient function of with respect to culated through

A. Modified Quasi-Newton Solver for Image Formation and Phase Adjustment Clearly, formation of the full-aperture resolution image by SA data is an ill-posed problem. Due to the data missing among the subapertures, the pulse amount is much less than that of Doppler bins of the high-resolution image. According to the theory of the compressive sensing, SA-ISAR imagery is possible by solving the -norm optimization problem in (13), and it is also widely accepted that the problem is equivalent to the -norm constraint optimization in compressive sensing [37], for which many efficient solvers are available [38]–[40]. Nevertheless, these methods are not directly applicable to (13) due to the phase errors induced by unexpected target motion. In this section, we present a modified Quasi-Newton solver for ISAR imaging from SA data, joint with correction of phase error from translational motion. At first, in order to overcome the nondifferentiability of the -norm around the origin in (13), a useful approximation [23], [41], [42] is employed by

is cal-

(23) where the Hessian matrix is approximately given by

(24) and

(25) , Because the Hessian approximation relies on the objective an iterative solver to (22) is presented through the following formula:

(19) (26) where stands for the modulus operator, and is a small nonnegative parameter. Clearly, to ensure the approximation as rigid as possible, should be set small. Thus, the MAP estimator of the image in (13) can be reformatted as

(20)

where and are the estimators of and in the th iteration, respectively. To accelerate the update, conjugate gradient algorithm (CGA) can be applied to avoid the matrix inverse calculation. In the case of no prior information about the phase error, starting from the initial value and , then we have the estimator of the phase error in the th iteration (denoted by ) (27)

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Here, the updating exponential term of the phase error is given by

(28)

and (29) In the sparstiy-driven SA-ISAR algorithm, we consider the phase error among the pulses. The phase error is corrected during the image formation in an iterative manner. It should be noted that there are no constraints on concrete form of phase errors in the solver, and even when the phase errors vary randomly from pulse to pulse, it is capable of achieving high quality SA-ISAR images. The computational load of the Quasi-Newton solver is a significant to its real applications. We note that major computational load of in each update sources from the matrix inversion calculation of in (24), which is implemented by CGA. However, due to the iterative property of CGA, its efficiency may be slow as one need to perform the calculation of the linear equation many times. Its major computational load lies in the multiplication of in . As the term corresponds to the partial Fourier matrix, allowing us to use fast Fourier transform (FFT) to implement ( denotes an -dimensional vector) efficiently: We perform the inverse FFT to and obtain , then set the components corresponding to the vacant apertures to zero and followed FFT. For simplicity, only the multiple operations are accounted. Therefore, taking only the multiple operations into account, can be implemented with only flops corresponding to two FFTs. For the number of the CGA iterations to solve (26) being case, the computational cost of CGA is about flops. Assuming there are times of iterations in the Quasi-Newton solver, the computational cost by using FFT is flops approximately. B. Efficiency Improvement by Combining Conventional Phase Adjustment The major problem of the Quasi-Newton solver in dealing with severe phase errors lies in its low efficiency. From the viewpoint of optimization, appropriate initialization of and is essential to improve the efficiency and accuracy of the Quasi-Newton solver. Precise initialization can dramatically reduce the iteration number to achieve a satisfactory solution to the optimization problem. However, precise prior information of and is usually not achievable. Herein, the initialization of is achieved by setting the vacant apertures to zero and applying FFT. Preprocessing of motion compensation should be carried out to suppress the phase errors as much as possible, which would put much less burden on the Quasi-Newton solver and enhance its convergence with much less iterations. Due to the data discontinuity, phase adjustment for the SA data is quite

different from the conventional ISAR scenarios. Herein, we propose two-step preprocessing for phase error reduction. In the first step, we utilize the Doppler tracking technique. Without assumption of even and full aperture, the Doppler tracking technique still works by multiple prominent points processing (PPP) [2]. The basic idea of the multiple PPP algorithm is to track the phase history of one or more isolated point-scatters in aligned range profiles in order to extract phase errors. The main challenge in applying the multiple PPP algorithm is the selection of the point-like prominent scatterers, which should be well isolated in their respective range cells. It is easily found that this method performs well in artificial targets with dominant scatters, for instance, airplanes, missiles and ships, etc. The PPP procedure includes three steps: 1) searching for one or several reference range cells by using some criteria like minimum variance; 2) taking conjugate phase at the reference range cells and combining them together by weighting; and 3) making phase correction for all range cells by the conjugate phase. However, in the presence of strong noise or absence of prominent points, the precision of the Doppler tracking may degrade. Therefore, Doppler tracking by the multiple PPP serves as a coarse step. In the second step, we perform conventional phase adjustment to each subaperture to suppress phase errors within subaperture effectively. Note that each subaperture has a self-organized structure where the pulses are distributed continuously and evenly. Therefore, we can apply precise phase adjustment to eliminate the residual phase errors for each subaperture. Many phase adjustment algorithms could be applied to implement this step, such as the weighted least-squares phase estimation (WLSPE) [18] and the time-frequency transform-based auto-focusing [19]. This step can be regarded as a fine step for the phase error correction. Nevertheless, phase correction is performed on each subaperture independently, and thus phase errors within a subaperture can be eliminated to a nominal level. WLSPE is robust to noise, and as it directly extracts phase error rather than phase gradient, there is no significant additional linear phase in each SA data. In real application, we always apply it to implement the fine correction. However, residual phase errors still exist among different subapertures. By two-step preprocessing for phase adjustment, the majority of phase errors are removed, and only a fraction of them are left for the sparsity-driven algorithm, improving its efficiency dramatically. Considering the phase error difference within a subaperture is removed in the second step, for the th subaperture, we have (30) In the phase error estimations (27) and (28) of the sparsitydriven SA-ISAR imaging, we may reformat the update of phase error for the th subaperture in a simpler way

(31)

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Fig. 4. Aligned range profiles.

Fig. 3. Flowchart of the SA-ISAR imaging.

Here, the updating exponential term of the phase error is reformatted as

(32) and (33) where denotes the operator to sum up all the matrix elements; is the conjugate operator, and represents Hadamard multiplication. The convergence for the iteration is straightforward. Let increase, then we may repeat the iterative procedure in the optimization problem until we have (34) where the constant is chosen as small as the predetermined threshold. Additionally, we can terminate the iteration, when exceeds a predetermined number. Due to the two-step phase adjustment, the residual phase error is small. Therefore, an optimal solution can be obtained by the Quasi-Newton algorithm with only several iterations. For example, in the following experiments, only five iterations are used for the SA with a quarter of pulses missing. To make it clear, a flowchart for the SA-ISAR imaging is given in Fig. 3. IV. PERFORMANCE ANALYSIS WITH EXPERIMENTS In this section, real ground-based measurements are used to generate synthetic data for carrying out a performance analysis of the SA-ISAR imaging by the sparsity-driven optimization. Accounting for the special cases of the ISAR imaging with noisy SA measurements, the performance analysis is carried out by considering two aspects: the phase error and the sparse aperture pattern. The experiments here are vital to validate the effectiveness of our approach.

Fig. 5. RD image after auto-focusing.

A. Data Set and Experimental Conditions In our algorithm for the SA-ISAR imaging, the sparsity of the target scattering field is exploited to overcome the model error and form a well-focused image. We believe that the inherent sparsity of a real ISAR target is difficult to be represented by simple simulated data. To make it convincing, we utilize the real measured ISAR data to perform different experiments. A data set of Yak-42 airplane is recorded by a C-band (5.52 GHz) ISAR experimental system. The system transmits 400-MHz linear modulated chirp signal with 25.6- s pulse width, providing a range resolution of 0.375 m. The received signal is dechirped and I/Q sampled for range compression. We notice that since tracking errors are involved in the reference distances for the dechirping on receiving, random initial phase is introduced for each pulse. Range alignment and phase adjustment are required before we perform azimuth compression to the full-aperture data. The pulse repetition frequency is 400 Hz without undersampling. The data set consists of 1024 pulses. Conventional range alignment is applied to the data set eliminating the MTRC. The aligned profiles are shown in Fig. 4. From the aligned profiles, we know that several prominent scattering centers are available. For comparison, we apply the WLSPE to the data set and then generate the RD image shown in Fig. 5. The generated image is well focused, which can be used as a standard image for evaluating experimental results from sparse apertures. The estimated phase errors from a full

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Fig. 6. Full aperture phase errors with WLSPE.

Fig. 7. Target region.

aperture are shown in Fig. 6, from which we note that due to the complex motion and dechirping on receiving, the phase errors are in a random pattern. In the following SA-ISAR imaging experiments, these phase errors are overcome by the two-step phase correction and the sparsity-driven imaging algorithm jointly. Phase error estimation in Fig. 6 is utilized as the criterion to evaluate the precision of the SA phase adjustment. To provide a quantitative evaluation for the following SA-ISAR imaging experiment, we consider two metrics. The first evaluation metric could be the target-to-background ratio (TBR). By applying an adaptive threshold on the full-resolution image in Fig. 5 to separate the target and the background regions and then counting the target energy (within the target region) and noise energy of the reconstructed image, the TBR is given by

(35)

and are the predetermined target and background where region shown in Fig. 7. It can also measure the target energy preservation with the help of the target region. Herein, we use the signal energy within the target region as the other metric, which is given by

(36) In the following SA imaging experiment, both TBR and SE are utilized as the quantitative metric to evaluate the SA image quality. B. SA-ISAR Imagery Comparison With GAPES In this section, we simulate the SA data of one target collected by a radar system observing multiple targets. In this scenario, the data amount corresponding to one target decreases along with the increase of the target amount. In the following experiment, we extract echoes from the complete aperture data

TABLE I ESTIMATED AND IDEAL IN SA1 PATTERN

set of Yak-42 plane as SA samples for simulation. The SAs with 512, 256, and 128 pulses are regarded as Case 1, 2, and 3 respectively. To test the robustness of the approach, we add complex-valued Gaussian noise into the SA data sets to generate different SNRs (20, 10, and 5 dB). Herein, the SNR is defined as the energy ratio between the original data set and the added noise. In our experiments, we consider that the pulse amount within a subaperture is 128. Collecting 128 pulses is achieved within a very short observation for a conventional ISAR system (0.32 s), which should not conflict with other radar activities including tracking and locating for multiple targets. In all experiments, the SA-ISAR imaging procedure in Fig. 3 is applied. The weights in the optimizations are estimated with . For the CFAR-based approach, and the CFAR is set to the purpose of comparison, we also provide the ideal sparsity coefficients calculated by using the ideal ML-estimated from the image in Fig. 5 and the real noise variances under different SNRs. The estimated sparsity coefficients and the ideal ones are all listed in Table I. Clearly, there is some difference between the estimated and ideal sparsity coefficients; one can note that the difference within a single SA are very small. By using the estimated sparsity coefficients, optimizations under different SNRs and SA cases are developed. For all cases above, we first exploit the two-step phase adjustment to reduce the phase error. However, by comparing the results to those via WLSPE in Fig. 6, there still exist residual phase errors, as plotted in red in Fig. 8. Note that we only show the outcomes of Cases 1 and 2 since the results of Case 3 are identical to those in the first SA of the other two cases correspondingly. Although the residual phase errors are small within a single SA, among different SAs they vary in a large range almost one radian difference. The proposed sparsity-driven algorithm with (32) is expected to correct the rest phase difference

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Fig. 8. Phase adjustment evaluation.

Fig. 9. Results with the proposed approach.

between SAs. The residual phase errors via sparsity-driven correction with 20 iterations are plotted in blue in Fig. 8. It is explicit to see that since the constant difference between different SAs is removed effectively via sparsity-driven correction, the residual phase errors are at the same level. Therefore, the phase difference becomes nominal which promises good performance of imaging. Fig. 9 shows the SA imaging results by using the proposed approach under different SNRs. The first column of Fig. 9 gives the sparse aperture patterns with different SA numbers (4, 2, and 1). Different rows in Fig. 9 correspond to imaging results with different cases. The second, third, and right columns give the

20, 10, and 5 dB, respectively. imaging results under One notes that, in all cases, well-focused images are achieved, which validates the effectiveness of our algorithm. For comparison, we also use the GAPES to process SA data under the same conditions. It should be emphasized that, as GAPES requires no phase error within the SA data, the phase error is precorrected before we extract SAs from original data in Fig. 6. The image results obtained by GAPES are given in Fig. 10. For both SA approaches, the decrease of measurements amount yields some noise increase in the reconstructed image, as we can see from Figs. 9 and 10. However, the sparsity-driven SA-ISAR imaging generally removes major noise producing image with

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Fig. 10. Results with GAPES.

TABLE II TBR OF IMAGES (dB)

TABLE III SE OF IMAGES (dB)

V. CONCLUSION In this paper, we present a sparsity-driven algorithm to generate high-resolution ISAR images with sparse apertures, in which SA-ISAR imaging problem is converted into a sparsity-constrained optimization based on Bayesian compressive sensing. By using conventional Doppler tracking and autofocus, a two-step preprocessing for phase adjustment is developed to improve the efficiency and precision of the sparsity-constrained SA-ISAR imaging effectively. Real data experiments and the results manifest the effectiveness of the proposed approach in different conditions. For the issue of SA-ISAR imaging, there are still some open problems. For example, the SA imaging for maneuvering targets may be much involved, and distributed ISAR can also generate SA data with very short CPI [43], but the synchronization is a significant problem. They remain to be carried out in the future work. ACKNOWLEDGMENT

much higher TBRs than GAPES as one can note from Table II. High TBR indicates that the sparsity-driven method has prominent capability of denoising. In Table III, we list the signal energy of the reconstructed images. Generally, GAPES via images has identical signal energy, while the sparsity-driven approach generates images with lower signal energy, which indicates that to achieve the denoising performance, the sparsity-driven approach pays a price of some signal energy loss. However, loss of some weak scatters will not affect the geometrical construction of ISAR image, therefore we believe the sparsity-driven approach for SA-ISAR imaging is useful in real applications.

The authors thank the anonymous reviewers for their valuable comments to improve the paper quality. REFERENCES [1] D. L. Mensa, High Resolution Radar Imaging. Dedham, MA: Artech House, 1981. [2] W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Synthetic Aperture Radar-Signal Processing and Algorithms. Boston, MA: Artech House, 1995. [3] Q. Zhang and Y. Q. Jin, “Aspects of radar imaging using frequencystepped chirp signals,” EURASIP J. Appl. Signal Process., vol. 2006, pp. 1–8, 2006, Article ID 85823. [4] E. G. Larsson, G. Q. Liu, P. Stoica, and J. Li, “High-resolution SAR imaging with angular diversity,” IEEE Trans. Aerosp. Electron. Syst., vol. 37, no. 4, pp. 1359–1372, Oct. 2001.

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[5] J. Salzman, D. Akamine, R. Lefevre, and J. C. Kirk, Jr., “Interrupted synthetic aperture radar (SAR),” IEEE Aerosp. Electron. Syst. Mag., vol. 17, no. 5, pp. 33–39, May 2002. [6] J. A. Hogbom, “Aperture synthesis with non-regular distribution of interferometer baselines,” Astron. Astrophys. Suppl., vol. 15, pp. 417–426, 1974. [7] J. Tsao and B. D. Steinberg, “Reduction of sidelobe and speckle artifacts in microwave imaging: The CLEAN technique,” IEEE Trans. Antennas. Propag., vol. 36, no. 4, pp. 543–556, Apr. 1988. [8] R. Bose, A. Freeman, and B. D. Steinberg, “Sequence CLEAN: A modified deconvolution technique for microwave images of contiguous targets,” IEEE Trans. Aerosp. Electron. Syst., vol. 38, no. 1, pp. 89–96, Jan. 2002. [9] E. G. Larsson, P. Stoica, and J. Li, “Amplitude spectrum estimation for two-dimensional gapped data,” IEEE Trans. Signal Process., vol. 50, no. 6, pp. 1343–1353, Jun. 2002. [10] E. G. Larsson, P. Stoica, and J. Li, “Spectral analysis of gapped data,” in Proc. 34th Asilomar Conf. Signals, Syst. Comput., 2000, vol. 1, pp. 207–211. [11] E. G. Larsson and J. Li, “Spectral analysis of periodically gapped data,” IEEE Trans. Aerosp. Electron. Syst., vol. 39, no. 3, pp. 1089–1097, Jul. 2003. [12] H. J. Li, N. H. Farhat, and Y. S. Shen, “A new iterative algorithm for extrapolation of data available in multiple restricted regions with applications to radar imaging,” IEEE Trans. Antennas. Propag., vol. AP-35, no. 5, pp. 581–588, May 1987. [13] S. D. Cabrera and T. W. Parks, “Extrapolation and spectral estimation with iterative weighted norm modification,” IEEE Trans. Signal Process., vol. 39, no. 4, pp. 842–851, Apr. 1991. [14] K. M. Cuomo, J. E. Piou, and J. T. Mayhan, “Ultrawide-band coherent processing,” IEEE Trans. Antennas Propag., vol. 47, no. 6, pp. 1094–1107, Jun. 1999. [15] X. Xu and X. Feng, “SAR/ISAR imagery from gapped data: Maximum or minimum entropy,” in Dig. IEEE AP-S Int. Symp. URSI Meeting, Washington, DC, Jul. 2005, vol. 4B, pp. 122–125. [16] X. Xu, P. Huang, and X. Feng, “An iterative algorithm for ultra wideband radar imaging from randomly fragmented spectral data,” in Proc. SEE Radar Conf., France, Oct. 2004, pp. 668–671. [17] D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and C. V. Jakowatz, Jr., “Phase gradient autofocus—A robust tool for high resolution SAR phase correction,” IEEE Trans. Aerosp. Electron. Syst., vol. 30, no. 3, pp. 827–834, Jul. 1994. [18] W. Ye, T. S. Yeo, and Z. Bao, “Weighted least-squares estimation of phase errors for SAR/ISAR autofocus,” IEEE Trans. Geosci. Remote Sens., vol. 37, no. 5, pp. 2487–2494, Sep. 1999. [19] V. C. Chen and S. Qian, “Joint time-frequency transform for radar range-Doppler imaging,” IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 2, pp. 486–499, Apr. 1998. [20] F. Berizzi and G. Cosini, “Autofocusing of inverse synthetic aperture radar images using contrast optimization,” IEEE Trans. Aerosp. Electron. Syst., vol. 32, no. 3, pp. 1185–1191, Jul. 1996. [21] X. Li, G. S. Liu, and J. L. Ni, “Autofocusing of ISAR images based on entropy minimization,” IEEE Trans. Aerosp. Electron. Syst., vol. 35, no. 4, pp. 1240–1251, Oct. 1999. [22] L. Zhang, M. Xing, C. Qiu, J. Li, and Z. Bao, “Achieving higher resolution ISAR imaging with limited pulses via compressed sampling,” IEEE Geosci. Remote Sens. Lett., vol. 6, no. 3, pp. 567–571, Jul. 2009. [23] M. Çetin and W. C. Karl, “Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization,” IEEE Trans. Image Process., vol. 10, no. 4, pp. 623–631, Apr. 2001. [24] N. O. Onhon and M. Çetin, “A nonquadratic regularization-based technique for joint SAR imaging and model error correction,” in Proc. SPIE, Algor. Synthetic Aperture Radar Imagery XVI, 2009, vol. 7337, p. 73370C. [25] A. C. Gurbuz, J. H. McClellan, and W. R. Scott, “A compressive sensing data acquisition and imaging method for stepped frequency GPRs,” IEEE Trans. Signal Process., vol. 57, no. 7, pp. 2640–2650, Jul. 2009. [26] E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006. [27] E. Candès, J. Romberg, and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006. [28] D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 5406–5425, Apr. 2006.

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[29] S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process., vol. 56, no. 6, pp. 2346–2356, Jun. 2008. [30] C. C. Chen and H. C. Andrews, “Target-motion-induced radar imaging,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-16, no. 1, pp. 2–14, Jan. 1980. [31] J. Wang and D. Kasilingam, “Global range alignment for ISAR,” IEEE Trans. Aerosp. Electron. Syst., vol. 39, no. 1, pp. 351–357, Jan. 2003. [32] T. Sauer and A. Schroth, “Robust range alignment algorithm via Hough transform in an ISAR imaging system,” IEEE Trans. Aerosp. Electron. Syst., vol. 31, no. 3, pp. 1173–1177, Jul. 1995. [33] D. Zhu, L. Wang, Y. Yu, Q. Tao, and Z. Zhu, “Robust ISAR range alignment via minimizing the entropy of the average range profile,” IEEE Geosci. Remote Sens. Lett., vol. 6, no. 2, pp. 204–208, Apr. 2009. [34] S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev., vol. 43, no. 1, pp. 129–159, 2001. [35] H. Rohling, “Radar CFAR thresholding in clutter and multiple target situations,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-19, no. 4, pp. 608–621, Jul. 1983. [36] G. Davidson, “Matlab and c radar toolbox,” 2003 [Online]. Available: http://www.radarworks.com [37] C. W. Zhu, “Stable recovery of sparse signals via regularized minimization,” IEEE Trans. Inf. Theory, vol. 54, no. 7, pp. 3364–3367, Jul. 2008. [38] J. F. Sturm, “Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones,” Tilburg Univ., Tilburg, The Netherlands, Tech. rep., Aug. 1998–Oct. 2001. [39] D. L. Donoho, I. Driori, V. C. Stodden, and Y. Tsaig, “Sparselab,” 2007 [Online]. Available: http://sparselab.stanford.edu/ [40] M. Grant, S. Boyd, and Y. Ye, “cvx: Matlab software for disciplined convex programming,” 2011 [Online]. Available: http://www.stanford. edu/~boyd/cvx/ [41] C. R. Vogel and M. E. Oman, “Fast, robust total variation-based reconstruction of noisy, blurred images,” IEEE Trans. Image Process., vol. 7, no. 6, pp. 813–824, Jun. 1998. [42] R. Acar and C. R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Prob., vol. 10, pp. 1217–1229, 1994. [43] D. Pastina, M. Bucciarelli, and P. Lombardo, “Multistatic and MIMO distributed ISAR for enhanced crossrange resolution of rotating Targets,” IEEE Trans. Geosci. Remote Sens., vol. 48, no. 8, pp. 3300–3317, Aug. 2010.

Lei Zhang was born in China in 1984. He received the B.S. degree in mechanism and electrical engineering from Chang’an University, Xi’an, China, in 2006, and is currently pursuing the Ph.D. degree in signal processing at the National Key Laboratory of Radar Signal Processing, Xidian University, Xi’an, China. His major research interest is radar imaging (SAR/ ISAR).

Zhijun Qiao (M’10) received the Ph.D. degree in applied math from the Institute of Mathematics, Fudan University, Shanghai, China, in 1997, wherein his dissertation was one of the first 100 excellent Ph.D. dissertations awarded in 1999. From 1999 to 2001, he was a Humboldt Research Fellow with the Department of Mathematics and Computer Science, University of Kassel, Kassel, Germany. From 2001 to 2004, he was a Researcher with the Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM. He was also a Professor with the Department of Mathematics, Liaoning University, Shenyang City, China, since 1997. Currently, he is the PI of two grants under the Department of Defense program and the Norman Hackerman Advanced Research Program. He is currently with the Department of Mathematics, The University of Texas-Pan American, Edinburg. He is currently the Editor-in-Chief of the Pacific Journal of Applied Mathematics. He has published two monographs and more than 90 articles in peer-reviewed international journals. His research interests include nonlinear partial differential equations and its application in radar imaging.

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Mengdao Xing (M’04) was born in China in 1975. He received the Bachelor’s and Ph.D. degrees in electrical engineering from Xidian University, Xi’an, China, in 1997 and 2002, respectively. He is currently a Full Professor with the National Key Laboratory for Radar Signal Processing, Xidian University. His research interests include SAR, ISAR, and over-the-horizon radar (OTHR).

Rui Guo was born in China in 1985. She received the B.S. degree in electrical engineering from Xidian University, Xi’an, China, in 2007, and is currently pursuing the Ph.D. degree in signal processing at the National Laboratory for Radar Signal Processing, Xidian University. Her major research interests are radar imaging and image processing, especially polarimetric synthetic aperture radar.

Jialian Sheng was born in China in 1987. She received the B.S. degree in electrical engineering from Xidian University, Xi’an, China, in 2010, and is currently pursuing the Ph.D. degree in signal processing at the National Laboratory for Radar Signal Processing, Xidian University. Her major research interests are radar signal processing and radar imaging.

Zheng Bao (M’80–SM’90) was born in Jiangsu, China. He received the Bachelor’s degree in radar engineering from Xidian University, Xi’an, China, in 1953. He is currently a Professor with Xidian University and the Chairman of the Academic Board of the National Key Laboratory of Radar Signal Processing. He has authored or coauthored six books and published over 300 papers. Currently, his research fields include space-time adaptive processing (STAP), radar imaging (SAR/ISAR), automatic target recognition (ATR) and over-the-horizon radar (OTHR) signal processing. Prof. Bao is a member of the Chinese Academy of Sciences.

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Nondestructive Material Characterization of a Free-Space-Backed Magnetic Material Using a Dual-Waveguide Probe Milo W. Hyde, IV, Member, IEEE, Michael J. Havrilla, Senior Member, IEEE, Andrew E. Bogle, Member, IEEE, and Edward J. Rothwell, Fellow, IEEE

Abstract—A free-space-backed dual-waveguide probe measurement technique is introduced to determine nondestructively the complex permittivity and permeability of an unknown material. The purpose of this new measurement technique is to complement the existing PEC-backed dual-waveguide probe material-characterization method. Provided in this paper is the theoretical development of the new technique and its experimental validation. It is shown, by applying Love’s equivalence theorem, that a system of coupled magnetic field integral equations can be formulated and subsequently solved for the dominant mode reflection and transmission coefficients using the method of moments. Also included in the theoretical development of the new technique is a derivation of the dyadic Green’s function for a magnetic-current-excited two-medium grounded-slab environment. Last, experimental complex permittivity and permeability parameters extracted for two magnetic-shielding materials are presented and analyzed to validate the new technique. Index Terms—Green’s function, integral equations, microwave measurements, moment methods, open-ended, parallel-plate waveguides, permeability measurement, permittivity measurement, waveguides.

I. INTRODUCTION HE properties of waveguide probes, whether they be rectangular, circular, or coaxial guides, have been the subject of significant research over the past several decades [1]–[7]. Published applications of waveguide probes include nondestructive evaluation/nondestructive inspection (NDE/NDI) characterization of solids and liquids [8]–[21], surface and subsurface crack detection and characterization [22]–[26], and in vivo characterization of biological tissues [27], [28]. The vast majority of the published literature regarding waveguide probes describes single-probe techniques. While very well suited to characterize dielectric media, i.e., determine permittivity , single-probe characterization techniques suffer when one

T

Manuscript received January 24, 2011; manuscript revised June 03, 2011; accepted August 22, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. The views expressed in this paper are those of the authors and do not reflect the official policy or position of the U.S. Air Force, the Department of Defense, or the U.S. Government. M. Hyde, M. Havrilla, and A. E. Bogle are with the Department of Electrical and Computer Engineering, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433 USA (e-mail: [email protected]). E. J. Rothwell is with the Department of Electrical and Computer Engineering, Michigan State University, East Lansing, Michigan 48824 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173136

wants to fully characterize a material with both magnetic and as well as electric properties (i.e., determine permeability ). A few techniques—most notably the two-thickness method [21], [29]–[31], two-layer method [17], [18], [20], [31], [32], frequency-varying method [31], [33], and the short/free-space method [17], [18], [20]—have been published to address this single-probe shortfall. Unfortunately, these techniques are not always applicable and, in some circumstances, are numerically unstable [32]. A better approach would be to use a material-characterization apparatus which permits both and transmission coefficients to be the reflection and are independent simultaneously measured. Since and of the material under test over all frequencies, the (MUT) can be determined unambiguously at every data point. Currently, two such measurement geometries exist, namely, the flanged-waveguide measurement geometry (utilizing a rectangular waveguide [34] or utilizing a coaxial waveguide [35]) and the dual-waveguide probe (DWP) geometry [36] (the focus of this paper). The DWP measurement geometry, as introduced in [36], is shown in Fig. 1(a). The structure consists of two rectangular waveguides attached to an infinite PEC flange. The MUT is assumed to be PEC backed. The authors derive theoretical expresand by replacing the waveguide apertures sions for with equivalent magnetic currents in accordance with Love’s equivalence principle. By enforcing the continuity of the transverse magnetic field at the waveguide apertures, a system of coupled magnetic field integral equations is derived, which when and . The solved via the method of moments, yields and of the MUT are then found via numerical inversion and using the Newton–Raphson method or nonof linear least squares. It is found via measurement that using the due to a strong inDWP provides very accurate values of terrogating magnetic field, but difficulty arises in determining . The -measurement sensitivity is attributed to the fact that since the MUT is typically electrically thin and the predominately transverse electric field in the MUT/parallel-plate region of the DWP is forced to zero at the PEC waveguide walls, only a small interrogating electric field exists in the MUT/parallel-plate region to measure permittivity [36]. In this paper, the structure in Fig. 1(b) is proposed to address the DWP -measurement sensitivity referenced above. This structure is very similar to the DWP geometry depicted in Fig. 1(a); however, in this approach, the MUT is assumed

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A. Electric and Magnetic Field Distributions and The first step in deriving expressions relating to and is to find expressions for the transverse electric and magnetic fields in the waveguide and MUT regions of Fig. 1(b). The transverse fields in the waveguide regions of Fig. 1(b) can and rectangular be expressed as a summation of waveguide modes [40]:

(2)

Fig. 1. (a) PEC-backed DWP measurement geometry as analyzed in [36]. (b) Free-space-backed (FS-backed) DWP measurement geometry analyzed in this paper.

to be free-space (FS) backed. This change permits a larger interrogating electric field to penetrate the MUT region and measurethus, it is hypothesized, provides a more accurate ment. In the section to follow, the theoretical expressions for and are derived (Section II). This is done in the same manner as outlined in the previous paragraph. However, the Green’s function for the geometry depicted in Fig. 1(b) (termed a two-medium grounded-slab geometry hereafter), does not exist in the literature and is, in general, of a much more complicated mathematical form. It is for these reasons that the derivation of the dyadic Green’s function for a magnetic-current-excited two-medium grounded-slab geometry is also included (Section III). Last, measurements found using the new free-space-backed (FS-backed) geometry for two magnetic-shielding materials are presented and compared to both a traditional destructive material-characterization technique (namely, the Nicolson Ross Weir technique (NRW) [37], [38]) and the original PEC-backed technique introduced in [36] (Section IV). II. METHODOLOGY The purpose of this section is to theoretically analyze the measurement structure depicted in Fig. 1(b). This entails finding and transmission theoretical expressions for the reflection coefficients, which are functions of frequency , the MUT thickness , , and . In most material-characterization problems (including the one presented in this paper), it is not possible and in to find closed-form mathematical expressions for and (a notable exception is the NRW method). terms of and Thus, it becomes necessary to numerically invert to find and . This is equivalent to solving

for probe 1, and

(3) for probe 2. Here is assumed and suppressed. In (2) , , , and are the transverse electric and and (3), , transverse magnetic field distributions (see Appendix); , , and are the transverse electric and transverse magnetic reflection and transmission coefficients ( and ); and is the -directed propagation constant. Note that because of the symmetry incident field and of the measurement apparatus, of the only and modes of odd index are excited [36]. are Of these, only higher order modes of the form significant [36], [41]. The effects of probe alignment, flangethickness, and MUT-thickness uncertainties on extracted and values were investigated in [36]. For brevity, this analysis is not repeated here. The transverse magnetic field in the MUT region of Fig. 1(b) is found by replacing the waveguide apertures with equivalent and , in accordance with Love’s magnetic currents, equivalence principle [40], [42]. The transverse magnetic field and via the electric vector potential , deis related to fined as

(4) by the expression (5)

(1) where is a specified tolerance, using, for instance, the Newton–Raphson method [39]. If, as is commonly done to and mitigate measurement noise, the reverse reflection coefficients are included in the system of transmission equations (thereby making the system overdetermined), nonlinear least squares can then be used [39].

In (4), fined by

;

; as deand as defined by are the waveguide-aperture cross-sectional areas over which and are distributed; is the dyadic Green’s function for a magnetic-current and excited two-medium grounded slab (6)

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See Section III for the derivation of

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.

B. Coupled Magnetic Field Integral Equations With the expressions for the transverse fields in the waveguide and MUT regions of Fig. 1(b) determined, a system of coupled magnetic field integral equations (MFIEs) can be derived by enforcing the continuity of the transverse magnetic fields at the waveguide apertures, namely,

(7) is given in (5). The unknowns in the above system where and , as well as are the equivalent magnetic currents, , , the modal reflection and transmission coefficients, , and . C. Method of Moments Solution The method of moments (MoM) [42], [43] is used to solve (7). To ensure fast convergence in the MoM, basis functions should be chosen which closely resemble the physical distribution of the currents they are representing. In the case of equivalent magnetic currents (8) where is the unit normal vector pointing into the MUT region. Because of (8), the transverse electric field distributions given in (2) and (3) are chosen as expansion functions for and , respectively. Substitution of (2) and (3) into (7) and subsequent simplificasystem of functional equations. To produce tion yields a a nonsingular system of linear equations, testing functions must be chosen and applied to (7). The testing functions chosen and , applied here are the transverse magnetic field distributions (23) and (24), given in the Appendix. These field distributions are utilized as testing functions because of the computational simplicity provided by waveguide-mode orthogonality. via the inner product and simplifying produces Applying matrix equation of the form where a is the MoM, or impedance matrix; is the vector containing , , , and (i.e., the desired modal reflection and transmission coefficients); and is a vector containing the incident field. contribution from the The MoM matrix takes the form .. .

..

.. .

..

.. .

.

.. .

.. .

..

.

.. .

.

.. .

.. .

..

.

.. .

(9) where the submatrices of the form describe how a source located at probe affects the field observed at probe . Because

M

Fig. 2. Magnetic-current-excited two-medium grounded-slab geometry. The is confined to medium 1. Medium 2 is unbounded, induced magnetic current h and extending to infinity. starting at z

=

of this physical meaning, the and submatrices are and equal due to the symmetry of the DWP, while the submatrices are equal due to reciprocity. The individual matrix elements describe how a source mode (specified by the second set of subscript indices) couples into an observation mode (specified by the first set of subscript indices). For instance, the element is physically interpreted as how the -source mode located at probe 1 couples into the 10-mode observed at probe 2. III. DYADIC GREEN’S FUNCTION FOR A MAGNETIC-CURRENT-EXCITED TWO-MEDIUM GROUNDED SLAB In this section, the dyadic Green’s function for a magneticcurrent-excited two-medium grounded-slab environment is derived. Recall that it is needed to compute the transverse magnetic field which exists in the MUT region of Fig. 1(b). The geometry for this Green’s function derivation is shown in Fig. 2. The figure depicts a PEC grounded-slab waveguide filled with two homogeneous, isotropic media excited by an impressed magconfined to medium 1. To find the Green’s funcnetic current tion for such a geometry, the solutions to the following forced and unforced Helmholtz wave equations, subject to boundary conditions at the interfaces, must be found:

(10) where . Physically, the particular, or forced represents the principle wave emanating from the solution magnetic source in unbounded space. The homogeneous, or unrepresents the waves scattered from the forced solution boundaries at and in the absence of the source. It is for this reason that only the unforced solution is needed to model behavior in medium 2, while both the forced and unforced solutions are needed to model behavior in medium 1:

(11) The geometry depicted in Fig. 2 is invariant along the and directions, thus prompting transformation of those variables using the two-dimensional Fourier transform

(12)

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where (10) produces

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. Applying the above Fourier transform to

(13) and is the -directed where spectral-domain wavenumber. The spectral-domain solutions to the differential equations in (13) are readily found to be

(14)

Fig. 3. Photograph of the DWP measurement apparatus including specially machined line and short calibration standards.

are the unknown positive - and negative -directed where complex wave amplitudes, respectively. Substituting (14) into the spectral representations of (11) yields

(18) for an observer in medium 1 and

(15) where the various functional dependencies have been omitted for notational convenience. Note that as a consequence of the root choice of and the Sommerfeld radiation condition [42], only a positive -directed wave exists in medium 2. This is physically expected since medium 2 is unbounded. To find the unknown complex wave amplitudes, boundary conditions must be applied to (15). The boundary condition at . In terms of the electric vector the PEC plane is potential, the boundary condition is

(19) for an observer in medium 2. Here

(20) (16) The remaining boundary conditions at the material interface are and . In terms of the vector potential, the material interface boundary conditions are

(17) where . Applying (16) and (17) to (15) and simplifying yields the spectral dyadic Green’s function

The spectral-domain dyadic Green’s function given above can be transformed to the spatial domain using the inverse Fourier transform relationship given in (12). However, it is computationally advantageous to perform the integration in (4) using the spectral-domain representation. The evaluation of six integrals—two basis function integrals, two testing function integrals, and two inverse Fourier transform integrals—is required to evaluate (4) using the spectral-domain representation. The four spatial integrals can be computed in closed form and the two inverse Fourier transform integrals, which have even and rapidly converging integrands, are readily calculated using numerical quadrature. IV. EXPERIMENTAL VERIFICATION A. Apparatus Description and Experimental Procedure Material measurements at X-band (8.2 to 12.4 GHz) of ECCOSORB® FGM-125 (a lossy silicon-based magnetic-shielding material approximately 3.175 mm thick) and ECCOSORB® FGM-40 (1.016 mm thick) [44] were made using an Agilent Technologies E8362B vector network analyzer [45]. The DWP, shown in Fig. 3, consists of two precision X-band rectangular

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Fig. 4. (a) Complex permittivity " results for FGM-125 using the PEC-backed (circle traces) and FS-backed (square traces) DWP measurement geometries (1, 5, and 10 modes traces are included). (b) Complex permeability  results for FGM-125 using the PEC-backed (circle traces) and FS-backed (square traces) DWP measurement geometries (1, 5, and 10 modes traces are included).

waveguides connected with screws to a 30.48 cm 30.48 cm 9.779 mm aluminum flange plate. To ensure sufficient coupling when measuring lossy materials, the rectangular waveguides are machined so that only a 3.810 mm spacing exists between their apertures. Prior to attaching the flange plate to the rectangular waveguides and making material characterization measurements, the apparatus is first calibrated using the thru, reflect, line calibration method (TRL) [46] utilizing the specially machined line and short standards depicted in Fig. 3. The TRL calibration places the forward and reverse phase-reference (calibration) planes at the probe 1 and probe 2 rectangular waveguide apertures. The flange plate is then attached to the rectangular waveguides using precision alignment pins. To shift the rectangular waveguide TRL calibration planes to the flange/MUT interface, a short calibration measurement (measurement made using the DWP pressed against a PEC plate) and an “empty” calibration measurement (measurement made using the DWP of a column of air approximately 9.703 mm thick) are made. For the empty calibration measurement, the measured -parameters are time gated to remove the unwanted reflections from the flange-plate edges. The measured -parameters of the MUT are also time gated to remove any possible flange-edge reflections. More detail on this -parameter time-gating technique can be found in [47]–[51]. The and of the MUT are found by solving

(21)

via the Levenberg–Marquardt algorithm [39]. with due to the symmetry of the Note that measurement geometry and due to reciprocity. B. ECCOSORB® FGM-125 Results [Fig. 4(a)] and Fig. 4 shows the complex permittivity [Fig. 4(b)] for FGM-125 extracted using the permeability PEC-backed (circle traces) and FS-backed (square traces) DWP measurement geometries. The material-characterization results are further segregated into 1, 5, and 10 higher-order modes traces. To serve as a reference, the permittivity and permeability for FGM-125 found using the NRW technique are also included. Table I shows the root-mean-square errors and results for the PEC-backed and FS-backed DWP assuming the NRW results are the true FGM-125 permittivity and permeability values. The results depicted in the figures as well as those in the table show that significantly worse permittivity and permeability values are obtained using the FS-backed DWP geometry compared to those obtained using the PEC-backed DWP geometry. The poorer performance of the FS-backed DWP in measuring permeability is expected since the FS-backed geometry does not permit a large interrogating magnetic field into the MUT region to effectively determine . Note that this -measurement difficulty is the complement of the PEC-backed DWP -measurement difficulty discussed in Section I. The poorer performance of the FS-backed DWP in measuring permittivity is rather unexpected since the geometry of the measurement implies that a strong interrogating electric field should exist in the MUT region to effectively determine . The only positive aspect of the FS-backed results, compared to the PEC-backed results, is that the dielectric loss remains

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Fig. 5. (a) Magnitudes of the measured and theoretical (calculated using the NRW FGM-125 " and  results) forward S -parameters for FGM-125 using the FS-backed DWP measurement geometry. (b) Magnitudes of the measured and theoretical (calculated using the NRW " and  results) forward S -parameters for is also FGM-125 using the PEC-backed DWP measurement geometry. The magnitude of the short calibration forward transmission measurement S included.

TABLE I PEC-BACKED AND FS-BACKED DWP ROOT-MEAN-SQUARE ERRORS

physically realizable (although significantly over predicted), , for the entire frequency range. i.e., The reason for the rather poor performance of the FS-backed DWP measurement geometry in measuring (as well as ) is depicted in Fig. 5. The figure shows the magnitudes of the measured and theoretical forward -parameters (calculated using and values) for both the FS-backed the NRW FGM-125 and PEC-backed DWP measurement geometries. Note that the reverse -parameters are not shown so as to not clutter the plots. and . Also inRecall that cluded on the figures is the magnitude of the short calibration forward transmission measurement . Theoretically, no energy should be transmitted from probe 1 to probe 2 when making this measurement; therefore, this quantity should be bezero. When actually measured, however, cause of measurement error/noise sources, such as small air gaps which exist between the flange plate and the MUT. Note that during data collects, weight (approximately 18 kg) is applied

to the top of the aluminum flange near where the rectangular waveguides attach to the plate in order to minimize these gaps. is a good indicator of the effective “noise” Thus, floor of the measurement apparatus. Note that in Fig. 5(a), the magnitude of the forward transmission coefficient is on the order for a majority of the frequency range, whereas of in Fig. 5(b) the magnitude of the transmission coefficient is several times larger than . This physically implies that stronger coupling into the surface-wave-mode spectrum occurs for the PEC-backed DWP than for the FS-backed DWP (when measuring a material like FGM-125). Little energy couples between probe 1 and probe 2 meaning that the FS-backed DWP behaves very similarly to a FS-backed single-waveguide probe, in this context. Based on this analysis, the FS-backed DWP transmission measurements for FGM-125 are practically useless being on the order of the measurement error/noise of the apparatus. Thus, one should expect that the and values returned using the FS-backed DWP measured -parameters will be inaccurate and unstable (with much greater variation than the PEC-backed DWP results). This is precisely the behavior depicted in Fig. 4. Before progressing to other experimental results and analysis, it should be stated that the and experimental results presented in this section are not without precedent. Historically, researchers have had difficulty obtaining accurate permittivity and permeability results using nondestructive techniques like the one presented in this paper with dielec, being the most error prone measurement (as is tric loss, the case here) [18], [20], [29], [31], [33]. C. ECCOSORB® FGM-40 Results While the FGM-125 results of the FS-backed DWP are discouraging, measurements of a different magnetic-shielding ma-

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Fig. 6. (a) Complex permittivity " results for FGM-40 using the FS-backed DWP measurement geometry (1, 5, and 10 modes traces are included). (b) Complex permeability  results for FGM-40 using the FS-backed DWP measurement geometry (1, 5, and 10 modes traces are included). Note that the " and  results for FGM-40 using the PEC-backed DWP measurement geometry cannot be presented because a solution to (21) is not found.

terial yielded better results. Fig. 6 shows the [Fig. 6(a)] and [Fig. 6(b)] results for FGM-40 using the FS-backed DWP. Like FGM-125, FGM-40 is a silicon-based magnetic-shielding material; however, it is more heavily dielectrically and magnetically loaded than FGM-125 and thinner. The magnitudes of the measured and theoretical forward -parameters (calculated using and values) for the FS-backed DWP the NRW FGM-40 are shown in Fig. 7(a). The magnitudes of the measured forward -parameters for the PEC-backed DWP are shown in Fig. 7(b). Note that only the magnitudes of the measured forward -parameters (and not the theoretical forward -parameters) for the PEC-backed DWP are plotted because numerical calculation of the theoretical -parameters is highly unstable due to FGM-40 being very electrically thin. Even if the theoretical -parameters could be calculated for FGM-40, it is very likely that the and values returned using the PEC-backed DWP would be inaccurate and unstable because, like the FS-backed DWP in the case of FGM-125, the magnitudes of the transmission coefficients are on the order of the measurement error/noise of the apparatus. Contrast this with the measured -parameter magnitudes depicted in Fig. 7(a). With the exception of the lower frequencies, the magnitude of the transmission coefficient using the FS-backed DWP is several times larger than . Thus, one should expect stable and values for FGM-40 for most of the frequency range using the FS-backed DWP measurement geometry. This is precisely what is shown in Fig. 6.

main reason for this is revealed in Fig. 5 which definitively shows that little energy couples between the two probes of the FS-backed DWP thereby making the permittivity and permeability results inaccurate and unstable. On the other hand, the and results for FGM-40 show the superiority of FS-backed DWP over the PEC-backed DWP since the PEC-backed DWP permittivity and permeability results cannot even be calculated. The question of when to use the FS-backed or PEC-backed DWP geometries still remains. A theoretical answer to this question is very difficult considering that the theoretical -parameters depend on the characteristics of the MUT in a very complex way, i.e., they are functions of , , , , and the thickness (a five-dimensional space). However, a simple experimental answer to the question is to compare the magnitudes of the measured transmission coefficients with . Whichever geometry yields the

D. Experimental Summary

etry had an average . In this case, the FS-backed measurements should be used to extract permittivity and permeability and the and results shown in Fig. 6 support this selection criterion. Note that this criterion does not provide any information on the overall accuracy of the values, only which DWP geometry can be extracted and expected to perform better.

It is important to summarize the key findings obtained from the measurements of FGM-125 and FGM-40 as they pertain to the PEC-backed and FS-backed DWP measurement geomeand results for FGM-125 clearly show that tries. The the FS-backed DWP struggles compared to the PEC-backed DWP in accurately determining constitutive parameters. The

higher should be used to extract and . For instance, in Fig. 5, the FS-backed geometry had an average ; the PEC-backed geometry had an average . Clearly, the PEC-backed measurements should be used to extract and and the permittivity and permeability results depicted in Fig. 4 support this conjecture. In Fig. 7, the FS-backed geometry had an average , while the PEC-backed geom-

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Fig. 7. (a) Magnitudes of the measured and theoretical (calculated using the NRW FGM-40 " and  results) forward S -parameters for FGM-40 using the FS-backed DWP measurement geometry. (b) Magnitudes of the measured forward S -parameters for FGM-40 using the PEC-backed DWP measurement geometry. Note that the theoretical S -parameters for FGM-40 using the PEC-backed DWP measurement geometry cannot be presented because the numerical calculation of is also included. the theoretical S -parameters is highly unstable. The magnitude of the short calibration forward transmission measurement S

Fig. 8. (a) Complex permittivity " results for FGM-125 using S -parameter measurements collected from both the PEC-backed and FS-backed (triangular traces) DWP measurement geometries (1, 5, and 10 modes traces are included). (b) Complex permeability  results for FGM-125 using S -parameter measurements collected from both the PEC-backed and FS-backed (triangular traces) DWP measurement geometries (1, 5, and 10 modes traces are included).

Before concluding, it is worth noting that the two geometries seem to behave as the complements of each other, both intuitively and experimentally. Therefore, combining -parameter measurements from both DWP geometries to determine and could provide better results than either alone. Fig. 8 shows the complex permittivity [Fig. 8(a)] and permeability [Fig. 8(b)] results for FGM-125 using -parameter measure-

ments collected from both the PEC-backed and FS-backed (triangular traces) DWP measurement geometries. Here and are found by solving

(22)

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TABLE II COMBINED PEC-BACKED AND FS-BACKED DWP ROOT-MEAN-SQUARE ERRORS

where . Table II shows the root-mean-square errors for the combined PEC-backed and FS-backed DWP and results assuming the NRW results are the true FGM-125 permittivity and permeability values. When one compares the results depicted in Fig. 8 and Table II with Fig. 4 and Table I, it is clear that the two DWP measurement geometries complement one another and a significant improvement (and quite extraordinary accuracy) in both permittivity and permeability can be gained by using both geometries and . Indeed, when it is possible to make to determine both measurements, both should be made and the measured -parameters from both measurements should be used to extract the permittivity and permeability regardless of MUT. and of FGM-40 were also extracted using Note that the both measurement geometries. Since numerical calculation of the theoretical -parameters using the PEC-backed geometry is highly unstable for very electrically thin MUTs like FGM-40, the permittivity and permeability results are nearly identical to those obtained using the FS-backed geometry alone (Fig. 6) and thus are not shown. V. CONCLUSION In this paper, the FS-backed DWP measurement geometry [Fig. 1(b)] is introduced and analyzed. Section II presents the theoretical derivation of the FS-backed DWP reflection and transmission coefficients necessary to determine the complex permittivity and permeability of the MUT. This is accomplished by replacing the waveguide-probe apertures with equivalent magnetic currents in accordance with Love’s field equivalence theorem [40], [42]. Enforcing the continuity of the transverse magnetic field at the waveguide-probe apertures (Section II-A) produces a system of coupled MFIEs (Section II-B), which when solved via the MoM (Section II-C), yields the theoretical reflection and transmission coefficients. Section III presents the theoretical derivation of the dyadic Green’s function for a magnetic-current-excited two-medium grounded-slab environment. This Green’s function is necessary to compute the magnetic field which exists in the MUT region of the FS-backed DWP. Last, the FS-backed DWP is verified experimentally via the measurement of two lossy magnetic-shielding materials (ECCOSORB® FGM-125 and ECCOSORB® FGM-40). It is found that the FS-backed DWP measurement geometry is the complement of the published PEC-backed DWP measurement geometry [Fig. 1(a)] [36] and that when the measurements of both DWP geometries are combined, significantly better permittivity and permeability results are obtained than using either geometry alone.

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APPENDIX The transverse electric and transverse magnetic field distributions , , , and are

(23)

when in probe 1, and

(24)

and are when in probe 2. Here, and the - and -directed wavenumbers, are the transverse electric and transverse magnetic wave impedances, and is the -directed propagation constant. REFERENCES [1] R. Mailloux, “Radiation and near-field coupling between two collinear open-ended waveguides,” IEEE Trans. Antennas Propag., vol. AP-17, no. 3, pp. 400–400, May 1969. [2] E. Jull, “Aperture fields and gain of open-ended parallel-plate waveguides,” IEEE Trans. Antennas Propag., vol. AP-21, no. 1, pp. 14–18, Jan. 1973. [3] A. Altintas, P. Pathak, and M.-C. Liang, “A selective modal scheme for the analysis of EM coupling into or radiation from large open-ended waveguides,” IEEE Trans. Antennas Propag., vol. 36, no. 1, pp. 84–96, Jan. 1988. [4] S. Maci, P. Ufimtsev, and R. Tiberio, “Equivalence between physical optics and aperture integration for radiation from open-ended waveguides,” IEEE Trans. Antennas Propag., vol. 45, no. 1, pp. 183–185, Jan. 1997. [5] F. Mioc, F. Capolino, and S. Maci, “An efficient formulation for calculating the modal coupling for open-ended waveguide problems,” IEEE Trans. Antennas Propag., vol. 45, no. 12, pp. 1887–1889, Dec. 1997. [6] K. Selvan, “Approximate formula for the phase of the aperture-reflection coefficient of open-ended rectangular waveguide,” IEEE Trans. Antennas Propag., vol. 52, no. 1, pp. 318–321, Jan. 2004. [7] R. Cicchetti and A. Faraone, “Analysis of open-ended circular waveguides using physical optics and incomplete Hankel functions formulation,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1887–1892, Jun. 2007. [8] M. Mostafavi and W.-C. Lan, “Polynomial characterization of inhomogeneous media and their reconstruction using an open-ended waveguide,” IEEE Trans. Antennas Propag., vol. 41, no. 6, pp. 822–824, Jun. 1993. [9] B. Sanadiki and M. Mostafavi, “Inversion of inhomogeneous continuously varying dielectric profiles using open-ended waveguides,” IEEE Trans. Antennas Propag., vol. 39, no. 2, pp. 158–163, Feb. 1991.

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[10] K. J. Bois, A. Benally, and R. Zoughi, “Multimode solution for the reflection properties of an open-ended rectangular waveguide radiating into a dielectric half-space: The forward and inverse problems,” IEEE Trans. Instrum. Meas., vol. 48, no. 6, pp. 1131–1140, Dec. 1999. [11] S. I. Ganchev, S. Bakhtiari, and R. Zoughi, “A novel numerical technique for dielectric measurement of generally lossy dielectrics,” IEEE Trans. Instrum. Meas., vol. 41, no. 3, pp. 361–365, Jun. 1992. [12] K. Folgerø and T. Tjomsland, “Permittivity measurement of thin liquid layers using open-ended coaxial probes,” Meas. Sci. Technol., vol. 7, no. 8, pp. 1164–1164, 1996. [13] M. Wu, X. Yao, and L. Zhang, “An improved coaxial probe technique for measuring microwave permittivity of thin dielectric materials,” Meas. Sci. Technol., vol. 11, no. 11, pp. 1617–1617, 2000. [14] M. Wu, X. Yao, J. Zhai, and L. Zhang, “Determination of microwave complex permittivity of particulate materials,” Meas. Sci. Technol., vol. 12, no. 11, pp. 1932–1932, 2001. [15] D. H. Shin and H. J. Eom, “Estimation of dielectric slab permittivity using a flared coaxial line,” Radio Sci., vol. 38, no. 2, 2003, 1034. [16] R. Olmi, M. Bini, R. Nesti, G. Pelosi, and C. Riminesi, “Improvement of the permittivity measurement by a 3D full-wave analysis of a finite flanged coaxial probe,” J. Electromagn. Waves Appl., vol. 18, pp. 217–232, 2004. [17] J. Baker-Jarvis, M. D. Janezic, P. D. Domich, and R. G. Geyer, “Analysis of an open-ended coaxial probe with lift-off for nondestructive testing,” IEEE Trans. Instrum. Meas., vol. 43, no. 5, pp. 711–718, Oct. 1994. [18] C.-L. Li and K.-M. Chen, “Determination of electromagnetic properties of materials using flanged open-ended coaxial probe—Full-wave analysis,” IEEE Trans. Instrum. Meas., vol. 44, no. 1, pp. 19–27, Feb. 1995. [19] C.-W. Chang, K.-M. Chen, and J. Qian, “Nondestructive measurements of complex tensor permittivity of anisotropic materials using a waveguide probe system,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 7, pp. 1081–1090, Jul. 1996. [20] O. Tantot, M. Chatard-Moulin, and P. Guillon, “Measurement of complex permittivity and permeability and thickness of multilayered medium by an open-ended waveguide method,” IEEE Trans. Instrum. Meas., vol. 46, no. 2, pp. 519–522, Apr. 1997. [21] C.-W. Chang, K.-M. Chen, and J. Qian, “Nondestructive determination of electromagnetic parameters of dielectric materials at X-band frequencies using a waveguide probe system,” IEEE Trans. Instrum. Meas., vol. 46, no. 5, pp. 1084–1092, Oct. 1997. [22] C.-Y. Yeh and R. Zoughi, “A novel microwave method for detection of long surface cracks in metals,” IEEE Trans. Instrum. Meas., vol. 43, no. 5, pp. 719–725, Oct. 1994. [23] C. Huber, H. Abiri, S. I. Ganchev, and R. Zoughi, “Modeling of surface hairline-crack detection in metals under coatings using an open-ended rectangular waveguide,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 11, pp. 2049–2057, Nov. 1997. [24] J. Nadakuduti, G. Chen, and R. Zoughi, “Semiempirical electromagnetic modeling of crack detection and sizing in cement-based materials using near-field microwave methods,” IEEE Trans. Instrum. Meas., vol. 55, no. 2, pp. 588–597, Apr. 2006. [25] F. Mazlumi, S. H. H. Sadeghi, and R. Moini, “Interaction of an openended rectangular waveguide probe with an arbitrary-shape surface crack in a lossy conductor,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 10, pp. 3706–3711, Oct. 2006. [26] A. McClanahan, S. Kharkovsky, A. R. Maxon, R. Zoughi, and D. D. Palmer, “Depth evaluation of shallow surface cracks in metals using rectangular waveguides at millimeter-wave frequencies,” IEEE Trans. Instrum. Meas., vol. 59, no. 6, pp. 1693–1704, Jun. 2010. [27] J.-Z. Bao, S.-T. Lu, and W. D. Hurt, “Complex dielectric measurements and analysis of brain tissues in the radio and microwave frequencies,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 10, pp. 1730–1741, Oct. 1997. [28] D. Popovic, L. McCartney, C. Beasley, M. Lazebnik, M. Okoniewski, S. C. Hagness, and J. H. Booske, “Precision open-ended coaxial probes for in vivo and ex vivo dielectric spectroscopy of biological tissues at microwave frequencies,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 5, pp. 1713–1722, May 2005. [29] C.-P. Chen, Z. Ma, T. Anada, and J.-P. Hsu, “Further study on twothickness-method for simultaneous measurement of complex EM parameters based on open-ended coaxial probe,” in Proc. Eur. Microw. Conf., Oct. 2005, pp. 505–508.

[30] J. W. Stewart and M. J. Havrilla, “Electromagnetic characterization of a magnetic material using an open-ended waveguide probe and a rigorous full-wave multimode model,” J. Electromagn. Waves Appl., vol. 20, pp. 2037–2052, 2006. [31] N. Maode, S. Yong, Y. Jinkui, F. Chenpeng, and X. Deming, “An improved open-ended waveguide measurement technique on parameters " and  of high-loss materials,” IEEE Trans. Instrum. Meas., vol. 47, no. 2, pp. 476–481, Apr. 1998. [32] G. D. Dester, E. J. Rothwell, M. J. Havrilla, and M. W. Hyde, “Error analysis of a two-layer method for the electromagnetic characterization of conductor-backed absorbing material using an open-ended waveguide probe,” PIER B, vol. 26, pp. 1–21, 2010. [33] S. Wang, M. Niu, and D. Xu, “A frequency-varying method for simultaneous measurement of complex permittivity and permeability with an open-ended coaxial probe,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2145–2147, Dec. 1998. [34] M. W. Hyde and M. J. Havrilla, “A nondestructive technique for determining complex permittivity and permeability of magnetic sheet materials using two flanged rectangular waveguides,” PIER, vol. 79, pp. 367–386, 2008. [35] J. Baker-Jarvis and M. D. Janezic, “Analysis of a two-port flanged coaxial holder for shielding effectiveness and dielectric measurements of thin films and thin materials,” IEEE Trans. Electromagn. Compat., vol. 38, no. 1, pp. 67–70, Feb. 1996. [36] M. W. Hyde, J. W. Stewart, M. J. Havrilla, W. P. Baker, E. J. Rothwell, and D. P. Nyquist, “Nondestructive electromagnetic material characterization using a dual waveguide probe: A full wave solution,” Radio Sci., vol. 44, no. 3, 2009, RS3013. [37] A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas., vol. 19, no. 4, pp. 377–382, Nov. 1970. [38] W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE, vol. 62, no. 1, pp. 33–36, Jan. 1974. [39] K. Madsen, H. B. Nielsen, and O. Tingleff, Methods for Non-Linear Least Squares Problems. Lyngby, Denmark: Technical Univ. of Denmark, 2004. [40] R. E. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEE Press, 1991. [41] G. D. Dester, E. J. Rothwell, and M. J. Havrilla, “An extrapolation method for improving waveguide probe material characterization accuracy,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. 5, pp. 298–300, May 2010. [42] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. New York: IEEE Press, 1998. [43] R. Harrington, Field Computation by Moment Methods. New York: IEEE Press, 1993. [44] ECCOSORB® FGM Permittivity and Permeability Data Emerson & Cuming Microwave Products, Inc., 2007 [Online]. Available: http:// www.eccosorb.com/file/717/fgm%20parameters.pdf [45] “Technical Specifications Agilent Techologies PNA Series Network Analyzers E8362B/C, E8363B/C, and E8364B/C,” Agilent Technologies, Inc., 2008. [46] G. F. Engen and C. A. Hoer, “Thru-reflect-line: An improved technique for calibrating the dual six-port automatic network analyzer,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, no. 12, pp. 987–993, Dec. 1979. [47] M. J. Havrilla and M. W. Hyde, “Nondestructive clamped waveguide lowloss material extraction technique,” in Proc. Appl. Comput. Electromagn. Soc. Conf., 2010, pp. 1–4. [48] M. J. Havrilla and M. W. Hyde, “Dual-probe lowloss material extraction technique,” in Proc. URSI Int. Symp. Electromagn. Theory, 2010, pp. 268–271. [49] M. J. Havrilla and M. W. Hyde, “Clampled waveguide lowloss material characterization technique,” in Proc. Microw. Mater. Their Applicat. Conf., 2010, pp. 100–100. [50] M. W. Hyde and M. J. Havrilla, “Reducing the measurement footprint in the characterization of low-loss materials using the flanged-waveguide measurement geometry,” in Proc. Int. Conf. Electromagn. Adv. Applicat., 2010, pp. 43–46. [51] M. W. Hyde, M. J. Havrilla, and A. E. Bogle, “A novel and simple technique for measuring low-loss materials using the two flanged waveguides measurement geometry,” Meas. Sci. Technol., vol. 22, no. 8, 2011, 085704.

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Milo W. Hyde IV (S’10–M’10) received the B.S. degree in computer engineering from the Georgia Institute of Technology, Atlanta, in 2001 and the M.S. and Ph.D. degrees in electrical engineering from the Air Force Institute of Technology, Wright–Patterson Air Force Base, Dayton, OH, in 2006 and 2010, respectively. From 2001 to 2004, he was a Maintenance Officer with the F-117A Nighthawk, Holloman Air Force Base, Alamogordo, NM. From 2006 to 2007, he was a Government Researcher with the Air Force Research Laboratory, Wright–Patterson Air Force Base. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, Air Force Institute of Technology, Wright–Patterson Air Force Base. His current research interests include electromagnetic material characterization, guided-wave theory, scattering, and optics. Dr. Hyde is a member of the IEEE Instrumentation and Measurement (IMS), Microwave Theory and Techniques (MTT-S), Antennas and Propagation (APS), Geoscience and Remote Sensing (GRSS), and Electromagnetic Compatibility Societies (EMC), as well as the International Society for Optical Engineering (SPIE) and the Optical Society of America (OSA).

Michael J. Havrilla (S’85–M’86–SM’05) received B.S. degrees in physics and mathematics, the M.S.E.E degree, and the Ph.D. degree in electrical engineering from Michigan State University, East Lansing, MI, in 1987, 1989, and 2001, respectively. From 1990 to 1995, he was with General Electric Aircraft Engines, Evendale, OH and Lockheed Skunk Works, Palmdale, CA, where he worked as an Electrical Engineer. He is currently an Associate Professor in the Department of Electrical and Computer Engineering at the Air Force Institute of Technology, Wright-Patterson AFB, OH. His current research interests include electromagnetic and guided-wave theory, electromagnetic propagation and radiation in complex media and structures and electromagnetic materials characterization. Dr. Havrilla is a member of URSI Commission B and the Eta Kappa Nu and Sigma Xi honor societies.

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Andrew E. Bogle (S’04–M’07) received B.S., M.S., and Ph.D. Degrees in electrical engineering from Michigan State University, East Lansing, MI, in 2001, 2004, and 2007, respectively. From 2007 to 2009, he was with Niowave, Inc., Lansing, MI, where he worked as an Electrical Engineer. He is currently an Research Engineer in the Sensor Systems Division at the University of Dayton Research Institute, Dayton, OH. His current research interests include electromagnetic materials characterization, electromagnetic and guided-wave theory, electromagnetic propagation, and radiation in complex media and structures.

Edward J. Rothwell (F’05) was born in Grand Rapids, MI, on September 8, 1957. He received the B.S. degree in electrical engineering from Michigan Technological University, Houghton, in 1979, the M.S. degree in electrical engineering and the degree of electrical engineer from Stanford University, Stanford, CA, in 1980 and 1982, respectively, and the Ph.D. degree in electrical engineering from Michigan State University, East Lansing, MI, in 1985, where he held the Dean’s Distinguished Fellowship. He worked for Raytheon Co., Microwave and Power Tube Division, Waltham, MA, from 1979 to 1982 on low-power traveling wave tubes, and for MIT Lincoln Laboratory, Lexington, MA, in 1985. He has been at Michigan State University from 1985 to 1990 as an Assistant Professor of electrical engineering, from 1990 to 1998 as an Associate Professor, and from 1998 as Professor. He is coauthor of the book Electromagnetics (CRC Press, 2001). Dr. Rothwell received the John D. Withrow award for teaching excellence from the College of Engineering at Michigan State University in 1991, 1996, and 2006, the Withrow Distinguished Scholar Award in 2007, and the MSU Alumni Club of Mid-Michigan Quality in Undergraduate Teaching Award in 2003. He was a joint recipient of the Best Technical Paper Award at the 2003 Antenna Measurement Techniques Association Symposium, and in 2005 he received the Southeast Michigan IEEE Section Award for Most Outstanding Professional. He is a member of Phi Kappa Phi, Sigma Xi, Commission B of URSI.

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Evaporation Duct Height Estimation and Source Localization From Field Measurements at an Array of Radio Receivers Xiaofeng Zhao

Abstract—Remote sensing of the atmospheric refractivity structure using signal strength measurements from a single emitter to an array of radio receivers has been proposed as a promising way for refractivity estimation. As a complement to the pioneers’ published works, this paper focuses on addressing the problem of simultaneously estimating the evaporation duct height and localizing the source’s position. The problem is organized as a multi-parameter optimization issue and genetic algorithm is adopted to search for the optimal solution from various trial parameters. The performance is determined via numerical simulations and mainly evaluated as a function of: 1) the geometry of the receiver array; 2) the transmitting frequency; and 3) the noise in the measurements. Index Terms—Antenna array, electromagnetic propagation, evaporation duct, optimization.

I. INTRODUCTION

A

PROMISING method for remotely sensing of the refractivity structure is based on inference from measurements of radar signal strength. The most popular approach is termed refractivity from clutter (RFC) technique which retrieves the refractivity profiles by taking advantages of the changes in radar clutter returns due to the changes in atmospheric environments. In the last decade, several RFC methods have been developed [1]–[6]. Detailed discussions about these different RFC algorithms can be found in the works completed by Yardim et al. [7], Vasudevan et al. [8] and Douvenot et al. [9]. An important issue of these new techniques is how to evaluate their performance under realistic conditions. Instead of using radar clutter returns, point-to-point microwave measurements have also been proposed as useful information for refractivity estimation. Gingras et al. introduced basic concepts of electromagnetic matched-field processing (EM-MFP) techniques and performed simulations for simultaneously localizing an EM emitter and estimating surface-based duct parameters from synthetic complex-valued (amplitude and phase) field measurements [10]. Tabrikian et al. proposed using point-to-point field measurements to estimate atmospheric duct parameters by the maximum a posteriori (MAP) method [11]. Manuscript received October 08, 2010; manuscript revised April 03, 2011; accepted July 15, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported in part by the National Natural Science Foundation of China under Grant 41175025. The author is with the Institute of Meteorology, PLA University of Science and Technology, Nanjing 210093, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2173115

Gerstoft et al. quantified the performance of EM-MFP method for remote sensing of the refractivity structure using signal strength measurements from a single emitter to an array of radio receivers [12]. Valtr et al. put forward the usage of field measurements at a receiver site of a terrestrial point-to-point link in terms of angle-of-arrival spectra to retrieve the vertical refractivity structure [13]. Zhao et al. investigated possibilities of refractive index profile retrieval using field measurements at an array of radio receivers in terms of variational adjoint approach [14]. In the above mentioned papers, however, only surface-based ducts and elevated ducts were investigated. Compared with these two kinds of duct, evaporation ducts are more prevalent in the marine environment and are more important for shipboard radar communications. Previous works about estimating evaporation duct profiles from radar sea echoes have been given in [1], [6] and [15]. This paper focuses on addressing the problem of simultaneously estimating the evaporation duct height and localizing the source’s position using signal strength measurements from a single emitter to an array of radio receivers. The problem is organized as a nonlinear optimization issue, and a global optimization technique referred to as genetic algorithm is adopted to search for the optimal solution from various trial parameters. The performance is determined via numerical simulations. The parabolic equation method is used to simulate the synthetic signal measurements and the replica fields. The validities of using parabolic equation method to model electromagnetic wave propagation from point-to-point measurements have been discussed in [16]–[19]. This paper will proceed in the following manner: Forward models including evaporation duct refractivity model and electromagnetic propagation model are introduced in Section II. Section III shows the numerical simulations of evaporation duct height estimation and source localization. Detailed discussions about the factors that affect the performance are presented in this section, including 1) the geometry of the receiver array; 2) the transmitting frequency; and 3) the measurement noise. In Section IV, the basic conclusions are summarized.

II. FORWARD MODEL In this paper, the electromagnetic wave propagating in the evaporation duct environments is the main considerations. Therefore, a proper refractivity parameter model and an accurate propagation model should be given first.

0018-926X/$26.00 © 2011 IEEE

ZHAO: EVAPORATION DUCT HEIGHT ESTIMATION AND SOURCE LOCALIZATION FROM FIELD MEASUREMENTS AT AN ARRAY OF RADIO RECEIVERS

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the radius of the earth. Owing to is very close to unity, for environmental inputs, modified refractivity , defined by , is commonly used to describe the information of the atmospheric environments. Let be the complex scalar component of the field at range and height . Then, the field at range and height , denoted by , could be given by the Fourier split-step solution to PE as

(3)

Fig. 1. Modified refractivity M versus height (a) Evaporation duct (b) Surfacebased duct (c) Elevated duct.

A. Evaporation Duct Refractivity Model In the marine environment, three major types of duct are frequently encountered. They are the evaporation duct, the surface-based duct and the elevated duct, as shown in Fig. 1. The evaporation duct is surface-based and is persistent over ocean areas because of the rapid decrease of moisture immediately above the surface. Evaporation ducts are small (typically less than 40 m high), but have a substantial effect on the propagation of radio waves above 3 GHz [12]. For thermally neutral conditions in which the air and sea temperature are equal, the modified refractivity for an evaporation duct can be determined at any height by the relationship [20] (1) where is the modified refractivity at sea surface and its typical value is 339 M-units, is evaporation duct height (EDH), is the natural logarithm and is Jeske’s roughness length of [21]. As shown in Fig. 1(a), EDH defines the upper boundary of the trapping layer where . B. Propagation Model For many years now the parabolic equation (PE) method has been used to model electromagnetic wave propagation in the troposphere. The biggest advantage of using the PE method is that it gives a full-wave solution for the field in the presence of range-dependent environments. PE can be derived from the scalar Helmholtz wave equation under certain assumptions, and it has the form

(2) where represents a scalar component of the electric field for horizontal polarization or a scalar component of the magnetic field for vertical polarization, is the free-space wave number, is the range and is the height. is the modified index of refraction, takes into account the earth’s curvature and is defined by , being the index of refraction and being

where and are the Fourier transform and Fourier inverse transform, respectively. is the transform variable, and is the range increment, given by . Detailed information about the Fourier split-step PE solution may be found in [22], [23]. III. NUMERICAL SIMULATIONS The evaporation duct height estimation, as well as source location, from radio signal strength measurements is an inverse problem. Matched-field processing (MFP) methods represent one approach for solving inverse problems through the usage of extensive forward model runs. The basic concepts of electromagnetic matched-field processing were introduced in [10] and the related genetic algorithms (GA) based on global optimization procedures were discussed in [10] and [12]. In [10], Gingras et al. separated the surface-based duct inversions into three cases, being case A: source-location estimation in a known environment, case B: estimation of environment parameters, source location known, and case C: joint source-location and environmental parameter estimation. Here, we will just deal with the case C, i.e., source localization and evaporation duct height estimation. In this paper, the problem is formulated as a parameter optimization issue, in which the observations are known to be related to an unknown parameter vector through a known nonlinear function. The observations are the receiver array data samples, the nonlinear function is the parabolic equation governing the troposheric electromagnetic wave propagation, and the parameter vector is comprised of the evaporation duct height and the source-location coordinates (source range and source height). This is a multi-parameter inversion problem, and an efficient global search algorithm should be employed. GA has been proved to be such an algorithm, which differs from other search techniques by the usage of concepts of natural selection and evolution. It uses simplified genetic operators and Darwinist principles such as that of survival of the fittest. In all simulations, the synthetic array data are generated for a scenario with the following general characteristics. Source Signal: The synthetic signal simulates a Gaussian antenna with vertical polarization at frequencies from 1 GHz to 10 GHz. Source range is 100 km and source height is 10 m. Receive Antenna: The receive antenna is a vertical array antenna. Three configurations of the receiver array are considered. 1) 10 elements from 1–10 m with spacing of 1 m, 2) 10 elements

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TABLE I PARAMETER SEARCH BOUNDS FOR THE THREE RETRIEVED PARAMETERS

from 21–30 m with spacing of 1 m and 3) 30 elements from 1–30 m with spacing of 1 m. Propagation Environment: The propagation environment used for all synthetic cases is an evaporation duct with 30 m height. In order to simplify the computation, the refractive conditions are considered constant with the distance for the entire propagation path. Propagation Code: The advanced propagation model (APM) [24] is used for all simulations. APM combines the capabilities of radio physical optics (RPO) and terrain parabolic equation model (TPEM) in a relatively fast code. Objective Function: In all cases, the objective function used is the Bartlett processor defined by [25]

Fig. 2. Coverage diagrams (dB) showing propagation loss as a function of height and range for a 30 m evaporation duct case. The transmitter height is 10 m, and the transmitting frequency is 1 GHz for (a) and 10 GHz for (b).

(4)

where is the objective function, is the parameter vector that should be retrieved, is a complex-valued vector of synthetic measurements, is a complex-valued vector of replica field, is the number of receiver elements and denotes conjugate transpose. Optimization Parameters: The genetic algorithm is used to perform global optimal search [26]. The GA search parameters are: population size 20, crossover probability 0.5, jump mutation probability 0.05, creep mutation probability 0.16. The termination is controlled by the computation time of 5 minutes. Our computation source is ThinkPad R400 equipped with dual CPUs (P8600, 2.40 GHz) and 2 GB EMS memory bank. Every running could get different forward model runs, depending on the operating frequency and the source to the receiver range [23]. Through a great deal of numerical experiments, we think that 5 minutes is enough to get a preferable retrieved parameter vector for the cases in this paper. The convergence of the algorithm is discussed in the appendix. The three retrieved parameters are the evaporation duct height (EDH), the source height and the source range. The parameters and their search bounds are given in Table I. Fig. 2 illustrates the propagation loss coverage diagrams for a 30 m evaporation duct case computed by APM. The source height is 10 m at a frequency of 1 GHz and 10 GHz, respectively, for Figs. 2(a) and 2(b). It is clear that in 1 GHz case, little energy is trapped in the duct layer. While in 10 GHz case, the dominant component of the energy is trapped. A. The Inversions of Different Geometries of the Receiver Array As mentioned above, three configurations of the receiver array are considered. 1) 10 elements from 1–10 m with spacing

Fig. 3. Detailed inversion results of the three retrieved parameters with 5%, 10% and 20% Gaussian noise at different transmitting frequencies (a) The inversion results of the evaporation duct height (b) the inversion results of the source height and (c) the inversion results of the source range.

of 1 m, 2) 10 elements from 21–30 m with spacing of 1 m and 3) 30 elements from 1–30 m with spacing of 1 m. The detailed inversion results at frequency of 1 GHz and 10 GHz are shown in Table II. From Table II, we could see that different receiver antenna position and different aperture size could result in different inversion results. When the transmitter operates at the frequency of 1 GHz, 1–10 m aperture size inversions are unsatisfied except for the evaporation duct height. Compared with 1–10 m aperture size, 21–30 m has a great improvement in the source range inversions. When the aperture size extends to 1–30 m, though the source range is not as well defined, the whole inversion results are acceptable. At 10 GHz frequency case, the retrieved results are better than that of 1 GHz. The reason might be that at 10 GHz frequency, more energy at the receiver array location that can be used for profile estimation purposes, see Fig. 2. In 1–30 m aperture size case, the results are very close to the actual parameter values. However, in any case depicted in Table II, the evaporation duct height retrievals are believable.

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TABLE II INVERSION RESULTS OF DIFFERENT GEOMETRIES OF THE RECEIVER ARRAY

TABLE III INVERSION RESULTS WITH DIFFERENT GAUSSIAN NOISE.

DETAILED PROCESS

TABLE IV DIFFERENT INVERSIONS,

OF TWO

B. The Inversions With Gaussian Noise Added to the Measurements In practical operations, errors could be stemmed from many sources: errors in describing the environment, errors in the forward model, instrument and measurement errors, and noise in the data [12]. Here, only measurement noise will be considered. The error term is assumed complex Gaussian distributed, stationary with zero mean, and the error at each receiver element is uncorrelated. In order to investigate the antinoise ability of GA for duct parameters inversion and source localization, three different quantities of Gaussian noise are considered, being 5% Gaussian noise, 10% Gaussian noise and 20% Gaussian noise. The configuration of the receiver array is with 30 elements from 1–30 m with spacing of 1 m. The detailed inversion results at frequency of 1 GHz and 10 GHz with different Gaussian noise are shown in Table III. From Table III, it is seen that at lower frequency (1 GHz), the inversion results are degraded. However, at higher frequency (10 GHz), the antinoise ability of using GA to retrieve evaporation duct height and localize source position is perfect. Even

with 20% Gaussian noise, the mean deviation between the inversion results and the true parameter values is just 0.714%. The detailed inversion results of the three retrieved parameters with 5%, 10% and 20% Gaussian noise at different transmitting frequencies (1–10 GHz) are given in Fig. 3. When the Gaussian noise is only 5%, the retrieved parameter values are acceptable above 4 GHz. In 10% Gaussian noise case, the frequency should above 7 GHz, and 20% with 8 GHz. IV. CONCLUSION This paper focuses on addressing the problem of simultaneously retrieving the evaporation duct height and localizing the source’s position. This is a nonlinear optimization problem and genetic algorithm is adopted to perform global optimal search. Three factors are mainly concerned, i.e., the geometry of the receiver array, the transmitting frequency, and the measurement noise. Numerical simulations indicate that larger aperture size and/or higher operating frequency can retrieve better results. The antinoise ability is very good when the frequency is above 8 GHz. Only 30 m evaporation duct case is investigated. It could be asserted that with lower duct height, the minimum frequency

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Fig. 4. A posteriori distributions for the evaporation duct height, the source height and the source range (a) 1–10 m aperture size operating at frequency of 1 GHz (b) 1–30 m aperture size operating at frequency of 10 GHz.

will increase. The preliminary results obtained in this paper might be helpful for practical applications, such as antenna distributions and frequency selections. Further work is necessary to collect real-field data and the corresponding environmental parameters, and use them to verify the preliminary conclusions in this paper. APPENDIX MODEL CONVERGENCE Douvenot et al. have pointed out that: 1) the aim of refractivity estimation is not to give the exact refractivity profile, but to propose a ’generic’ model able to render an approximation of the real atmospheric conditions; 2) operational applications of the estimation necessitate short computation time, less than 10 minutes, to avoid errors due to temporal evolution of atmosphere [27]. Based on these two points, the convergence of the algorithm is discussed. From Table II, it is seen that the worst and best inversions are 1–10 m aperture size operating at 1 GHz and 1–30 m aperture size operating at 10 GHz, respectively. Table IV gives the detailed process of these two different inversions. For 1–10 m aperture size operating at frequency of 1 GHz case, the inversions are converged in 5 minutes computations. For 1–30 m aperture size operating at 10 GHz case, the best inversions are obtained when the computation time achieves 9 minutes. However, the results are just a little improvement compared with 4 minute ones. Through a great deal of numerical experiments, we found that 5 minutes is enough to get preferable retrieved parameters. Therefore, the termination condition is set to be 5 minutes for all of the inversions discussed in this paper. Figs. 4(a) and 4(b) give the statistical results for the above two experiments. For each experiment, GA is executed 100 times with different initial random seed and starting values for the model parameters. REFERENCES [1] L. T. Rogers, C. P. Hattan, and J. K. Stapleton, “Estimating evaporation duct heights from radar sea echo,” Radio Sci., vol. 35, no. 4, pp. 955–966, 2000. [2] P. Gerstoft, L. T. Rogers, J. L. Krolik, and W. S. Hodgkiss, “Inversion for refractivity parameters from radar sea clutter,” Radio Sci., vol. 38, no. 3, pp. 1–22, 2003.

[3] A. E. Barrios, “Estimation of surface-based duct parameters from surface clutter using a ray trace approach,” Radio Sci., vol. 39, pp. 1–15, 2004. [4] C. Yardim, “Statistical Estimation and Tracking of Refractivity From Radar Clutter, Electrical Engineering,” Ph.D, University of California, San Diego, 2007. [5] S. X. Huang, X. F. Zhao, and Z. Sheng, “Refractivity estimation from radar sea clutter,” Chin. Phys. B, vol. 18, no. 11, pp. 5084–5090, 2009. [6] B. Wang, Z. S. Wu, Z. W. Zhao, and H. G. Wang, “Retrieving evaporation duct heights from radar sea clutter using particle swarm optimization algorithm,” Prog. Electromagn. Res. M, vol. 9, pp. 79–91, 2009. [7] C. Yardim, P. Gerstoft, and W. S. Hodgkiss, “Statistical maritime radar duct estimation using hybrid genetic algorithm-Markov chain monte carlo method, 10.1029/2006RS003561,” Radio Sci., vol. 42, pp. 1–15, 2007. [8] S. Vasudevan, R. H. Anderson, S. Kraut, P. Gerstoft, L. T. Rogers, and J. Krolik, “Recursive bayesian electromagnetic refractivity estimation from radar sea clutter,” Radio Sci., vol. 42, pp. 1–19, 2007. [9] R. Douvenot, V. Fabbro, P. Gerstoft, C. Bourlier, and J. Saillard, “Real time refractivity from clutter using a best fit approach improved with physical information,” Radio Sci., vol. 45, pp. 1–13, 2010. [10] D. F. Gingras, P. Gerstoft, and N. L. Gerr, “Electromagnetic matched field processing: Basic concepts and tropospheric simulations,” IEEE Trans. Antennas Propag., vol. 45, no. 10, pp. 1536–1545, Oct. 1997. [11] J. Tabrikian and J. L. Krolik, “Theoretical performance limits on tropospheric refractivity estimation using point-to-point microwave measurement,” IEEE Trans. Antennas Propag., vol. 47, no. 11, pp. 1727–1734, Nov. 1999. [12] P. Gerstoft, D. F. Gingras, L. T. Rogers, and W. S. Hodgkiss, “Estimation of radio refractivity structure using matched-field array processing,” IEEE Trans. Antennas Propag., vol. 48, no. 3, pp. 345–356, Mar. 2000. [13] P. Valtr and P. Pechac, “Novel method of vertical refractivity profile estimation using angle of arrival spectra,” presented at the XXVIIIth General Assembly of International Union of Radio Sci., New Delhi, India, 2005. [14] X. F. Zhao, S. X. Huang, and H. D. Du, “Theoretical analysis and numerical experiments of variational adjoint approach for refractivity estimation,” Radio Sci., vol. 46, pp. 1–12, 2011. [15] C. Yardim, P. Gerstoft, and W. S. Hodgkiss, “Sensitivity analysis and performance estimation of refractivity from clutter techniques,” Radio Sci., vol. 44, pp. 1–16, 2009. [16] A. E. Barrios, “Parabolic equation modeling in horizontally inhomogeneous environments,” IEEE Trans. Antennas Propag., vol. 40, no. 7, pp. 791–797, Jul. 1992. [17] R. Akbarpour and A. R. Webster, “Ray-tracing and parabolic equation methods in the modeling of a tropospheric microwave link,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3785–3791, Nov. 2005. [18] S. D. Gunashekar, E. M. Warrington, D. R. Siddle, and P. Valtr, “Signal strength variations at 2 GHz for three sea paths in the british channel islands: Detailed discussion and propagation modeling,” Radio Sci., vol. 42, pp. 1–13, 2007. [19] P. Valtr, P. Pechac, V. Kvicera, and M. Grabner, “A terrestrial multiplereceiver radio link experiment at 10.7 GHz—comparisons of results with parabolic equation calculations,” Radioengineering, vol. 19, no. 1, pp. 117–121, 2010. [20] H. V. Hitney and R. Vieth, “Statistical assessment of evaporation duct propagation,” IEEE Trans. Antennas Propag., vol. 38, no. 6, pp. 794–799, 1990. [21] H. Jeske, “State and limits of prediction methods of radar wave propagation conditions over the sea,” in Modern Topics in Microwave Propagation and Air-Sea Interaction, A. Zancla, Ed. New York: Reidel, 1973, pp. 131–148. [22] G. D. Dockery, “Modeling electromagnetic wave propagation in the troposphere using the parabolic equation,” IEEE Trans. Antennas Propag., vol. 36, no. 10, pp. 1464–1470, Oct/ 1988. [23] J. R. Kuttler and G. D. Dockery, “Theoretical description of the parabolic approximation/Fourier split-step method of representing electromagnetic propagation in the troposphere,” Radio Sci., vol. 26, no. 2, pp. 381–393, 1991. [24] A. E. Barrios and W. L. Patterson, “Advanced propagation model (APM) ver. 1.3.1 computer software configuration item (CSCI) documents,” Technical Document 3145, 2002. [25] P. Gerstoft, SAGA User Manual 5.4: An Inversion Software Package, 2007. [Online]. Available: http://www.mpl.ucsd.edu/people/gerstoft/saga

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[26] D. L. Carroll, FORTRAN Genetic Algorithm Version 1.7a, 2001. [Online]. Available: http://cuaerospace.com/carroll/ga.html [27] R. Douvenot, V. Fabbro, P. Gerstoft, C. Bourlier, and J. Saillard, “A duct mapping method using least squares support vector machines,” Radio Sci., vol. 43, pp. 1–12, 2008.

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Xiaofeng Zhao was born in Jiangsu, China, on November 2, 1983. He received the M.Sc. degree from PLA University of Science and Technology, Nanjing, China, in 2009, where he is currently working towards the Ph.D. degree at the Institute of Meteorology. His research interests include electromagnetic wave propagation modeling and atmospheric refractivity estimation.

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Extrapolation of Wideband Electromagnetic Response Using Sparse Representation Huapeng Zhao, Student Member, IEEE, and Ying Zhang

Abstract—Wideband electromagnetic response can be extrapolated using combined low frequency and early time information, which can substantially reduce the computational load. Most existing extrapolation methods are based on orthogonal polynomials, but selecting optimal parameters of orthogonal polynomials is not straightforward. This work proposes to extrapolate wideband electromagnetic response using sparse representation. The electromagnetic response is expressed as linear combination of atoms from an overcomplete dictionary. Optimal linear combination of atoms is then sought through the affine scaling transformation and the support vector regression. By increasing the data length step by step, convergence of the sparse solution is used as a criterion to determine the sufficient data length. Performance analysis shows that our proposed extrapolation method retains lower computational complexity and renders more flexibility in reconstructing a signal. Numerical examples are presented to show the efficacy and advantages of the proposed extrapolation method. Index Terms—Extrapolation, overcomplete dictionaries, sparse representation, wideband electromagnetic response.

I. INTRODUCTION

I

N COMPUTATIONAL electromagnetics (CEM), electromagnetic response from an arbitrary structure can be sought by numerically solving Maxwell’s equations in either frequency- or time-domain. Though accurate, CEM methods are usually computation intensive, especially when wideband information is requested, for which one has to conduct computation at many frequency points. On the other hand, early time and low frequency information is relatively easy to be obtained using CEM methods. In order to reduce the computational burden in wideband analysis, various extrapolation techniques have been proposed to obtain wideband response using the information at a few low frequency points. Furthermore, low frequency and early time information are mutually complementary, and they can be combined to obtain better extrapolation accuracy [1]. Manuscript received October 14, 2010; revised June 15, 2011; accepted August 10, 2011. Date of publication October 24, 2011; date of current version February 03, 2012. This work was supported by a grant from the National Natural Science Foundation of China for Young Scholars (No. 61101094 ). H. P. Zhao was with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, and is now with the Department of Electronics and Photonics, Institute of High Performance Computing, Singapore 138632, Singapore (e-mail: [email protected]). Y. Zhang was with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, and is now with the School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173116

In extrapolation methods, extrapolating bases influence the accuracy a lot. Orthogonal polynomials are usually adopted as extrapolation bases, because they are complete and provide compact support. Meanwhile, it has been found that the extrapolation accuracy is affected by the number of basis functions and the time scaling factor [2]. and should be optimized in order to obtain accurate and stable extrapolation. In [3], the bounds of and the minimum of were given, and the range of convergence was derived. Nevertheless, optimal choice of and is still not straightforward. Furthermore, the data length also affects the extrapolation accuracy, where and are lengths of time- and frequency-domain data, is a necessary respectively. It was stated that condition for accurate extrapolation [4]. However, in practice, it is interesting to know the sufficient values of and , with which one can decide when to stop the expensive CEM simulation. In [5], [6], the genetic algorithm was adopted to optimize the values of and . A criterion was proposed to be used together with genetic algorithm to determine the sufficient data length [5], [6]. Though genetic algorithm automated the , and the data length, it may converge to selection of local minimums for non-convex problems. Therefore, it is highly desirable to develop new extrapolation methods which and automatically determine the sufficient value of efficiently select optimal values of and . The optimization of and is essentially a process of choosing suitable basis functions. In this paper, a new extrapolation method is developed based on sparse representation. The sparse representation method is utilized for its ability in automatically selecting suitable basis functions. An overcomplete dictionary is designed to represent the electromagnetic response. Elements of the dictionary are called atoms. Since the sparse representation method chooses optimal atoms automatically and efficiently, the computational load in optimizing and is reduced substantially. Meanwhile, a criterion is proposed to determine the sufficient value of . Thus, one can stop the CEM simulation once the criterion is satisfied. The rest of this paper is organized as follows. For completeness, a brief introduction to sparse representation is presented in Section II. The proposed method is detailed in Section III, where performance analysis of the proposed method is also given. Numerical examples are provided in Section IV, and concluding remarks are presented in Section V. II. SPARSE REPRESENTATION Giving an space

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th order tensor , where denotes the vector spanning the Hilbert is named as a dictionary, and

ZHAO AND ZHANG: EXTRAPOLATION OF WIDEBAND ELECTROMAGNETIC RESPONSE USING SPARSE REPRESENTATION

is called an atom. For an observed quantity decomposed, the approximation of in terms of is given by

to be atoms of

(1) where denotes the approximation coefficient. The approximation error is defined as (2) where inf means the infimum. Because , approximation by (1) is called sparse representation. Due to the underlying linear dependency of atoms in (i.e., the dictionary is overcomplete), solution of subject to minimizing (2) is not unique. From the perspective of sparse representation, one aims to find a constituted by the smallest number of atoms in , i.e., the value of is the smallest. When is an orthogonal basis of , the optimal approximation is given by the weighted sum of the basis functions yielding the first largest values of , where denotes the inner product. However, for an overcomplete dictionary, finding the optimal is an nondeterministic polynomial time (NP)-hard problem. There have been many literature discussing how to solve such an NP-hard problem. In general, given the observation , the problem of finding can be formulated as (3a) (3b) where represents the expectation,1 the tensor is the tolerance, and defined as

is the matrix form of is the -norm of

(4) which is a conventionally used sparsity measurement [7], [8]. (3b) is used to measure the noise so that the derived can be more accurate. With different choices of , the solution methods and results can be different. Details on the solution of (3) can be found in [7]–[15] and references therein. Sparse representation has been widely used in various engineering problems, including the direction-of-arrival estimation [16], [17], signal compression [18], blind source separation [19], channel equalization [20], etc. More recently, sparse representation has been introduced to the antenna and propagation community for power angle spectrum estimation [21]. In most applications, it has been shown that sparse representation can greatly enhance the performance of conventional algorithms. III. PROPOSED EXTRAPOLATION METHOD In this section, an overcomplete dictionary is developed in order to extrapolate wideband electromagnetic response efficiently. The affine scaling transformation and support vector 1Expectation

is only needed for statistical application.

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regression (SVR) are adopted to find the approximation coefficients. For proper choice of and , convergence of approximation coefficients is utilized as a sufficient condition. Performance of the proposed method is analyzed and compared with existing methods. A. Proposed Overcomplete Dictionary Define two sets and , which contain all possible values of polynomial order and time scaling factor , respectively. An overcomplete dictionary is constructed using , where denotes the th order orthogonal polynomial, and it can be any of the three polynomials studied in [2], which provide similar performance. In this work, is chosen to be the weighted Lagurre polynomial. The time-domain signal is then expanded as (5) and correspondingly, the frequency-domain signal expansion

has the

(6) is the Fourier transform of , and . The difference of our expansion from that using orthogonal polynomials lies in the introduction of and as variables, while orthogonal expansion represents the signal with fixed values of and . It is known that and construct two sets of orthogonal bases for a fixed value of . However, any element of is not necessarily orthogonal to when . The same holds for and if . It means that expansions using (5) and (6) are not orthogonal expansion. Instead, the atoms with different values of and are linearly dependent. When most elements of are zero, (5) and (6) can be considered as sparse representation of and , respectively. It should be noted that when contains a single element, and , (5) and (6) degenerate to orthogonal expansions. For practical implementation, and are constructed as follows. First, the minimum of is determined using (51) of [3]. is then chosen to be 1.5 times its minimum. is defined as . Second, the convergence range of can be found from (50) of [3]. Denote the convergence range as . According to [2], can be written as (7) is the time step size. From the values of and where , the minimum and maximum of can be found from (7), and they are denoted by and , respectively. A new interval is then created, where is the largest integer satisfying , and is the smallest integer satisfying . The set of is now defined as

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. The sampling step size of will be discussed in Section IV. Constructing two overcomplete dictionaries as shown in (8)

(8a)

where denotes the th row of , and . By using different error penalty functions, (14) is able to handle different measurement noises. When quadratic loss function is used, the measurement noise is assumed to be Gaussian. When Huber penalty function is used, the effect of outliers on the regression can be reduced. Considering as the target for the input pattern , (14) is identical to the optimization problem of SVR formulated as [23] (15a)

(8b) and defining the approximation coefficients

(15b)

(9) (5) and (6) can be equivalently expressed as (10) (11) early time sequences and low frequency data are When available, can be computed from the following linear equation (12) where .. .

.. .

.. .

.. .

are slack variables, and where determines the trade-off between finding a sparse solution and retaining small residual error. Empirical results suggest that the value of has negligible effect on the regression performance [24]. A robust choice of is given by [25] (16) denotes their stanwhere denotes the mean of targets, and dard deviation. The dual problem of (15) is given by

(17) .. .

.. .

and

is given by

(13) . When In (12), the dimension of matrix is , (12) is an overcomplete problem which can be solved by imposing sparse constraint on . After is derived, only atoms corresponding to the nonzero elements of contribute to the expansion of and . Therefore, only these atoms are used to extrapolate the signals, and they are called active atoms. B. Solution of the Sparse Representation Problem In this paper, we propose to solve (12) using the SVR. Different from (3) which assumes that the measurement noise is Gaussian, we may settle a wider range of problems by choosing different penalty functions in the SVR. Using the affine scaling transformation [22], can be found via

(18) which is called support vector expansion. In this work, the input pattern is unknown. Therefore, the following iterative algorithm is proposed to find the sparse solution of (12). Step 1: Initialize using a randomly generated vector, , and . Step 2: Solve (17). Step 3: Compute by (18). Step 4: Update . Step 5: Evaluate whether the termination criterion is satisfied. If yes, stop; else, go to Step 2. In this paper, the proposed iterative algorithm is terminated when is satisfied. C. A Criterion To Determine Sufficient Data Length

(14)

For practical application of extrapolation methods, it is desirable to know the sufficient value of , with which one

ZHAO AND ZHANG: EXTRAPOLATION OF WIDEBAND ELECTROMAGNETIC RESPONSE USING SPARSE REPRESENTATION

will be able to decide when to stop the expensive CEM simulation. Naturally, the larger is, the better accuracy one can obtain. Nevertheless, there must be a sufficient value of , where enough information have been provided to obtain accurate extrapolation. Beyond this sufficient value, very little new information will be added by increasing the values of and . As a result, as long as reaches its sufficient value, the solved approximation coefficients will change very little with increasing . Therefore, by increasing and step by step, the convergence of can be utilized as a criterion to determine the sufficient value of . In order to measure the convergence of , a sequence of can be generated in ascending order. The relative variation of is then defined as (19) where represents the approximation coefficients calculated using the th value of in the ascending sequence. Based on this criterion, the following iterative scheme is proposed. Step 1: Initialize , and set . Set the threshold and the initial value of . Step 2: Solve (12) using the solution method presented in Section III-B. Step 3: If m is zero, go to step 4, otherwise, compute using and . Step 4: If has been continuously satisfied for three times, sufficient value of has been found, and iteration can be stopped. Otherwise, , go to Step 2. and denote the step sizes of and , respectively. Initial values of and should not affect the final extrapolation accuracy a lot, because the aforementioned iteration is repeated until convergence. However, more iterations will be required if initial values of and are too small. In practice, and can be chosen to be around one sixth to one third of the total time- and frequency-domain data lengths, respectively. is the tolerance for the relative variation of approximation coefficient . Numerical examples in Section IV show that satisfactory accuracy can be obtained with of 0.05. D. Performance Analysis A favorable property of using sparse representation is that a high order atom with may be linearly dependent with low order atoms with , where . Namely, an atom can be approximated by linear combination of lower order atoms with time scaling factors other than . This renders more flexibility compared to conventional orthogonal bases, and it is beneficial for signal representation using atoms with limited orders. For orthogonal expansions, a signal can be accurately represented if and only if the order of the signal is not higher than the order of the polynomials used. If the signal contains high order components, the number of basis functions has to be increased in orthogonal expansion, whereas the proposed sparse representation method is able to approximate the

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signal using low order atoms with different values of . This is helpful in enhancing the extrapolation accuracy. Besides, for existing orthogonal expansion methods, the optimal values of and are determined by global searching. For the searching method, if the optimal value of is large, the searching range also becomes very large. Supposing orthogonal expansion methods search from order to order with time scaling factors, the computational complexity is about , which is very high for large values of and . In the proposed method, the computational complexity of SVR is and when and , respectively. denotes the number of support vectors [26]. Therefore, the computational complexity of the proposed method can be estimated to be , where is the number of iterations required to find the sparse solution.2 As long as is not very large, the proposed method retains lower computational complexity compared to orthogonal expansion methods. Although it is difficult to estimate , the convergence rate of the iteration algorithm in Section III-B can be shown to be quadratic when quadratic loss function is used [7], [26]. One may refer to [26] for more details on the convergence of the iteration method. IV. NUMERICAL EXAMPLES Numerical examples are presented in this section to illustrate the efficacy and advantages of the proposed method. Choice of parameters in our proposed method is discussed at the end of this section. The following Gaussian pulse is used as the timedomain excitation (20) where is used to control the bandwith, and is the time shift to make the pulse negligible at . The combined error is adopted as a criterion to measure the extrapolation accuracy, and it is computed by (21) where , and symbols with hat denote the extrapolated results. 1) Example 1: Scattering from a conducting plate is first considered. Following the convention, the unit of time adopted for this example is light meter (lm). The time-domain induced current is calculated using an in-house developed time-domain integral equation solver, and the frequency-domain data is obtained using a frequency-domain integral equation (FDIE) solver. Figs. 1 and 2 show the extrapolated results of the induced current in time- and frequency-domain, respectively, both of which are validated by directly computed results. Also shown in these two figures are results obtained using orthogonal polynomial basis. One can observe the good agreement between extrapolated and directly computed results. Fig. 3 illustrates the expansion coefficients for the overcomplete dictionary and the 2This computational complexity is overestimated because ally satisfied in sparse representation method.

is gener-

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Fig. 1. Induced current at the center of a square conducting plate in time-do0.15 lm). The side length of the plate is 2 m. A main (time step size plane wave is illuminating the conducting plate. The incident wave is polarized V/m. Its temporal in negative -direction, and its magnitude is 120 lm, and lm. variation is governed by (20) with

Fig. 3. Values of expansion coefficient solved using different approaches. (a) Overcomplete dictionaries. (b) Orthogonal polynomials.

Fig. 2. Induced current at the center of the conducting plate in frequency-do1.3 MHz). (a) Real part. (b) Imaginary part. main (frequency step size

orthogonal polynomials. Fig. 3(a) shows that expansion coefficients using the overcomplete dictionary are very sparse. Furthermore, comparing Fig. 3(a) with Fig. 3(b), one can easily see that the solution from sparse representation is searched in

a two-dimensional space, whereas the expansion coefficients of orthogonal polynomials are only solved in a one-dimensional space. This means the sparse representation renders another degree of freedom, which enlarges the possibility to find a better signal reconstruction. Therefore, one can expect better extrapolation accuracy using sparse representation. Table I presents the CPU time required by different extrapolation approaches. The proposed method requires much less CPU time than the orthogonal extrapolation method. It should also be noted that 12 seconds will be required to obtain the induced current at one frequency point if the direct FDIE solver is utilized. On the other hand, it only takes 1.59 seconds to conduct extrapolation once using the proposed method. Extrapolation accuracy using different methods is listed in Table I. It is seen that the proposed method obtains better accuracy than the orthogonal expansion method. Fig. 4 illustrates the variation of and as and increase. drastically decreases when and are less than 60, after which it keeps at a small level, indicating that there is little change in . also decreases as the data length increases. However, it decreases much slower after and exceed 60. Therefore, there is a sufficient value of and , at which

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TABLE I CPU TIME AND ACCURACY COMPARISON BETWEEN OUR METHOD AND ORTHOGONAL EXPANSION METHOD FOR THE FIRST TWO EXAMPLES

Fig. 5. Reflection coefficient of a microstrip bandpass filter in the time-domain ns).

Fig. 4. Variations of conducting plate.

and

as the data length increases for the example of

enough information has been provided for accurate extrapolation. Fig. 4 also shows that there is a connection between and . Hence, it is reasonable to utilize as a criterion to determine sufficient values of and . The value of depends on user’s accuracy requirement. An empirical choice is to set it to 0.05, and increase and until has been continuously satisfied for three times. The effect of on the accuracy will be discussed at the end of this section. 2) Example 2: The reflection coefficient of a microstrip bandpass filter is considered as the second example. The filter is the same as the one used in [3]. The time- and frequency-domain information are obtained from [3]. In this example, the iterative method in Section III-C has been adopted to determine the sufficient values of and . With the empirical choice of , sufficient values of and are found to be 70. Figs. 5 and 6 show the time- and frequency-domain reflection coefficients of the bandpass filter, respectively, where good agreement is observed between extrapolated and directly computed results. For this and next examples, the unit of time is nanoseconds (ns). Fig. 7 illustrates and versus data length. One can have the same observations in Fig. 7 as in Fig. 4. CPU time for this example is presented in Table I. Compared with the orthogonal expansion method, the proposed method obtains better extrapolation accuracy with less CPU time. 3) Example 3: The third example is a planar dipole antenna, whose time-domain and frequency-domain driving-point current is shown in Figs. 8 and 9, respectively. The directly computed results are obtained using the CST Microwave Studio. A discrete current port of amplitude 1 A is used to excite the

Fig. 6. Reflection coefficient of a microstrip bandpass filter in the frequencyMHz). (a) Real part. (b) Imaginary part. domain (

antenna. Parameters for the Gaussian pulse are ns, and ns. With the criterion presented in Section III-C, and are chosen to be 100 and 230, respectively. Good agreement is observed between extrapolated and directly computed results. Fig. 10 illustrates the variation of and against

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Fig. 7. Variations of bandpass filter.

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and

as the data length increases for the example of

Fig. 8. Time-domain driving-point current of a planar dipole antenna ( ns). Geometry of the dipole antenna is shown in the inset. Parameters of the dipole antenna are the same as the one used in [3] except for that the width and length of the substrate are 3.6 cm and 5.6 cm, respectively.

and . It is seen that after the chosen values of and keeps at a small levle, and does not decrease a lot, which indicates that and are sufficient for accurate extrapolation. This validates the effectiveness of the proposed criterion in determining sufficient values of and . It should be mentioned that the proposed method takes 11 seconds to conduct extrapolation once, while 180 seconds are needed to obtain the driving-point current at one frequency point using the frequency-domain CEM solver. 4) Parameter Choice: There are mainly three parameters requiring presetting in our proposed method, i.e., the value of , the number of basis functions , and the interval . Table II lists the relative extrapolation error corresponding to different choices of . It is seen that the accuracy with of 0.05 is satisfactory. Reducing the value of will improve the accuracy with the price of more CEM simulations. Therefore, 0.05 is suggested as an empirical choice of . Guidelines for setting and have been given in Section III-A.

Fig. 9. Frequency-domain driving-point current of the dipole ( (a) Real part. (b) Imaginary part.

Fig. 10. Variations of planar dipole antenna.

and

=25 MHz).

as the data length increases for the example of

ZHAO AND ZHANG: EXTRAPOLATION OF WIDEBAND ELECTROMAGNETIC RESPONSE USING SPARSE REPRESENTATION

TABLE II RELATIVE ERROR WITH DIFFERENT VALUES FOR THE THREE EXAMPLES

OF

TABLE III VALUES OF PARAMETERS IN DESIGNING THE OVERCOMPLETE DICTIONARY FOR THE THREE EXAMPLES

Table III lists the values of parameters in designing the overcomplete dictionary for the three examples. is sampled uniformly in with a step size of . In this work, is set to 0.5, though it may be determined dynamically using the grid refinement technique [17]. It should be noted that instability may occur if atoms of the dictionary are highly correlated. Hence, in order to avoid instability, the value of should not be too small. Besides the aforementioned parameters, and also need initialization. As noted in Section III-C, initial values of and may be set to one sixth to one third of the total data lengths in time- and frequency-domain, respectively. V. CONCLUDING REMARKS In this paper, a sparse representation method has been developed to obtain wideband response from combined low frequency and early time information. A criterion has been proposed to determine the sufficient data length, with which one can decide when to stop the expensive CEM simulation. Performance analysis has shown the lower computational complexity and higher flexibility of the proposed method. Numerical examples have been provided to illustrate the efficacy and advantages of the new extrapolation method. It has been shown that the sparse representation method not only reduces the time in selecting optimal values of and , but also provides more flexibility and better accuracy. Furthermore, for orthogonal expansion, it is required that . If is increased beyond during its optimization, or should be enlarged, which means additional expensive CEM simulation is required. However, the proposed algorithm does not have this restriction. Finally, simulation results showed the effectiveness of the proposed criterion in determining the sufficient data length. The proposed method can be used together with CEM methods to greatly save computation time in obtaining wideband electromagnetic response. REFERENCES [1] M. M. Rao, T. K. Sarkar, T. Anjali, and R. S. Adve, “Simultaneous extrapolation in time and frequency domains using Hermite expansions,” IEEE Trans. Antennas Propagat., vol. 47, no. 6, pp. 1108–1115, Jun. 1999.

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[2] M. Yuan, J. Koh, T. K. Sarkar, W. Lee, and M. Salazar-Palma, “A comparison of performance of three orthogonal polynomials in extraction of wide-band response using early time and low frequency data,” IEEE Trans. Antennas Propagat., vol. 53, no. 2, pp. 785–792, Feb. 2005. [3] M. Yuan, A. De, T. K. Sarkar, J. Koh, and B. H. Jung, “Conditions for generation of stable and accurate hybrid TD-FD MoM solutions,” IEEE Trans. Antennas Propagat., vol. 54, no. 6, pp. 2552–2563, Jun. 2006. [4] T. K. Sarkar, J. Koh, W. Lee, and M. Salazar-Palma, “Analysis of electromagnetic systems irradiated by ultra-short ultra-wide-band pulse,” Meas. Sci. Technol., vol. 12, pp. 1757–1768, Oct. 2001. [5] J. M. Frye and A. Q. Martin, “Extrapolation of time and frequency responses of resonant antennas using damped sinusoids and orthogonal polynomials,” IEEE Trans. Antennas Propagat., vol. 56, no. 4, pp. 933–943, Apr. 2008. [6] J. M. Frye and A. Q. Martin, “Time and frequency bias in extrapolating wideband responses of resonant structures,” IEEE Trans. Antennas Propagat., vol. 57, no. 12, pp. 3934–3941, Dec. 2009. [7] I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using FOCUSS: A re-weighted minimum norm algorithm,” IEEE Trans. Signal Process., vol. 45, no. 3, pp. 600–616, Mar. 1997. [8] K. Kruetz-Delgado and B. D. Rao, “Application of concave/Schur-concave functions to the learning of overcomplete dictionaries and sparse representations,” in Conf. Rec. 32nd Asilomar Conf. Signals, Systems and Computers, 1998, vol. 1, pp. 546–550. [9] D. P. Wipf and B. D. Rao, “Bayesian learning for sparse signal reconstruction,” in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing, Apr. 2003, vol. 6, pp. 601–604. [10] M. E. Tipping, “Sparse Bayesian learning and relevant vector machine,” J. Mach. Learn. Res., pp. 211–244, 2001. [11] S. Chen and D. Donoho, “Basis pursuit,” in Proc. 28th Asilomar Conf. Signals, Systems, Computers, Nov. 1994, vol. 1, pp. 41–44. [12] S. G. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionary,” IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3397–3412, Dec. 1993. [13] S. F. Cotter, B. D. Rao, K. Engan, and K. Kruetz-Delgado, “Sparse solutions to linear inverse problems with multiple measurement vectors,” IEEE Trans. Signal Process., vol. 53, no. 7, pp. 2477–2488, Jul. 2005. [14] B. Wohlberg, “Noise sensitivity of sparse signal representations: Reconstruction error bounds for the inverse problem,” IEEE Trans. Signal Process., vol. 51, no. 12, pp. 3053–3060, Dec. 2003. [15] J. J. Fuchs, “Recovery of exact sparse representations in the presence of bounded noise,” IEEE Trans. Inf. Theory, vol. 51, no. 10, pp. 3601–3608, Oct. 2005. [16] Y. Zhang, Q. Wan, M.-H. Wang, and W.-L. Yang, “A partially sparse solution to the problem of parameter estimation of card model,” Signal Process., vol. 88, pp. 2483–2491, 2008. [17] D. Malioutov, M. Cetin, and A. S. Willsky, “A sparse signal reconstruction perspective for source localization with sensor arrays,” IEEE Trans. Signal Process., vol. 53, no. 8, pp. 3010–3022, Aug. 2005. [18] R. Demirli and J. Saniie, “Denoising and compression of ultrasonic signals using model-based estimation techniques,” in Proc. 2004 IEEE Int. Ultrasonics, Ferroelectrics, and Frequency Control Joint 50th Anniversary Conf., 2004, vol. 1, pp. 2306–2309. [19] T.-W. Lee, M. S. Lewicki, M. Girolami, and T. J. Sejnowski, “Blind source separation of more sources than mixtures using overcomplete representations,” IEEE Signal Process. Lett., vol. 4, no. 4, pp. 87–90, Apr. 1999. [20] D. Luengo, I. Santamaria, J. Ibanez, L. Vielva, and C. Pantaleon, “A fast blind SIMO channel identification algorithm for sparse sources,” IEEE Signal Process. Lett., vol. 10, no. 5, pp. 148–151, May 2003. [21] J. W. Wallace and M. A. Jensen, “Sparse power angle spectrum estimation,” IEEE Trans. Antennas Propagat., vol. 57, no. 8, pp. 2452–2460, Aug. 2009. [22] B. D. Rao and I. F. Gorodnitsky, “Affine scaling transformation based methods for computing low complexity sparse solutions,” in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing, Aug. 1996, vol. 3, pp. 1783–1786. [23] V. Vapnik, The Nature of Statistical Learning Theory. New York: Springer, 1995. [24] D. Basak, S. Pal, and D. C. Patranabis, “Support vector regression,” Neural Inf. Process. Lett. Rev., vol. 11, no. 10, pp. 203–224, Oct. 2007.

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[25] V. Cherkassky and Y. Ma, “Selection of meta-parameters for support vector regression,” Lecture Notes in Computer Science, vol. LNCS 2415, pp. 687–693, 2002. [26] C. J. C. Burges, “A tutorial on support vector machines for pattern recognition,” Data Mining and Knowledge Discovery, vol. 2, no. 2, pp. 121–167, June 1998.

niques in electromagnetics, and measurements in electromagnetic reverberation chamber. Mr. Zhao was a recipient of the Outstanding Master’s Thesis Award by the Office of Education of Sichuan Province, China, and the Science and Technology Advancement Award (First Class) by the Ministry of Education, China, in 2009 and 2005, respectively.

Huapeng Zhao (S’08) was born in Hebei province, China, in 1983. He received the B.Eng. and M.Eng. degrees in electronic engineering from the University of Electronic Science and Technology of China, Chengdu, China, in July 2004 and March 2007, respectively. From January 2008 to August 2011, he was working towards the Ph.D. degree in the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He is now a Research Scientist with the Institute of High Performance Computing, Singapore. His research interests include computational electromagnetics, signal processing tech-

Ying Zhang was born in Chengdu, China, in 1981. She received the B.Eng. degree from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 2004, and the Ph.D. degree from the Nanyang Technological University, Singapore, in 2011, all in electronic engineering. She is currently an Associate Professor with the School of Electronic Engineering, UESTC, Chengdu, China. Her research interests include array signal processing, sparse signal representation, and wireless communication. Dr. Zhang was a recipient of the 2010 Best Student Paper Award of IEEE Singapore MTT/AP Chapter.

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A Wearable Two-Antenna System on a Life Jacket for Cospas-Sarsat Personal Locator Beacons Andrea A. Serra, Paolo Nepa, Member, IEEE, and Giuliano Manara, Fellow, IEEE

Abstract—A wearable two-antenna system to be integrated on a life jacket and connected to Personal Locator Beacons (PLBs) of the Cospas-Sarsat system is presented. Each radiating element is a folded meandered dipole resonating at 406 MHz and includes a planar reflector realized by a metallic foil. The folded dipole and the metallic foil are attached on the opposite sides of the floating elements of the life jacket itself, so resulting in a mechanically stable antenna. The metallic foil improves antenna radiation properties even when the latter is close to the sea surface, shields the human body from EM radiation and makes the radiating system less sensitive to the human body movements. Prototypes have been realized and a measurement campaign has been carried out. The antennas show satisfactory performance also when the life jacket is worn by a user. The proposed radiating elements are intended for the use in a two-antenna scheme in which the transmitter can switch between them in order to meet Cospas-Sarsat system specifications. Indeed, the two antennas provide complementary radiation patterns so that Cospas-Sarsat requirements (satellite constellation coverage and EIRP profile) are fully satisfied. Index Terms—Cospas-Sarsat system, folded dipole, life-jacket antennas, life vest antennas, meandered dipole, wearable antennas.

I. INTRODUCTION

T

HE Cospas-Sarsat system [1] is intended to provide a earth-to-satellite SOS communication in case of shipwrecks or similar crashes. The basic Cospas-Sarsat system is composed of [1] the following. • Distress radio beacons like ELTs (Emergency Locator Transmitters) for aviation use, EPIRBs (Emergency Position Indicating Radio Beacons) for maritime use, and PLBs (Personal Locator Beacons) for personal use, which transmit signals during distress situations; • Instruments on board satellites in geostationary and lowaltitude earth orbits, which detect the signals transmitted by distress beacons; • Ground receiving stations, referred to as Local Users Terminals (LUTs), which receive and process the satellite downlink signal to generate distress alerts; • Mission Control Centers (MCCs) which receive alerts produced by LUTs and forward them to Rescue Coordination

Manuscript received April 19, 2011; revised July 14, 2011; accepted August 26, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. A. A. Serra is with the Department of Information Engineering, University of Pisa, Pisa IT-56100, Italy (e-mail: [email protected]). P. Nepa and G. Manara are with the Department of Information Engineering, University of Pisa, Pisa IT-56100, Italy. They are also with Consortium Ubiquitous Technologies (CUBIT), Navacchio, Pisa IT-56100, Italy (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173151

Centers (RCCs), Search and Rescue Points Of Contacts (SPOCs) or other MCCs. ELTs are cumbersome and heavy devices because they are supposed to resist heavy impacts and high temperatures. EPIRBs are intended to be released by boats at the shipwreck and they are usually mounted on buoys. PLBs are similar to portable radios both in terms of size and weight and are supposed to be carried in pockets or attached to safe vests. They are sold as self standing transmitters. Typical antennas used for distress radio beacons are monopole/whip antennas; a fractal antenna layout has been presented in [2]. To the best of the authors’ knowledge, transmitting systems for Cospas-Sarsat applications fully integrated on emergency equipment are not available yet. Transport companies usually provide floating life vests to be worn in emergencies. Usually, they consist of a wide and flat front part and a back element around the neck intended to support the survivor’s head (like a collar). Some of them are inflatable, especially on aircrafts, and some are filled with floating elements. The latter are plastic foam blocks, such as polyvinyl chloride and polyethylene, inserted into the main jacket’s parts. Apart from what required by whistles, emergency lights, reflective strips and lashing straps, enough room is available for additional devices on both the main parts of the aforementioned vests, where PLB antennas can be attached to. Life-vest integrated PLB antennas belong to the class of wearable antennas and some effort has been recently made in this research field [3]–[7]. They should be as light and small as possible and they should not interfere with the human body to ensure good radiation performance. Apart from reflection coefficient specifications, radiation patterns should not be impaired from survivors’ movements and radiation efficiency should be kept at acceptable values even when the antenna is close to the sea surface. Moreover, the antenna should be shielded from the body to guarantee low SAR (Specific Absorption Rate) levels, although the latter is probably the least problem in distress situations. In this paper, the performance of a Cospas-Sarsat two-antenna system operating at 406 MHz and integrated on a real floating life jacket is investigated through both simulations and measurements. First, a comparison between three dipole-like antenna prototypes (a meandered dipole, a bow-tie dipole, and a folded meandered dipole) is done in order to outline some important antenna features, and results are shown in Section II. In Section III, the effect of human body movements on the antenna reflection coefficient is experimentally checked. Finally, in Section IV the radiation properties of the folded meandered dipole are analyzed. In order to maximize the satellite radio coverage, a two-antenna system installed in the front part of

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Fig. 2. Prototypes of the meandered dipole (top) and bow-tie (bottom) antennas. Fig. 1. Typical floating life vest: (a) front view, (b), (c) two possible antenna placements.

a life jacket is proposed and its compliance with the CospasSarsat system specifications (coverage requirements and Effective Isotropic Radiated Power, EIRP, profiles) is verified. Final remarks are discussed in the Conclusions. II. ANTENNA DESIGN Fig. 1(a) shows a typical floating life vest. It is composed of two main parts: a front/chest one (in the continuous line box) and the head/neck one (in the dotted line box). The chest part is supposed to be fasten to the human trunk and the head one should provide support to the head and keep it away from the sea surface. Modern life jackets are mostly designed to automatically orientate themselves in a way that the survivor chest faces the sky (and his back lies on the water surface). So, the highlighted parts in Fig. 1(a) should face upward. In the bottom of Fig. 1(a), one of the floating element is partially extracted from the jacket and it is clearly visible. It is made of several rectangular soft PVC slices stacked-up to form a thick block and its size is around cm . A similar PVC block is inside the life jacket in the neck area and its size is around cm . Enough room is available to realize a planar antenna on the largest face of each floating block. Fig. 1(b) and Fig. 1(c) show a sample of how two antennas could be attached to these floating elements in the front and in the head placements, respectively. In a preliminary phase, three different dipole-like antennas resonating at 406 MHz have been designed and prototyped to be attached to the floating blocks: a meandered dipole (MD), a bow-tie dipole (BTD) antenna, and a folded meandered dipole with a conducting reflector on its back (FMD). Fig. 2 shows the first two of the above mentioned radiating elements. The MD is realized with a copper adhesive foil and it is stuck directly on the 9 cm thick floating block. It is 30 cm long and 2 cm wide and the meandering has been necessary to reduce size in order to fit the available room in the life vest (strip width is equal to 5 mm and the meanders are all equal). In Fig. 2 the BTD is also shown, which is realized with the same technique; it is 27 cm long and 6 cm wide. In Fig. 3 the reflection coefficient for the MD and the BTD antennas are shown, when the antennas are either in free-space or on a life jacket worn by a user. Each antenna is considered as the group of

Fig. 3. Measured reflection coefficient for (a) the bow-tie dipole (BTD) and (b) the meandered dipole (MD) antenna. Both the chest and head placements have been considered when the antennas are attached to a life jacket worn by a user. Measured reflection coefficient for the free-space antennas is also shown as a reference.

the metallic strip and the foam block, and the free space condition just indicates that it is not mounted on the jacket. Both the chest and head placements (as illustrated in Fig. 1) are considered. Both free-space antennas exhibit a reflection coefficient less than 20 dB in the Cospas-Sarsat system frequency band (406–406.1 MHz). However, it is apparent from measurements shown in Fig. 3 that, when the life jacket is worn, antenna return loss performance is impaired by the presence of the survivor’s body. Indeed, human body conductivity and losses modify the antenna input impedance, as they are only 9 cm far apart (the 9 cm PVC block thickness corresponds to around 1/8 of the

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Fig. 5. Measured reflection coefficient for the bow-tie dipole (BTD) and the meandered dipole (MD) antenna when a metallic shield is placed on the opposite side of the floating block with respect to the dipoles. Both the chest and head placements have been considered when the antennas are attached to a life jacket worn by a user. Fig. 4. Measured impedance of the meandered dipole (MD) antenna when the antenna is in free space, placed on the body and when a conductive shield is placed on the opposite side of the floating block with respect to the dipole (neck case). Data plotted in the frequency range between 350 MHz and 450 MHz.

free-space wavelength at 406 MHz). When the BTD is mounted on the worn life jacket, the reflection coefficient is minimally degraded with respect to the free-space case if the antenna is placed in the front part. When it is mounted around the neck, performances are a little worse, reasonably because of the near presence of the head; however, the reflection coefficient is still less than 10 dB around 406 MHz. From Fig. 3(b) it can be observed that, when the MD is on the worn life jacket, the operating frequency is down shifted but the reflection coefficient keeps values lower than 12 dB. For the specific application [8], the VSWR is required to be smaller than 1.5 (reflection coefficient less than 14 dB). A shielding element could helpfully be interposed among the radiating element and the user’s body in order to reduce the coupling between the antenna and the human body. On the other hand, this electrically small separation will result in an input impedance reduction as it is well known for horizontal dipole antennas close to a metallic plane [9]. As an example, Fig. 4 shows the input impedance for the MD in free space, mounted on the jacket and when a metallic foil is stuck on the foam face opposite to the dipole. In the last case, the metallic sheet acts like a shield between the radiating element and the human body. It is apparent from Fig. 4 how the presence of the human body, and more of the conductive shield, affects the input impedance values. In the free space case, the impedance trace is inside the in the frequency range of the COSPAS SARSAT system. When the antenna is placed close to the body (with and without the shield) the impedance traces do not pass inside the and matching is no more satisfied (as apparent also from Fig. 3(b), where the MD_worn_neck trace is always higher than the 14 dB threshold). As expected, the higher value of the shield conductivity with respect to that of the body leads to lower impedance values (shorting effect). Fig. 5 shows the reflection coefficient for the MD and BTD when a metallic shield is placed on the opposite side of the

floating block with respect to the dipoles. As expected the reflection coefficient is affected by the presence of the shield and system requirements are not satisfied. The reflection coefficient is greater than 8 dB. This effect can be mitigated if a folded dipole is used, since folding determines an input impedance increasing with respect to a conventional dipole [9]. The FMD is shown in Fig. 6(a) and is essentially a printed folded dipole whose arms are meandered in order to save space according to the available room on the life vest. The FMD is composed of two identical parallel printed elements which exhibit the same topology of the meandered dipole in Fig. 2. The two elements are stuck, facing each other, on the two sides of a 1 mm thick Rohacell substrate and electrically connected at their ends (bottom left detail of Fig. 6(a)). The dimensions of the antenna and the floating block are also indicated in Fig. 6. The antenna has been optimized in order to resonate at 406 MHz when the folded meandered dipole and the metallic shield are mounted on two opposite sides of the 9 cm thick floating element, as shown in the bottom of Fig. 6. The shielding element is realized with a 30 cm long and 10 cm wide conductive sheet. Fig. 7 shows the reflection coefficient for the FMD with the shielding sheet when the life jacket is worn by a user. To show the beneficial contribution of the conductive sheet, the reflection coefficient for the free-space FMD is also shown for comparison. Similar to the MD and BTD, the FMD antenna is considered as comprising the metallic strip, the foam block and the metallic shield. The free-space notation indicates that the antenna is not mounted on the jacket. As expected, results for the folded meandered dipole are better than those obtained for the simpler meandered dipole and the bow-tie antenna. Indeed, the resonance is kept around the operating frequency and the reflection coefficient is less than 14 dB even when the life jacket is worn. The antenna reflection coefficient was also measured in a wet condition. First, the FMD, was made waterproof by means of a wrapping foil. Then, it was placed (floating) in a large basin filled with sea water. The FMD was got wet in order to create some water layer on the antenna, in order to simulate a more

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Fig. 8. Measured reflection coefficient for the folded meandered dipole—FMD antenna, in free space, when the height h of the floating block is reduced (to simulate an unintentional compression of the floating block).

Fig. 6. Prototype of the Folded Meandered Dipole—FMD (top) and details of the feeding gap (top right) and of the shorting at one of its ends (top left). In the bottom of the figure, the metallic shield on the opposite side of the floating element is also shown.

As it can be observed from Fig. 8, the antenna is slightly detuned and the reflection coefficient minimum is shifted toward higher frequencies. Nonetheless, the bandwidth requirement is still fully satisfied. It is worth mentioning that the floating block used in the measurement campaign (extracted from a real life jacket) is very light but also difficult to compress. For instance, a hundred kilograms load causes a thickness variation of less than 1 cm and the thickness rises back up to 9 cm when the load is removed. III. ANTENNAS PERFORMANCE ON A LIFE JACKET: RETURN LOSS FLUCTUATIONS

Fig. 7. Measured reflection coefficient for the folded meandered dipole—FMD antenna, in free space, worn and floating on sea water. Both the chest and head placements have been considered when the antennas are attached to a life jacket worn by a user. The measured reflection coefficient for the free-space antenna is also shown as a reference.

realistic operating condition. The reflection coefficient remains less than 14 dB at the operating frequency, this means that both the water surface beyond the antenna reflector and the thin water layer on the dipole do not strongly affect its performance. The FMD, which forms the antenna composed of the dipole and the metallic shield, was designed and tuned on the actual dimensions of the available floating block (9 cm). One of the practical inconveniences that could occur in an emergency is that the floating element could be compressed and its thickness reduced. Fig. 8 shows the effect of the thickness reduction on the antenna reflection coefficient.

In realistic emergency situations, the wearable antenna impedance tuning can be degraded by body activities due to user’s excitement or simply to movements induced by the sea waves. A measurement campaign has been conducted when the life jacket is worn by a user who performs a series of random body movements. The investigated activities are related to the upper limbs and head movements, which could introduce the main perturbations. Samples of the reflection coefficient amplitude at 406 MHz were collected in a three minute time sweep. In this interval the arms were waved in the front of the trunk and around the head, the head was swung ahead and behind and the chest was slightly leaned forward. During these activities the jacket is displaced from its nominal position, being slightly bent and deformed. Fig. 9 shows the collected samples when the antenna is mounted in the front part of the jacket. As apparent from Fig. 9, reflection coefficient fluctuations are smaller for the FMD than for the BTD and the MD antennas. Moreover, the reflection coefficient is always less than 12 dB for the FMD, while it can become greater than 10 dB for the other two antennas. Each of the above antennas has been separately considered, while trying to maintain an analogous mobility degree during measurements. Fig. 10 shows the probability density function of the collected samples and Table I summarizes some statistics (for ten data acquisition, for each antenna), including the mean values and the relative standard deviation (RSD%), that is the standard deviation divided by the mean value. In the third column of Table I the occurrence of VSWR values higher than 1.5 is also shown.

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TABLE I MEAN VALUES AND RELATIVE STANDARD DEVIATION OF THE REFLECTION COEFFICIENT FOR THE BOW TIE DIPOLE, THE MEANDERED DIPOLE AND THE FOLDED MEANDERED DIPOLE ANTENNAS

Fig. 9. Samples of the measured reflection coefficient at 406 MHz collected in a three minute time sweep when the life jacket is worn and the user moves the arms, the trunk and the head.

Fig. 10. Probability density function of the measured reflection coefficient amplitude at 406 MHz for the bow-tie dipole (BTD), meandered dipole (MD) and folded meandered dipole (FMD) antennas, during a series of repetitive body movements.

The mean value of the reflection coefficient amplitude results to be approximately equal to 16 dB, while it is around 12.5 dB for the MD antenna. As already noted in Fig. 9, the folded meandered dipole exhibits the lowest RSD% and it is also significant that it shows a very low percentage (less than 3%) of VSWR values higher than 1.5. On the basis of the results shown in Fig. 9 and Fig. 10, the FMD was chosen because its reflection coefficient is more stable (traces in Fig. 7 are very similar and they almost overlap) both in a stationary condition and when some movements are performed (Fig. 9, 10 and Table I). IV. ANTENNAS PERFORMANCE ON A LIFE JACKET: RADIATION PATTERNS In this section radiation patterns are analyzed to check their compliance with the Cospas-Sarsat specifications [8]. The Cospas-Sarsat System includes two types of satellites [1]: • Satellites in low-altitude Earth orbit (LEO) which form the LEOSAR System;

• Satellites in geostationary Earth orbit (GEO) which form the GEOSAR System. The GEOSAR system can provide almost immediate alerting in the footprint of the GEOSAR satellite, whereas the LEOSAR system provides coverage of the polar regions (which are beyond the coverage of geostationary satellites). The latter can locate the distress event using Doppler processing techniques and it is less vulnerable to obstructions which may block a beacon signal in a given direction because the satellite is continuously moving with respect to the beacon. On the other side, the LEOSAR satellites are not always visible by the beacons as their orbit time is around ninety minutes long. Then, in order to see a LEOSAR satellite as soon as it arises from its polar orbit, the radiation beam of the beacon antenna is required to be relatively wide. System specifications [8] require the EIRP to be between 32 dBm and 43 dBm for at least 90% of the elevation region between 5 and 60 (radiation patterns are required to be hemispherical) when the power transmitted to the antenna is 37 dBm 2 dB. It obviously follows that high efficiency antennas could allow to satisfy EIRP requirements with significant beacon power saving. Due to the better performance of the folded meandered dipole antenna in terms of reflection coefficient robustness with respect to the body presence and movements, radiation performance will be analyzed just for this antenna. The radiation patterns were evaluated both numerically and through measurements. The antenna was simulated in the presence of a simple model of the human trunk when placed in the above mentioned available spots (front chest and head). The human model was composed of a box for emulating the chest and an ellipsoid for the head, as shown in Fig. 11(a). Electrical characteristics for a 2/3 muscle equivalent phantom (available at [10]) were applied to the chest and head homogeneous volumes. The resulting properties at the system operating frequency correspond to , . In order to consider the effect of the sea surface, an infinite planar surface with sea water electric characteristics ( , ) was imposed just behind the model back as it was supposed to be floating on it. Numerical simulations were performed with CST Microwave Studio [11]. The folded meandered dipole exhibits a quite large beam in its H-plane (the H-plane pattern would be omni-directional without the metal shield and the body model). The simulated gain is around 7 dB when the antenna is placed on the chest (in either vertical or horizontal configuration) and around 1 dB when the antenna is at the head position. It follows that the head has a

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Fig. 11. Human trunk and head models used in numerical simulations mm . The chest and the head are modeled as homogeneous volumes with electric characteristics equal to 2/3 of the muscle’s one at 406 MHz. The two-antenna configurations consist of two folded meandered dipoles are perpendicularly oriented: (a) chest and head placements; (b) both antennas are located at the chest.

Fig. 12. Photo of the Satimo SG 3000F measurement facility at Ce.R.Ca., Calearo’s research centre in Vicenza, Italy.

significant effect on the antenna gain with a reduction of more than 6 dB. In order to reduce the front end gain requirement at the satellite, it is necessary that the life jacket EIRP is as close as possible to the upper limit of 43 dBm. This makes the folded meandered dipole placed on the head spot an inefficient solution. Then, a more efficient practical implementation of a two-antenna system was considered where both radiating elements are orthogonally placed on the front part of the life jacket (Fig. 11(b)). It is worth noting that the receiving antennas at the Cospas-Sarsat satellites are circularly polarized antennas. A simple phantom model of the human body was prepared by filling a plastic tank mm with water and a proper salt concentration to simulate human body complex permittivity at 406 MHz [10], [12]. Measurements on the FMD prototype were carried out in a Satimo SG 3000F system (available at Ce.R.Ca., Calearo’s research centre in Vicenza, Italy). A photo of the measurement system is shown in Fig. 12, where the tank and the life jacket are also visible.

Fig. 13. Simulated and measured gain patterns for the horizontal (a) and the vertical (b) folded meandered dipoles, arranged as shown in Fig. 11(b).

The effect of the sea surface was added in a post processing phase through the electromagnetic numerical solver Feko [13]. The Satimo system was first used to measure radiation patterns in free space. Radiation pattern data were then imported in the simulation software FEKO and a radiation boundary in the form of a flat surface was added under the antenna to account for the effect of the sea surface. With this post processing setup, the final radiation patterns (Fig. 13) were computed. The radiation patterns for the configuration corresponding to Fig. 11(b) were measured and results are shown in Fig. 13, together with the simulated ones. Simulated and measured gain patterns are in a good agreement in both principal planes. Fig. 13(a) shows patterns relevant to the horizontal FMD antenna, while Fig. 13(b) shows those for the vertical FMD antenna. As expected, both antennas exhibit a relatively wide beam in the H-plane (y-z plane for the horizontal dipole and x-z plane for the vertical dipole) and a narrower beam in the E-plane (x-z plane for the horizontal dipole and y-z plane for

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the vertical dipole). The E-plane pattern of the horizontal dipole is slightly asymmetric as a consequence of the presence of the vertical dipole antenna in the direction. As aforementioned, EIRP limits are between 32 dBm and 43 dBm and must be satisfied for the 90% of the elevation angles between 5 and 60 . The proposed idea is to use the two above described antennas and to alternatively transmit with one of the two, in order to cover the desired satellite region. Fig. 14(a) shows the EIRP contour plot (measured data) for the vertically mounted folded meandered dipole placed on the front chest as in Fig. 11(b), when the transmitted power is set to 37 dBm (the nominal value). The 5 to 60 elevation circles are highlighted by two dottedlines, the white region indicates allowed EIRP values between 32 dBm and 43 dBm, while the green (red) color indicates those regions where EIRP values are below 32 dBm (above 43 dBm). If a single antenna is considered (vertical dipole), the green region covers around 89% of the elevation angle space and, even if for a very small percentage (less than 1%), system requirements are not fully satisfied. Moreover, it is worth considering that, for a number of reason related to the emergency situation (natural body movements, sea wave induced movements), the performance of a single antenna could be degraded and it could not radiate as required by the system or as it was measured during the Cospas-Sarsat testing procedure [8]. An increased reliability can be gained with a two-antenna system where the transmitter periodically switches between the two dipoles (it is worth noting that fast switching is not required, so that the switch implementation is not a complicate and expensive task). Thus, the radiation patterns of the horizontal and vertical FMD in Fig. 13(a) were combined by selecting the antenna with the highest gain between the two. With this technique the personal locator beacon uses alternatively the two antennas to transmit the emergency request by increasing the likelihood that at least one of the two exhibits a relatively high gain toward the direction the satellite is arising from. In Fig. 14(a) and Fig. 14(b) the white areas indicates the directions where the EIRP requirements are met. It is apparent that lower EIRP values are located along the directions parallel to each dipole (dipole null directions). On the other hand, when the results are combined, the green region (EIRP values less than 32 dBm) is quite reduced and the elevation profile is covered for 97% of the direction space, with a wide margin with respect to the 90% requirement. V. CONCLUSIONS A wearable two-antenna system to be integrated on a life jacket and connected to Personal Locator Beacons (PLBs) of the Cospas-Sarsat system has been presented. A set of dipole-like configurations resonating at 406 MHz have been prototyped and compared. The most promising radiator resulted to be a folded meandered dipole which includes a planar reflector made of a metallic foil. The folded dipole and the metallic foil could be attached on the opposite sides of the floating elements of a typical commercial life jacket, so resulting in a mechanically stable antenna. Prototypes have been realized and a measurement campaign has been carried out when the life jacket with the antenna was worn by a user. It has been shown that the

Fig. 14. EIRP contour plot (measured data) for the two-antenna system as shown in Fig. 11(b) when the transmitted power is set to 37 dBm: (a) vertical dipole, (b) horizontal dipole, (c) combined data for a system where the transmitter continuously switches between the horizontal and the vertical FMD antennas.

metallic foil makes the radiating system less sensitive to the human body movements and improves antenna radiation properties even when it is close to the sea surface (it also shields the human body from EM radiation). A waterproof prototype

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was arranged to test its performance when the antenna floats on water and a thin water film is placed on the dipole. Results showed that bandwidth requirements are still fully satisfied. The effect of the floating block height variation (compression) was also analyzed and the resulting antenna was very robust even for heavy compressions. The proposed radiating elements are intended for their use in a two-antenna scheme in which the transmitter can switch between them in order to cover most of the Cospas-Sarsat satellite constellation. Indeed, the two antennas provide almost complementary radiation patterns so that Cospas-Sarsat requirements (satellite constellation coverage and EIRP profile) can be fully satisfied. ACKNOWLEDGMENT The authors would like to thank Ce.R.Ca., Calearo’s research centre in Vicenza, Italy (http://www.calearo.com/) for making available the measurement site and E. Toniolo for his precious help in the measurement campaign. The authors also acknowledge the support of CST for providing additional resources and technical assistance for the parallel version of CST Microwave Studio. REFERENCES [1] [Online]. Available: http://www.cospas-sarsat.org/ [2] R. Azaro, M. Donelli, D. Franceschini, E. Zeni, and A. Massa, “Optimized synthesis of a miniaturized SARSAT band pre-fractal antenna,” Microw. Opt. Technol. Lett., vol. 48, no. 11, pp. 2205–2207, 2006. [3] A. A. Serra, A. R. Guraliuc, P. Nepa, G. Manara, I. Khan, and P. S. Hall, “Dual-polarisation and dual-pattern planar antenna for diversity in body-centric communications,” IET Microw. Antennas Propag., vol. 4, no. 1, pp. 106–112, 2010. [4] P. S. Hall and Y. Hao, Antennas and Propagation for Body-Centric Wireless Communications. Norwood, MA: Artech House, 2006. [5] F. Declercq and H. Rogier, “Active integrated wearable textile antenna with optimized noise characteristics,” IEEE Trans. Antennas Propag., vol. 58, no. 9, pp. 3050–3054, 2010. [6] B. Sanz-Izquierdo, J. C. Batchelor, and M. I. Sobhy, “Button antenna on textiles for wireless local area network on body applications,” IET Microw. Antennas Propag., vol. 4, no. 11, pp. 1980–1987, 2010. [7] G.-Y. Lee, D. Psychoudakis, C.-C. Chen, and J. L. Volakis, “Omnidirectional vest-mounted body-worn antenna system for UHF operation,” IEEE Antennas Wireless Propag. Lett., vol. 10, pp. 581–583, 2011. [8] [Online]. Available: http://www.cospassarsat.org/images/stories/SystemDocs/Current/t7oct28.10completedoc.pdf [9] C. Balanis, Antenna theory. Analysis and Design. New York: Wiley, 2005, ch. 4. [10] [Online]. Available: http://niremf.ifac.cnr.it/tissprop/htmlclie/htmlclie.htm#atsftag [11] [Online]. Available: http://www.cst.com/ [12] C.-K. Chou, G. W. Chen, A. W. Guy, and K. H. Luk, “Formulas for preparing phantom muscle tissue at various radiofrequencies,” Bioelectromagnetics, vol. 5, pp. 435–441, 1984. [13] [Online]. Available: http://www.feko.info/ Andrea A. Serra received the Laurea degree in telecommunications engineering and the Ph.D. degree from the University of Pisa, Pisa, Italy, in 2003 and 2007, respectively. Since September 2003, he has been with the Microwave and Radiation Laboratory, Department of Information Engineering at the University of Pisa. In 2006, he was a Visiting Ph.D. Researcher at the Electronic, Electrical and Computer Engineering, University of Birmingham. He is currently conducting his postdoctoral research at the University

of Pisa. His research interests are about the design of wideband and multiband antennas for base station and mobile terminal, the implementation of diversity schemes for mobile communications. He is also involved in the characterization of channel propagation for body centric communication systems and in the design of wearable antennas.

Paolo Nepa received the Laurea degree in electronics engineering (summa cum laude) from the University of Pisa, Italy, in 1990. In 1993 he became Researcher at the Department of Information Engineering of the University of Pisa. From July to December 1998, he was at the ElectroScience Laboratory (ESL), The Ohio State University, Columbus, as a Visiting Scholar supported by a grant of the Italian National Research Council. At the ESL, he was involved in research on efficient hybrid techniques for the analysis of large antenna arrays. In April 2002 he became Associate Professor at the University of Pisa, where he currently teaches courses on Electromagnetic Fields, Antennas and Propagation, EM Radiations and Biological Interactions. His research interests include the extension of high-frequency techniques to electromagnetic scattering from material structures and its application to the development of radio propagation models for indoor and outdoor scenarios of wireless communication systems. His research group his mainly involved in the design of wideband and multiband antennas, for both base stations and user terminals of modern communication systems, and in the performance analysis of spatial/polarization diversity techniques, as well in radiolocation systems. More recently he is working on channel characterization and wearable antenna design for body-centric communication systems, in collaboration with the University of Birmingham, U.K., and the Queen Mary University of London, U.K. He has been Visiting Professor at the University of Oviedo, Spain, and at the Yuan-Ze University, Taiwan. He has coauthored over 60 technical papers in international journals and more than 120 refereed conference papers. Dr. Nepa received the Young Scientist Award from the International Union of Radio Science, Commission B, in 1998.

Giuliano Manara was born in Florence, Italy, on October 30, 1954. He received the Laurea degree in electronics engineering (summa cum laude) from the University of Florence, Italy, in 1979. Currently, he is a Professor at the College of Engineering of the University of Pisa, Italy. Since 2000, he has been serving as the President of the Bachelor and the Master Programs in Telecommunication Engineering at the same University. Since 1980, he has been collaborating with the Department of Electrical Engineering of the Ohio State University, Columbus, Ohio, where, in the summer and fall of 1987, he was involved in research at the ElectroScience Laboratory. His research interests have centered mainly on the asymptotic solution of radiation and scattering problems to improve and extend the uniform geometrical theory of diffraction. In this framework, he has analyzed electromagnetic wave scattering from material bodies, with emphasis on the scattering from both isotropic and anisotropic impedance wedges. He has also been engaged in research on numerical, analytical and hybrid techniques (both in frequency and time domain), scattering from rough surfaces, frequency selective surfaces (FSS), and electromagnetic compatibility. More recently, his research has also been focused on the design of microwave antennas with application to broadband wireless networks, and on the development and testing of new microwave materials (metamaterials). Prof. Manara was elected an IEEE Fellow in 2004 for “contributions to the uniform geometrical theory of diffraction and its applications.” Since 2000, he has been serving as the Secretary/Treasurer of the Italian Society on Electromagnetics (Società Italiana di Elettromagnetismo, SIEm). Since 2002, he has been working as a member of the IEEE Italy Section Executive Committee. In May 2004, he was the Chairman of the Organizing Committee for the International Symposium on Electromagnetic Theory of Commission B of the International Union of Radio Science (URSI). He also served as a Convenor for several URSI Commission B international conferences, and URSI General Assemblies. Since August 2011, he is serving as a Chairman of the International Commission B of URSI.

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Analysis of Cellular Antennas for Hearing-Aid Compatible Mobile Phones Pekka M. T. Ikonen, Member, IEEE, and Kevin R. Boyle, Member, IEEE

Abstract—The Federal Communications Commission ensures that a certain portion of mobile phones sold in the United States are hearing-aid compatible. When the phone is tested for compatibility, the spatial near-field distribution generated by the phone in the vicinity of its acoustic output is used as the assessment criteria. Certain types of cellular antennas are known partial solutions for the radio-frequency related challenges of hearing-aid compatibility. We briefly summarize the characteristics of these antennas, introducing a new figure-of-merit based on a radiating and balanced mode analysis. We then introduce a class of dual-feed, dual-radiator cellular antennas as promising new candidates for enabling hearing-aid compatibility. It is shown that the proposed antennas, utilized with proper matching circuits, have inherent characteristics that make them attractive solutions for hearing-aid compatible mobile phones. Index Terms—Cellular antenna, GSM, hearing-aid compatibility, mobile phone, radiating and balanced mode analysis, spatial near-field profile.

I. INTRODUCTION

T

HE Federal Communications Commission (FCC) requires that a certain portion of mobile phones offered to consumers in the United States of America (USA) are hearing-aid compatible (HAC) according to the ANSI C63.19 standard [1], [2]. A typical hearing-aid device is used either in microphone mode or in T-coil (induction coil) mode [2]. Devices operating in the microphone mode are susceptible to radio-frequency (RF) electromagnetic disturbances, whereas those operating in the T-coil mode are disturbed by audio-frequency magnetic field phenomena stemming from transient currents. Here we focus only on the microphone mode, and use the term HAC to refer to phone compatibility in this mode. We bear in mind, however, that a real HAC phone must also meet the compatibility requirements when the hearing-aid device is used in the T-coil mode: audio designers are typically responsible for this. The root cause of compatibility challenges in the microphone mode is the emission of pulsed RF signals by the cellular antenna. The signals are picked up and demodulated at the input of hearing-aid device. They are then further amplified and filtered by the remaining stages of the device and may produce audible interference if the signal is strong enough [3], [4]. Antenna Manuscript received April 11, 2011; revised June 28, 2011; accepted August 03, 2011. Date of publication October 24, 2011; date of current version February 03, 2012. P. M. T. Ikonen is with TDK-EPC Corporation, Espoo 02600, Finland (e-mail: [email protected]). K. R. Boyle is with Bracknell, Berkshire, RG12 2XH, U.K. (e-mail: kevin. [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173149

and RF designers are typically responsible for ensuring that the near-field amplitudes in the vicinity of the phone’s acoustic output are within allowed limits, referred to as M-limits in the ANSI standard [2]. Pulsed signals are generated by time-division multiple-access (TDMA) systems such as GSM. The fundamental frequency (and several of the harmonics) associated with the GSM pulse repetition rate of 217 Hz fall in the audible frequency region. In addition to this, the peak transmission power is high, at 2 W and 1 W in the low-band (bands V and VIII) and high-band (bands I, II, III, IV) respectively. In other TDMA systems such as TD-SCDMA and LTE TDD the transmission power is noticeably lower than in GSM. It is therefore predicted that HAC challenges are less likely with these systems. The allowance limits for emissions at frequencies below 960 MHz are 10 dB higher (more relaxed) than at frequencies greater than 960 MHz [2]. This, added to the fact that the GSM peak transmission power at frequencies below 960 MHz is only 3 dB higher than at frequencies above 960 MHz, implies that the biggest real-life challenge to meet HAC RF emission limits are over the transmission (Tx) frequencies of band II (GSM1900). Similar arguments are presented also in [5]–[7]. From a standard-specified compatibility point of view, only GSM1900 is important, since currently HAC is a regulatory requirement only in the USA. Some initial works devoted mainly to general modeling and measurements of interaction between hearing-aid devices and wireless devices can be found in [3], [4], [8], [9]. Recently, the focus has turned more towards real antenna and RF solutions leading to dedicated HAC mobile phones. The techniques proposed to manipulate the near-field distribution of a HAC phone can be roughly divided into two categories: i) manipulate the mechanics of the phone, ii) utilize certain types of cellular antennas. Wavetraps (typically grounded quarter-wave resonators) [5], [6], [10], [11] or band-stop filters implemented in the chassis [12] are some proposals aimed to solve HAC by manipulating the mechanics of the phone. Whilst in most cases the results are rather good with “bare” printed-wiring-board (PWB) prototypes, the operation of the proposed parasitic high-Q resonators might be disturbed when placed inside a real phone packed full of lossy (for radio-frequency wave interaction) mechanics components. An alternative approach is to utilize cellular antennas, mounted at the bottom of the PWB, that are able to excite (quasi-) balanced (differential) spatial near-field profile over certain frequencies relevant for HAC [7], [13]. The majority of such antennas proposed so far either have parasitic elements or are variants of folded loops [14], [15]. Implementation examples of folded

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Fig. 2. Loop antenna with simplified equivalent radiating and balanced modes.

Fig. 1. Loop antenna with equivalent radiating and balanced modes.

loops in real mobile phones can be found, for example, in [16], [17]. We demonstrate in this paper how folded loops and antennas with parasitic elements can operate quasi-independently of the phone PWB (which in turn leads to low fields at the phone speaker location and good HAC performance). We then show that a class of dual-feed bottom-mounted cellular antennas, utilized with proper matching circuits, can also excite quasi-independent modes over the frequencies relevant for HAC. We explain the HAC performance of these antennas using a radiating and balanced mode analysis [18]. It is shown that the proposed dual-feed antennas are volume-versus-performance competitive, when compared against the known single-feed antennas that enable HAC. The rest of the paper is organized as follows. In Sections II and III we analyze the basic performance of loop and parasitic antennas respectively. We utilize a radiating and balanced mode analysis to determine the extent to which radiation is independent of the PWB. In Section IV a proposed dual-feed antenna is introduced, analyzed and shown to have HAC properties similar to those of parasitic structures. Input matching and total efficiency are verified by measurements in Section V and conclusions are drawn in Section VI. All 3-D electromagnetic simulations are performed using Ansoft HFSS. II. ANALYSIS OF “LOOP” ANTENNAS The “loop” antennas under consideration and their radiating and balanced mode decomposition are shown in Fig. 1. It is assumed that the feed and shorting pins of the antenna (shown vertically in Fig. 1) are electrically close and therefore strongly coupled. The same assumption is also made for the horizontal sections of the antenna. These assumptions are normally met due to the size constraints of typical commercial implementations and the need to obtain resonance close to 900 MHz. The feed and shorting pins are denoted by 1 and 2 respectively. and are the radiating mode currents on the feed and shorting pins respectively, whereas and are the balance mode currents. One of the balanced mode voltage sources is multiplied by a current sharing factor, given by (1)

Fig. 3. Example “loop” antenna. All dimensions are in millimeters.

This is necessary to ensure balance, i.e., . In this case, the radiating and balanced modes can be further simplified as shown in Fig. 2. Consider first the radiating mode, the impedance of which is given by . It consists of a “T” shaped monopole fed directly against the PWB. Within the monopole is a slot that is excited at its centre. When the slot is half-wave long it has a low current adjacent to the feed, forcing a high impedance. In turn, this reduces the currents on both the monopole and the PWB (since current continuity must be maintained between the two). The balanced mode impedance is given by . Nominally no current flows on the PWB and the “loop” antenna acts as a folded dipole. At the input, the radiating and balanced modes add according to [18] (2) is impedance seen at the antenna feed due to the radiating mode, where it simply adds in parallel with . Consider the implementation shown in Fig. 3. This is similar to the folded-loop analyzed in [7], but with the parasitic element removed for analysis and illustration purposes. The loop is made

IKONEN AND BOYLE: ANALYSIS OF CELLULAR ANTENNAS FOR HEARING-AID COMPATIBLE MOBILE PHONES

Fig. 4. Radiating and balanced mode impedances of the structure shown in , (b) , (c) , (d) , (e) of and in dB Fig. 3: (a) versus frequency. The frequency range is 0.8–3 GHz for all plots and Smith Chart trace symbols are separated by 20 MHz.

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of copper strips that are folded to conform to the outer casing of the handset. The antenna is predominantly over ground and we model the PWB as a 40 100 mm copper plate. The height of the antenna element in z-direction is 5 mm, and the maximum dimension in the y-direction is 21.5 mm. Throughout the rest of the paper we study the near-field distributions generated by the antennas over the grey area shown in Fig. 3 below the PWB. The dimensions of the area are 50 110 mm , and it is located 15 mm below the PWB. A similar area has been used for near-field studies also in [7]. It is to be noted that this area is not the same measurement area as specified in the standard [2]1. We deliberately study the near fields over a larger area to better illustrate the quasi-balanced mode related characteristics of near-field distributions. The radiating and balanced mode impedances of this structure are shown in Fig. 4. They combine to give exactly the correct impedance at the feed. It can be seen that the radiating mode impedance, is that of a “T” shaped monopole fed against PWB that is impedance transformed to at the antenna input due to the current sharing of the feed and shorting strips. In this case the structure is symmetric, so is real and equal to 1, such that the impedance transformation factor is equal to 4. Importantly, an antiresonance is imposed by the slot within the monopole which minimizes the radiating mode currents. This occurs when the slot is electrically half-wave: in this case at 1.88 GHz, as shown in Fig. 4(a). In the balanced mode, the antenna feed is isolated from the PWB and the impedance is that of a folded dipole (which is also impedance transformed by the action of the fold). It is resonant at 1.935 GHz, as shown in Fig. 4(c). The physical requirements for the slot and the dipole to be half-wave resonant at a particular frequency are approximately the same. At this frequency, the radiating mode is at a high impedance level, drawing a small current, whereas the balanced mode is close to the system impedance, drawing a relatively much higher current. Hence, the antenna is well matched, but independent of the ground plane—exactly what is required for good HAC performance. A figure-of-merit (FOM) for this can be derived by comparing the radiating mode current with the total current, as follows: (3) and is used to indicate a real comwhere ponent. This is necessary to disallow localized reactive currents from the FOM. This is plotted for the structure of Fig. 3 in Fig. 5. It is shown that the relative magnitude of the radiating mode current is low over a moderate range of frequencies centered about 1.9 GHz, corresponding approximately to the resonant frequency of the balanced mode and the antiresonant frequency of the radiating mode. The FOM indicates the extent to which source currents are decoupled from the PCB. However, it does not take account of

Fig. 5.

versus frequency (GHz) for the structure shown in Fig. 3.

1The distance between the antenna and the near-field measurement area is according to the standard.

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Fig. 6. Near-field distributions generated by the antenna shown in Fig. 3 at 1.0 GHz. (The radiator location corresponds to the bottom of the distributions).

Fig. 8. Near-field distributions generated by the antenna shown in Fig. 3 at 2.75 GHz.

Fig. 9. Parasitic antenna with equivalent radiating and balanced modes.

Fig. 7. Near-field distributions generated by the antenna shown in Fig. 3 at 1.95 GHz.

the coupling between the antenna and the PCB, so should not be used as an absolute indication of HAC performance. Fig. 4(d) shows the impedance at the antenna feed and indicates that three resonances are achieved. In a typical industrial design the second resonance of the radiating mode is often lowered to form a dual-resonance in the 2 GHz region. This will not significantly affect the balanced mode, so the HAC performance will remain good compared to a conventional antenna (without a mode that is isolated from the PWB). However, the radiating mode antiresonance may not be maintained within band II, which will lessen the improvement achievable. Figs. 6–8 show the near field distributions at 1.0, 1.95 and 2.75 GHz, respectively. All the numerical near-field plots presented in the paper have been simulated with 1 W accepted power at the active antenna port. The simulated radiation efficiencies at the above mentioned frequencies are dB, dB and dB respectively. At the lowest resonant frequency (Fig. 6) the field distributions are dominated by the radiating mode, with sine-like E-field and cosine-like H-field distributions. The near-field distribution corresponding to the second resonant frequency (Fig. 7) is a clear quasi-balanced mode as can be seen when comparing

the distributions to ([7], Figs. 2 and 5). The H-field distribution corresponds to balanced-mode current distributions presented, for example, in ([15], Figs. 3 and 9) and ([19], Fig. 5). At the highest resonant frequency the near-field distribution (Fig. 8) again shows a clear radiating-mode distribution as is apparent from the standing-wave patterns. Fig. 7 illustrates the benefit of the quasi-balanced mode for HAC: “cool” near-field areas are observed at the top of the near-field analysis region that corresponds to the location of the phone acoustic output. III. ANALYSIS OF PARASITIC ANTENNAS A parasitic antenna and an equivalent radiating and balanced mode decomposition are shown in Fig. 9. With chosen such that , the radiating and balanced modes can be further simplified as shown in Fig. 10. The radiating mode is a simple “T” monopole fed against the PWB, whereas the balanced mode is a dipole fed in isolation from the PWB. Consider the implementation shown in Fig. 11. Here the structure has the same dimensions as the “loop” antenna shown in Fig. 3 and the driven and parasitic elements are symmetric: we have closed the slots in the loop antenna to generate the driven and parasitic elements. The impedances of the radiating and balanced modes are as given in (2) and shown in Fig. 12. The is shown in

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Fig. 10. Parasitic antenna with simplified equivalent radiating and balanced modes.

Fig. 11. Parasitic antenna (of the same dimensions as the “loop” antenna shown in Fig. 3).

Fig. 13. Here the is clearly not as good as for the “loop” antenna, though at 1.71 GHz some improvement in HAC performance should be expected. Away from their resonant frequency, parasitic elements increase the antenna quality factor. This is particularly the case in dual-band antenna designs: a parasitic element used to increase the high frequency bandwidth tends to reduce the low frequency bandwidth. The reason for this has not received much attention and can be explained by the addition of the radiating and balanced modes. The radiating mode has a lower quality factor (Q) than balanced mode. The summation (radiating plus balanced modes) also has a higher Q than the radiating mode. We define the Q as [20]

Fig. 12. Radiating and balanced mode impedances of the structure shown in , (b) , (c) , (d) , (e) of and in dB Fig. 11: (a) versus frequency. The frequency range is 0.8–3 GHz for all plots and Smith Chart trace symbols are separated by 20 MHz.

(1) where is impedance, is resistance and is the angular frequency. The Q versus frequency of the parasitic arrangement is shown in Fig. 14. It can be seen that the radiating mode has the lowest Q at all frequencies. Hence, using part of the available space for a parasitic element is only worthwhile if a resonant loop is introduced in the impedance response (the driven and parasitic elements must have different dimensions for this to occur), that, in turn,

Fig. 13.

versus frequency (GHz) for the structure shown in Fig. 11.

aids matching over a particular frequency band. For frequencies outside this band, the parasitic element degrades the antenna bandwidth (compared to an antenna that also utilizes the

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Fig. 16. Matching circuits used with the antenna shown in Fig. 15.

Fig. 14. versus frequency for the structure shown in Fig. 11 in the radiating (red, solid), balanced (green, long dash) and radiating plus balanced (blue, short dash) modes.

Fig. 15. Simulation model for the considered dual-feed antenna. All dimensions are in millimeters.

volume occupied by the parasitic element). This is because the balanced mode contribution effectively increases the antenna Q. Near field distributions for a similar parasitic arrangement are shown in the section that follows, hence, for brevity they are omitted here.

Fig. 17. Input impedance of the low-band radiator without the matching circuit over 824–960 MHz.

IV. PROPOSED DUAL-FEED ANTENNA The proposed dual-feed antenna is shown in Fig. 15. The low-band radiator is the meandered radiator on the left and the high-band radiator is the shorter radiator on the right. The PWB dimensions are 45 105 mm . Metallization has been removed from the PWB for an area 45 7 mm at the bottom end of the PWB (there is 7 mm ground clearance below the radiators). PWB corner rounding, with a rounding radius of 5 mm, is used to more realistically model the main board of a real mobile phone. The dimensions of the PWB and ground clearance are different as compared to the antenna structures presented in Sections II and III. The PWB dimensions chosen for the dual-feed antenna correspond to the main board of a real smartphone available on the market. We simulate the plastic carrier, shown semi-transparent in Fig. 15, as a pc-abs plastic block having permittivity . The height of the carrier is 5 mm. The PWB conductor and the radiators have simulated conductivity S/m, and the C-clips (connecting the radiators to PWB) are modeled as copper blocks having bulk copper conductivity. Example matching circuits used with the proposed antenna are shown in Fig. 16. The principal operation of the antenna and

matching circuit is straightforward. Consider, for example, the low-band radiator. The electrical length of the radiator at around 900 MHz is where is the effective wavelength (in the given prototype mechanics setup). Thus, over bands V and VIII the radiator obeys normal quarter-wave monopole type seriesresonant input impedance behavior, as shown in Fig. 17. The same holds for the high-band radiator, as shown in Fig. 18 (the “pure” series resonant impedance behavior is slightly disturbed by finite coupling between the first harmonic of the low-band radiator and the fundamental mode of the high-band radiator, and by the influence of the PWB). In the PWB prototype setup the real part of input impedance is mainly determined by the ground-clearance portion. When the antenna is located inside a real phone, close by mechanics components such as battery, possible metal back cover, hands-free speakers below the antenna carrier, and display typically lower the real part of input impedance. In this case the purpose of the matching circuit is to first increase the input resistance close to 50 Ohms. This can be done, for example, with the tapped inductors shown in Fig. 16. Small inductor values can be implemented as narrow lines on the PWB. The residual reactance is tuned with the shunt capacitor to create a double-resonant matching peak.

IKONEN AND BOYLE: ANALYSIS OF CELLULAR ANTENNAS FOR HEARING-AID COMPATIBLE MOBILE PHONES

Fig. 18. Input impedance of the high-band radiator without the matching circuit over 1710–2170 MHz.

Fig. 19. Simulated S-parameters of the radiators in Fig. 15 through the matching circuits in Fig. 16. Solid blue line is the input matching of the low-band radiator, dashed magenta line is the input matching of the high-band radiator, and the brown dash-dotted line (visible at around 1.9 GHz) is the dB, the coupling between the radiators. The horizontal line marker is at vertical line markers are at 824 and 960 MHz, and at 1710 and 2170 MHz.

The input matching through the matching circuits is shown in Fig. 19. A conceptually similar antenna and matching configuration is considered in [21]. Also [22] shows a conceptually rather similar antenna implementation. The novelty of the present paper is to analyze the HAC performance of the proposed antenna. RF front-end architectures that are suitable for use with dual-feed cellular antennas are presented in [23]–[25]. The proposed antennas and matching circuits behave as bandpass filters. This filtering behavior ensures good isolation between the radiators, which is a mandatory requirement for the

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Fig. 20. Impedance seen towards the RF front-end when looking from the low-band radiator connection point through the matching circuit. The plotted frequency range is 1850–1910 MHz.

proper operation of the rest of the RF in the phone. The low-band matching circuit also plays an important role in the HAC performance of the antenna. The impedance seen looking towards the RF front end at the low-band radiator connection point through the matching circuit is shown in Fig. 20 over the Tx band of GSM1900. The matching circuit effectively short circuits the low-band radiator at these frequencies. This means that heuristically the antenna in Fig. 15 can be considered around 1900 MHz as an asymmetrical dipole, where the high-band radiator is the active dipole arm, and the low-band radiator is an electrically long parasitic arm. Hence, the radiating and balanced mode analysis of Section III is appropriate. The impedances of the radiating and balanced modes are as given in (2) and shown in Fig. 21. The is shown in Fig. 22. It is not as good as for the “loop” antenna, though at 1.9 GHz (and other frequencies) some improvement in HAC performance over more conventional antennas is expected. At 1850 MHz there is a peak in the due to resonance in the radiating mode (causing a peak in the radiating mode current). This is closely followed by a minima in the at 1930 MHz due to resonance in the balanced mode. The peak can be suppressed by reducing the resonant frequency of the radiating mode. This can be achieved by making the high frequency resonator electrically longer. Doing so also reduces the resonant frequency of the balanced mode, as indicated by the modified result shown in Fig. 22. More work is required to find ways of independently controlling the radiating and balanced mode impedances, either by antenna or matching circuit design. The above observations are partly validated by the near-field distribution shown in Fig. 23. We have implemented lossless 3-D models for the matching circuits into HFSS (losses in the

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Fig. 23. Near-field distributions of the antenna shown in Fig. 15 at 1.9 GHz.

Fig. 21. Radiating and balanced mode impedances of the structure shown in , (b) ,. The frequency range is 0.8–3 GHz for all plots and Fig. 15: (a) trace symbols are separated by 20 MHz.

Fig. 24. Near-field distributions of the antenna shown in Fig. 15 at 0.9 GHz.

Fig. 22. versus frequency for the structure shown in Fig. 15 with and without modification to alter the frequencies of the minima/maxima.

matching circuits are accounted for in the experimental verification where we present measured total and radiation efficiency results). The high-band radiator feed port is active, and the low-band radiator feed port is replaced with a 50 Ohm load. The near-field distribution has a close resemblance to the near field distribution shown in Fig. 7, implying a quasi-balanced mode excitation. The slight asymmetry observed in Fig. 23 is explained by the structural asymmetry (of the effective dipole). The simulated radiation efficiency at 1.9 GHz is dB. For reference, a clear common-mode distribution is illustrated in Fig. 24 at 0.9 GHz (the low-band radiator is active, and the highband radiator is terminated in 50 Ohms). The corresponding simulated radiation efficiency is dB. It is noted that the matching circuits shown in Fig. 16 are example proposals aimed to enable HAC. The antenna in Fig. 15 also works for HAC with other matching circuits, provided that

IKONEN AND BOYLE: ANALYSIS OF CELLULAR ANTENNAS FOR HEARING-AID COMPATIBLE MOBILE PHONES

Fig. 25. Photograph of the implemented prototype.

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Fig. 27. Measured total efficiency (solid line) and radiation efficiency (dashed line) in free space (dB).

lower than the simulated radiation efficiency, indicating lower realized conductivity of radiators and C-clips, and higher realized loss tangent of the antenna carrier. Also, we observe that there is a total efficiency minimum at high-band around 1950 MHz. This is mainly due to the corresponding matching notch, as the radiation efficiency is rather flat at the middle of 2 GHz region. Thus, the excitation of the quasi-balanced mode itself does not noticeably lower the efficiency of the antenna, even though the corresponding mode has slightly higher quality factor as compared to common mode. VI. CONCLUSION Fig. 26. Measured S-parameters of the prototype radiators. Solid blue line is the input matching of the low-band radiator, dashed magenta line is the input matching of the high-band radiator, and the brown dash-dotted line is the coupling between the radiators. The horizontal and vertical line markers are at same locations as in Fig. 19.

the low-band matching circuit effectively short circuits the lowband radiator at around 1900 MHz. V. PROTOTYPE A photograph of the implemented prototype is shown in Fig. 25. We have chosen discrete matching components whose values are closest to the values shown in Fig. 16. Component values for the low-band radiator are 2.2 nH and 15 pF, and for the high-band radiator 1.5 nH and 5.1 pF. In this case the matching circuits have been implemented as parallel LC circuits (values of series inductors are practically zero). The measured input matching and total efficiency are shown in Figs. 26 and 27. We observe that the radiators are well matched over the target bands. There is a slight total efficiency challenge at the highest edge of band VIII. It is to be noted that the total efficiency is somewhat sensitive, especially at band edges, to the matching component values (impedance transformation). Thus, it is most likely the component values (transformation) used in the prototype implementation are not optimum for total efficiency, though they optimize the input matching. The measured radiation efficiency is also slightly

This paper provides detailed analysis of the HAC performance of “loop” and parasitic based mobile phone antennas. The analysis uses a radiating and balanced mode decomposition to isolate currents that flow on the PWB and solely on the antenna structure itself. This allows a HAC FOM to be derived that indicates the degree to which currents are suppressed on the PWB. It is shown that “loop” antennas have a particularly good FOM (over a narrow frequency band). Parasitic antennas also suppress PWB currents, and a similar mechanism can be utilized in the design of a dual antenna system (a low- and high-band antenna pair) with appropriate matching. Such a system has the advantages that the antennas can be simple and the matching specific to each band. ACKNOWLEDGMENT The authors would like to thank Mr. Naoaki Utagawa for assistance in making the prototype, and carrying out the measurements. REFERENCES [1] Hearing Aid Compatibility for Wireless Telephones. New York: FCC Consumer Publications [Online]. Available: http://www.fcc.gov/cgb/ consumerfacts/hac_wireless.html [2] American National Standard for Method of Measurements of Compatibility Between Wireless Communication Devices and Hearing Aids, ANSI C63.19-2007, Amer. Nat. Standards Inst., 2007, New York. [3] K. Caputa, M. A. Stuchly, M. Skopec, H. I. Bassen, P. Ruggera, and M. Kanda, “Evaluation of electromagnetic interference from a cellular telephone with a hearing aid,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pp. 2148–2154, Nov. 2000.

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[4] R. E. Schlegel and F. H. Grant, “Modeling the electromagnetic response of hearing aids to digital wireless phones,” IEEE Trans. Electromagn. Compat., vol. 42, no. 4, pp. 347–357, Nov. 2000. [5] P. Lindberg, A. Kaikkonen, and M. Sudow, “Improvement of hearing aid compatibility (HAC) of terminal antennas using wavetraps,” in Proc. 2009 IEEE Int. Workshop Antenna Technology IWAT, Santa Monica, CA, Mar. 2009, pp. 1–4. [6] P. Bahramzy, L. Azzinnari, K. B. Jakobsen, and M. Sager, “Near-field reduction techniques in the speaker area of slide mobile phones for improved HAC performance,” in Proc. 2009 Antennas Propag. Soc. Int. Symp., Charleston, SC, Jun. 2009, pp. 1–4. [7] P. Hui, “Near fields of phased antennas for mobile phones,” in Proc. 2009 Asian Pacific Microw. Conf., Singapore, Dec. 2009, pp. 2718–2721. [8] M. Skopec, “Hearing aid electromagnetic interference from digital wireless telephones,” IEEE Trans. Rehabilitation Engineering, vol. 6, no. 2, pp. 235–239, Jun. 1998. [9] M. Okoniewski and M. A. Stuchly, “Modeling of interaction of electromagnetic fields from a cellular telephone with hearing aids,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 11, pp. 1686–1693, Nov. 1998. [10] J. Holopainen, J. Ilvonen, O. Kivekäs, R. Valkonen, C. Icheln, and P. Vainikainen, “Near-field control of handset antennas based on inverted-top wavetraps: Focus on hearing-aid compatibility,” IEEE Antennas Wireless Propag.Letters, vol. 8, pp. 592–595, 2009. [11] C.-T. Lee and K.-L. Wong, “Internal WWAN clamshell phone antenna using a current trap for reduced ground plane effects,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3303–3308, Oct. 2009. [12] A. Zhao, J. Ollikainen, J. Thaysen, and T. Bodvarsson, “Design of HAC compatible mobile phone with LC band-stop filter embedded in PWB,” in Proc. 2008 Asian Pacific Microw. Conference, Macau, Dec. 2008, pp. 1–4. [13] J. Ilvonen, J. Holopainen, O. Kivekäs, R. Valkonen, C. Icheln, and P. Vainikainen, “Balanced antenna structures of mobile phones,” in Proc. 2010 Eur. Conf. Antennas Propagatiion (EuCAP 2010), Barcelona, Spain, Apr. 2010, pp. 1–5. [14] H. Morishita, H. Furuuchi, and K. Fujimoto, “Performance of balance-fed antenna system for handsets in the vicinity of a human head or hand,” in IEE Proc. Microw., Antennas Propag., Apr. 2002, vol. 149, no. 2, pp. 85–91. [15] S. Hayashida, H. Morishita, and K. Fujimoto, “Self-balanced wideband folded loop antenna,” in IEE Proc. Microw., Antennas Propag., Feb. 2006, vol. 153, no. 1, pp. 7–12. [16] C. Di Nallo and A. Faraone, “Multiband internal antenna for mobile phones,” IEE Electron. Lett., vol. 41, no. 9, pp. 1–2, Apr. 2005. [17] I. Szini, C. Di Nallo, and A. Faraone, “The enhanced bandwidth folded inverted conformal antenna (EB FICA) for multi-band cellular handsets,” in Proc. 2007 Antennas Propag. Soc. Int. Symp., Honolulu, HI, Jun. 2007, pp. 4697–4700. [18] K. R. Boyle and L. P. Ligthart, “Radiating and balanced mode analysis of PIFA antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 231–237, Jan. 2006. [19] B. S. Collins, S. P. Kingsley, J. M. Ide, S. A. Saario, R. W. Schlub, and S. G. O’Keefe, “A multi-band hybrid balanced antenna,” in Proc. 2006 IEEE Int. Workshop Small Antennas Metamaterials IWAT, Mar. 2006, pp. 1–4.

[20] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” in Proc. 2003 IEEE Antennas Propag. Soc. Int. Symp., Columbus, OH, Jun. 2003, vol. 1, pp. 501–504. [21] T. Ranta, “Dual-Resonant Antenna,” U.S. Patent 7 242 364 B2, Jul. 10, 2007. [22] P. Bahramzy and M. Sager, “Dual-feed ultra-compact reconfigurable handset antenna for penta-band operation,” in Proc. 2010 Antennas Propag. Soc. Int. Symp., Toronto, ON, Jul. 2010, pp. 1–4. [23] R. Whatley, T. Ranta, and D. Kelly, “RF front-end tunability for LTE handset applications,” in 2010 Proc. Compound Semiconductor Integrated Circuit Symp., Monterey, CA, Oct. 2010, pp. 1–4. [24] T. Ranta, D. Pilgrim, and R. Whatley, “RF front end adapts for increased mobile data demand,” EE Times, pp. 1–9, Oct. 2010. [25] The Prismark Wireless Technology Report, Sep. 2010.

Pekka M. T. Ikonen (S’04–M’07) was born on December 30, 1981, in Mäntyharju, Finland. He received the M.Sc. and D.Sc. degrees in communications engineering (with distinction) from the Helsinki University of Technology, Helsiinki, Finland, in 2005 and 2007, respectively. From 2007 to 2009 he was first with Nokia Research Center and then with Nokia Devices R&D working as Antenna Researcher and Antenna Technology Manager, respectively. Since August 2009 he has been with TDK-EPC defining strategies for RF and antenna system development for future handset devices.

Kevin Boyle (M’05) was born in Chelmsford, U.K., on January 23, 1966. He received the B.Sc. (hons.) in electrical and electronic engineering from City University, London, U.K., the M.Sc. degree in microwaves and optoelectronics (with distinction) from University College, London, U.K., and the Doctor of Technology degree from Delft University of Technology, Delft, The Netherlands. He was with Marconi Communications Systems Ltd. until to 1997, working on all aspects of antenna system design. He then joined Philips Research Laboratories (which became NXP Semiconductors Research in 2006) where he was a Principal Research Scientist and a Project/Cluster Leader for antenna and propagation related activities. In 2008 he joined EPCOS (which has since become TDK-EPC) working as an Antenna Systems Architect. His main areas of interest include antenna design for mobile communication systems, adaptive RF systems, MIMO/diversity, propagation modeling and related areas of mobile system design. Dr. Boyle has actively participated in COST 259 and 273, is a member of the IET—where currently serving on the Antennas and Propagation Executive Committee—and a Chartered Engineer. He has published more than 30 papers in refereed international journals and conferences, has contributed to two books and holds more than fifteen patents.

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A Mobile Communication Base Station Antenna Using a Genetic Algorithm Based Fabry-Pérot Resonance Optimization Dongho Kim, Member, IEEE, Jeongho Ju, and Jaeick Choi

Abstract—We proposed a high-gain wideband resonant-type mobile communication base station antenna using a Fabry-Pérot cavity (FPC) technique. To overcome inherent narrow radiation bandwidth of FPC-type antennas while keeping relatively high gain, we introduced a new superstrate structure composed of square patches and loops, which satisfies an FPC resonance condition at a target frequency region. To do that, we optimized the superstrate geometry with the help of a real-value coding hybrid genetic algorithm (RHGA). The optimized superstrate is very thin, and therefore, it can be fabricated with a single dielectric substrate, which is a fairly strong point in practical applications. Moreover, we enclosed four openings of the antenna in lateral directions to increase antenna gain with a limited aperture area. Therefore, a modified prediction method of an FPC resonance is used, which reduced the effort of complicated three-dimensional antenna optimization. Consequently, our antenna is able to operate in a wide bandwidth with a relatively high realized gain. Furthermore, good agreement between measured results and prediction ones confirms the validity of our design approach. Index Terms—Base station antenna, Fabry-Pérot cavity antenna, hybrid genetic algorithm, high-gain antenna, wideband antenna.

I. INTRODUCTION

R

ECENTLY, in accordance with the growth of mobile communication industry, the usage of a personal mobile phone has been explosively increased. For that reason, mobile base station antenna techniques also have been rapidly developed to keep up with the increased number of users within a service area using limited frequency resources [1], [2]. Many sorts of a base station antenna employ an array of a dipole antenna or a microstrip patch antenna, which is ready to increase overall antenna gain and to control a beam shape according to a frequency reuse plan [3]–[5]. However, signal feeding networks from a power input port to wave radiating structures are generally long and complicated, which might cause an unwanted energy loss during signal transportation.

Manuscript received September 14, 2010; revised May 12, 2011; accepted June 16, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. D. Kim is with the Department of Electronic Engineering, Sejong University, Seoul 143-747, Korea (e-mail: [email protected]). J. Ju and J. Choi are with the Antenna Research Team, Electronics and Telecommunications Research Institute, Daejeon, 305-700, Korea. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173108

Recently, highly directive antennas using resonance of a partially reflective surface (PRS) such as Fabry-Perot cavity (FPC) or electromagnetic band gap (EBG) structures have been introduced [6]–[12]. The FPC antenna makes use of resonance of a cavity generally consisting of a ground plane and a superstrate. By appropriately adjusting the cavity height and the reflection magnitude and phase of the superstrate, the FPC antenna can provide very high gain at and near the resonant frequency [13]. One strong point of the FPC antenna lies in its simple feeding structure. Practically, the FPC antennas provide high gain with a single feeding antenna such as a dipole or a microstrip patch antenna. It is matter of course that array signal feeding can more increase antenna gain compared to a single feeding case. In addition to a horizontally arranged PRS structure, cylindrical EBG structures have also been proposed for base station antenna applications [14], [15]. However, because the cavity resonance condition is satisfied only at one frequency, a radiation bandwidth of the FPC antenna is usually very narrow; in other words, the cavity resonates with a very high quality factor. Therefore, impedance matching and radiation bandwidths of FPC antennas are also inherently very narrow due to the nature of a cavity operation, which are not appropriate to commercial applications. To overcome the narrow radiation bandwidth problem, an FPC antenna with a single-layer frequency selective surface (FSS) superstrate consisting of dissimilar size square conducting patches was proposed [16], [17]. In [16], [17], antenna bandwidth was increased by tapering cells printed on the superstrate, which spread resonant frequencies around a center frequency of a target bandwidth. In the meantime, some techniques of adjusting reflection phase of a superstrate unit cell to meet the resonance condition of a FP cavity were proposed in [18], [19]. Instead of tapering superstrate unit cells, they introduced two individual conductive patterns printed on a single or double dielectric layers, which provides relatively large reflection magnitude with reflection phase similar to an ideal phase response satisfying the FPC resonance. In this paper, we provide a broadband high gain mobile base station antenna. Our antenna has a superstrate composed square patches and loops, which meets an FPC resonance condition in a target Korean personal communication service (PCS) band from 1 750 MHz to 1 870 MHz. The superstrate is very thin, and therefore, it is quite comfortable to fabricate and to apply for practical antennas. To optimize reflection behavior of the superstrate, we use a real-value coding hybrid genetic algorithm

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Fig. 1. Photographs of (a) the inside and (b) the outside of the fabricated FPC antenna.

(RHGA) providing fast convergence with a relatively small size of population [20]–[23]. Initially, we design the superstrate based on a modified FPC resonance prediction formulation, which is able to consider the effect of four metallic side walls. And overall performance of the entire antenna structure is tuned by using a commercial 3-D full wave simulator of CST microwave studio (MWS) [24]. Experimental data show good agreement with the simulation result, which proves the validity of our design approach.

Fig. 2. Description of (a) the patch antenna with mm, mm, mm, mm, mm, mm, mm, and , and (b) a unit cell of the superstrate with mm, mm, mm, and mm.

II. ANTENNA DESIGN AND MEASUREMENT Photographs of a proposed FPC antenna are shown in Fig. 1. A 50 coaxial probe-fed wideband patch antenna is located inside the FP cavity that is enclosed with four lateral metallic walls in the x- and y-direction, respectively. A ground plane of the patch antenna is the bottom face of the cavity. The superstrate consisting of 19 5 unit cells covers the entire upper opening of the FP cavity shown in Fig. 1(a), which is supported with eight acrylic posts. Overall dimension of the FPC antenna is 590 mm 170 mm 98 mm in the x-, y-, and z-direction. A detailed description of the patch antenna and the unit cell geometry are shown in Fig. 2. To extend an impedance matching bandwidth of the patch antenna, we have inserted two rectangular slits as shown in Fig. 2(a) [25], [26]. An inner conductor of a signal feeding coaxial cable is directly connected to the patch. And an air gap has been placed between the substrate of the patch and the ground plane. As for the unit cell of the superstrate, on one side of a dielectric substrate is printed with a square patch and the opposite side with a square loop. The lower side of the superstrate composed of square loops is confronting the bottom side of the cavity. Because our superstrate is very thin, where the thickness is about 1.5 mm, it is directly applicable to practical antennas. We can determine the FP resonance condition by considering reflection phases of two faces of the cavity, which consist of

the superstrate and the ground plane. However, with the help of a modified resonance prediction formula based on a dispersion relation of a classical metallic rectangular cavity, we can more accurately estimate FPC resonance including the effect of four metallic side walls [17]:

(1) are integer numbers corresponding to possible where eigen-modes inside the cavity, and and are lengths of the cavity shown in Fig. 1(b), is the speed of light in air, is the reflection phase of the superstrate, and is the resonant height from the ground to the bottom face of the superstrate. It is well known that waves satisfying the FP resonance condition with relatively large magnitude of reflection can collimate outgoing waves toward a specific direction, and therefore, enhance the directivity and gain of antennas [6]–[10]. Generally, the large reflection can be easily obtained with various types of

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Fig. 5. Reflection phase and magnitude responses of the unit cell composed of a square loop and patch.

Fig. 3. Flow chart of a real-value coding hybrid genetic algorithm (RHGA).

Fig. 4. Optimization process of the unit cell geometry using a RHGA, which shows the best and the average costs, respectively.

conducting patterns within a certain frequency bandwidth. Consequently, the FP resonance condition is also readily acquired by changing the geometry of conducting patterns of a superstrate or by varying the distance between the superstrate and a ground plane. However, the FP cavity resonance condition is exactly satisfied at only one frequency, which restricts expansion of a radiation bandwidth of FPC antennas. Therefore, to obtain a wide radiation bandwidth and high gain properties at the same time, reflection phase of the superstrate should satisfy the FPC resonance condition at more than one frequency. To do that, we have optimized reflection behavior of the superstrate, which might provide an ideal phase-like response (see a broken line in Fig. 5) in a target PCS frequency region. From (1), the ideal phase response depicted in Fig. 5 is derived by (2)

In the frequency region of interest, the lowest mode inside the cavity is a mode, so we set and , respectively [17]. To optimize the geometry of superstrate unit cell, we used a real-value coding hybrid genetic algorithm (RHGA) [20]–[23]. As shown in Fig. 3, the RHGA is equipped with a gradient-like selector based reproduction, modified simple crossover, and dynamic mutation. And, we applied elitism to prevent a loss of the best individual from the preceding generation, which might occur on account of inherent nature of a genetic algorithm. Four target optimization parameters are the height of the FPC cavity, and lengths ( ) and width of the square patch and the square loop patterns. The population size of the RHGA is 10. The optimization process of the superstrate unit cell geometry is shown in Fig. 4. A fitness or cost function to be minimized is defined by

(3) and are reflection phases of an ideal rewhere sponse and the proposed unit cell, is a reflection magnitude of the proposed cell, and are optimization starting and ending frequencies, and are weighting coefficients for each angular and magnitude component of the fitness function. We set MHz, MHz, and , respectively. The total number of frequency points for the calculation of the fitness function is 25. To prevent the gain decrease, we selected the weighting coefficient of , which is five times larger than . Consequently, we could minimize the gain reduction caused by a small magnitude of throughout the relatively wide target frequency range. Using the (1) and (2), and the optimized reflection behavior of the superstrate, the resonant height is determined as 96.4 mm. Computed reflection behaviors of the superstrate unit cell are shown in Fig. 5. In the figure, the broken line denotes an ideal reflection phase satisfying the FP resonance at each frequency. Therefore, it can be said that we could acquire the necessary

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Fig. 7. (a) Magnitude and (b) phase distribution inside the cavity at (right below the bottom face of the superstrate). Fig. 6. Comparison of simulated and measured (a) input reflection coefficients, . The behaviors of the same patch antenna without and (b) realized gain at metallic walls and superstrate are also shown here.

reflection phase values from 1.76 GHz to 1.88 GHz, which are very similar to the ideal phase response. But, a reflection magnitude is reduced to about 0.65, which may not be helpful to enhance antenna gain. As shown in Fig. 1, we fabricated the wideband FPC antenna based on the optimization result of the superstrate geometry and the prediction of FPC resonance. Fig. 6 shows performance of the proposed antenna. As for the input reflection coefficient , 10 dB bandwidth is from 1.74 GHz to 1.84 GHz, which corresponds to a fractional bandwidth of about 5.6%. In regard to antenna gain, the maximum measured gain is 13.8 dB and the 3 dB radiation bandwidth is about 180 MHz, which corresponds to a fractional bandwidth of 10%. The antenna gain shown in Fig. 6 is realized gain in the superstrate surface normal direction including overall mismatch and efficiency parameters of the antenna. Hence, it is undoubtedly clear that our antenna well operates with relatively flat gain within the target PCS frequency band. The patch antenna behavior without the FPC is also shown in Fig. 6. The maximum gain of the patch antenna is about 9.4 dB. Accordingly, we could increase the overall antenna gain about 4.5 dB by introducing the FPC technique. In Fig. 6(a), we can see that there is another impedance matched frequency band near 2.06 GHz, which exists because of a generation of higher modes inside the cavity. To more clearly show the existence of the higher mode, we compute the magnitude and phase distribution in the cavity,

mm

which are shown in Fig. 7. At a fundamental radiation mode, i.e., at 1.75 GHz and 1.85 GHz, the magnitude and phase distribution is approximately symmetric with respect to the center of the cavity. Moreover, the overall phase contrast shown in Fig. 7(b), namely, the maximum phase difference between the largest and the smallest values, does not exceed 100 degrees, which tells us that the signal distribution inside the cavity is not destructive. Therefore, the antenna stably and strongly radiates energy toward a normal direction (z-direction) in the fundamental mode frequencies. However, at the second radiation mode near 2.1 GHz, overall phase varies from 0 to 340 degrees, which is indicating the existence of destructive interference inside the cavity. In fact, we can see several magnitude peaks at 2.1 GHz resulting from the interference of waves at the higher mode. Accordingly, different from the radiation behavior in the fundamental mode, there exist several main beams distributing in the x-direction. Measured radiation properties are compared with computed values, which are depicted in Fig. 8. To obtain practical beam shapes that are narrow in the elevation direction (the xz-plane) and wide in the azimuthal direction (the yz-plane), we intentionally make the aperture as a rectangular shape, which is narrower in the azimuthal direction. Consequently, the half-power beam width in the E-plane is more than 2 times narrower than that in the H-plane. As for the H-plane radiation pattern, the antenna structure including the feeding patch antenna is perfectly symmetric with respect to the xz-plane, so the radiation pattern in the azimuthal direction is also symmetric. However, the patch is not symmetric with respect to the yz-plane. That is the reason why the E-plane radiation pattern is not symmetric.

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azimuthal direction, radiation aperture was made as a rectangular shape. For wide beam width corresponding to a target personal communication service, we optimized the superstrate structure consisting of the square patches and loops, which satisfies an FPC resonance condition in the target frequency band. We used a hybrid genetic algorithm for the optimization of the superstrate geometry. Our antenna radiates well in the target band with relatively high-gain. And, there exists only a fundamental mode inside cavity, which is important for high gain behavior radiating only toward the aperture-normal direction. Consequently, it was also shown that radiation behaviors at each frequency are also appropriate for the application of base station antenna. Predicted antenna performance showed good agreement with experimental data gathered in a fully anechoic chamber, which confirms validity of our design approach. REFERENCES

Fig. 8. Radiation patterns (realized gain) at (a) 1.765 GHz, and at (b) 1.855 GHz. TABLE I PERFORMANCE OF THE FABRICATED FPC ANTENNA

It is also important to note that a front-to-back radiation ratio (FBR) of the proposed antenna is relatively high, which is one significant design parameter required for base station and repeater antennas of today. We can see that the measured and predicted radiation properties agree very well, which confirms the validity and accuracy of our design approach. The detailed antenna performance is described in Table I. III. CONCLUSION A base station antenna for mobile communication was proposed. We chose an FPC-type antenna as our prototype antenna to obtain relatively high-gain property. A single wide band patch-antenna fed energy into the FP cavity, which is enclosed with four metallic side-walls. To get a wider beam width in an

[1] L. C. Godara, Handbook of Antennas in Wireless Communications. Boca Raton, FL: CRC Press, 2002. [2] S. C. Swales, M. A. Beach, D. J. Edwards, and J. P. McGeehan, “The performance enhancement of multibeam adaptive base-station antennas for cellular land mobile radio systems,” IEEE Trans. Vehicular Technol., vol. 39, no. 1, pp. 56–67, 1990. [3] R. J. Mailloux, J. F. McIlvenna, and N. P. Kemweis, “Microstrip array technology,” IEEE Trans. Antennas Propag., vol. AP-29, no. 1, pp. 25–37, 1981. [4] L. C. Godara, “Applications of antenna arrays to mobile communications, Part I: Performance improvement, feasibility, and system considerations,” Proc. IEEE, vol. 85, no. 7, pp. 1031–1060, 1997. [5] L. C. Godara, “Applications of antenna arrays to mobile communications, Part II: Beam-forming and direction-of-arrival considerations,” Proc. IEEE, vol. 85, no. 8, pp. 1195–1245, 1997. [6] N. Guerin, S. Enoch, G. Tayeb, P. Sabouroux, P. Vincent, and H. Legay, “A metallic Fabry-Perot directive antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 220–223, 2006. [7] J. Ju, D. Kim, and J. I. Choi, “Fabry-Perot cavity antenna with lateral metallic walls for WiBro base station applications,” Electron. Lett., vol. 45, no. 3, pp. 141–142, 2009. [8] D. Kim and J. I. Choi, “Analysis of antenna gain enhancement with a new planar metamaterial superstrate: an effective medium and a FabryPerot resonance approach,” J. Infrared, Milli. Terahertz Waves, vol. 31, no. 11, pp. 1289–1303, 2010. [9] J. Yeo and D. Kim, “Novel design of a high-gain and wideband FabryPerot cavity antenna using a tapered AMC substrate,” Int. J. Infrared Milli. Waves, vol. 30, no. 3, pp. 217–224, 2009. [10] D. Kim, “Novel dual-band Fabry-Perot cavity antenna with low frequency separation ratio,” Microw. Opt. Tech. Lett., vol. 51, no. 8, pp. 1869–1872, 2009. [11] Y. J. Lee, J. Yeo, J. Mittra, and W. S. Park, “Application of electromagnetic bandgap (EBG) superstrates with controllable defects for a class of patch antennas as spatial angular filters,” IEE Antennas Propag., vol. 53, no. 1, pp. 224–235, 2005. [12] R. Chantalat, C. Menudier, M. Thevenot, T. Monediere, E. Amaud, and P. Dumon, “Enhanced EBG resonator antenna as feed of a reflector antenna in the Ka band,” IEEE Wireless Propag. Lett., vol. 7, pp. 349–353, 2008. [13] G. V. Trentini, “Partially reflecting sheet arrays,” IEEE Trans. Antennas Propagat., vol. 7, pp. 666–671, 1956. [14] G. A. Palikaras, A. P. Feresidis, and J. C. Vardaxoglou, “Cylindrical electromagnetic bandgap structures for directive base station antennas,” IEEE Wireless Propag. Lett., vol. 3, pp. 87–89, 2004. [15] H. Chreim, E. Pointereau, B. Jecko, and P. Dufrane, “Omnidirectional electromagnetic band gap antenna for base station applications,” IEEE Wireless Propag. Lett., vol. 6, pp. 499–502, 2007. [16] Z. Liu, W. Zhang, D. Fu, Y. Gu, and Z. Ge, “Broadband Fabry-Perot resonator printed antennas using FSS superstrate with dissimilar size,” Microw. Opt. Tech. Lett., vol. 50, no. 6, pp. 1623–1627, 2008. [17] D. Kim, J. Ju, and J. I. Choi, “A broadband Fabry-Pérot cavity antenna designed using an improved resonance prediction method,” Microw. Opt. Tech. Lett., vol. 53, no. 5, pp. 1065–1069, 2011.

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[18] A. F. Feresidis and J. C. Vardaxoglou, “A broadband high-gain resonant cavity antenna with single feed,” in Proc. EuCAP 2006, Nice, France, 2006, vol. 626SP. [19] L. Moustafa and B. Jecko, “EBG structure with wide defect band for broadband cavity antenna applications,” IEEE Wireless Propag. Lett., vol. 7, pp. 693–696, 2008. [20] J. J. Grefenstette, “Optimization of control parameters for genetic algorithms,” IEEE Trans. Syst. Man, Cybern., vol. SMC-16, no. 1, pp. 122–128, 1986. [21] D. T. Pham and G. Jin, “Genetic algorithm using gradient-like reproduction operator,” Electron. Lett., vol. 31, no. 18, pp. 1558–1559, 1995. [22] D. T. Pham and G. Jin, “A hybrid genetic algorithm,” in Proc. 3rd World Congr. Expert Systems, Seoul, Korea, 1996, vol. 2, pp. 748–757. [23] G. G. Jin, Genetic Algorithms and Their Applications. Seoul, Korea: Kyo-Woo Sa Press, 2002. [24] CST Microwave Studio: Workflow & Solver Overview. CST Studio Suite 2009, CST-GmbH, 2009. [25] K. L. Wong and W. H. Hsu, “A broadband rectangular patch antenna with a pair of wide slits,” IEEE Trans. Antennas Propagat., vol. 49, pp. 1345–1347, 2001. [26] N. Fayyaz and S. Saravi-Naeini, “Bandwidth enhancement of a rectangular patch antenna by integrated reactive loading,” in IEEE Trans. Antennas Propagat. Soc. Int. Symp. Dig., 1998, pp. 1100–1103.

Dongho Kim (M’08) received the B.S. and M.S. degrees in electronics engineering from Kyungpook National University, Daegu, Korea, in 1998 and 2000, respectively, and the Ph.D. degree in electrical and electronics engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2006. From 2000 to 2011, he was a Senior Researcher with the Electronics and Telecommunications Research Institute (ETRI), Daejeon, Korea, where he was involved with the development of various antennas including RFID and mobile communication antennas, and artificially engineered structures such as electromagnetic band-gap (EBG) structures, frequency selective surfaces (FSS), and artificial magnetic conductors (AMC). In 2011, he joined the Department of Electronic Engineering, Sejong University, Seoul, Korea, where he is now an Assistant Professor. His research interests include advanced electromagnetic wave theory and application, design of highly efficient and miniaturized antennas using artificially engineered materials, design of EBG structures, FSS, and AMC, platform-tolerant special RFID antenna design, and development of a variety of metamaterials with negative permittivity and permeability. Prof. Kim is a life member of the Korean Institute of Electromagnetic Engineering and Science (KIEES).

Jeongho Ju received the B.S. and M.S. degrees in information and telecommunication engineering from Incheon University, Incheon, Korea, in 2006 and 2008, respectively. Since 2008, he has been with ETRI, Daejeon, Korea, where he currently works in the antenna research team as a member of the engineering staff. His current research interests include passive components, filters, and antenna design based on metamaterials.

Jaeick Choi received the B.S., M.S., and Ph.D. degrees from the Korea University, Seoul, Korea, in 1981, 1983, and 1995, respectively. Since 1983, he has been with ETRI, Daejeon, Korea. He had been involved in the RF/antenna development of the earth station, especially the SCPC and VSAT systems, TT&C ground station (of Arirang satellite), IMT2000 system, and digital DBS. He was in charge of electromagnetic environment esearch and development of EMI/EMC technologies and EMF Exposure Assessment from 2004 to 2005. Currently, he is researching and developing metamaterials and their application technologies for antenna/RF sensors, RF components, and radio transmission technologies.

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Development of Novel 3-D Cube Antennas for Compact Wireless Sensor Nodes Ibrahim T. Nassar, Student Member, IEEE, and Thomas M. Weller, Senior Member, IEEE

Abstract—3-D antennas for narrow band, wireless sensor node applications are described herein. The antennas were designed on the surface of a cube which makes available the cube interior for sensor electronics placement. The layout of each antenna consists of a dipole fabricated on two sides of the cube and connected to a balanced-to-unbalanced line transition on the third side. The base of the cube serves as a ground plane for the microstrip feed line. The first cube antenna was designed for an operating frequency of the of 2.4 GHz and its 10 dB return loss bandwidth is 2%. proposed design is 0.55 and its measured gain is 1.69 dBi with 78% measured radiation efficiency. The second cube antenna is similar to the first one but it was loaded with high dielectric constant superstrates. of the second proposed antenna is 0.45, its measured gain is 1.25 dBi with 73% measured radiation efficiency and the bandwidth is 1.5%. The designs compare well with high efficiency, electrically small antennas that have been described in the open literature. A Wheeler Cap was used to measure the efficiency and the 3-antenna method was used for measuring the gain. Index Terms—3-D antennas, balanced, dipole antennas, electrically small antennas, Wheeler cap method, wireless sensor networks (WSNs).

I. INTRODUCTION

I

T is expected that the use of distributed wireless sensor networks will undergo continuous growth in the future with numerous applications such as environmental and biomedical monitoring. The motivation for this paper is to develop antennas for wireless sensor nodes used for embedded and through-life structural health monitoring of civil infrastructures. Important design goals in such applications are to minimize power consumption and the size of the sensor node [1]. Embedding sensor nodes into man-made objects or natural environments also introduces certain antenna design challenges. One of the most significant challenges is miniaturizing the antenna size, as the antenna usually occupies the majority of the overall sensor node volume. Most of the antennas currently in use for wireless sensor node applications are planar [2]–[11], due to their low cost, ease of fabrication, and relatively high radiation efficiency. However, efficient planar antennas tend to have large cross-sectional areas. Therefore, 3-D antennas Manuscript received November 08, 2010; revised February 13, 2011; accepted August 09, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported by the National Science Foundation under Grant #ECS-0925929. The authors are with the WAMI Lab, University of South Florida, Tampa, FL 33620 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173121

are often preferred for applications that require high efficiency concurrently with small size, since these antennas make more efficient use of the available volume by realizing relatively long antenna lengths. 3-D antennas are also beneficial in opening up internal volume for other uses, such as storage for batteries or other circuit elements. The design of the presented cube antennas aims to produce omnidirectional, simple, and low-cost antennas that are directly integrated into the structural packaging with the capability of housing the sensor electronics within the structure. The interest in a low-cost, omnidirectional solution motivated the use of a dipole antenna. A dipole antenna can also be easily fabricated in different shapes and configurations. In [12], a dipole antenna has been fabricated on a sphere and provided very good performance as the occupied volume was utilized to the greatest extent. In [13], a dipole antenna has been printed on a pyramid configuration in a manner which is similar to the presented design. The antenna was easily fabricated; however, the pyramid configuration provided low gain even though the antenna exhibited large electrical size. The low gain is due to the high percentage of canceled radiated fields related to the shape and relative orientation of the dipole arms. In contrast, a cubical structure has proven to exhibit relatively high gain and efficiency, as demonstrated previously for MIMO applications [14]–[18]. In these implementations, multiple antennas were fabricated on the cube surface and each one was fed by an isolated port. One potential drawback to this design is that it would be difficult to achieve a compact feeding configuration. The cube approach has also been used with a single input [19]; however, the antenna has a low gain and a relatively low G/Q (gain/quality factor) ratio, and may be difficult to manufacture. In addition, the design in [19] may present challenges in terms of integration with other circuit elements as it does not have a ground plane or an integrated balun. In this paper, a new approach to the design of dipole antennas on a cube configuration, with and without superstrate loading, is presented (Fig. 1). The approach is based on the meandered line dipole antenna [20], fabricated on two sides of the cube and connected to a balanced-to-unbalanced line transition on a third side. The dipole arms were meandered and wrapped around the cube faces in a manner that provided a well-balanced antenna with high gain relative to the electrical size. The base of the cube serves as a ground plane for the microstrip feed line, leaving the other sides of the cube available for uses such as another integrated antenna. The measured and simulated data prove that the cube antennas have very good performance relative to their occupied volumes, with gain and efficiency that are comparable to that

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Fig. 1. Fabricated cube antennas: dielectrically loaded design (left) and nonloaded design (right). Fig. 3. Geometry of the meandered dipole antenna.

Fig. 2. Parallel plate balun.

of the spherical antennas. Also, a comparison of the G/Q ratio for different designs shows that the presented cube antennas are among the more efficient electrically small antennas known in the literature. A comparison with a 2-D version of the antennas shows that the radiation efficiency is not significantly degraded in the transformation to the 3-D form factor. In addition, the presented cube antennas are relatively insensitive to the presence of dielectric and metallic inclusions within the antenna structure. The sections that follow begin with a discussion of the transition between unbalanced to balanced structures using a parallel plate balun, along with the meandered-line, half-wave dipole that was the starting point for the 3-D antenna design. The arms of the dipole were rotated and placed on two sides of a cube to form the first cube antenna. The second iteration of the cube antenna incorporates high dielectric constant superstrates, resulting in smaller antenna size. In order to determine the radiation efficiencies the Wheeler Cap method was used. II. CUBE ANTENNA DESIGN A. Parallel Plate Balun Symmetric dipole antennas require a balanced feed. In cases where the connection to the signal source is unbalanced (e.g., using a coaxial or microstrip feed), a balun is needed to transition between the source and the antenna input. In this work, a parallel plate waveguide transmission line is used as a balun (Fig. 2). The design consists of two conducting strips of a width larger than the separation between them, in order to minimize the fringing fields [21]. The two strips of the balun need to be in length in order to provide a high impedance at the dipole antenna side, canceling the unbalanced current coming from the ground of the unbalanced microstrip feed line [13]. This is the approach used with the Bazooka balun [22].

Fig. 4. Meandered dipole antenna arm dimensions.

B. Meandered Line Dipole Antenna Design In this section, a meandered line half-wave dipole antenna operating at 2.4 GHz is described. The antenna geometry, shown in Figs. 3 and 4, is formed by two symmetric, non-meandered rectangular strips of dimensions and and two meandered sections. Each of the two arms is fabricated on opposite sides of the substrate. The dipole is center-fed by the parallel plate balun that is perpendicular to the feed line. Between the balun and the feed point are an impedance-matching line and a microstrip line with a characteristic impedance of 50 . It was shown via electromagnetic simulation that the balanced current distribution at the antenna input was not affected by bending the balun. The substrate is Rogers/RT Duroid 6010 with a nominal relative dielectric constant of 10.2, and a thickness of 50 mil. This high permittivity substrate will reduce the antenna size, although higher permittivity is unfortunately often equivalent to higher dielectric losses [23]. The meandered line approach was employed to minimize the length of the antenna. Ansoft HFSS 11 was used to determine the total length of the meandered portion of the arms, the slot size between sections, and the number of meandered sections in order to minimize the antenna size without degrading the gain. To match the input impedance to 50 , the width of the meander line, the width of the parallel plate transformer, and the width and length of the matching line were adjusted. Table I shows the meandered dipole antenna dimensions. The characteristic impedances of the parallel plate transformer and the matching line are 48 and 70 , respectively. An approximate equivalent circuit model of the antenna is given in Fig. 5; here is the characteristic impedance and is the antenna input impedance at 2.4 GHz.

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TABLE I MEANDERED DIPOLE ANTENNA DIMENSIONS IN mm

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TABLE II FIRST CUBE ANTENNA DIMENSIONS IN mm

Fig. 5. Approximate equivalent circuit model for the meandered dipole antenna; the 48 transmission line represents the balun.

Fig. 8. Measured and simulated

Fig. 6. Measured and simulated

for the meandered dipole antenna.

Fig. 7. Geometry of the first cube antenna.

Fig. 6 shows the measured and simulated of the meandered dipole antenna. As seen, the measured data fits the simulated data well, although there is a small shift in the frequency which could be attributed to fabrication errors. The measured 10-dB return loss bandwidth is 4.7%. C. First Cube Antenna Design After studying the meandered line antenna, the cube antenna was designed as shown in Fig. 7. The antenna consists of a half-wave dipole printed on three sides of the cube and connected to the parallel plate balun in the center. The left-hand arm is attached to the microstrip feed line and the right-hand arm is attached to the ground plane of the microstrip line. By wrapping the dipole arms around the cube, i.e., transforming from

for the first cube antenna.

the initial meandered line antenna to the cubical form factor, the effective volume is reduced by a factor of 1.27. The meandered sections were rotated in a clockwise/counter clockwise fashion and placed along the -axis as shown in Fig. 7. This orientation preserves the balanced current on the dipole and balun; when the meandered sections are not rotated with respect to each other the simulated radiation patterns exhibit distorted, non-dipole like characteristics. Improved gain is achieved with longer non-meandered sections, requiring a compromise between antenna size and performance. The cube antenna input impedance at 2.4 GHz without the feeding network is . In order to compensate for this change in the input impedance relative to the initial meandered line dipole, the characteristic impedance of the quarter-wave parallel plate transformer was decreased to 33 . The width and length of the matching line were subsequently optimized to obtain the best match to 50 . The characteristic impedance of the matching line is 73 with an electrical length of . Table II shows the first cube antenna dimensions. The of the antenna is shown in Fig. 8. As seen, a good match between the measured and simulated data over a wide frequency range was obtained. The bandwidth was decreased relative to the first design by a factor of 2.35 due to the reduction of the antenna size and the lower input resistance. D. Second Cube Antenna Design In order to minimize the size of the cube antenna, the dielectric loading technique was used. As with the substrate, the superstrate material is 50-mil-thick Rogers/RT Duroid 6010. The superstrate was placed over the dipole arms and over the both sides of the parallel plate balun (Fig. 9). Since the presence of the

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Fig. 9. Geometry of the second cube antenna design. (a) Side view and (b) top view. The light layer is the superstrate and the dark layer is the substrate. TABLE III SECOND CUBE ANTENNA DIMENSIONS IN mm

Fig. 11. Measured E-plane (solid) and H-plane (dashed) patterns for (a) the meandered antenna, (b) the first cube antenna, and (c) the second cube antenna.

III. RADIATION PERFORMANCE AND DISCUSSION MEASURED RESULTS

Fig. 10. Measured and simulated

for the second cube antenna.

superstrate reduces the guided wavelength, this loading caused shrinkage of the occupied volume by a factor of 1.3 relative to the first cube antenna design. The loaded cube antenna input impedance without the feeding network at the frequency of operation is 5.5–3i . The antenna input impedance was matched to 50 using a quarter wave parallel plate transformer with a characteristic impedance of 22 , and a matching line with a characteristic impedance of 83 . Table III shows the second cube antenna dimensions. Fig. 10 shows the measured and simulated of the second cube antenna. As seen the measured data fit the simulated data, but the resonant frequency shifted up by 0.05 GHz, which could be attributed to the air gaps between the substrates and the superstrates. Relative to the first cube design, the return loss bandwidth reduced by a factor of 1.3 due to the reduction in the physical size.

OF

Fig. 11 shows the measured E- and H-plane radiation patterns for the three antenna designs. These antennas are linearly polarized in the horizontal direction ( -axis), relative to the coordinate system in Fig. 3. The E-plane tests were carried out by rotating the antennas in the azimuth plane from 0 to 360 at an elevation angle of 0 in the -plane. For the H-plane pattern the antennas were rotated along the -plane. As seen, all antenna designs demonstrate similar omnidirectional patterns with the maximum radiation occurring broadside to the non-meandered portions of the dipole arms. The gain has been measured using the 3-antenna method, following the same procedures as above for the E-plane radiation patterns. The gain measurement has an uncertainty of dB. In order to determine the efficiency of the designed antennas, a cubical configuration of copper measuring was used as a Wheeler Cap (Fig. 12). The size of the Wheeler Cap was selected to push the interior modes to higher frequencies resulting in a sparse mode spectrum [24]. These measurements were found to have a variability of approximately . The measured and simulated antenna parameters at the resonant frequency are listed in Table IV. The data shows that the gain did not deteriorate significantly with the 3-D orientation. However, this orientation did result in a decrease of the axial ratio by 12 dB, which is due to the difference in the direction of radiation between the meandered and non-meandered sections. Despite this change, the antennas remain linearly polarized as the axial ratio exceeds 16 dB. It was also found that the antenna polarization was not significantly affected by the changes in the

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Fig. 12. Cubical Wheeler cap. TABLE IV MEASURED AND SIMULATED ANTENNA PARAMETERS Fig. 13. Comparison of G/Q ratio of the presented designs and other miniaturized antennas.

antenna proposed in [19], and performance that is similar to the 4-arm folded spherical helix antenna presented in [12]. IV. WIRELESS SENSOR NODE APPLICATIONS

dipole arm configuration, since the current distribution is concentrated on the non-meandered sections of the arms. As a result, the gain was not considerably impacted by cross-polarization effects. The measured return loss bandwidths are relatively narrow compared to typical dipole antennas of these sizes due to the low antenna input impedance; however, this bandwidth range is suitable for the narrowband sensing application that motivated this work. As expected, loading the antenna with superstrates resulted in a smaller design. However, the loading also reduced the efficiency as the high permittivity material introduces additional dielectric loss and concentrates the electric field inside the substrate [23]. The loading also resulted in a 0.3 dB gain decrease for the same reasons. Fig. 13 shows the gain over quality factor ratio (G/Q) for the designs presented in this work and others from the literature, compared with an approximated G/Q limit. As small, narrowband antennas with a ground plane can have a linear gain of 3, the G/Q limit was calculated using and the expression for the minimum radiation Q for a linearly polarized antenna given in [26, Eqn. (20)]. The G/Q ratio was calculated for each of the presented designs based on the measured maximum gain and the Q at 2.4 GHz. The Q was found from the measured 10 dB return loss bandwidth according to [27, Eqn. (7)]. A sequential comparison of the G/Q ratio from the meandered line dipole antenna to the second cube antenna design shows that this new method for minimizing the total occupied volume retains the relative separation from the G/Q limit. Fig. 13 also shows that the cube antennas in this work provide a larger G/Q ratio than the pyramidal antenna proposed in [13] and the cube

For the application of interest to this work, the small, 3-D form factor is attractive. As previously noted, the study is primarily geared toward sensor nodes that can be embedded in civil infrastructure, such as bridges, pilings or pavement for structural health monitoring. Very low, or zero power sensor transceiver designs such as described in [28] and [29] will be used, allowing long-term monitoring but also heightening the need for highly efficient antennas. The planar antennas commonly used for wireless sensor nodes would result in a relatively large cross-sectional area in comparison to the 3-D configuration. Another potential advantage of a 3-D antenna for sensor nodes is the relative ease of rapid dispersal (e.g., air-drop) into a preferred orientation that facilitates wireless interrogation from above or from the sides, by simply weighting the bottom of the package. While the fabrication process required for a 3-D antenna will be more complex than that for a typical 2-D antenna, significant advances are being made in stereolithography and 3-D printing methods that will address this challenge [30], [31]. The manual assembly method used for the designs presented herein is currently being transitioned to 3-D printing. As the available volume for the antennas in wireless sensor nodes is limited, a compromise between the bandwidth and the antenna radiation efficiency has to be made. As noted above, radiation efficiency is the more important performance metric in this work. For the harmonic sensor transceivers that will be used with these antennas, the operational bandwidth is typically less than 1% [29], a result of maximizing the efficiency of the transceiver that is not system-performance limiting because the data rates will be very low. As the dimensional control of current-day 3-D printing is on the order of 1 micrometer, it is expected that sufficient repeatability in the manufacturing process is achievable. If that is not found to be the case, approaches for frequency tuning, such as thin spray-on dielectric coatings or trimming the length of the meandered sections, could be investigated. If the presented cube antenna were to be used in the harmonic sensor design, the antenna would need to be re-optimized to achieve a conjugate match to the complex impedance of the diode multiplier for optimal frequency conversion efficiency [29]. In this respect, an advantage of the 3-D antenna is that

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Fig. 14. First cube antenna and an inserted block representing internal sensor electronics.

Fig. 16. Simulated for the first cube antenna with an inserted lossy dielectric block of different heights.

located beneath the ground plane if necessary to minimize the impact on antenna performance. V. CONCLUSION

Fig. 15. Simulated of different heights.

for the first cube antenna with an inserted metallic block

its input impedance is readily adjusted without increasing the total occupied volume. The desired complex impedance can be obtained by optimizing the meandered arm parameters, the parallel plate transformer width, as well as the width and the length of the microstrip matching line. The 3-D antenna also provides the capability of housing the sensor electronics within the antenna structure while being relatively insensitive to the presence of such inclusions (dielectric, metallic or a combination of both). To demonstrate, the first cube antenna was tested with blocks of conductive and lossy dielectric material ( of 10 and loss tangent of 0.1) inside the structure, as seen in Fig. 14. Figs. 15 and 16 show the effect of the block size on the antenna resonant frequency. As seen, the conductive and dielectric blocks, with a height up to 8 mm, have little impact on the resonant frequency frequency shift due to the weak coupling. This frequency shift could be accommodated by small adjustments in the antenna design. Also, the metallic block did not affect the radiation pattern or the antenna gain, while the lossy dielectric block with a height of 8 mm (equaling the height of the meandered arm section) decreased the maximum gain by 0.2 dB without affecting the radiation pattern. The insignificant variation in the radiation pattern and gain is important, since these parameters are more difficult to adjust than the center frequency. As the zero power sensor nodes of interest will have minimal electronics inside the cube, these results demonstrate that the antenna approach is suitable for the application of interest. If a battery is required for sensor operation, it could be

Designs of 3-D cube antennas have been developed that are good candidates to work efficiently for narrowband, wireless sensor applications where the available volume is constrained. The performance of the antennas has been validated theoretically and experimentally, and shown to approach the theoretical performance limits for electrically small antennas. The 3-D designs exhibit sensitivity to the dimensions of the ground plane that is being investigated further, along with the integration of a second antenna on the opposing side of the cube. However, measurement and simulation data have proven that conductive and lossy dielectric objects, of sizes up to mm , can be placed inside the cube without significantly degrading the antenna performance, suggesting that this design approach may be a good candidate for use with wireless sensor nodes. Direct-write printing techniques which enable conductor deposition on flexible and non-planar surfaces are a possible approach for high-volume manufacturing. ACKNOWLEDGMENT This work was conducted in partnership with the University of Vermont and SRI International. The authors would like to thank Rogers Corporation for providing substrate material, Diamond Engineering Company for providing support for our antenna measurement system, and nScrypt, Inc., for advice regarding 3-D printing. They would also like to thank Dr. G. Mumcu for helpful discussions on electrically small antennas and design suggestions. REFERENCES [1] G. Whyte, N. Buchanan, and I. Thayne, “An omnidirectional, low cost, low profile 2.45 GHz microstrip fed rectaxial antenna for wireless sensor network applications,” presented at the IEE and IEEE Conf. Antennas Propag., Loughborough, U.K., 2006.

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[2] K. L. Wong, Planar Antennas for Wireless Communications. New York: Wiley, 2003. [3] D. Liao and K. Sarabandi, “Optimization of low-profile antennas for applications in unattended ground sensor networks,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 2006, pp. 783–786. [4] W. Hong and K. Sarabandi, “Design of low-profile omnidirectional antenna for ground sensor networks,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jun. 2007, pp. 6007–6010. [5] A. Babar, L. Ukkonen, and L. Sydanheimo, “Dual UHF RFID band miniaturized multipurpose planar antenna for compact wireless systems,” in Proc. Int. Workshop Antenna Tech. (iWAT), Mar. 2010, pp. 1–4. [6] S. Genovesi, S. Saponara, and A. Monorchio, “Parametric design of compact dual-frequency antennas for wireless sensor networks,” IEEE Trans. Antennas Propag., vol. 59, no. 7, pp. 2619–2627, Jul. 2011. [7] C. G. Kakoyiannis and P. Constantinou, “Co-design of antenna element and ground plane for printed monopoles embedded in wireless sensors,” in Proc. IEEE Int. Conf. Sens. Technol. Applicat. (SENSORCOMM ’08), Cap Esterel, France, Aug. 2008, pp. 413–418. [8] C. G. Kakoyiannis, G. Stamatiou, and P. Constantinou, “Small square meander-line antennas with reduced ground plane size for multimedia WSN nodes,” in Proc. 3rd Eur. Conf. Antennas Propag. (EuCAP 2009), Berlin, Germany, Mar. 2009, pp. 2411–2415. [9] C. G. Kakoyiannis, P. Gika, and P. Constantinou, “Compact antennas with reduced mutual coupling for wireless sensor networks,” National Technical University of Athens, Athens, Greece. [10] D. T. Phan and G. S. Chung, “Design and optimization of reconfigurable inset-fed microstrip patch antennas with high gain for wireless sensor networks,” in Proc. Int. Conf. Comput. Commun. Technol. (RIVF ’09), Jul. 2009, pp. 1–4. [11] J. Carter, J. Saberin, T. Shah, P. R. Ananthanarayanan, and C. Furse, “Inexpensive fabric antenna for off-body wireless sensor communication,” in Proc. IEEE Antennas Propag. Soc. Int. Symp. (APSURSI), Jul. 2010, pp. 1–4. [12] S. R. Best, “Low Q electrically small linear and elliptical polarized spherical dipole antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1047–1053, Mar. 2005. [13] S. E. Melais, T. M. Weller, C. M. Newton, R. W. Smith, and C. A. Gamlen, “Origami packaging—Novel printed antenna technology for ad-hoc sensor applications,” presented at the 40th Int. Symp. Microelectron., Oct. 2007. [14] C. Y. Chiu and R. D. Murch, “Experimental results for a MIMO cube,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Albuquerque, NM, Jul. 2006, pp. 2533–2536. [15] C. Y. Chiu and R. Murch, “Design of a 24-port MIMO cube,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Honolulu, HI, Jun. 2007, pp. 2397–2400. [16] A. Nemeth, L. Sziics, and L. Nagy, “MIMO cube formed of slot dipoles,” in Proc. IST Mobile and Wireless Commun. Summit, Jul. 2007, pp. 1–5. [17] B. N. Getu and J. B. Andersen, “The MIMO cube—A compact MIMO antenna,” IEEE Trans. Wireless Commun., vol. 4, no. 3, pp. 1136–1141, May 2005. [18] W. I. Son, W. G. Lim, M. Q. Lee, S. B. Min, and J. W. Yu, “Design of compact quadruple inverted-F antenna with circular polarization for GPS receiver,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1503–1510, May 2010. [19] C. M. Kruesi, R. J. Vyas, and M. M. Tentzeris, “Design and development of a novel 3-D cubic antenna for wireless sensor networks (WSNs) and RFID applications,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3293–3299, Oct. 2009. [20] H. Nakano, H. Tagami, A. Yoshizawa, and J. Yamauchi, “Shortening ratios of modified dipole antennas,” IEEE Trans. Antennas Propag., vol. 32, no. 4, pp. 385–386, Apr. 1984.

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[21] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005, ch. 3, pp. 98–106. [22] T. A. Milligan, Modern Antenna Design, 2nd ed. New York: Wiley, 2005, ch. 5, pp. 251–260. [23] A. K. Skrivervik, J. F. Zurcher, O. Staub, and J. R. Mosig, “PCS antenna design: The challenge of miniaturization,” IEEE Antennas Propag. Mag., vol. 43, no. 4, pp. 12–27, Aug. 2001. [24] H. Choo, R. Rogers, and H. Ling, “On the Wheeler cap measurement of the efficiency of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 7, pp. 2328–2332, Jul. 2005. [25] H. A. Wheeler, “The radiansphere around a small antenna,” Proc. IRE, vol. 47, no. 8, pp. 1325–1331, Aug. 1959. [26] J. S. McLean, “A re-examination of the fundamental limits on the radiation Q of electrically small antennas,” IEEE Trans. Antennas Propag., vol. 44, no. 5, pp. 672–676, May 1996. [27] R. Bancroft, “Fundamental dimension limits of antennas ensuring proper antenna dimensions in mobile device designs,” Centurion Wireless Technologies, Westminster, CO. [28] S. M. Presas, T. M. Weller, S. Silverman, and M. Rakijas, “High efficiency diode doubler with conjugate- matched antennas,” in Proc. IEEE Eur. Microw. Conf., Oct. 2007, pp. 250–253. [29] S. M. Aguilar and T. M. Weller, “Tunable harmonic re-radiator for sensing applications,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2009, pp. 1565–1568. [30] X. Chen, K. Church, and H. Yang, “High speed non-contact printing for solar cell front side metallization,” in Proc. IEEE Photovoltaic Specialists Conf. (PVSC), Jun. 2010, pp. 001343–001347. [31] J. Czyzewski, P. Burzynski, K. Gawel, and J. Meisner, “Rapid prototyping of electrically conductive components using 3D printing technology,” J. Mater. Process. Technol., vol. 209, no. 12–13, pp. 5281–5285, Jul. 2009.

Ibrahim T. Nassar (S’09) was born in Irbid, Jordan, on May 16, 1986. He received the B.S. degree from Jordan University of Science and Technology, Irbid, and the M.S. degree from University of South Florida, Tampa, in 2008 and 2010, respectively, all in electrical engineering. He is now a Graduate Research Assistant with the WAMI Lab, the University of South Florida. His research is focuses on design and development of RF microwave circuits and small, low-cost, and low-profile antennas for wireless sensor applications. Mr. Nassar ranked within the top ten among his colleagues in his Bachelor degree and was one of the students selected to participate in the NSF IRES program at the University of Central Florida in 2008.

Thomas M. Weller (S’92–M’95–SM’98) received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Michigan, Ann Arbor in 1988, 1991, and 1995, respectively. From 1988 to 1990 , he was with Hughes Aircraft Company, El Segundo, CA. He joined the University of South Florida, Tampa, in 1995, where he is currently a Professor in the Electrical Engineering Department. He cofounded Modelithics, Inc., in 2001. His current research interests are in the areas of RF micro electromechanical systems, development and application of microwave materials, and integrated circuit and antenna design. He has thirteen U.S. patents and over 150 professional journal and conference publications. Dr. Weller was a recipient of the Outstanding Young Engineer Award from the IEEE Microwave Theory and Techniques Society in 2005, the USF President’s Award for Faculty Excellence in 2003, IBM Faculty Partnership Awards in 2000/2001, a National Science Foundation CAREER Award in 1999, and the IEEE MTT Society Microwave Prize in 1996.

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Influence of the Hand on the Specific Absorption Rate in the Head Chung-Huan Li, Member, IEEE, Mark Douglas, Senior Member, IEEE, Erdem Ofli, Member, IEEE, Benoit Derat, Member, IEEE, Sami Gabriel, Nicolas Chavannes, Member, IEEE, and Niels Kuster, Fellow, IEEE

Abstract—The influence of the user’s hand holding a mobile phone to the ear on the peak spatial-average Specific Absorption Rate (psSAR) averaged over any 1 g and 10 g of tissue in the head is investigated. This study is motivated by recent reports that found substantial increases in psSAR by the presence of the hand in some cases. Current measurement standards prescribe the measurement of SAR in a head phantom without a hand present. The mechanisms of interaction between the hand and mobile phone models are studied. Simulations and measurements at 900 and 1800 MHz have been conducted to complement the understanding of the hand grip parameters leading to higher SAR in the head. Numerical simulations were conducted on four mobile phone models, and parameters such as the palm-phone distance and hand position were varied. Measurements of 46 commercial mobile phones were made, and the maximum psSAR with different hand positions and palm-phone distances was recorded. Both simulations and measurements have found increases in the psSAR in the head of at least 2.5 dB due to the presence of the hand. Furthermore, the psSAR is sensitive to the hand grip, i.e., the variations can exceed 3 dB. Index Terms—Dosimetry, FDTD methods, interaction with continuous media, mobile communication, numerical simulation, SAR.

I. INTRODUCTION

D

URING a voice call, a mobile phone is typically held by the user’s hand next to the side of the head. The hand and head are therefore likely to be in the reactive near field of the antenna and can significantly influence the radiation pattern, efficiency, radiofrequency (RF) current coupling within the device and antenna impedance [1]–[6]. Human tissues are lossy dielectric materials at mobile phone frequencies [7] and therefore absorb RF power. The absorbed power is quantified in terms of the Specific Absorption Rate (SAR). SAR limits are established in international exposure standards for the whole-body averaged SAR and peak spatial-average SAR (psSAR) averaged over 1 gram or 10 grams of tissue [8], [9]. Measurement standards and national regulations have been established to evaluate Manuscript received June 03, 2010; revised March 29, 2011; accepted August 08, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. C.-H. Li and N. Kuster are with IT’IS Foundation and the Swiss Federal Institute of Technology (ETH) CH-8092, Zurich, Switzerland (e-mail: [email protected]. ch). M. Douglas and N. Chavannes are with IT’IS Foundation, CH-8004 Zurich, Switzerland. E. Ofli is with Schmid & Partner Engineering AG (SPEAG), CH-8004 Zurich, Switzerland. B. Derat is with Field Imaging, Meudon 92048, France. S. Gabriel is with the Vodafone Group, Newbury RG14 2FN, U.K. Digital Object Identifier 10.1109/TAP.2011.2173102

the psSAR in the user when using a mobile phone so that compliance with the exposure standards can be demonstrated [10], [11]. The measurement standards specify the use of the Specific Anthropomorphic Mannequin (SAM), a homogeneous head phantom having a size, ear thickness and dielectric parameters that result in a conservative over-estimate compared to the SAR in a person [12]. For this reason, the simulated results in this paper are conducted primarily using SAM. These standards do not specify the use of a hand model when measuring the SAR in the head because previous studies cited by the standards concluded that the SAR in the head is generally reduced when the hand is introduced [13]–[16]. In 1995, Balzano et al. reported that the change in psSAR in the head due to the hand was negligible for large phones having sleeve dipole antennas and 10–30% lower for flip phones if the palm is in direct contact with the casing [13]. In 1997, Kuster et al. used real hands in three fixed positions on twenty mobile phones at 900 MHz and 1800 MHz and found that the psSAR in the head did not increase above the measurement uncertainty [15]. A numerical study by Meyer et al. in 2001 with two simplified mobile phone models found an increase in psSAR in the head in one of the two cases of only 7% [16]. Since the publication of these studies, several changes have taken place. Mobile phones have become smaller and antenna designs and locations have changed. The integration of CAD modeling of human anatomy into simulation tools and advanced algorithms for posing the anatomy have made it possible to study the influence of different hand grips. Simulations of large parameter sets can be conducted in a reasonable time due to the dramatic increase in processor speed. The development of fast SAR measurement systems greatly reduces the time to measure large sample sizes of mobile phones [17]. Additionally, standardized hand phantoms have recently become available. These hand phantoms have been developed for an over-the-air test plan for certification of mobile phones, developed by the CTIA [8]. The hand phantoms have homogeneous dielectric properties representing dry palm [19] (900 , ; 1800 MHz: , MHz: ). User studies were conducted for [18] to define four different hand grips for the majority of mobile phones, depending on the width, form factor and usage mode [20]. Two of the grips are used in this study, as shown in Fig. 1. The size of the hand phantoms is an average value of the 50th percentile sizes for men and women, taken from [21]–[24]. The phantoms therefore represent average hands for radio-frequency analysis of mobile phone performance. The hand models include a low loss and low permittivity spacer for accurate mobile phone positioning.

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Fig. 2. Mobile phone models used to investigate the effect of the user’s hand on SAR in the head are the generic clamshell model with a single-band PIFA at (a) 900 MHz and (b) 1800 MHz, (c) a CAD model having a clamshell style with an external helical antenna operating at 1750 MHz and (d) a CAD model having a bar style with an internal PIFA operating at 1750 MHz (phone 4). (a) Phone 1. (b) Phone 2. (c) Phone 3. (d) Phone 4. Fig. 1. CTIA-defined anthropomorphic hand phantoms for (a) bar style and (b) clam-shell style mobile phones. The spacer attached on the palm is for phone positioning.

II. NUMERICAL SIMULATIONS A. Models and Method

Recent studies have found cases where the hand can significantly increase the psSAR in the user. A study of bodyworn mobile phones by Francavilla and Schiavoni found that the psSAR in the body can increase by between 30% and 40% when the user’s hand is holding the mobile phone [25]. Measurements were made on four mobile phones, and simulations were conducted on simplified mobile phone models with helical or whip antennas. Limited studies showing significant influence of the hand on psSAR in the head [4] and radiated antenna performance [4], [5], [26] have also been conducted. Following these studies, it is necessary to do a thorough investigation, including a wide range of realistic hand positions. The objective of this study is to investigate the hand effect over a large sample of mobile phones used in several different hand grips and positions. Numerical simulations are conducted using a wide range of hand grips and positions with four mobile phone models. Conclusions are drawn regarding the types of hand grips and positions that result in increased psSAR in the head. Measurements are made of 46 commercial mobile phones using a human hand in different grips. Finally, the considerations of the hand effect on the head psSAR are discussed. In this paper, the change of the head psSAR due to the hand is defined as

(1) Throughout the paper, the psSAR value is normalized to the antenna forward power, except where indicated. Each psSAR value in (1) is determined by averaging the SAR over a 1-gram or 10-gram cubical mass centered at each point, then selecting the maximum value over all points. Therefore, (1) represents the ratio of the highest psSAR values regardless of location. This is important, as the presence of the hand may cause a shift in the location of the psSAR. To specifically refer to the change in the 1-gram or 10-gram psSAR, the variables and are used. The absolute values of psSAR are not presented in this paper as these are dependent on the power level of the device and the operating mode. The focus of the paper is the investigation of the mechanisms that change the .

Numerical simulations are performed with the software package SEMCAD X that has been and continuously is co-developed with several university partners and Schmid & Partner Engineering AG (SPEAG, Zurich, Switzerland). The RF solver is based on the Finite-Difference Time-Domain (FDTD) method enhanced with multiple method extensions that enhance the accuracy and speed [27]–[29]. The simulation frequencies selected in this study are 900 MHz, 1750 MHz and 1800 MHz. The model resolution is about 0.1 to 0.2 mm, depending on the phone models. Five to seven layers of UPML/CPML are used as the absorbing boundary surround the modeling space [30], with at least 0.25 wavelengths of free space between the model and the absorbing layers. Four mobile phone models have been selected for this study, including two generic designs and two computer-aided design (CAD) models (Fig. 2). The generic mobile phones both have a clamshell style with three metal parts representing the top half, the bottom half and a conductive element (flexible PCB) joining the two halves at the hinge. These models have a single-band planar inverted-F antenna (PIFA) at 900 MHz (phone 1) or 1800 MHz (phone 2). The two CAD mobile phone models have either a clamshell style with an external helical antenna (phone 3) or a bar style with a PIFA (phone 4). The details of the CAD models are reported and have been validated with measurements in previous studies [4], [31]. The antennas in phone 3 and phone 4 were re-designed using genetic algorithm optimization and thus their performance may be different from those of commercially-available mobile phones [32]. The mobile phone models are placed in the cheek position against the SAM head, as defined in [11]. For phone 4, heterogeneous head models were also used. A previous study of the hand effect on Over-The-Air (OTA) parameters reports that the mobile phone antenna performance is sensitive to the hand position [4]. Thus, it is important to define a rigorous process in order to observe the range of that may occur. The process makes use of a generic block model of the hand as shown in Fig. 3. This generic hand consists of three bricks which represent the palm spaced away from the back face of the mobile phone and the fingers holding the sides. The generic hand is used because it can be well defined and controlled by scripts for

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Fig. 3. Generic hand model shown holding phone 2 to the SAM head. The generic hand consists of three blocks representing the palm, the thumb and the other fingers gripping the sides of the phone. The hand phantom is moved over a wide range of positions along the length of the phone (Y-direction) and palm distances to the back of the phone (Z-direction).

automated simulations and optimization. To find the hand position that maximizes , the generic hand is moved in the Y-direction from the top to the bottom of the phone, and the palm spacing is moved in the Z-direction over a 60 mm range, as shown in Fig. 3. The fingers are touching the cheek, and the finger length L varies according to the hand position. In the coordinate system, corresponds to the top of the inside of the palm touching the top of the outer surface of the antenna. The movement resolution is 10 mm in both Yand Z-directions. The hand position that maximizes is then selected. Next, the generic hand model is substituted with a CTIA-defined homogeneous anthropomorphic hand. The anthropomorphic hand is used in two different grips. • Grip 1: identical to the CTIA-defined grips shown in Fig. 1 for the specific mobile phone type (bar or clamshell). The position of the hand on the mobile phone model is based on the position of the generic hand that maximizes . • Grip 2: the fingers of the hand are posed using the Poser tool in SEMCAD X [33] so that the hand grip corresponds as close as possible to the grip of the generic hand that maximizes while still conforming to the range of realistic hand grips [34].

Fig. 4. for phone 2 at 1800 MHz is represented as a function of the generic hand position.

Fig. 5. of phone 4 at 1750 MHz is represented as a function of the generic hand position.

B. Results is to The simulation results quantify how sensitive the hand position. is shown as a function of hand position in Fig. 4 for phone 2 and Fig. 5 for phone 4. The total variation of is about 3.5 dB for phone 2 and 2.1 dB for phone 4. For phone 2, 30% of the hand positions result in positive values of , while for phone 4, it was found for 95% of the hand positions. The 90th percentile value of is 0.5 dB and 1.8 dB for phone 2 and phone 4, respectively. Based on the position of the generic hand that maximizes , the phones were simulated with the anthropomorphic hand phantom. For phones 1, 2 and 3, both grip 1 and grip

Fig. 6. Generic hand at the position that maximizes and the corresponding anthropomorphic hand phantom in (a) grip 1 and (b) grip 2 on phone 1.

2 are applied. Due to the small size of phone 4, it was not possible to position the anthropomorphic hand in grip 1 and use a realistic grip (the index finger would be above the top of the phone). Thus, only grip 2 is used for phone 4. Fig. 6 and Fig. 7 show the hand grips applied on phone 1 and phone 4, respectively. Phone 4 with grip 2 is also simulated with anatomical head models, as shown in Fig. 8. The setup is identical to that of

LI et al.: INFLUENCE OF THE HAND ON THE SPECIFIC ABSORPTION RATE IN THE HEAD

Fig. 7. Generic hand at the position on phone 4 that maximizes and the corresponding anthropomorphic hand phantom in grip 2. The views show (a) the front of the mobile phone model, looking through a transparent SAM, and (b) the back of the mobile phone model.

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The same results are shown normalized to the feed point current in Table I. The increase of psSAR is also over 2 dB for phone 3 and phone 4. is not as high for phone 1 and phone 2. Note that the hand grips have been chosen to maximize based on power normalization rather than current normalization. For all but one case in Table I, the antenna radiation efficiency, , and the total radiated power, TRP, drop by at least 3 dB when the hand models are included. The hand absorbs a significant amount of power as expected [4]. Table II shows the results when anatomical head models are used instead of SAM for phone 4. The psSAR in the external ear (pinna) is excluded from the 1-gram and 10-gram averaging for the results shown. RF exposure standards from ICNIRP [8] and IEEE [9] have different approaches for how the psSAR limits in the pinna are taken into account. It is outside the scope of this paper to deal with the complexities of this pinna issue. in SAM also excludes the ear, as the ear is lossless. Table II shows that for the anatomical heads is similar or higher to those for SAM, regardless of whether the psSAR is normalized to current or power. The use of anatomical heads therefore do not change the conclusions of this study. III. MEASUREMENT A. Models and Method

Fig. 8. Phone 4 with grip 2 as seen in Fig. 7 is simulated with anatomical head models. In addition to the Duke model seen here, the heads of Ella and Visible Human model are also used in this study.

Fig. 7 except that the SAM head is replaced by the heads of the Visible Human (VH) [35] and the Duke (adult male) and Ella (adult female) models of the Virtual Family [36]. The ears of the models are compressed to represent the force exerted by the mobile phone [37]. Table I shows the simulation results of the four mobile phone models with the hand grips where the highest values of were observed. For the results normalized to the same forward power to the antenna, maximum values higher than 2 dB are consistently observed for both frequencies and all four phone models investigated. The data shows how the antenna match, and therefore the delivered power, are influenced by the presence of the hand. As the hand is in the antenna near field, it can in general strongly affect the delivered power, depending on its size and position, and on the antenna and mobile phone design. It is interesting to note that the results presented in Table I are not consistently or strongly biased by the anthropomorphic hands in the grips presented. The change in the delivered power ranges from 0.8 dB (phone 3, grip 2) to 0.7 dB (phone 2, grip 1), and it is less than the change in in all cases. This leads to the conclusion that the increase in presented in Table I is more strongly affected by the disturbance of the fields than the change in the antenna match. For other cases not presented in Table I, the effect of antenna match could be substantial.

SAR measurements were made with 46 commercial mobile phones in order to determine the effects of the hand on the psSAR in commercial mobile phones. All of the general styles are included: bar (including personal digital assistants), clamshell and slide, as shown in Table III. All of the mobile phones were operated in GSM mode aside from two in WCDMA mode. A base station simulator was used to establish the call at the center channel of the band and maintain a fixed output power of the mobile phone. The measurements with each mobile phone were repeated and psSAR variations less than 5% were observed. Measurements were made using an iSAR system (SPEAG, Zurich, Switzerland) [17]. The top surface of the system follows the ear-to-mouth line of the SAM, as specified in [10], [11]. This line is extruded in an orthogonal direction (see Fig. 12). The thickness of the outer shell is 2 mm, except at the ear spacer where it is 6 mm, as specified in [10], [11]. The 256 electric field probes are arranged in a planar array which is conformal and 4 mm below the shell. Cubic spline interpolation is applied between the measured points, and the psSAR is estimated using the algorithms derived from [38], [39]. The probes are embedded in a lossy dielectric material having dielectric parameters over a wide frequency range, at least 600–6000 MHz, within 10% of the target values for human tissue that are standardized in [10], [11]. The short measurement time ( 1 s) compared to conventional systems ( 300 s) makes it practical to perform the large number of measurements in this paper. The position of the mobile phone is secured by a dielectric phone holder. A comparison study between iSAR and a standard SAR measurement system, DASY52 (SPEAG, Zurich, Switzerland) shows that the difference in psSAR is within 0.5 dB for most transmitters [17]. The repeatability of iSAR measurements is within 0.2 dB.

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TABLE I SIMULATION RESULTS OF THE FOUR MOBILE PHONE MODELS WITH DIFFERENT HAND SCENARIOS

TABLE II IN ANATOMICAL HEAD MODELS WITH PHONE 4 AND THE CORRESPONDING ANTHROPOMORPHIC HAND PHANTOM IN GRIP 2

TABLE III NUMBER OF EACH TYPE OF COMMERCIAL MOBILE PHONE MEASURED IN THIS STUDY

B. Results The maximum values obtained for the mobile phone models is shown in Fig. 10. Previous studies suggest that the increase of psSAR due to hand is more pronounced at higher frequencies [25]. Indeed, the data show that a narrower range of maximum values is observed at 900 MHz than at 1800 MHz, and that the percentage of mobile phones exhibiting a significant increase in psSAR is less at 900 MHz than at 1800 MHz. For example, the number of mobile phones exhibiting values above 0.5 dB is 5 out of 21 at 900 MHz and 21 out of 46 at 1800 MHz. IV. DISCUSSION A. Influence of Hand Position

A human right hand, as seen in Fig. 9 is used, with dimensions shown in Table IV. The hand dimensions are close to the CTIAdefined hand phantom [18]. The original hand grip applied in this study is based on the grip studies [18], [34] (Fig. 9(a) and Fig. 9(b)). The hand is moved vertically (Fig. 9(c)) and horizontally (Fig. 9(e)) with different palm-phone distances (Fig. 9(d)). This does not cover the full range of hand positions possible, but it is intended to represent a subset of realistic hand positions. The maximum psSAR among all the hand grips and positions is recorded for each mobile phone.

The proximity of the user’s hand significantly perturbs the near-field distribution around the mobile phone. To illustrate this point, Fig. 11 shows the SAR distribution in SAM from phone 2 with and without the anthropomorphic hand phantom in grip 1. In the absence of the hand, the SAR is more evenly spread out, while the presence of the hand results in a more concentrated absorption pattern for this case. The observed changes in the pattern have several non-independent causes 1) reflections by the hand resulting in a confinement of the RF energy between head and hand, 2) detuning of the antenna, 3) modification of the RF coupling between the electrical components in the phone and therefore on the current distribution inside the phone. In the simulated cases, if the palm of the hand is very close to the antenna (less than 15 mm), the strongly increased

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Fig. 9. Hand grip and movement applied to obtain the maximum different palm-phone distances and moved to the left (e) and right sides.

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value. The original grip (a), (b) is shifted (c) to different positions, posed (d) with

TABLE IV DIMENSIONS OF THE TESTER’S HAND AND THE CTIA-DEFINED HAND PHANTOM. HAND LENGTH IS THE DISTANCE FROM THE CENTER OF WRIST TO THE TIP OF THE MIDDLE FINGER. PALM LENGTH IS THE DISTANCE FROM THE MIDDLE CREASE TO DISTAL PALM CREASE

Fig. 11. Distribution of 1 g-averaged SAR of phone 2 in the SAM (a) without the hand and (b) with the anthropomorphic hand in grip 1. (The white squares are the hotspots.

hand are electrically larger and there are more opportunities to influence both the near field distribution and the currents on the mobile phone. This result is consistent with the general finding of Francavilla and Schiavoni [25]. Additional studies at other frequencies would be needed to see if this trend continues. B. Suitability of the Hand Model

Fig. 10. for (a) 21 mobile phones at 900 MHz and (b) 46 Mobile phones at 1800 MHz.

absorption inside the hand reduces the psSAR in the head (as seen in Fig. 4 and Fig. 5). At palm-phone distances of 15 mm to 35 mm, the psSAR in the head increases for the cases at 1750 and 1800 MHz. The psSAR then drops at larger palm-phone distances. The psSAR in the head is also sensitive to the hand location, with highest psSAR values when the top of the palm is directly over the antenna. The higher range of values and more frequent occurrence of high values may be due to the shorter wavelength of 1800 MHz. At shorter wavelengths, the mobile phone and the

To examine if a homogeneous hand can approximate the effect of a real hand, a comparison is made between the CTIA-defined hand and the tester’s hand for one mobile phone at a frequency near 1800 MHz (Fig. 12). SAR measurements are made on the iSAR for both left and right hands. Efforts were made by the tester to pose the hand as closely as possible to the grip used by the CTIA-defined hand. The measurement results as seen in Table V show that the difference between the hand phantom and the human hand is small (within 0.25 dB). This suggests that a human hand can be approximated by the homogeneous hand model for this purpose. An interlaboratory comparison study between seven laboratories found that the reproducibility of psSAR measurements with a CTIA-defined hand phantom at 900 and 1800 MHz was within acceptable levels, resulting in only a modest increase in the measurement uncertainty compared to psSAR measurements without a hand model [40]. C. Antenna Design Considerations It was shown in Fig. 4 and Fig. 5 that psSAR in the head can be very sensitive to the hand position. Given that the psSAR is

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TABLE VI IN SAM OVER ALL POSITIONS OF THE GENERIC MAXIMUM HAND. RESULTS ARE SHOWN FOR THE DESIGNS OF FIG. 13

V. CONCLUSION Fig. 12. psSAR measurement of a mobile phone using (a) the tester’s hand and (b) the hand phantom. TABLE V FOR THE MEASUREMENT SETUP SHOWN IN FIG. 12 WITH THE TESTER’S HAND AND THE CTIA-DEFINED HAND MODEL

Fig. 13. (a) Original model of phone 2 and modifications to change the location of (b) the antenna and (c) the flexible PCB.

caused by currents on the radiating structure [41], the psSAR is also sensitive to the design of the antenna and mobile phone. However, the complexity of mobile phone design makes it difficult to devise simple antenna design rules that are guaranteed to mitigate the hand effect. Proposals to keep the current from flowing on the mobile phone chassis have been proposed such as creating a current choke [1]. Modifications to the mobile phone chassis can be made to keep the current from flowing on areas influenced by the hand [3]. To suppress regions of high electric field concentration on the antenna, such as slots on PIFA antennas, the slots can be replaced by lumped elements [6]. A practical approach for a specific mobile phone design is to identify and change the primary design features that cause the psSAR increase. This approach is demonstrated for phone 2, where high values of psSAR in the head have been identified with enhanced currents on the flexible PCB. The current on the flexible PCB can be reduced by changing the locations of the antenna or the flexible PCB, as shown in Fig. 13. Each configuration was simulated with the generic hand in the same range of hand positions as described in Section II-A. The results were used to produce the distribution of with position as shown in Fig. 4, and the highest value of was recorded. These maximum values, given in Table VI show how sensitive the hand effect can be to a small number of mobile phone parameters.

Significant increases (2.5 dB or more) in psSAR in the user’s head have been observed when the hand is considered. The hand has a more pronounced effect at 1800 MHz than at 900 MHz. Simulations at 1800 MHz over a wide range of hand positions found that higher psSAR in the head occurs when the palm is over the antenna with a minimum palm-antenna spacing. At closer distances, the psSAR in the head decreases. The psSAR in the head is sensitive to the hand position, with variations of more than 3 dB observed. The CTIA-defined hand model gives similar results as a real hand, and psSAR increases in the SAM head have been replicated in anatomical heads. A main objective of this work was to investigate the psSAR increases in SAM, as this is the head phantom used by international measurement standards. The results support the conclusion that significant and reproducible psSAR increases in SAM are possible when the hand is introduced. Therefore, the influence of the hand is an important factor to consider for future revisions of these standards. Possible considerations, such as the addition of hand models in SAR measurement procedures or the application of scaling factors to account for hand effects, require further work. This study reports the highest increases in psSAR in the head, but it does not investigate the likelihood of such an increase among the user population. It also does not address compliance with regulatory limits, as it is outside the scope of this investigation. ACKNOWLEDGMENT The authors would like to thank the Vodafone Group for providing the mobile phones for this study. REFERENCES [1] M. Okoniewski and M. A. Stuchly, “A study of the handset antenna and human body interaction,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 10, pp. 1855–1864, Oct. 1996. [2] M. A. Jensen and Y. Rahmat-Samii, “EM interaction of handset antennas and a human in personal communications,” Proc. IEEE, vol. 83, no. 1, pp. 7–17, Jan. 1995. [3] S.-J. Kim, K.-H. Kong, M.-J. Park, Y.-S. Chung, and B. Lee, “Design concept of a mobile handset antenna to mitigate user’s hand effect,” Microw. Opt. Technol. Lett., vol. 50, pp. 2696–2698, Jul. 2008. [4] C.-H. Li, E. Ofli, N. Chavannes, and N. Kuster, “Effects of hand phantom on mobile phone antenna performance,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2763–2770, Sep. 2009. [5] J. Krogerus, J. Toivanen, C. Icheln, and P. Vainikainen, “Effect of the human body on total radiated power and the 3-D radiation pattern of mobile handsets,” IEEE Trans. Instrum. Meas., vol. 56, no. 6, pp. 2375–2385, Dec. 2007. [6] K. R. Boyle, Y. Yun, and L. P. Ligthart, “Analysis of mobile phone antenna impedance variations with user proximity,” IEEE Trans. Antennas Propag., vol. 55, pp. 364–372, Feb. 2007. [7] S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz,” Phys. Med. Biol., vol. 41, pp. 2251–2269, 1996. [8] ICNIRP, “Guidelines for limiting exposure to time-varying electric, magnetic and electromagnetic fields (up to 300 GHz),” Health Phys., vol. 74, pp. 494–522, 1998.

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[9] IEEE Standard for Safety Levels with Respect to Human Exposure to Radio Frequency Electromagnetic Fields, 3 kHz to 300 GHz, IEEE Standard C95.1, 2005. [10] “Human Exposure to Radio Frequency Fields from Handheld and Body-Mounted Wireless Communication Devices—Human models, Instrumentation, and Procedures Part 1: Procedure to Determine the Specific Absorption Rate (SAR) for Hand-Held Devices Used in Close Proximity to the Ear (Frequency Range of 300 MHz to 3 GHz),” IEC 62209-1, 2005. [11] IEEE Recommended Practice for Determining the Peak Spatial-Average Specific Absorption Rate (SAR) in the Human Head From Wireless Communications Devices: Measurement Techniques, IEEE Std. 1528, 2003. [12] A. Drossos, V. Santomaa, and N. Kuster, “The dependence of electromagnetic energy absorption upon human head tissue composition in the frequency range of 300–3000 MHz,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pp. 1988–1995, 2000. [13] Q. Balzano, O. Garay, and T. Manning, “Electromagnetic energy exposure of the users of portable cellular telephones,” IEEE Trans. Vehic. Technol., vol. 44, no. 3, pp. 390–403, Aug. 1995. [14] K. A. Meier, “Scientific Bases for Dosimetric Compliance Tests on Mobile Telecommunications Equipment,” Ph.D. dissertation, Swiss Federal Institute of Technology Zurich, ETHZ, Switzerland, 1996. [15] N. Kuster, R. Kastle, and T. Schmid, “Dosimetric evaluation of handheld mobile communications equipment with known precision,” IEICE Trans. Commun., pp. 645–652, 1997. [16] F. Meyer, K. Palmer, and U. Jakobus, “Investigation into the accuracy, efficiency and applicability of the method of moments as numerical dosimetry tool for the head and hand of a mobile phone user,” Appl. Comput. Electromagn. Soc. J., vol. 16, pp. 114–125, 2001. [17] M. Douglas, S. Gabriel, C. Bucher, D. Iliev, J. Kastrati, C. Leubler, M. Meili, K. Pokovic, and N. Kuster, “Fast SAR methods for EM exposure evaluation of wireless devices,” in Proc. Eur. Conf. Antennas Propag., Apr. 2011, pp. 2786–2789. [18] “CTIA Test Plan for Mobile Station Over the Air Performance,” Revision 3.0, CTIA Wireless Association, Apr. 2009. [19] C. Gabriel, “Tissue equivalent material for hand phantoms,” Phys. Med. Biol., vol. 52, pp. 4205–4210, Jul. 2007. [20] M. D. Foegelle, K. Li, A. Pavacic, and P. Moller, “The development of a standard hand phantom for wireless performance testing: Part 2,” presented at the Wireless Design Development, 2010. [21] T. M. Greiner, “Hand Anthropometry of U.S. Army Personnel,” U.S. Army Natick Research, Development and Engineering Center, Dec. 1991. [22] A. R. Tilley and H. D. Associates, The Measure of Man & Woman: Human Factors in Design, Revised ed. New York: Wiley, 2002. [23] B. Buchholz, T. J. Armstrong, and S. A. Goldstein, “Anthropometric data for describing the kinematics of the human hand,” Ergonomics, vol. 35, no. 3, pp. 261–273, Mar. 1992. [24] W. D. Bugbee and M. J. Botte, “Surface anatomy of the hand: The relationships between palmar skin creases and osseous anatomy,” Clinical Orthopaedics Related Res., no. 296, pp. 122–126, 1993. [25] M. Francavilla and A. Schiavoni, “Effect of the hand in SAR compliance tests of body worn devices,” presented at the Appl. Comput. Electromagn. S. Conf., 2007. [26] O. Kivekas, J. Ollikainen, T. Lehtiniemi, and P. Vainikainen, “Bandwidth, SAR, and efficiency of internal mobile phone antennas,” IEEE Trans. Electromagn. Compat., vol. 46, no. 1, pp. 71–86, Feb. 2004. [27] N. Chavannes, “Computational electrodynamics: The finite-difference time-domain method,” in Nonuniform Grids, Nonorthogonal Grids, Unstructured Grids, and Subgrids. Norwood, MA: Artech House, 2005, ch. 11.8, pp. 463–515. [28] S. Benkler, N. Chavannes, and N. Kuster, “A new 3-D conformal PEC FDTD scheme with user-defined geometric precision and derived stability criterion,” IEEE Trans. Antennas Propag., vol. 54, pp. 264–272, May 2006. [29] S. Schild, N. Chavannes, and N. Kuster, “A robust method to accurately treat arbitrarily curved 3-D thin conductive sheets in FDTD,” IEEE Trans. Antennas Propag., vol. 55, pp. 3587–3594, Dec. 2007. [30] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Norwood, MA: Artech House, Jun. 2005.

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[31] N. Chavannes, R. Tay, N. Nikoloski, and N. Kuster, “Suitability of FDTD based TCAD tools for RF design of mobile phones,” IEEE Antennas Propag. Mag., vol. 45, pp. 52–66, Dec. 2003. [32] E. Ofli, C.-H. Li, N. Chavannes, and N. Kuster, “Analysis and optimization of mobile phone antenna radiation performance in the presence of head and hand phantoms,” Turkish J. Electr. Eng. Comp. Sci., vol. 16, pp. 67–77, 2008. [33] Schmid & Partner Engineering AG, SEMCAD X: EM/T Simulation Platform [Online]. Available: www.semcad.com [34] M. Pelosi, O. Franek, M. B. Knudsen, M. Christensen, and G. F. Pedersen, “A grip study for talk and data modes in mobile phones,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 856–865, Apr. 2009. [35] M. Ackerman, “The visible human project,” Proc. IEEE, vol. 86, no. 3, pp. 504–511, 1998. [36] A. Christ, W. Kainz, E. G. Hahn, K. Honegger, M. Zefferer, E. Neufeld, W. Rascher, R. Janka, W. Bautz, J. Chen, B. Kiefer, P. Schmitt, H.-P. Hollenbach, J.-X. Shen, M. Oberle, D. Szczerba, A. Kam, J. W. Guag, and N. Kuster, “The virtual family-development of surface-based anatomical models of two adults and two children for dosimetric simulations,” Phys. Med. Biol., vol. 55, 2010. [37] A. Christ, M. Gosselin, S. Kühn, and N. Kuster, “Impact of pinna compression on the RF absorption in the heads of adult and juvenile cell phone users,” Bioelectromagnetics, vol. 31, no. 5, pp. 406–412, 2010. [38] M. Y. Kanda, M. G. Douglas, E. Mendivil, M. Ballen, A. V. Gessner, and C.-K. Chou, “Faster determination of mass-averaged SAR from 2-D area scans,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 2013–2020, Aug. 2004. [39] M. G. Douglas and C.-K. Chou, “Accurate and fast estimation of volumetric SAR from planar scans from 30 MHz to 6 GHz,” presented at the Bioelectromagnetics Soc. Ann. Meeting, Jun. 2007. [40] M. G. Douglas, B. Derat, C.-H. Li, X.-W. Liao, E. Ofli, N. Chavannes, and N. Kuster, “Reliability of specific absorption rate measurements in the head using standardized hand phantoms,” in Proc. Eur. Conf. Antennas Propagation, Apr. 2010, pp. 1–4. [41] N. Kuster and Q. Balzano, “Energy absorption mechanism by biological bodies in the near field of dipole antennas above 300 MHz,” IEEE Trans. Vehic. Technol., vol. 41, no. 1, pp. 17–23, Feb. 1992.

Chung-Huan Li was born in May, 1979 in Taipei, Taiwan. He received the B.Sc. and M.Sc. degrees in electronic engineering from the National Taiwan University of Science and Technology (NTUST), Taipei, Taiwan, in 2002 and 2004, respectively. He joined the Foundation for Research on Information Technologies in Society (IT’IS), Swiss Federal Institute of Technology (ETH), Zurich, Switzerland, in April 2007 as a Ph.D. student. His interests include antenna design, as well as the study of electromagnetic waves and theory.

Mark Douglas (S’86–M’98–SM’05) received the B.Eng. degree from the University of Victoria, Victoria, BC, Canada in 1990, the M.Sc. degree from the University of Calgary, Calgary, AB, Canada in 1993, and the Ph.D. degree from the University of Victoria in 1998, all in electrical engineering. He joined the IT’IS Foundation in 2009 as a Project Leader in the area of electromagnetic dosimetry. From 2002 to 2009, Mark was an engineering manager in the Corporate Electromagnetic Energy (EME) Research Laboratory at Motorola, where he led advancements in radiofrequency dosimetry research and testing. Before joining Motorola, he was a Senior Technical Leader with the Antenna Development Group at Ericsson and a member of the Ericsson EMF Research Group, Stockholm, Sweden. His research work has resulted in over 60 papers for scientific conferences and peer-reviewed journals. He also holds 5 patents.

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Erdem Ofli (M’02) received the B.Sc. and M.Sc. degrees in electrical engineering from Bilkent University, Ankara, Turkey, in 1995 and 1997, respectively, and the Ph.D. degree in electrical engineering from ETH Zurich, Switzerland, in 2004. In 2005, he joined Schmid and Partner Engineering AG (SPEAG), Zurich, Switzerland, and is currently working as a Senior Engineer. He is interested in numerical techniques in electromagnetics, microwave and millimeter wave components and systems design, wireless communications.

Benoit Derat was born in Drancy, France, in 1979. He received the Engineer degree from the Ecole Superieure d’Electricite (Supelec), Gif-sur-Yvette, France, in 2002, and the Ph.D. degree in physics from the University of Paris XI, Orsay, France, in 2006, in collaboration with the mobile phones R&D Department of SAGEM Communication, Cergy-Pontoise, France. From 2006 to 2008, he worked for SAGEM Mobiles R&D as a research engineer and expert in analytical and numerical modeling of electromagnetic radiation and near-field interactions. In 2009, he founded the FIELD IMAGING S.A.R.L. company, providing services in his areas of expertise. His research interests include small antenna design and measurement, 3-D electromagnetic simulation, near-field power dissipation mechanisms and Specific Absorption Rate (SAR) measurement and computation. Dr. Derat is currently an active member of the IEC MT62209 and ICES TC34 SC2.

Sami Gabriel received the B.Eng. (hons) and M.Sc. degrees in London, U.K., in 1990 and 1992, respectively. He joined the Dielectrics Research Group at King’s College London to investigate the dielectric properties of biological tissues then moved to Imperial College London to investigate the microwave heating of organic compounds. His work is published in peer-reviewed journals with over 3,000 citations. He joined Vodafone Group in 2003 as Chief Engineer in Research and Development. He is a member of the IEEE International Committee on Electromagnetic Safety, TC34 contributing for over 14 years, he represents the U.K. as an expert delegate to the International Electrotechnical Committee, PT62209 as well as the European Committee for Electrotechnical Standardization TC106 WG1. He is an expert advisor to the UK Mobile Operators Association and GSM Association on mobile device SAR evaluation matters. Mr. Gabriel is a Fellow of the Institute of Engineering and Technology and member of its policy advisory group BEPAG. He .

Nicolas Chavannes was born in Bern, Switzerland, in April 1972. He received the M.Sc. and Ph.D. degrees in electrical engineering from the Swiss Federal Institute of Technology (ETH), Zurich, Switzerland, in 1998 and 2002, respectively. In 1996, he joined the Bioelectromagnetics/EMC Group (BIOEM/EMC) at ETH Zurich where he was involved in computational electrodynamics and related dosimetric applications. From 1998 to 2002, he was with the Laboratory for Electromagnetic Fields and Microwave Electronics (IFH) as well as the Lab-

oratory for Integrated Systems (IIS), both located at ETH Zurich. There, his research activities were focused on the development of FDTD local refinement techniques and their application to numerical near-field analysis. In late 1999, he joined the Foundation for Research on Information Technologies in Society (IT’IS), Switzerland, where he is currently in charge of the development and extension of a simulation platform targeted for antenna modeling and MTE design in complex environments, dosimetry and optics applications. In early 2002, he joined Schmid & Partner Engineering AG (SPEAG), Zurich, as head of the software R&D team. His primary research interests include the development, implementation and application of computational modeling and simulation techniques to electromagnetics in general, and antennas as well as bioelectromagnetic interaction mechanisms in particular.

Niels Kuster (F’11) received the M.S. and Ph.D. degrees from the Swiss Federal Institute of Technology, ETH Zurich (ETHZ), Zurich, both in electrical engineering. Between 1993 and 1999, he was an Assistant Professor at the Department of Electrical Engineering, ETHZ. He was awarded Professor at the Department of Information Technology and Electrical Engineering, ETHZ, in 2001. From 1999 until now, he has served as the Founding Director of the Foundation for Research on Information Technologies in Society (IT’IS), Switzerland. In 2010, he initiated the sister institute IT’IS USA, a nonprofit research unit incorporated in Maryland, of which he is currently the President. During his career, he has held invited Professorships at the Electromagnetics Laboratory of Motorola, Inc., FL, and at the Metropolitan University, Tokyo, Japan, in 1998. He also founded several spin-off companies Schmid & Partner Engineering AG, MaxWave AG, NFT Holding AG, Zurich MedTech AG and advises other companies as board member such as IMRICOR, Inc., TheraBionic LLC, etc. He has published more than 600 publications (books, journals, and proceedings) on measurement techniques, computational electromagnetics, dosimetry, exposure assessments, and bioexperiments. His primary research interests include safe and beneficial applications of electromagnetic fields in health and information technologies. He is particularly interested in measurement technology; computational electrodynamics for the evaluation of close near fields in complex environments (e.g., handheld or body-mounted transceivers, residential/work environments, etc.); safe and reliable wireless communication links within the body or between implanted devices and exterior equipment for biometric applications; development of exposure setups and quality control for bioexperiments to evaluate interaction mechanisms, therapeutic effects and potential health risks; exposure assessments; EM safety of medical devices; medical diagnostic and therapeutic applications of EM, in particular EM cancer treatment modalities; and virtual patient applications. He is currently building up a new research team in computational life science in biology. Dr. Kuster is a member of several standardization bodies and acts as a consultant to government agencies around the globe on the safety of mobile communications. He was a board member of various scientific societies and boards, was Bioelectromagnetics Society (BEMS) president in 2008–2009. He is delegate of the Swiss Academy of Science and is currently an Associate Editor of the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY.

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Demonstration of a Cognitive Radio Front End Using an Optically Pumped Reconfigurable Antenna System (OPRAS) Youssef Tawk, Joseph Costantine, Sameer Hemmady, Ganesh Balakrishnan, Keith Avery, and Christos G. Christodoulou

Abstract—A cognitive radio front end using an optically pumped reconfigurable antenna system (OPRAS) is investigated. The scheme consists of a ultrawidebhand antenna and a reconfigurable narrowband antenna in close proximity to one another. The narrowband reconfigurability is achieved by a integratinglaser diodes within the antenna structure to control the switching state of photoconductive silicon switches. This scheme has the advantage of eliminating the use of optical fiber cables to guide light to the switches, and enables easier integration of the reconfigurable antenna in a complete communication system. The performance of the proposed technique is presented, and comparisons are made to other commonly used switching techniques for reconfigurable antennas, such as techniques based on PIN diodes and RF microlectromechanical systems integration. The application of this antenna design scheme serving as the receive channel in a cognitive radio communication link is also demonstrated. Index Terms—Cognitive radio, laser diodes, photoconductivity, reconfigurable antenna, silicon, ultrawideband (UWB).

I. INTRODUCTION

A

COGNITIVE radio system minimizes interference with other wireless systems in its operating band and maximizes throughput by dynamically altering its transmit/receive characteristics to occupy unused frequency channels [1], [2]. The basic architecture of a cognitive radio system is comprised of a “UWB sensing antenna” that continuously monitors the wireless channel and searches for unused frequency channels; and a “reconfigurable transmit/receive antenna” to perform the required communication within those unused frequency channels [3], [4].

Manuscript received February 05, 2011; revised June 07, 2011; accepted August 16, 2011. Date of publication October 24, 2011; date of current version February 03, 2012. This work was supported by the Air Force Research Lab under Contract FA9453-09-C-0309. Y. Tawk, S. Hemmady, and C. G. Christodoulou are with the Electrical and Computer Engineering Department, University of New Mexico, Albuquerque NM 87131 USA (e-mail: [email protected]; [email protected]; [email protected]). J. Costantine is with the Electrical Engineering Department, California State University Fullerton, Fullerton, CA 92834-9480 USA (e-mail: [email protected]). G. Balakrishnan is with the Center for High Technology Materials, University of New Mexico, Albuquerque, NM 87131 USA (e-mail: [email protected]). K. Avery is with the Air Force Research Laboratory, Space Vehicles Directorate, Kirtland Air Force Base, Albuquerque NM 87117 USA (e-mail: keith. [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173139

One of the challenges in a cognitive radio RF front end is the design of the reconfigurable antenna. So far, reconfigurable antennas for cognitive radio communications have been implemented using PIN diodes, RF MEMS or some physical alteration of the antenna structure using a rotational movement [5]–[10]. For example, in [5] a co-located wideband and narrowband antenna is fabricated. The wideband antenna is a continous planar waveguide (CPW)-fed printed hour-glass-shaped monopole which operates from 3 to 11 GHz. The narrowband antenna is a microstrip patch printed on the reverse side of the substrate, and connected to the wideband antenna via a shorting pin and designed to operate from 5.15 to 5.35 GHz. A reconfigurable C-slot microstrip patch antenna is proposed in [6]. The reconfigurability is achieved by switching on and off two patches using PIN diodes. The antenna can operate in dual band or in very wideband mode. In [7], a quad-antenna with a directional radiation pattern is presented. The operating frequency can be adjusted by the use of a microelectromechanical-systems (MEMS) switch, making it suitable for cognitive radio applications. The authors in [8] incorporate the sensing and the reconfigurable antennas into the same substrate. The reconfigurable antenna is able to tune between 3–5 GHz and 5–8 GHz via a rotating circular patch. With each rotation, a different triangular shape is fed. Some research has also been conducted on the design of optically reconfigurable antennas [11]–[15]. In [11], the authors used an -type silicon switch doped with phosphorus to increase its conductivity. The authors implemented the photoconductive switch on a printed dipole antenna which was fed by optical-fiber cables in order to create frequency and radiation pattern reconfigurability by effectively changing the dipole arm length. An optically controlled frequency reconfigurable microstrip antenna was implemented in [12] as well. The authors shorten the slot inside the antenna patch in order to make the antenna change its electrical length. This has the effect of producing a different resonant frequency. In [13], planar arrays of electrically small metallic patches are connected by switches. The field-effect-transistor (FET)-based electronic switches are used with optical control. The drawback in this design is that the energy lost in the switches reduces the radiating efficiency of the antenna to a point where it might prove unbeneficial for some applications. In this paper, we focus on demonstrating a cognitive radio application by utilizing a new technique to achieve frequency reconfigurability in photoconductive-switch based antennas. The

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design is based on integrating laser diodes within the antenna structure. This technique negates the need for optical-fiber cables for delivering light to the photoconductive switches [16], thereby reducing the complexity of the system and allowing for easier integration of such antennas in future wireless handheld devices. This technique does not require any biasing lines for switch activation purposes in the antenna radiating plane, as is the case with RF MEMS [17] or PIN diodes [18]. This paper is divided into the following sections: In Section II, a comparison between the proposed technique in this paper and the previous work done on reconfigurable antenna is discussed. In Section III, the architecture of the investigated cognitive antenna structure is shown. Section IV presents the integration of the laser diodes into the antenna substrate structure. The experimental RF performance of the antenna in comparison with numerical simulations is shown in Section V. In Section VI, we present an algorithm to implement a cognitive radio receive channel using the designed antenna from Section III. Finally, we conclude in Section VII by recapitulating the salient results presented through this work and proposing future work. II. COMPARISON BETWEEN “OPRAS” AND RF MEMS/PIN-DIODE-BASED RECONFIGURABLE ANTENNA SYSTEMS In this paper, an -type silicon (Si) piece with an initial carrier cm is used as the switching element. It concentration of mm. By illuminating the has physical dimensions of mm silicon switches by light from the laser diode, the mobility of charges in the silicon decreases but their density increases. This increase in the charge carrier density results in a general increase in the conductivity of the switch [19]–[22]. In the design of reconfigurable antennas and reconfigurable arrays, it is always desired to minimize the power required to activate the switching elements. The laser diode used in this paper requires a supply voltage in the range of 1.8–2.07 V and a driving current in the range of 0–129 mA in order to generate an output laser power in the range of 0–90 mW. The relation between the output power level (in miilwatts) and the current/ voltage for this laser diode is shown in Fig. 1, which shows that a voltage of 1.9 V and a current of 87 mA are required to drive the laser diode to produce an output laser power of 50 mW. This power level is sufficient to make the silicon switch transition from the OFF state to the ON state, and this transition is manifested by the increase in the silicon total conductivity (dc + RF). The silicon substrate used in this paper has a bandgap en1.12 eV, which corresponds to an activation bandgap ergy wavelength: 1.24 1.107 m 785 nm. determines the cutoff wavelength of the incident photons that are necessary to excite electrons from the silicon valence band to the conduction band. RF-MEMs/PIN diodes-based reconfigurable antenna systems require the design of appropriate bias lines which lie in the plane of the antenna. Bias lines affect the antenna radiation pattern and increase the complexity of the structure by adding additional RF components. On the other hand, “OPRAS” is based on integrating laser diodes within the antenna substrate where

Fig. 1. Relation between the optical power and the voltage/current for the laser diode.

no bias lines are needed to be printed on the plane of the radiating structure of the antenna. A copper piece is attached to the back of the antenna ground. This piece has a minimal effect on the antenna radiation pattern since it has a small depth and the same (or smaller) width/height as the antenna ground plane. Also, this technique eliminates the use of optical-fiber cables for light delivery which enables easier integration of the reconfigurable antenna. RF MEMs also suffer from poor reliability [23]. The deployment of RF MEMs into commercial and defense applications is very limited. PIN diodes exhibit nonlinear behavior at RF frequencies since the stored charge can be insufficient to control the RF current [24]. The nonlinear behavior of PIN diodes manifests itself as undesired antenna resonances. All of the problems produced by RF MEMs and PIN diodes’ integration can be overcome by implementing the “OPRAS” technique. The copper piece that is used as a heat sink for the laser diode improves the reliability of the proposed technique compared to RF MEMs, by increasing the laser diode lifetime. Also, the activation/deactivation of the photoconductive switch by shining light from the laser diode does not produce harmonics and intermodulation distortion as with the case of PIN diodes. The relation between the input power and the output power from a photoconductive switch shows a linear behavior [25], [26]. Table I shows a comparison between the three different techniques in terms of voltage/current requirement, amount of power consumption, and the switching speed [27]–[30]. The “OPRAS” performs faster than RF MEMs but needs more driving current. The PIN-diodes-based reconfigurable antenna acts faster than “OPRAS, ” but needs a higher level of driving voltage. The estimated power consumption of “OPRAS” lies between that of RF MEMS and PIN diodes. III. ANTENNA STRUCTURE The cognitive radio front end described in this paper consists of an ultrawideband (UWB) and a reconfigurable narrowband antenna placed next to each other. The antenna top view is shown in Fig. 2(a); its bottom view is shown in Fig. 2(b).

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TABLE I COMPARISON BETWEEN DIFFERENT SWITCHING TECHNIQUES

Fig. 3. (a) Integration of the laser diode into the antenna structure. (b) Drilled copper piece which supports the laser diodes.

The reconfigurable antenna is a modified printed monopole. It has an elliptical slot that contains a triangular arm. Both structures are connected together via a silicon switch (S1). At the end of the modified monopole, a hexagonal patch is attached via another silicon switch (S2).

IV. LASER DIODE INTEGRATION

Fig. 2. Antenna structure. (a) Top view. (b) Bottom view.

The cognitive antenna is printed on a Taconic TLY substrate with a dielectric constant of 2.2 and a height of 1.6 mm. The sensing and the reconfigurable structures are fed via a stripline. They both have a partial ground in order to allow radiation above and below the substrate. The reconfigurable antenna ground is 36 mm 9 mm. It has a longer length than the ground of the sensing antenna in order to accommodate the copper fixture. The separation between the two ground planes is chosen to be 8 mm. All of the dimensions shown in Fig. 2 are in millimeters. The UWB sensing antenna is a modified elliptical-shaped monopole. It covers the band from 3 to 11 GHz. It has a major 0.38 and a minor axis of 22 mm axis of 25.2 mm 0.35 , where corresponds to the lowest frequency (3 GHz). A small tapered microstrip section is used to match the UWB sensing antenna to the feed point.

A semiconductor laser diode is a device that converts electrical energy into optical radiation. The laser radiation is highly monochromatic and it produces highly directional beams of light. The laser action is produced by simply passing forward current through the diode itself. On account of its compact size and capability for high-frequency modulation, the semiconductor laser is one of the most important light sources for optical-fiber communication [20], [21]. The laser diode used in this paper operates at 785 nm and has a maximum output power of 90 mW. It has a can-type architecture and a part number L785P100 [31]. The integration of the laser diode is achieved by attaching it to the back of the ground of the reconfigurable antenna as shown in Fig. 3(a). In order to couple the light from the laser diodes efficiently, two holes of diameter 1 mm are drilled through the substrate. The copper piece used to integrate the laser diodes with the antenna structure is shown in Fig. 3(b). It is included into the simulation environment with the antenna structure to take its effect on the antenna performance into consideration. Its width is the same as the ground of the reconfigurable antenna 9 mm). It has a length of 21.5 mm and depth of 6.5 mm. This copper piece has two holes where inside each hole a laser diode is fixed 4.6 GHz The reconfigurable antenna radiation pattern at 5.2 GHz in the YZ plane for the case when the copper and piece is removed and when it is present is shown in Fig. 4. One can notice that the inclusion of this piece has a minimal effect on the antenna radiation pattern. Also, the laser diode cables are shielded and, therefore, do not interfere with the antenna-

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Fig. 4. Antenna radiation pattern in the YZ plane (8 = 90 ).

Fig. 5. Fabricated antenna structure. (a) Top view. (b) Bottom view.

radiated field. These two frequencies are chosen only as a proof of concept. V. FABRICATION AND RESULTS The fabricated antenna structure is shown in Fig. 5(a) and (b). Two photoconductive switches are integrated within the reconfigurable antenna. A. UWB Antenna The comparison between the measured and the simulated reflection coefficient (in decibels) for the UWB sensing antenna is shown in Fig. 6(a). The antenna is able to cover the spectrum

Fig. 6. (a) Measured and simulated jS11j for the sensing antenna when both switches are OFF. (b) Contour map of the normalized radiation pattern for the UWB antenna in the XZ plane. (c) YZ plane.

from 3 to 11 GHz. Good agreement is noticed between the simulated and the measured data for the UWB sensing structure.

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This data correspond to the case when both switches are OFF. We observed that the same UWB response is maintained for the different states of the two switches. The normalized UWB antenna /YZ plane radiation pattern in the XZ plane is shown in Fig. 6(b) and (c) as a colored contour plot for the case when both switches are OFF. The color code in the right side of the plot corresponds to the normalized values (in decibels) of the total radiated electric field for different frequency/angle values. It is observed that the UWB antenna has less variation in the XZ plane compared to the YZ plane for the radiated E-field across the majority of the frequency bands/angle values. B. Reconfigurable Antenna For the narrowband reconfigurable antenna, when the two silicon switches (S1 and S2) are not illuminated by a laser light (OFF state), only the modified monopole is fed. This results in an antenna resonance between 4.15 and 5.1 GHz. Upon activation of the first switch (S1) by driving the laser diode via a current of 87 mA and a voltage of 1.9 V (this corresponds to 50-mW optical power), the antenna shifts its resonance to the 4.8–5.7 GHz band. By illuminating the second switch (S2) with the same amount of pumped power, the 3.2–4.3 GHz band is covered. The case when both switches are ON produces a resonance outside the band of the UWB sensing antenna, and is not considered for our application. The simulated and the measured reflection coefficient for the reconfigurable narrowband antenna are summarized in Fig. 7(a) and (b). C. Coupling Since both structures are incorporated into the same cognitive antenna substrate, it is essential to look at the coupling between the UWB sensing and the reconfigurable narrowband antenna. This coupling or “cross-talk” is quantified by the transmission between the two antenna ports. The coupling between the two radiating structures is a function of their physical separation. In order to find the optimum distance between the two radiating structures without making the overall cognitive antenna too large, an optimization study was performed in HFSS. It was observed that for separation distances of less than 6 mm, there was considerable crosstalk between the two radiating structures. between A separation of 8 mm ensured that the measured the two radiating structures was less than 20 dB across the band from 3 to 11 GHz. The comparison between the simulated and the measured for the case when S1:OFF-S2:OFF and S1:OFF-S2:ON for a separation of 8 mm between the UWB and the reconfigurable narrowband structures is shown in Fig. 8. A measured coupling of less than 20 dB was achieved throughout the whole band of the UWB sensing antenna. The case when S1:ON-S2:OFF also gives a similar response.

Fig. 7. (a) Simulated and (b) measured reflection coefficient for the reconfigurable antenna.

D. Radiation Pattern The comparison between the simulated and the measured rafor the reconfigurable diation pattern in the XZ plane antenna is shown in Fig. 9. The radiation pattern is taken at 3.6 GHz (S1: OFF-S2: ON), 4.6 GHz (S1: OFF-S2: OFF) and

Fig. 8. Simulated and measured coupling for the case when both switches are off and when S1:OFF/S2:ON where the distance between the UWB and the reconfigurable antenna ground is 8 mm.

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Fig. 11. Cognitive radio receive channel experiment workflow.

Fig. 9. Simulated and measured radiation pattern for the three different cases of the switches in the XZ plane ( = 0 ).

Fig. 12. Experiment setup.

Fig. 10. Generic cognitive radio work-flow diagram.

at 5.2 GHz (S1: ON-S2: OFF). A reasonable omnidirectional radiation pattern was achieved. VI. IMPLEMENTATION OF A COGNITIVE RADIO RECEIVE ALGORITHM A work-flow diagram indicating the operation of a cognitive radio system is shown in Fig. 10. The sensing antenna is generally a UWB antenna with an operating band that spans the entire frequency spectrum over which the wireless communication is expected to occur. The “Spectrum Sensing” module of the cognitive radio engine continuously searches for unused frequency channels within this operating band. This information is fed to the “Spectrum Decision” module which determines the corresponding band for communication. The “Switch Controller” module then performs the required electronic operation (switching, multiplexing, etc.) to tune the operating frequency of the reconfigurable antenna which performs the data communication over the unused wireless frequency channels determined by the “Spectrum Decision” module.

In this section, we demonstrate the applicability of our cognitive antenna, described previously, by incorporating it into a mockup of a cognitive radio receive channel. The experiment flowchart is shown in Fig. 11. The setup of this experiment is shown in Fig. 12. The experiment workflow consists of the following steps: Step 1) Controlling the frequency sweeper via LABVIEW: A frequency sweeper is used as a model of the wireless channel in a cognitive radio environment. It is programmed to generate a continuous wave (CW) every 3 s at a randomly chosen carrier frequency between 3 and 6 GHz. The RF output of the frequency sweeper is connected to the broadband TX horn antenna that is placed at a distance of 3 m away from the cognitive radio antenna. We consider the transmitted randomly chosen carrier frequency by the horn antenna as the “unused” frequency channel in the cognitive radio environment, to which the reconfigurable receive antenna should tune its operating frequency. Step 2) Sensing the channel: The UWB sensing antenna of the cognitive antenna structure discussed in Section IV is connected to the spectrum analyzer.

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ceive the incoming signal. A diagnostic algorithm is also introduced within our workflow controller code (written in LABVIEW) to ensure that the reconfigurable antenna switches are appropriately commanded by the laser drivers. The reconfigurable antenna is connected to a network analyzer (NA). The signal acquired by the NA is analyzed by the controlling computer to determine the operating frequency band of the reconfigurable antenna. Step 6) Resume: The whole process (Steps 1)– 6)) is repeated every 3 s. The tuning of the reconfigurable antenna is in the microsecond range [21]. The entire analysis is done instantaneously in a way that the reconfigurable antenna immediately changes its operating frequency once the unused frequency band is determined. Fig. 13. Sensing antenna spectrum.

VII. CONCLUSION

Fig. 14. Switching decision tree that determines which switch will be activated.

The spectrum analyzer continuously measures the power-spectral density of the received signal from the UWB sensing antenna. Step 3) Spectrum decision: The signal acquired by the spectrum analyzer is analyzed by a controlling computer, which then determines the dominant frequency component within the signal. Fig. 13 shows an example of the power spectral density of the received signal from the UWB antenna, in the case when the TX 4.238 GHz. horn was transmitting at Step 4) Switch controller: Once the dominant frequency component in the received signal is determined, the controlling computer will then activate the appropriate laser-current driver. This is done by providing the appropriate inputs to the laser-current driver through a digital-to-analog converter. The switching decision tree is shown in Fig. 14. Step 5) Reconfigurable antenna status: Based on the defined combination of the activated switches (S1 OFF; S2 ON/S1 OFF; S2 OFF/S1 ON; S2 OFF), the narrowband reconfigurable antenna will then tune its operating frequency to the appropriate frequency band to re-

In this paper, a new antenna scheme for cognitive radio communications is presented. The cognitive antenna consists of a UWB antenna and a frequency-reconfigurable antenna incorporated into the same substrate. The reconfigurable antenna is based on photoconductive switches. A novel approach for switch activation is proposed, which allows the laser diodes to be incorporated directly within the antenna structure. This approach will enable easier integration of such antennas into commercial wireless devices. A prototype cognitive antenna was fabricated to test the suggested method. Good agreement was observed between the simulated and the measured RF performance of the antenna. A cognitive radio receive channel experiment was also conducted to demonstrate the applicability of the proposed scheme. For future work, we are investigating techniques to reduce the transition thresholds of the switching elements. Also, an approach is under investigation to use a field-programmable gate array controller to provide all of the decision logic required in the cognitive radio system workflow where the CW tones that are generated by the frequency sweeper are replaced by modulated signals of a given bandwidth. This will enable fabrication of a cognitive radio front-end module on a chip. REFERENCES [1] FCC Spectrum Policy Task Force, “Report of the spectrum efficiency working group, FCC” 2002. [2] J. Mitola, “Cognitive radio: An integrated agent architecture for software defined radio,” Ph.D. dissertation, Royal Inst. Technol. (KTH), Stockholm, Sweden, 2000. [3] Y. Tawk, M. Bkassiny, G. El-Howayek, S. K. Jayaweera, and C. G. Christodoulou, “Reconfigurable front-end antennas for cognitive radio applications,” Inst. Eng. Technol. Microw., Antennas Propag., vol. 5, no. 8, pp. 985–992, Jun. 2011. [4] Y. Tawk, J. Costantine, K. Avery, and C. G. Christodoulou, “Implementation of a cognitive radio front-end using rotatable controlled reconfigurable antennas,” IEEE Trans. Antennas Propag., vol. 59, no. 5, pp. 1773–1778, May 2011. [5] E. Ebrahimi and P. S. Hall, “A dual port wide-narrowband antenna for cognitive radio,” in Proc. 3rd Eur. Conf. Antennas Propag., Mar. 2009, pp. 809–812. [6] H. F. AbuTarboush, S. Khan, R. Nilavalan, H. S. Al-Raweshidy, and D. Budimir, “Reconfigurable wideband patch antenna for cognitive radio,” in Proc. Loughborough Antennas Propag. Conf., Nov. 2009, pp. 141–144.

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[7] G. T. Wu, R. L. Li, S. Y. Eom, S. S. Myoung, K. Lim, J. Laskar, S. I. Jeon, and M. M. Tentzeris, “Switchable quad-band antennas for cognitive radio base station applications,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1466–1476, May 2010. [8] Y. Tawk and C. G. Christodoulou, “A new reconfigurable antenna design for cognitive radio,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1378–1381, 2009. [9] Y. Tawk, J. Costantine, and C. G. Christodoulou, “A frequency reconfigurable rotatable microstrip antenna design,” in Proc. IEEE Int. Symp. Antennas Propag., Jul. 2010, pp. 1–4. [10] Y. Tawk and C. G. Christodoulou, “A cellular automata reconfigurable microstrip antenna design,” in Proc. IEEE Int. Symp. Antennas Propag., Jun. 2009, pp. 1–4. [11] C. J. Panagamuwa, A. Chauraya, and J. C. Vardaxoglou, “Frequency and beam reconfigurable antenna using photoconductive switches,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 449–454, Feb. 2006. [12] R. N. Lavalle and B. A. Lail, “Optically-controlled reconfigurable microstrip patch antenna,” in Proc. IEEE Int. Symp. Antennas Propag., Jul. 5–11, 2008, pp. 1–4. [13] L. N. Pringle, P. H. Harms, S. P. Blalock, G. N. Kiesel, E. J. Kuster, P. G. Friederich, R. J. Prado, J. M. Morris, and G. S. Smith, “A reconfigurable aperture antenna based on switched links between electrically small metallic patches,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1434–1445, Jun. 2004. [14] A. S. Nagra, O. Jerphagnon, P. Chavarkar, M. VanBlaricum, and R. A. York, “Bias free optical control of microwave circuits and antennas using improved optically variable capacitors,” in Proc. IEEE Int. Symp. Microw. Theory Tech., 2000, vol. 2, pp. 687–690. [15] R. L. Haupt and J. R. Flemish, “Reconfigurable and adaptive antennas using materials with variable conductivity,” in Proc. 2nd NASA Conf. Adaptive Hardware Syst., 2007, pp. 20–26. [16] Y. Tawk, A. R. Albrecht, S. Hemmady, G. Balakrishnan, and C. G. Christodoulou, “Optically pumped reconfigurable antenna system (OPRAS),” in Proc. IEEE Int. Symp. Antennas Propag., Jul. 2010, pp. 1–4. [17] D. E. Anagnostou, G. Zheng, M. T. Chryssomallis, J. C. Lyke, G. E. Ponchak, J. Papapolymerou, and C. G. Christodoulou, “Design, fabrication, and measurement of an RFMEMS-based self-similar reconfigurable antenna,” IEEE Trans. Antennas and Propag., vol. 54, no. 2, pp. 422–432, Feb. 2006. [18] M. I. Lai, T. Y. Wu, J. C. Hsieh, C. H. Wang, and S. K. Jeng, “Design of reconfigurable antennas based on an L-shaped slot and PIN diodes for compact wireless devices,” Inst. Eng. Technol. Microw., Antennas Propag., vol. 3, pp. 47–54, 2009. [19] Y. Tawk, A. R. Albrecht, S. Hemmady, G. Balakrishnan, and C. G. Christodoulou, “Optically pumped frequency reconfigurable antenna design,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 280–283, 2010. [20] S. M. Sze, Physics of Semiconductor Devices. New York: Wiley, 1981. [21] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics. Hoboken, NJ: Wiley, 2007. [22] C. A. Balanis, Advanced Engineering Electromagnetic. New York: Wiley, 1989. [23] A. Carton, C. G. Christodoulou, C. Dyck, and C. Nordquist, “Investigating the impact of carbon contamination on RF MEMS reliability,” in Proc. IEEE Antennas Propag. Int. Symp., Jul. 2006, pp. 193–196. [24] R. H. Caverly and G. Hiller, “Distortion properties of MESFET and PIN diode microwave switches,” in Proc. IEEE MTT-S Int. Microw. Symp., Jun. 1992, vol. 2, pp. 533–536. [25] Y. Kaneko, T Takenaka, T. S. Low, Y. Kondoh, D. E. Mars, D. Cook, and M. Saito, “Microwave switch: LAMPS (light-activated microwave photoconductive switch),” Electron. Lett., vol. 39, no. 12, pp. 917–919, Jun. 2003. [26] E. K. Kowalczuk, C. J. Panagamuwa, R. D. Seager, and J. Vardaxoglou, “Characterizing the linearity of an optically controlled photoconductive microwave switch,” in Proc. Loughborough Antennas Propag. Conf., Nov. 2010, pp. 597–600. [27] Y. Yashchyshyn, “Reconfigurable antennas by RF switches technology,” in Proc. 5th Int. Conf. Perspective Technol. Meth. MEMS Design, Apr. 2009, pp. 155–157. [28] G. M. Rebeiz, RF MEMS Theory, Design and Technology. Hoboken, NJ: Wiley, 2003. [29] Y. Tawk, S. Hemmady, G. Balakrishnan, and C. G. Christodoulou, “Measuring the transition switching speed of a semiconductor-based photoconductive switch using RF techniques,” in Proc. IEEE Int. Symp. Antennas Propag., Jul. 2011, pp. 972–975.

[30] Y. Tawk, J. Costantine, S. Hemmady, G. Balakrishnan, and C. G. Christodoulou, “Measuring the switching time of an optically pumped reconfigurable antenna system (OPRAS),” IEEE Trans. Antennas Propag., unpublished. [31] [Online]. Available: http://www.thorlabs.com/thorProduct.cfm?partnumber=L785P100

Youssef Tawk received the B.Sc. degree in computer and communications engineering from Notre Dame University, Louaize, Lebanon, in 2006, the M.Sc. degree in computer and communications engineering from the American University of Beirut, Beirut, Lebanon, in 2008, and the Ph.D. degree in electrical and computer engineering from the University of New Mexico, Albuquerque, in 2011. He has published several journal and conference papers. His research areas include reconfigurable antenna systems, cognitive radio, as well as RF electronic design and photonics. Dr. Tawk is the recipient of many awards during his studies.

Joseph Costantine received the B.Sc. degree in electrical, electronics, computer and communications engineering from the second branch of the Faculty of Engineering, Lebanese University, Beirut, in 2004, the M.Sc. degree in computer and communications engineering from the American University, Beirut, in 2006, and the Ph.D. degree in electrical and computer engineering from the University of New Mexico, Albuquerque, in 2009, where he also completed his Postdoctoral Fellowship in 2010. When he was with the American University, he was awarded a six-month research scholarship at Munich University of Technology (TUM), Munich, Germany, as part of the TEMPUS program. Currently, he is an Assistant Professor in the Electrical Engineering Department, California State University Fullerton. He has also published many research papers and is a co-author of an upcoming book on reconfigurable antennas. His research interests are in the areas of reconfigurable systems and antennas, antennas in wireless communications, electromagnetic fields, RF electronic design, and communication systems. Dr. Costantine received many awards during his studies and career.

Sameer Hemmady received the B.S. degree in electronics engineering from the University of Mumbai, Mumbai, India, in 2002, and the M.S. degree in telecommunications engineering and the Ph.D. degree in applied physics from the University of Maryland-College Park in 2004 and 2006, respectively. Currently, he is an Applied Physicist with more than seven years of experience in the planning, design, implementation, and technical assessment of advanced directed energy weaponized systems. He has been a Program Manager and Principal Investigator on several U.S. Department of Defense programs pertaining to nonlethal-directed energy weapons, counterelectronics, and radar technologies. His technical expertise includes intentional electromagnetic interference and compatibility, low-observable phase-array antennas, radar systems, Terahertz and optical beam transport systems, and lasers. He is also a Research Professor in the Applied Electromagnetics Group, Electrical and Computer Engineering Department, University of New Mexico, Albuquerque. He has authored one book on statistical electromagnetism, several journal papers, and conference proceedings covering applied research in wave propagation, statistical electromagnetism, electromagnetic interference/electromagnetic compatibility, and quantum electronics. He holds a U.S. patent on wave imaging and a pending patent on reconfigurable low-observable stealth antennas.

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Ganesh Balakrishnan received the B.E. degree in electronics and communications engineering from the University of Madras, Madras, India, in 2000, the M.S. degree in engineering with a specialization in communication engineering from the University of Toledo, Toledo OH, in 2001, and the Ph.D. degree in optical sciences from the University of New Mexico (UNM), Albuquerque, in 2006. Currently, he is an Assistant Professor at the Center for High Technology Materials, UNM. His research interests include high-power vertical external-cavity surface-emitting laser development using quantum-dot-based and antimonide quantum-well-based active regions. He has co-authored many journal articles and conference presentations.

Keith Avery (M’04) received the B.S. degree from DeVry Institute of Technology, in 1983. Currently, he is the Program Lead for the Integrated Microsystems program at the Air Force Research Laboratory (AFRL), focusing on advanced packaging and optoelectronics for space. For the first 12 years of his career, he was with the commercial sector, designing digital and analog circuits for commercial, industrial, and telephony applications. Prior to joining AFRL he was a government contractor performing design activities for space experiments, advanced packaging techniques, and radiation effects on microelectronics. During his career, he has increased his level of responsibility for design activities and program management. He has authored or co-authored numerous papers on designs for space and radiation effects. Mr. Avery is a member of NPSS and AIAA.

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Christos G. Christodoulou (F’02) received the Ph.D. degree in electrical engineering from North Carolina State University, Raleigh, in 1985. He was a faculty member with the University of Central Florida, Orlando, from 1985 to 1998. In 1999, he joined the faculty of the Electrical and Computer Engineering Department, University of New Mexico, Albuquerque, where he was Chair of the Department from 1999 to 2005. Currently, he is the Director of the Aerospace Institute, University of New Mexico (UNM), and the Chief Research Officer for the Configurable Space Microsystems Innovations and Applications Center (COSMIAC), UNM. He was appointed IEEE AP-S Distinguished Lecturer from 2007 to 2010, and elected as the President for the Albuquerque IEEE Section in 2008. He was an Associate Editor for the IEEE TRANSACTION ON ANTENNAS AND PROPAGATION for six years as a Guest Editor for a special issue on “Applications of Neural Networks in Electromagnetics” in the Applied Computational Electromagnetics Society (ACES) journal, and as Co-Editor of the IEEE Antennas and Propagation Special issue on “Synthesis and Optimization Techniques in Electromagnetics and Antenna System Design” (2007). He has published about 400 papers in journals and conferences, has 14 book chapters, and has co-authored four books. His research interests are modeling of electromagnetic systems, reconfigurable antenna systems, cognitive radio, and smart RF/photonics. Dr. Christodoulou is a member of Commission B of URSI. He was the General Chair of the IEEE Antennas and Propagation Society/URSI 1999 Symposium, Orlando, FL, and the Co-Technical Chair for the IEEE Antennas and Propagation Society/URSI 2006 Symposium, Albuquerque. He is the recipient of the 2010 IEEE John Krauss Antenna Award for his work on reconfigurable fractal antennas using microelectromechanical switches, the Lawton-Ellis Award, and the Gardner Zemke Professorship at the University of New Mexico.

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Evaluation of a Statistical Model for the Characterization of Multipath Affecting Mobile Terminal GPS Antennas in Sub-Urban Areas M. Ur Rehman, Member, IEEE, X. Chen, Senior Member, IEEE, C. G. Parini, Member, IEEE, and Z. Ying, Senior Member, IEEE

Abstract—This paper describes and validates a technique to characterize the environmental effects on mobile terminal GPS antennas using statistical model. This method requires the knowledge of 3-D free space antenna gain patterns and average angular distribution of incident power in the environment. The power distribution must be known in both elevation and azimuth and separately for parallel and perpendicular polarizations. The antenna performance is assessed in terms of GPS Mean Effective Gain and GPS Coverage Efficiency . Angle of Arrival distributions of incident GPS radio waves arriving at the mobile terminal are assumed to be randomly uniform in both the azimuth and elevation planes. It effectively replicates the open field (sub-urban) working conditions for the mobile terminal GPS antennas. The method could be adapted to an urban environment by introducing the information of distributions. A lengthy open field measurement campaign based on received signal-to-noise ratio (SNR) and mean number of tracked GPS satellites is carried out to validate the statistical model. Index Terms—Environment effects, GPS antennas, multipath interference, wireless networks.

I. INTRODUCTION

Fig. 1. GPS environment and multipath signals.

T

HE introduction of built-in GPS in portable Personal Navigation Devices (PNDs) especially the mobile phones has revolutionized the navigation industry. The ever growing demand of availability of the navigation facilities in these devices has made the GPS an essential part of the modern Wireless Personal Area Network (WPAN) and Wireless Body Area Network (WBAN) applications. The portable mobile terminals are affected by multipath due to reflections, diffractions and scattering of the incident radio waves. It depends on the structure of surrounding environment, as shown in Fig. 1. This phenomena has been studied well over the time. Analysis of this complex and random in nature problem is complicated. Statistical modelling is a powerful tool that offers simple and flexible solutions to such problems.

Manuscript received January 18, 2011; revised June 28, 2011; accepted September 19, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. M. Rehman, X. Chen, and C. G. Parini are with the School of Electronic Engineering and Computer Science, Queen Mary University of London, London E1 4NS, U.K. (e-mail: [email protected]). Z. Ying is with Sony Ericsson Mobile Communications AB, Lund 22188, Sweden. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173134

A number of researchers has used this approach to study and characterize the multipath effects on mobile terminals and GPS antennas [1]–[10]. Axelrad et al. have used signal-to-noise ratios (SNR) to predict and remove multipath errors [7]. Wu et al. have used siderial filtering of time-series range residual variations based on least-square model to mitigate the multipath [8]. It requires a comprehensive data set containing multipath delay samples from yesterday and the preceding day. Spangenberg et al. have modelled the multipath as variance change in received signals in LOS conditions and as mean value jumps in NLOS conditions using extended Kalman filter [9]. Hannah has proposed a Parabolic Equation based propagation model for GPS multipath employing the concept of coupled polarization reflection coefficient [10]. However, these studies usually need large measurement data sets and mainly deal with quantification of multipath effects on stand-alone, static GPS receivers with antennas designed according to the theoretical guidelines. Ideally, the GPS antennas should have good Right Hand Circular Polarization (RHCP) with a uniform radiation pattern over entire upper hemisphere to receive the incoming GPS signal efficiently. A good rejection of Left Hand Circular Polarization (LHCP) is also required to avoid multipath [11]–[13]. However,

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these requirements are difficult to fulfil in portable devices, especially the mobile terminals that are required to allow maximum mobility of the user and flexibility of use with multiple functions such as Wi-Fi, Bluetooth, FM radio, digital camera, mobile TV and GPS [14], [15]. In the common scenarios of cluttered environments including indoors and city streets, Line-of-Sight (LOS) GPS signals arriving at the mobile terminal is weak while the reflected signals may have arbitrary polarizations. Moreover, the mobile phones are hardly used in a fixed position and the “up” direction of the antenna changes depending on the used orientation. Furthermore, the antenna suffers from electromagnetic absorptions and shielding of clear sky view in hand-held positions. Establishing a quick GPS link with good satellite lock is therefore, a difficult task in such devices. Use of wide-beam linearly polarized GPS antennas could help to address uncertainty of antenna orientation, blockage of LOS signal and clear sky view, and losses due to user’s body. Hence, linearly polarized antennas are a preferred choice for mobile terminal GPS as they give better performance compared to the conventional RHCP antennas [16]–[18]. Moreover, it enables the use of the multipath signal constructively in order to establish a quick GPS link. Once a GPS lock is achieved, positioning errors can be estimated and removed using software approaches. It necessitates the characterization of mobile terminal GPS antennas in multipath environment. Mean Effective Gain (MEG) has been used as an important performance metric for mobile handsets in multipath land mobile propagation environments [19], [20]. It statistically describes the impact of the antenna on the link budget considering Angle of Arrival and polarization of the incident waves and antenna gain characteristics [1]–[4]. Effects of the presence of human body can also be evaluated using this technique [5], [6]. The applicability of this idea to the portable GPS antennas in conjunction with a new parameter of antenna Coverage Efficiency has been proposed and developed by Ur Rehman et al. [21]–[23]. This paper investigates the usefulness of this method presenting a thorough analysis of the mobile terminal GPS antennas. The content of this paper is organized as follows: In Section II, the statistical model for the GPS multipath environment is introduced. In Section III, the open field measurement procedure for the GPS antennas is detailed. In Section IV, performance of various GPS mobile terminal antennas in the multipath environment is evaluated and results of the proposed statistical model and open field measurements are compared. Section V discusses the effects of change in antenna orientation on its performance in open field multipath environment. Finally, conclusions are drawn in Section VI. II. STATISTICAL MODELLING OF GPS MULTIPATH ENVIRONMENT Currently, the performance assessment of a mobile terminal GPS antenna is mostly done by field tests. However, it has drawback of longer procedures where weather, temperature and location hazards make it hard to control the test environment. It results in lack of accuracy due to poor repeatability and efficiency. The statistical models representing the real multipath

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Fig. 2. Spherical coordinates system and representation of a hypothetical incident wave distribution model.

scenarios provide an excellent alternative to the field tests, predicting the antenna performance while avoiding the shortcomings of the field tests. A statistical method is developed to characterize the environmental factors on the performance of the GPS antennas, introducing the parameters of GPS Mean Effective Gain and Coverage Efficiency . The MEG equation derived by Taga [1] is re-formulated to suit the GPS environment with RHCP incoming waves and environmental reflections. An overview of the expected performance of the GPS antenna in the multipath environment can be achieved using this method having the knowledge of 3-D free space antenna gain patterns and average angular distribution of incident power in the environment. A. GPS Mean Effective Gain The MEG is the average gain of the antenna performance in a multipath radio environment. The MEG of an antenna in a mobile terminal is defined as [1] (1) For spherical coordinates (Fig. 2), as [24]

can be expressed

(2) and are and components of the where antenna power gain pattern respectively, and indicate the and components of angular density functions of the incoming waves respectively. is the mean power that would be received by an isotropic antenna in polarization while is the mean power received by an isotropic antenna in polarization. The total mean incident power arriving

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at the antenna would be the summation of the mean powers in the two polarizations. The incident wave in the GPS mobile environment can be split into two components, perpendicular polarized component and parallel polarized component. Therefore, and components respectively correspond to the perpendicular and parallel polarized components. Re-formulating (2) to suit the GPS environment results as follows:

(3) and are the mean received powers in the perpenNow, dicular and parallel polarizations with respect to the ground plane while and represent the perpendicular and parallel components of the angular density functions of the incoming waves respectively, as shown in Fig. 1. arriving at the mobile GPS terminal is then (4)

Fig. 3. Multipath environment model around the mobile terminal GPS receiver antenna [22].

direction of arrival of the GPS waves, the angular density function is assumed to be uniform in the incident region. In the reflection region, it is no longer uniform and reduced by a factor governed by the reflection coefficients. Also, the favourable use of the multipath signal in the mobile terminal GPS antennas for quick link establishment leads to use sum of the received powers in the incident and reflected regions. In accordance with the preceding assumptions, the statistical model for the GPS antenna in an open field multipath environment with ground reflections is proposed as follows [22], [23]:

The ratio between the mean powers received in the two polarizations is called XPR (Cross Polarization Ratio) and described as

(7) (8)

(5) Using (3)–(5), the MEG expression for the GPS antenna can be formulated as [22], [23]

depends upon the reflection coefficients for the perpendicular and parallel components that varies with angle of incidence as [25]

(9) (6) Since, XPR governs the polarization of the incoming wave in this model; the circular polarized nature of the incoming GPS satellite signal is accumulated by making dB. It employs the fact that simultaneous transmission of two linearly polarized waves that have a phase difference of (radian) results in the generation of a circularly polarized wave. B. GPS Angle of Arrival Distribution The gives statistical definition of direction of arrival of the incident radio waves, arriving at the mobile terminal GPS antenna. Both the azimuth and elevation planes should be considered separately to replicate the multipath environment. The incident radio waves are reflected, diffracted and scattered from the objects located in the surroundings of the receiving antenna. Since, these objects vary in height and shape, the direction of arrival of the incident waves is random. This random occurrence is dealt with a uniform angular density function in azimuth, similar to the case of the land mobile environment [1]–[3]. For the GPS environment, reflections from the ground (Fig. 3) should also be considered in the elevation plane. The elevation plane is therefore, divided into incident and reflection regions. Due to lack of available measurement data for

(10)

In this study, the open field ground is considered to be of a semi-grassy semi-concrete type with a relative permittivity of 4.5 [26], [27]. It makes the model a replication of open field working environment. Having the knowledge of , the model could be easily adapted to an urban environment. C. GPS Coverage Efficiency Coverage Efficiency of the receiving GPS antenna is another important parameter. It defines the capability of the antenna to receive the signals coming directly from the satellites. The GPS antenna can receive signals from all directions that lie within its coverage area. However, the performance of a GPS antenna is currently characterized by its ability to receive the signals for elevation angles higher and lower than 10 (from the horizon) [12]. Defining the antenna coverage based on this criteria fails to describe its overall coverage in the upper hemisphere. The concept of the Coverage Efficiency overcomes this drawback by giving information about the antenna coverage in the whole upper hemisphere.

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been selected to investigate the worst case scenarios. This selection is validated in the following sections by a close agreement between values calculated using the proposed model and observed in the open field tests. III. OPEN FIELD TEST PROCEDURE A detailed open field test procedure is adopted to verify and validate the model’s predictions. A. Measurement of GPS Mean Effective Gain

Fig. 4. Illustration of calculations based on RHCP radiation pattern of a GPS antenna with cross-hatched regions indicating coverage area (where signal dBi). is above

The Coverage Efficiency of the antenna-under-test is calculated as the ratio of the solid angles subtended by its coverage area to the total area [22], [23]

For , the received signal power should be known. The signal-to-noise ratio (SNR) is a measure that is used to evaluate the performance of the designed GPS antennas. It indicates how strongly the satellite’s radio signal is being received. It is computed as a ratio of the signal power to the noise power corrupting the signal dB

(13)

Also [29] dB

(11)

(14)

These calculations are based on RHCP gain pattern of the antenna-under-test to suit the RHCP incoming radio waves. The coverage area depends on a carefully calculated received signal threshold level. Signals below this level are considered too weak to make an impact and hence, wasted. The maximum coverage that can be obtained by a reference GPS antenna is termed as the total area. It is considered to be the half hemispherical solid angle of for an isotropic antenna. Fig. 4 illustrates calculations. The box encloses the incident region (upper hemisphere) with the horizon at 0 and the zenith at 90 . Cross-hatched part indicates the coverage area. An appropriate threshold level is worked out using GPS link budget for L1 (1575.42 MHz) frequency band [28]

Here, NF is noise figure representing the noise generated within the GPS receiver while is the temperature dependent source resistance noise power. At a temperature of 25 C and a system bandwidth of 1 Hz [31]

(12)

It implies that the signal strength delivered to the GPS receiver is linearly dependent on the SNR if NF is constant in (16). Hence, Mean Received Power of a GPS antenna in (1) can be calculated using the mean SNR level for that antenna. Mean SNR level of the reference antenna (typically dipole antenna) gives the total Mean Incident Power . Finally, is calculated by taking the ratio of the mean SNR levels of the two antennas.

is the receiver sensitivity while is transmitted where output power including the transmitter losses. and indicate gain of the transmitting and receiving antennas respectively, while is the free space loss and indicates the miscellaneous losses including polarization mismatch and atmospheric losses. The link budget is calculated as follows: • Satellite km • dBi [28], [29] • dBW (corresponds to 50 W typical) [28] However, in practical scenarios could be reduced to 14.3 dBW due to impedance mismatches and circuit losses [29]. dB dB [28], [29] dBW (corresponds to dBm typical) Different GPS vendor specifications indicate that could be as high as dBW [30]. Submitting these values in (12) gives dBi. It shows that a minimum threshold level of dBi is required to calculate the Coverage Efficiency of the GPS antenna. However, a threshold level of dBi has

(15) where is the Boltzmann’s constant and is temperature in Kelvins. Now, putting these values in (13), the following expression is obtained: dB

dB

(16)

B. Measurement of GPS Coverage Efficiency The Coverage Efficiency describes how well the antenna can view the sky and receive the satellite signal. In the field, this quality corresponds to the number of tracked GPS satellites. It is obtained by taking the ratio of the mean value of the tracked satellites (representing the coverage area of the antenna-undertest in (11)) and the maximum number of the tracked satellites observed during the whole measurement process (representing the total area in (11)) (17)

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D. Tested Generic GPS Antennas

Fig. 5. Open field test set up for measurement of antennas.

and

for GPS

C. Measurement setup DG-100 GPS receiver from GlobalSat Technology is used in the measurements. This receiver has an embedded SIRF Star-III chipset module with 20 channels (can track 20 satellites). The sensitivity of the receiver is dBm. The data of number of tracked satellites and received SNR by the antenna-under-test is collected using satellite status chart. The field test set up is illustrated in Fig. 5. The antennas are connected to the GPS receiver via a MMCX-to-SMA jumper cable. The designed antennas are tested for the GPS signal reception in outdoor open environment in both horizontal and vertical orientations with respect to the ground. In this sub-urban environment, height of the buildings in the vicinity of the test point ranges from 10– m and are located at a distance of 30– m. The GPS receiver and the antennas are placed at a height of 1 m from the ground. The antennas are rotated horizontally and eight readings of satellite status chart are recorded for the following angles (18) It effectively provides the average reception of the signal in the environment. The information of the eight best values of the received SNR are used to calculate the mean SNR level for each antenna. Hence (19) is then calculated by dividing of the antenna-under-test to of the reference antenna. is calculated using mean number of the tracked satellites and dividing by 12, which appeared to be the maximum number of the tracked satellites during the measurements.

Three types of generic GPS antennas; dipole, microstrip patch and PIFA; are analysed in the open field multipath environment for this study. 1) Dipole: The use of a standard simple antenna with known characteristics is needed to bench-mark the model. A half wavelength dipole antenna working at 1575.42 MHz is chosen for this purpose due to its simplicity and wide usage as a standard antenna. The fabricated prototype of the antenna is shown in Fig. 6(a). The antenna performs well in L1 frequency band with dB bandwidth of 153 MHz as depicted by the measured S11 response in Fig. 6(b). The 3-D gain patterns of the antennas are measured in an anechoic chamber using Satimo’s Stargate 64 measurement system. The patterns for the perpendicular and parallel polarizations of the antenna in horizontal and vertical orientations are illustrated in Figs. 6(c) and (d). 2) Microstrip Patch Antenna: A circular polarized (CP) truncated corner microstrip patch antenna is used in this study. Fig. 7(a) shows the geometry of the fabricated antenna. The free space S11 response of the antenna in Fig. 7(b) shows good impedance matching in L1 band with centre frequency at 1578 MHz. The measured 3-D radiation patterns for perpendicular and parallel polarization of the antenna for both the horizontal and vertical orientations are illustrated in Figs. 7(c) and (d). 3) PIFA: The planar inverted F antenna (PIFA) is a popular choice for a wide range of GPS applications. The antenna used in this study is shown in Fig. 8(a). The measured S11 curve of the antenna shown in Fig. 8(b) indicates that the PIFA operates well in the L1 band having dB bandwidth of 45 MHz. Figs. 8(c) and (d) give an account of the measured 3-D radiation patterns for perpendicular and parallel polarizations of the antenna in both horizontal and vertical orientations. IV. EVALUATION OF GPS ANTENNA PERFORMANCE MULTIPATH ENVIRONMENT

IN

Two approaches have been adopted to establish accuracy and efficiency of the proposed statistical model to predict the GPS antenna performance in the multipath environment as described in the following sections. A. Comparison Based on Simulated and Measured 3-D Radiation Patterns First, statistical calculation results of the model are compared for two different input methods. The 3-D gain patterns of the three antennas, obtained through the simulations and the anechoic chamber (Satimo Stargate 64) measurements, are used for the comparison. The results summarized in Table I show good agreement between the calculated results of and using the simulated and the measured 3-D gain patterns. A maximum difference of 0.6 dB has been observed in values in the actual reflection environment while 4% in values. This difference is assigned to antenna fabrication errors. Theoretically, the performance of a GPS antenna in terms of its in actual reflection environment (ARE) should lie between the two extreme ideal environments i.e., total reflection environment (TRE) and no reflection environment (NRE)

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Fig. 6. Geometrical structure of dipole antenna with measured S11 and 3-D power gain patterns for perpendicular and parallel polarizations in horizontal and vertical orientations. (a) Dipole antenna geometry (all lengths are in mm), (b) Measured S11, (c) Gain patterns for horizontal orientation, (d) Gain patterns for vertical orientation.

Fig. 7. Geometrical structure of truncated corner microstrip patch CP antenna with measured S11 and 3-D power gain patterns for perpendicular and parallel polarizations in horizontal and vertical orientations. (a) CP Patch antenna geometry (all lengths are in mm), (b) Measured S11, (c) Gain patterns for horizontal orientation, (d) Gain patterns for vertical orientation.

as part of the incident wave is reflected back, depending upon the reflection coefficients for the ground permittivity of 4.5 [26], [27]. The results in Table I indicate that the proposed model works well exhibiting the expected theoretical behaviour. B. Comparison Based on Measured 3-D Radiation Patterns and Actual Field Tests The comparison of the model’s calculations to open field measurements serves as a crucial step in the validation process. The open field tests have been performed both at the Sony Ericsson Communications, Sweden and QMUL, London. The horizontal dipole antenna is used as a reference antenna in this study. Both the calculated (obtained through the proposed

model using 3-D measured gain patterns) and measured (mean SNR level observed in the actual field test) values of are normalized to the corresponding values for the horizontal dipole antenna and described in dBd. Hence, dB (in the actual reflection environment given in Table I) and 40.2 dB (in the open field test) is being used as the reference level for the calculated and measured results, respectively. The antennas are placed horizontally with respect to the ground. The results are summarized in Table II. In these assessments, the repeatability of the measurement procedure must also be known. Therefore, in these as well as the following investigations, and are calculated performing three sets of measurements for each antenna and

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Fig. 8. Geometrical structure of PIFA with measured S11 and 3-D power gain patterns for perpendicular and parallel polarizations in horizontal and vertical orientations. (a) PIFA geometry (all lengths are in mm), (b) Measured S11, (c) Gain patterns for horizontal orientation, (d) Gain patterns for vertical orientation.

TABLE I AND OF TESTED GPS ANTENNAS IN DIFFERENT REFLECTION ENVIRONMENTS USING SIMULATED AND MEASURED COMPARISON OF CALCULATED 3-D POWER GAIN PATTERNS FOR VALIDATION OF PROPOSED GPS MULTIPATH MODEL (TRE TOTAL REFLECTION ENVIRONMENT, NRE NO REFLECTION ENVIRONMENT, ARE ACTUAL REFLECTION ENVIRONMENT)

TABLE II COMPARISON OF CALCULATED

AND

OF TESTED GPS ANTENNAS IN HORIZONTAL AND VERTICAL GAIN PATTERNS TO THE ACTUAL FIELD TEST MEASUREMENTS

mean values are reported. The standard deviation of these three measurements (averaged over various tested scenarios) is 0.6 dB and 5% for and , respectively. The field test results show a good agreement with the model’s predictions for and values for the three mobile ter-

ORIENTATION USING MEASURED 3-D POWER

minal GPS antennas. Fig. 9 indicates that a similar performance ranking of the three antennas has been achieved both in the calculations and measurements. It confirms that the model can successfully translate and predict the working of the GPS antennas in the multipath environment. A maximum difference of 0.2 dB

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Fig. 9. Comparison of calculated and measured values of and showing performance ranking of the GPS dipole, CP patch and PIFA antennas taking horizontal dipole as a reference in horizontal orientation. (a) (0 dBd), (b) .

Fig. 10. Comparison of calculated and measured values of and showing performance ranking of the GPS dipole, CP patch and PIFA antennas taking horizontal dipole as a reference (0 in vertical orientation. (a) dBd), (b) .

is noted for values and 2% for values between the model’s calculations and open field measurements. These differences are well below the accepted levels reported in literature [1], [2], [5]. They are mainly attributed to random factors arising from atmospheric errors and weather conditions. The results also indicate that the two parameters of and do no rely tightly on each other. An antenna with good may exhibit poor and vice versa, for example in the case of the PIFA antenna. However, the multipath performance of the GPS antennas could only be characterized by a combined consideration of the two parameters. incorporates the whole environment taking into account both the direct as well as multipath signals, especially the ground reflections. On the other hand, only considers the direct link. An antenna having low values of and would be unable to establish a quick GPS link as the direct signal is weak while multipath impact is less significant. The performance of the antenna with high values of should be analysed further in terms of its . A high shows that the direct signal is stronger than the multipath signal enabling the antenna to achieve an overall good performance with quick GPS link and low multipath errors. On contrary, low shows a weaker direct signal with greater impact of the multipath signal. An antenna exhibiting such performance could build satellite link quickly but with high errors. It is evident from the results that the CP patch belongs to the first category while the PIFA lies in the second category. However, an optimal performance could only be achieved with an antenna exhibiting good and .

V. PERFORMANCE DEPENDENCE ON ANTENNA ORIENTATION The antenna orientation plays a vital role in multipath wireless communications. Varying orientation changes the antenna main lobe direction inflicting link losses. The mobile terminals operate in a dynamic environment with ever-changing orientation of the antennas depending on the user’s holding position. The effects of these changes on the mobile terminal GPS antennas are characterized in this section. The antennas are placed in vertical orientation (with respect to the ground) and performance is studied in comparison to the horizontal orientations. The calculated and measured results using the proposed statistical model and open field test are presented in Table II. Fig. 10 shows the comparison of the antenna rankings in terms of their and based on the two methods. It is further established that the model delivers precise results with a close agreement to the field test observations. A maximum relative difference of 0.4 dB in and 4% in has been noted. These results also show that change in the antenna orientation has a profound effect on the performance of the GPS antennas. Comparing the vertically oriented antennas to those in horizontal orientation (Fig. 11), the horizontal configurations show an overall better performance in terms of . These variations in are associated to the antenna gain patterns. incorporates overall changes in the antenna gain pattern for both polarizations and its response to the multipath environment in terms of (that also include ground reflections). Hence, it describes that the antenna gain and polarization responds better to the nature of the incident plane waves when

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Fig. 11. Performance comparison of the GPS antennas with effects of change and . (a) taking in orientation in terms of calculated horizontal dipole as a reference (0 dBd), (b) .

placed horizontally. The higher values of the antenna gain, especially in the upper hemispherical space, in both the perpendicular and parallel polarizations (as presented in the gain pattern figures) are a key contributor. The majority of the tested antennas also exhibit better in the horizontal orientation as more open sky view is available. It increases the number of the tracked satellites and hence, level of the received signal in the incident region . As a result, the wasted signal is reduced improving the overall . The change in antenna with changing orientation (with respect to the ground) is attributed to the sensitivity of the antenna to receive the incoming GPS signal. Since, the incoming GPS signal is RHCP, antenna RHCP gain patterns are evaluated in order to study the relation between and RHCP gain with change in antenna orientation. The RHCP gain patterns of the tested antennas are measured using Satimo’s Stargate 64 measurement system. Fig. 12 presents the comparison of measured RHCP gains in the incident region for the tested antennas in both the horizontal and vertical orientations. In these figures, cross-hatched area indicates the useful angles having a gain level above dBi, contributing to . The comparison of the presented plots clearly indicates that of the antenna depends upon the strength of the RHCP gain in the incident region. For example, in the case of the CP patch antenna, the vertical orientation exhibits much lower of 72% as compared to 99% for the horizontal orientation. It describes that the antenna has more clear sky view and a larger coverage area while placed horizontally. The RHCP gain patterns also

support this theory. Fig. 12(b.i) shows that all of the incident region is above the required threshold level of dBi when the antenna is working in the horizontal orientation. A decreased for the vertically orientated antenna is caused by comparatively less area meeting this threshold. Presence of the RHCP gain levels lower than dBi (non-hatched area) in Fig. 12(b.ii), particularly in the angles and , gives rise to the wasted signals resulting in a lower . In most of the tested cases, value increases with increase in . However, in case of PIFA, an opposite trend is observed for the vertical orientation. The vertical PIFA has shown a decreased value due to reduction in overall gain levels (Fig. 8). The non-hatched regions in Fig. 12(c), show the angles that are not covered by the antenna due to gain levels below dBi. This uncovered region is larger for the horizontal orientation (Fig. 12(c.i)) as compared to the vertical orientation (Fig. 12(c.ii)). Therefore, the PIFA exhibits an improved in vertical orientation. The results show that and is an efficient measure to characterize the antenna performance in the multipath environment. It simplifies the practical evaluation of the antenna performance as it is based on the antenna gain pattern measurements in anechoic chamber. It describes the antenna performance incorporating both the polarization properties of the antenna-under-test and the directional properties of the radio environment. For example, from the perspective of antenna efficiency and maximum gain, the CP patch should out-perform the dipole and PIFA for the GPS operation. However, and results show that these methods are not enough to describe the performance of the antennas in practical scenarios. It has been observed that the CP patch is more vulnerable to change in the orientation as and varies significantly. The vertical CP patch has lost 1.6 dB of its and 27% of its coverage as compared to the horizontal CP patch. It indicates that the multipath signal has little impact on the CP patch antenna and it relies more on the direct signal lowering its capability to establish a quick satellite link in arbitrary orientations. On the other hand, of the vertical PIFA has reduced by 0.8 dB but its has improved by 3%. It depicts that the direct signal is playing a greater part in the antenna performance. The ability of the PIFA to make good use of the multipath signal in the horizontal orientation and the direct signal in the vertical orientation enables it to establish a faster GPS link regardless of the orientation (at the expense of comparatively higher multipath error in horizontal orientation). Hence, a combined consideration of antenna gain, polarization, the radio environment (i.e., the distributions) and the orientation in terms of and depicts that PIFA could deliver better performance in the multipath environment as compared to the CP patch antenna. VI. CONCLUSION A statistical model to evaluate the performance of the mobile terminal GPS antennas in a multipath environment is presented with two novel concepts of the GPS Mean Effective Gain and GPS Coverage Efficiency . The model is implemented and verified through extensive numerical studies

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Fig. 12. Measured RHCP gain patterns in the incident region for dipole, CP patch and PIFA GPS antennas in horizontal and vertical orientations (cross-hatched dBi). (a) Dipole (i) Horizontal orientation (ii) Vertical orientation, (b) CP Patch (i) Horizontal orientation (ii) Vertical regions indicate where signal is above orientation, (c) PIFA (i) Horizontal orientation (ii) Vertical orientation.

and its precision is established through experimental procedures using a dipole, CP patch and PIFA. The model efficiently predicts the GPS antenna performance in an open field with multipath. Therefore, it provides a means to analyse antenna operation in actual working scenarios without doing open field measurements. Role of the antenna orientation in the multipath GPS operation is also investigated. The horizontal orientations has appeared to be more efficient due to availability of a wider clear sky view and higher gain values in the upper hemisphere. However, it is not a generalized trend and varies from antenna to antenna. It is also observed that the linear polarized GPS antennas could perform better establishing quick GPS link in open field working conditions as compared to the circularly polarized antennas. However, further investigations are needed to support the argument. Besides its precise predictive capabilities, the proposed statistical model is only suitable for open field (sub-urban) GPS environment due to the assumption of uniform in azimuth and elevation planes. For urban area operation with greater number of reflecting objects, more practical should be considered. Therefore, inclusion of representing greater probability of arrival of the GPS signal near the zenith as compared to the horizon angles, for example Gaussian and elliptical, is a future extension of this study. Furthermore, effects of human presence in the vicinity of the mobile terminal GPS antennas would also be investigated.

ACKNOWLEDGMENT The authors would like to thank Sony Ericsson Mobile Communications for their support of this study. REFERENCES [1] T. Taga, “Analysis for mean effective gain of mobile antennas in land mobile radio environments,” IEEE Trans. Veh. Tech., vol. 39, no. 2, pp. 117–131, May 1990. [2] K. Kalliola, K. Sulonen, H. Laitinen, O. Kivekas, J. Krogerus, and P. Vainikainen, “Angular power distribution and mean effective gain of mobile antenna in different propagation environments,” IEEE Trans. Veh. Tech., vol. 51, no. 5, pp. 823–838, Sep. 2002. [3] P. Carro and J. de Mingo, “Mean effective gain of compact WLAN genetic printed dipole antennas in indoor-outdoor scenarios,” in Proc. Int. Conf. Personal Wireless Commun. (PWC), Sep. 2006, pp. 275–283. [4] A. Ando, T. Taga, A. Kondo, K. Kagoshima, and S. Kubota, “Mean effective gain of mobile antennas in line-of-sight street microcells with low base station antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3552–3565, Nov. 2008. [5] J. Nielsen and G. Pedersen, “Mobile handset performance evaluation using radiation pattern measurements,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 2154–2165, Jul. 2006. [6] J. Krogerus, C. Ichelun, and P. Vainikainen, “Dependence of mean effective gain of mobile terminal antennas on side of head,” in Proc. Eur. Conf. Wireless Technology (ECWT), Oct. 2005, pp. 467–470. [7] P. Axelrad, J. Comp, and P. MacDoran, “SNR-based multipath error correction for GPS differential phase,” IEEE Trans. Aerosp. Electron. Syst., vol. 32, no. 2, pp. 650–660, Apr. 1996. [8] J. Wu and C. Hsieh, “Statistical modeling for the mitigation of GPS multipath delays from day-to-day range measurements,” J. Geodesy, vol. 84, no. 4, pp. 223–232, 2006. [9] M. Spangenberg, J. Tourneret, V. Calmettes, and G. Duchteau, “Detection of variance changes and mean value jumps in measurement noise for multipath mitigation in urban navigation,” in Proc. Asilomar Conf. Signals, Syst. Comput., Oct. 2008, pp. 1193–1197.

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[10] B. Hannah, “Modelling and Simulation of GPS Multipath Propagation,” Ph.D. dissertation, Queensland University of Technology, Australia, Mar. 2001. [11] R. Bancroft, Microstrip and Printed Antenna Design, 2nd ed. New York: SciTech Publishing, Inc., 2009. [12] G. Moernaut and D. Orban, “GNSS antennas,” GPS World, Feb. 2009. [13] L. Boccia, G. Amendola, and G. Di Massa, “A shorted elliptical patch antenna for GPS applications,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 6–8, 2003. [14] G. Miller, “Adding GPS applications to an existing design,” RF Design, pp. 50–57, Mar. 1998. [15] R. Langley, “A primer on GPS antennas,” GPS World, pp. 50–55, Jul. 1998. [16] V. Pathak, S. Thornwall, M. Krier, S. Rowson, G. Poilasne, and L. Desclos, “Mobile handset system performance comparison of a linearly polarized GPS internal antenna with a circularly polarized antenna,” in Proc. IEEE Antennas Propag. Soc. Int. Symp. (APS), Jun. 2003, vol. 3, pp. 666–669. [17] S. Kingsley, “GPS antenna design for mobile phones,” Electronics Weekly, vol. 11, Apr. 2007. [18] T. Haddrell, N. Ricquier, and M. Phocas, “Mobile-phone GPS antennas: Can they be better?,” GPS World, Feb. 2010. [19] K. Fujimoto and J. R. James, Mobile Antenna Systems Handbook, 2nd ed. Norwood, MA: Artech House Publishers, 2001. [20] Z. N. Chen, Antennas for Portable Devices. New York: Wiley, 2007. [21] M. Ur Rehman, Y. Gao, X. Chen, C. Parini, and Z. Ying, “Analysis of GPS antenna performance in amultipath environment,” in Proc. Antenna Propag. Soc. Int. Symp. (AP-S), Jul. 2008, pp. 1–4. [22] M. Ur Rehman, Y. Gao, X. Chen, C. Parini, and Z. Ying, “Environment effects and system performance characterization of GPS antennas for mobile terminals,” IET Electron. Lett., vol. 45, no. 5, pp. 243–245, Feb. 2009. [23] M. Ur Rehman, Y. Gao, X. Chen, C. Parini, and Z. Ying, “Characterisation of system performance of GPS antennas in mobile terminals including environmental effects,” in Proc. Eur. Conf. Antennas Propag. (EuCap), Mar. 2009, pp. 1832–1836. [24] W. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [25] D. Cheng, Field and Wave Electromagnetics, 2nd ed. Reading, MA: Addison Wesley, 1989. [26] J. Jemai, T. Kurner, A. Varone, and J. Wagen, “Determination of the permittivity of building materials through WLAN measurements at 2.4 GHz,” in Proc. IEEE Int. Symp. Personal, Indoor Mobile Radio Commun., Sep. 2005, pp. 589–593. [27] G. Klysza, J. Balayssaca, and X. Ferriresb, “Evaluation of dielectric properties of concrete by a numerical FDTD model of a GPR coupled antennaparametric study,” NDT & E International, vol. 41, no. 8, pp. 621–631, Dec. 2008. [28] J. Reed, Software Radio: A Modern Approach to Radio Enginnering. Englewood Cliffs, NJ: Prentice-Hall, Inc., 2002. [29] J. Tsui, Fundamentals of Global Positioning System Receivers: A Software Approach, 2nd ed. New York: Wiley, 2000. [30] Ultra Low Power Superior Sensitivity GPS Modules. [Online]. Available: http://www.starsnav.com/MTI-8T.htm URL: [31] D. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005.

Masood Ur Rehman (M’11) received the B.Sc .(hons.) degree in electronics and communication engineering from University of Engineering and Technology, Lahore, Pakistan, in 2004 and the M.Sc. degree in wireless networks and the Ph.D. degree in electronic engineering from Queen Mary University of London, U.K., in 2006 and 2010, respectively. He then joined the School of Electronic Engineering and Computer Science, Queen Mary University of London as Research Assistant. His main research interests include electromagnetic interaction of antennas and human body, multipath environment effects on mobile terminal antennas and UWB communications.

Xiaodong Chen (SM’96) received the B.Sc. degree in electronic engineering from the University of Zhejiang, Hangzhou, China in 1983, and the Ph.D. degree in microwave electronics from the University of Electronic Science and Technology of China, Chengdu, in 1988. In September 1988 he joined the Department of Electronic Engineering at King’s College, University of London, as a Postdoctoral Visiting Fellow. In September 1990 he was employed by the King’s College London as a Research Associate and was appointed to an EEV Lectureship later on. In 1999 he joined the School of Electronic Engineering and Computer Science at Queen Mary University of London where he is currently a Professor. His research interests are in the fields of wireless communications, microwave devices and antennas. He has authored and co-authored over 300 publications (book chapters, journal papers and refereed conference presentations) and is currently a member of UK EPSRC Review College and Technical Panel of IET Antennas and Propagation Professional Network.

Clive G. Parini (M’96) received the B.Sc. degree in 197 and the Ph.D. degree 1976 from Queen Mary College, University of London, U.K. He then joined ERA Technology Ltd (UK) working on the design of microwave feeds and offset reflector antennas. In 1977 he returned to Queen Mary and is currently Professor of Antenna Engineering and heads the Antenna and Electromagnetics Research Group. He has published over 300 papers on different research topics including communications, antenna and electromagnetics. Prof. Parini is a Fellow of the IET and a member and past Chairman of the IET Antennas and Propagation Professional Network Executive Team. He is a member of the editorial board and past Honorary Editor for the IET Journal Microwaves, Antennas and Propagation. In 2009 he was elected a Fellow of the Royal Academy of Engineering. He is currently the Director of Research for the School of Electronic Engineering and Computer Science.

Zhinong Ying (SM’04) received the B.Phys. degree from Zhejiang Normal University, China, in 1982;, the M.S.E.E. degree from Beijing University of Post and Telecommunications, China, in 1986, and the Ph.D. degree from Chalmers University of Technology, Sweden, in 1995. He is an expert of antenna technology in Sony Ericsson Mobile Communication AB, Lund, Sweden. He joined Ericsson as a Senior Engineer, became Senior Specialist in 1997 and Expert in 2003. He was an Adjunct Professor of Electromagnetic Wave Centre at Zhejiang University, China. His main research interests are small antennas, broad and multi-band antenna, multi-channel antenna systems, near-field human body effects and measurement techniques. He has authored and co-authored over 50 papers in various of journal, conference and industry publications and contributed a book chapter to the well known “Mobile Antenna Handbook”. He holds more than 60 patents in the field of antennas and mobile terminals. He was the supervisor for terminal antenna technology and concepts in Ericsson. His most significant contributions are the development of non-uniform helical antenna and multi-band integrated antenna whose innovative designs are used worldwide in mobile industry. His patented designs have reached a commercial penetration of more than several hundred million products worldwide. He was also involved in the evaluation of Bluetooth Technology invented by Ericsson. Dr. Ying received the Best Invention Award at Ericsson Mobile in 1996, Key Performer Award at Sony Ericsson in 2002 and nominated for President Award at Sony Ericsson in 2004 for his innovative contributions. He also served as TPC Co-chairmen in International Symposium on Antenna Technology (iWAT) 2007. He was a member of scientific board of ACE program (Antenna Centre of Excellent in European 6th frame) from 2004 to 2007.

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A Mixed Rays—Modes Approach to the Propagation in Real Road and Railway Tunnels Franco Fuschini and Gabriele Falciasecca

Abstract—These days, security and efficiency of transportation networks are usually supported by wireless communication systems. In all cases, the radio-link reliability strongly depends on propagation properties and, therefore, effective prediction tools and models are requested in the design and planning phases of the radio system. In this paper, a mixed rays—modes approach to the propagation modeling in real tunnels is presented. The propagating field is computed as the superimposition of many characteristic modes, whose amplitudes are properly estimated thanks to a limited and, therefore, fast ray-tracing procedure; the geometrical optics rules are also used to model the main effects of the possible tunnel curvature. Moreover, an equivalent wall roughness is introduced in order to approximately account for the actual tunnel transversal shape and for the presence of inner elements and objects. The model is compared with some other different existing models and with measurements carried out inside an underground line in a neighborhood of Naples. The achieved performance is in line with the published scientific data. Index Terms—Field prediction, modal theory, propagation in tunnels, ray models.

I. INTRODUCTION

S

ECURITY and efficiency of transportation networks have frequently benefitted from the support of wireless communications. Primary/secondary surveillance radar (PSR/SSR) and instrumental landing systems (ILS) represent probably the most popular among many useful applications for naval/air freight. The management and the safety of railway transport is also supported by wireless systems, such as the European Train Control System (ETCS) and the Automatic Train Operation and Control (ATO/ATC) systems present in some existing driverless subways [1]. Wireless transmissions for vehicle-to-vehicle communications have been recently investigated as well [2]. In all cases, the radio-link reliability strongly depends on propagation properties and, therefore, effective prediction tools and models are requested in the design and planning phases of the radio system. Since tunnels are usually present along road and railway routes, the modelling of propagation inside tunnels has been extensively addressed, and the corresponding existing models can be grouped into three main classes: Manuscript received January 31, 2011; revised June 07, 2011; accepted August 29, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. The authors are with the Department of Electronics, Computer Sciences and Systems (DEIS), University of Bologna—Villa Griffone, 40037 Pontecchio Marconi, Bologna, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173137

1) Ray Models [3]–[7]: Optical rays are traced between the transmitter and the receiver, according to the geometrical theories of propagation (Geometrical Optics and Uniform Theory of Diffraction [8]). Each ray tool must be fed by a proper description of the propagation environment. The more detailed and accurate the description, the more reliable the field prediction is, but also the higher the requested computation time. Besides, some errors and imprecision are always unavoidably present in the databases containing the environmental information to input to the model, since some objects are commonly mispositioned, misdimensioned, or even missing at all; other inaccuracies may be introduced in the values of the electromagnetic (EM) parameters. Therefore, the ray approach is always promising and effective in theory, but, in practice, its cost-benefit ratio can be sometimes questionable. Moreover, ray models usually consider plane surfaces and, therefore, cannot immediately account for possible tunnel curvatures, which on the contrary can often be present along the tunnel route. 2) Modal Approach [9]–[12]: Tunnels are regarded as an oversized dielectric waveguide and, therefore, the propagating field is described through the superimposition of proper characteristic modes. The tunnel transversal dimensions are usually , so that many modes are exmuch larger than the wavelength pected to be fully involved in the propagation process [9], [10]. In order to associate the tunnel with its characteristic modes, the transversal shape and dimensions of the tunnel must be known, as well as the EM parameters of the tunnel walls. If analytical, closed-form expressions for the tunnel modes are available, predictions can be achieved with a limited computation time. In practice, this occurs only if the tunnel has a simple transversal shape (rectangular/circular), which is therefore usually assumed [9]–[12]. Unfortunately, this assumption can turn out to be too coarse in several real cases (especially in the railway one), and this obviously reduces the model reliability. Even in cases when analytical formulas are considered for the modes, they must be weighted anyway by proper (complex) amplitudes, whose evaluation is not always trivial. In most cases, therefore, the modal approach is limited just to the fundamental mode (i.e., the least attenuated mode) [12], and a full multi-modal description of the propagating field is seldom considered [10]. Finally, the possible presence of bends within the tunnel can be addressed through the introduction of the characteristic modes of a curved waveguide [9], [13]–[15]. This approach often leads to quite complex expressions for the EM field, thus requiring some numerical computation for their solution.

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3) Heuristic Models [16]: The radio link inside the tunnel is described through a simple, ready-to-use analytical formula, based on some reasonable considerations and often supported by several measured data. Heuristic models usually provide an estimate of the signal path loss as a function of some synthetic parameters, such as the link distance, the tunnels dimensions, the operating frequency, the antennas positions, and their main radiation properties. Dual-slope behavior is often assumed [16] (i.e., the existence of a near and a far region with quite different propagation characteristics). In particular, the propagation of just one dominant mode is assumed in the far region. (with r being This assumption is strictly true in the limit the link distance), but could be doubtful otherwise (the higher the frequency, the rougher the assumption). Therefore, heuristic models can be affected by a slight underestimate of the received power for large distance. Moreover, the passage from the near to the far region in [16] occurs at the so-called breakpoint distance, whose value corresponds to the intersection between the curves related to the near model (based on the Friis formula) and the far model (based on the propagation of the only dominant mode). Unfortunately, this intersection may not exist, depending on the positions of the antennas inside the tunnel (for instance, it can disappear if the antennas are moved close to the tunnel walls). In this paper, the electromagnetic (EM) propagation inside a real railway and road tunnel is modelled through a sort of mixed rays—modes approach. The entire propagating field description is fundamentally based on the superimposition of proper characteristic modes, but it also benefits from some hints inspired by a ray representation of the tunnel propagation. In particular, a fast, preliminary ray-tracing procedure is used for an effective evaluation of the complex amplitudes of the propagating modes, which can be therefore properly weighted and combined. Moreover, the possibility of associating each mode with proper optical rays bouncing on the tunnel walls allows the introduction of some simple corrective elements to the modal approach in order to quickly but effectively account for the possible route curvature and the real tunnel section. With respect to the existing models, the adopted approach succeeds (within certain limits) in combining simplicity and reliability. Starting from a rather simple description of the environment (through a list of some synthetic, easy-to-know parameters), performance in line with the published scientific data can be achieved even in real cases with limited computational effort and moderate consumption of time. The paper is organized as follows. Tunnel modes are briefly introduced in Section II, whereas the suggested procedure for the evaluation of propagation properties in the ideal and real tunnel is addressed in Sections III and IV, respectively. Section V shows some comparisons between measured and simulated results, and conclusions are then drawn in Section VI. II. DESCRIPTION OF THE TUNNEL MODES The tunnel characteristic modes are shortly introduced here. In particular, the analytical expressions of the modes in the case of rectangular tunnel are recalled in Section II-A, whereas the possibility of associating each mode with optical rays properly bouncing on the tunnel walls is outlined in Section II-B.

Fig. 1. Tunnel section and reference system.

A. Analytical Description Real tunnels often exhibit an arched cross-section, which can be further complicated by the presence of inner, structural elements, such as walkways, cables, pipes, etc. Starting from such a complex transversal shape, a useful, easy-to-handle modal characterization is impracticable. Therefore, simpler, approximated rectangular or circular sections are usually considered [9]–[12]. With reference to the straight rectangular tunnel with transversal h (height) (Fig. 1), any field dimensions w (width) can be expressed as the superimposition of X- and Y-polarized modes with indexes (m,n) (1) (2) with being the complex amplitudes, being the mode functions, and being the mode propagation constant. Under the following restrictions [17]: (3) (4) the X-polarized modes can be characterized by the following approximated expressions (similar formulas hold for the Y-polarization) [17]:

(5)

(6)

(7) (8)

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with (m even) or (m odd), (n even) or 0 (n odd); is the complex, relative permittivity of the tunnel walls, i.e., (9) According to the usual values of the real relative permittivity , the wall conductivity and the communication frequency of the modern wireless systems GHz), is often assumed in the following sections. Equations (1)–(8) clearly show that the field associated with the (m, n) mode behaves as a progressive wave in the tunnel axis direction, whereas it is a stationary wave in the transversal ( ) plane. Moreover, it is also worth noticing that the EM field of each mode vanishes on the tunnel walls. , each mode undergoes a propagation loss as Since it travels along the tunnel; the mode attenuation is often represented by the modal attenuation factor (MAF) also introduced in [9]

It can now be easily noted that the least attenuated modes (i.e., with the lowest MAF values) are characterized by grazing angles values quite close to zero. Moreover, since directive antennas pointed in the tunnel axis direction are the likeliest radiators to be used, the most powerful rays launched by the transmitter impinge on the tunnel walls with rather small grazing angles and, therefore, the corresponding modes can be expected to also have the largest amplitudes. In conclusion, the modes (and rays) with high grazing angles values can often be considered negligible, at least to a first approximation; in that case, cos and, therefore, (13) and (14) can be further simplified (15) (16) • number of reflections on the vertical and the horiwalls over a tunnel stretch of length zontal

(10)

(17)

At mobile communications frequencies (hundreds of MHz and greater), the MAF values increase quite slowly with m, n and, therefore, strong multimodal propagation can be expected. B. Geometrical Description The possible modes in a rectangular tunnel can be approximately determined also by a ray theory approach. To this end, the (m, n) mode can be geometrically described (irrespective of its polarization) as the cluster of rays which proceed into the tunnel bouncing on the vertical/horizontal walls with proper and , respectively. These rays produce grazing angles an EM field which is a progressive propagating wave along the tunnel axis and on the contrary a full standing wave in the plane, where the rays superimposition must reproduce the (m,n) mode behavior. Hence, the following relations for the grazing angles can be derived:

(18) According to this geometrical, equivalent description, the MAF value is the result of the energy loss due to the multiple reflections of the mode wavefronts on the tunnel walls; after and consecutive reflections, the field intensity along the , ray is reduced by a factor and being the reflection coefficients related to with the vertical and horizontal walls, respectively. Therefore, the MAF can also be written according to the following expression, obviously equivalent to the previous (10):

(11) (12) Moreover, analytical expressions for the following quantities associated with the (m,n) mode can be immediately drawn from the mode geometrical description: • distance between two consecutive bounces on the vertical and the horizontal walls: by means of some trigonometrical considerations, the following expressions can be achieved: (13) (14)

(19)

III. PROPAGATION IN IDEAL TUNNEL Equations (5)–(8) provide an analytical description for the characteristic modes of a rectilinear tunnel with a rectangular transversal shape (“ideal” tunnel in the following text). Propagation inside such a tunnel can be therefore modelled through the modal expansion described by (1) provided that the modes’ are properly evaluated. complex amplitudes Assuming the knowledge of the electric field over the transversal reference sec-

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Fig. 2. Ray approach to the estimation of the field distribution on the reference section.

tion , the values can be immediately estimated through the following expression [18]: (20)

with S being the transversal tunnel section. Equation (20) can be immediately achieved on the base of the orthogonality between the modes functions (7)–(8), i.e., (21) with being the Kronecker delta. For the sake of precision, it is , then (1) must be necessary to observe that in case of slightly modified as follows:

(22) In order to achieve the electric-field distribution on the reference section, the geometrical ray-approach outlined in Fig. 2 is adopted here. Starting from the transmitting antenna radiation on the excitaproperties and the transmitter position , the multipath field contributions in each tion plane point of the reference section can be easily computed by means of a standard ray-tracing algorithm [19]. In order to achieve a satisfactory but simple and fast evaluation of the required field distribution, only the direct and the once-reflected rays can be ; in this case, considered, provided that the reflection coefficients amplitudes are rather small (since the incidence directions are close to the normal to the side walls) and, therefore, contributions undergoing more than one reflection can be approximately neglected. After the computation of the modes amplitudes, the propagating field inside the tunnel can be evaluated through (22) and, hence, the power received by an antenna placed in can be achieved as

(23)

Fig. 3. Simulation results for the ideal tunnel: w = 7.8 m, h 5, (x = x = 1.95 m, y = y = 0.65 m), z P =34 dBm.

0

0

j

= 5.3 m, " =

0 z j =1.5 m,

being the receiving antenna gain in the with direction of the tunnel axis, and being the polarization and the power matching coefficients (both ranging from 0 to 1). A multimodal procedure for the computation of the received power inside an ideal tunnel is also proposed in [10], where an analytical expression for the mode amplitudes is achieved. Nevertheless, the approach adopted in [10] is restricted to a sort of isotropic source, since the emitted field is independent of the rays departure directions; on the contrary, the actual antenna radiation pattern is here taken into account, since it is considered by the ray-tracing procedure which provides the field distribution on the reference plane. Power measurements carried out inside a straight tunnel with nearly smooth walls have been presented in [9] and are here compared with simulations performed in approximately the same conditions (Fig. 3). Different radiators are used at different frequencies (half-wave dipoles at 450 MHz, horn antennas at 900/2100 MHz), whereas the radiated power and the antennas positions are the same in all cases. The considered parameters values are listed in the figure caption; the are related transmitter/receiver coordinates to the reference system introduced in Fig. 1. The clear agreement between Fig. 3 and [9, Figs. 18–19] suggests that the modal approach to the propagation inside an ideal tunnel is rather satisfactory, provided that the modes’ amplitudes are reliably estimated. IV. PROPAGATION IN REAL TUNNEL The actual geometric properties of a real tunnel may considerably differ from the ideal case considered in the previous section. The presence of structural elements, such as walkways, cables, and conduits can complicate the shape of the tunnel transversal contour (especially in the railway case), so that the rectangular (or circular) approximation can represents a too rough simplification. The possible inner elements usually act as energy scatterers, thus producing an increase in the signal path loss [9]. Moreover, the presence of bends along the tunnel route is also rather frequent and, therefore, the effects of the curvature on the propagation should be properly modelled in

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order to avoid an unsatisfactory prediction. These aspects are shortly addressed in the following subsections. A. Walls Equivalent Roughness Part of the scientific literature on radio communication inside tunnels is dedicated to the mine environment [12], [20]–[22], where an important role in the propagation is played by the walls’ roughness. Because of the surface roughness, the energy impinging on the tunnel walls does not undergo just specular reflection but it is on the contrary spread in many different directions. Therefore, the energy carried by the principal modes (characterized by small grazing incidence and, therefore, by the lowest MAF values) is partially transferred to the higher order with more attenuated modes, thus producing an additional signal loss with respect to the case of perfectly smooth walls [12]. The rough surface is often assumed to have a Gaussian distribution with zero mean and standard deviation [12], [13], [21], so that the intensity of a ray impinging with a grazing angle is reduced after a single reflection by a factor [12] (24) Therefore, the intensity reduction suffered by the (m,n) mode after and bounces on the vertical and horizontal walls, respectively, and can be expressed as follows: (25) where and describe the roughness of the vertical and horizontal walls, respectively; obviously, the values of and are provided by (17) and (18), as well as (11) and (12). Since the ray intensity is proportional to the square of the field of the (m,n) mode propagating amplitude, the magnitude from the reference plane to the z transversal section must satisfy the following relation:

(26) with being the field amplitude that would be present at the distance z in the absence of any roughness , and, . therefore, it would be proportional to Hence

(27) This expression shows that the effect of the walls roughness on each mode can be taken into account by means of a proper increase ( in (27)) of the mode attenuation coefficient.

Fig. 4. Effects of the tunnel curvature on the optical rays.

Actually, the walls of road and railway tunnels can be assumed much smoother than those of mines, so that the scattering effects due to the surface roughness is not expected to have a great impact on propagation. Nevertheless, some energy scattering can be anyway originated by the possible, inner structural elements. Therefore, it is here assumed that each real tunnel can be associated with a somehow equivalent, distributed, surface roughness, which is supposed to produce approximately the same mean, additional loss due to the actual, discrete scattering sources. On the base of this assumption, the effects on propagation due to the actual tunnel contour and the presence of inner scattering objects can be approximately accounted in the rectangular model by a proper increase in the modes loss coefficients (acand pacording to (27)), provided that the values of the rameters have been properly evaluated. B. Tunnel Curvature According to previous studies [9], [13]–[15], [23], [24], the main impact of the tunnel curvature on propagation usually consists of an increase in the modes attenuation and, therefore, in the overall received signal loss. Starting from the geometrical interpretation of each mode, this effect can be immediately explained, since the incidence angles of the rays associated with the modes on the tunnel walls are usually decreased by the presence of a curve ( in Fig. 4). This leads to lower reflection coefficients and, hence, to a larger refraction loss (19) and MAF values (10). The effects produced by the tunnel curvature on the propagation properties have been investigated in some previous work adopting an analytical approach based on the introduction of the characteristic modes of a curved rectangular waveguide [9], [13]–[15]. The corresponding results are quite fair but also rather complicated to handle, since the achieved analytical expressions for the EM field inside the curve are usually unavailable in closed form and, therefore, some numerical computation is often required for their solution. In order to avoid any complicated formalism and complex computation, an extremely simplified, heuristic approach is considered here, which aims at a quick estimate of the additional loss produced by the curve and whose effectiveness should be evaluated a posteriori on the base of the provided results. In particular, the bends are here limited to the side walls of the tunnel (i.e., the roof and the floor are assumed to lie on horizontal planes) and are supposed to be circular, that means they

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are fully described through their radius and angular amplitude in Fig. 4). The only effect produced by the reflec(R and tions on the curved vertical walls is then considered, leaving the computation for the horizontal walls unchanged. According to this (quite coarse) approximation, the analyses are restricted to the 2–D plan represented in Fig. 4. The (curvature) additional loss is then evaluated through the following procedure: of the • for each (m,n) mode, an effective grazing angle corresponding rays on the curved tunnel walls is estimated by means of some simple geometrical considerations; the values basically depend on the curvature radius , the value related to the straight tunnel (12), and the tunnel width (w); suf• the number of reflections on the vertical walls fered by the modes inside a curved tunnel stretch of length is then evaluated; • the total refraction loss inside a curve with length is then computed for each mode through the following expression, which comes directly from (19):

TABLE I CURVATURE EFFECT ON THE MODE ATTENUATION FACTOR

Example with w

= 8 m, h = 4 m,  = 8.5 m, f = 2 GHz, R = 25 w.

(28) The additional loss provided by the tunnel curvature is therefore equal to

Fig. 5. Curvature effects on propagation losses—w " = 8.5.

= 4.26 m, h = 2.13 m,

(29) with (in decibels) being the modal loss in the case of the straight tunnel; substituting (19) into (29), the following expression of the mode additional loss in the linear unit can be immediately achieved: (30)

• finally, the extra loss to be considered for the (m,n) mode from the beginning of the bend is at a distance achieved as (31) Further details on the adopted procedure can be found in the Appendix.

Fig. 6. Dominant mode (m = n =) 1 MAF value for different curvature radius in a rough walled tunnel (w = 10 m, h = 5 m) at the frequency of 925 MHz.

Table I shows a comparison between the MAF values of some different modes in the straight/curved case; the considered main system parameters are again listed in the table caption. The curvature impact on the propagation losses is displayed in Fig. 5 in the case and for some values and both the horizontal ( ) and the vertical ( ) polarization. The tunnel dimensions are those considered in [9] and are reported in the figure caption. The results in Fig. 6 are instead related to the case investigated in [23], where the tunnel curvature (for different R values) and the walls roughness ( cm) are considered together; the transversal dimensions are 10 m (w) 5 m (h) and the frequency is equal to 925 MHz.

FUSCHINI AND FALCIASECCA: MIXED RAYS—MODES APPROACH

Fig. 7. Transversal section of the tunnel considered for measurement.

Fig. 8. Transmitting antenna radiation patterns.

The rather satisfactory agreements between Fig. 5 and [9, Figs. 23–24] and between Fig. 6 and [23, Fig. 3] suggest that the adopted approach to the evaluation of the curvature effects on propagation should be considered accurate enough. V. COMPARISON WITH MEASUREMENTS A measurement campaign has been carried out inside the underground line connecting Naples and Aversa (Italy); it is a “cut & cover” tunnel with an approximately rectangular shape 8.4 m, 5, 1 m), as schematically represented in Fig. 7. ( Some structural elements and objects are present inside the tunnel, such as the walkways along the vertical walls, the railway tracks on the floor (Fig. 7) and some lamps, and signs and cables on the roof and the lateral walls. All of these objects act as energy scatterers and make the transversal shape slightly different from the ideal, rectangular case.

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Fig. 9. Measurement routes inside the tunnel.

The measurement setup is constituted of a fixed transmitter (Tx) and a moving receiver (Rx). The Tx is placed at a distance of about 0.2 m from the vertical wall, and its height with respect to ground is equal to 3.4 m. (In the reference system of 4 m, 0.85 m, 0). The transFig. 1, this means 100 mW mitter output power and the carrier frequency are and 2.4 GHz, respectively; the radiating antenna is dual-po45 and bidirectional (i.e., with two opposite larized 45 main lobes, oriented in the direction of the tunnel axis), with a gain of 10.5 dB. (The vertical/horizontal normalized radiation patterns are shown in Fig. 8.) The receiver is mounted on a pole1 of height 2.9 m and 1.8 moves along a track going away from the transmitter ( 0.35 m, ); the receiving antenna is vertically m, polarized and approximately omnidirectional in the horizontal plane with a gain equal to nearly 7.5 dB. The overall cable losses amount to 2 dB ( 0.45 dB between the Tx and the radiating antenna and 1.55 dB between the receiving antenna and the receiver). Four different measurement routes having a length of some hundreds of meters have been considered inside the tunnel and are indicated with letters A–D in Fig. 9. The tunnel is approximately straight in all cases except for route B which exhibits a mild curvature with a radius of about 780 m. The comparison between measurements and simulations related to route A is shown in Fig. 10. The black-dashed line has been achieved assuming a perfectly rectangular and smoothwalled tunnel (i.e., with the model described in Section III). The agreement with measurement (blue-dotted line) is clearly quite poor, according to the idea that some correctives should be introduced into the “ideal model” in order to somehow account for the real tunnel properties and to get, therefore, a more satisfactory prediction. The possible scattering effects provided by the inner elements (walkways, railways tracks, etc.) have been therefore modeled according to the equivalent roughness approach (Section IV-A), thus strongly improving the prediction accuracy (red continuous line in Fig. 10). 1The possible presence of trains inside the tunnel is therefore neglected for the time being, according to most of the available scientific literature [9]–[11], [21], [22].

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TABLE II MEAN ERRORS AND ERRORS STD. DEVIATIONS FOR THE MEASUREMENT ROUTES

Fig. 10. Comparison simulation—measurements, route A.

simple but rather approximated approach to account for the actual tunnel shape and its real properties and, therefore, it unavoidably introduces some further inaccuracies. Finally, the introduced roughness is supposed to be distributed over all of the tunnel walls and, therefore, it may be fit to describe the effect of structural elements which are actually present along the entire tunnel length (such as cables, walkways, tracks, etc.). On the contrary, it could be rather unsuited to model the presence of “spot scatterers” such as air shafts or railways points which can be differently positioned along the routes producing “local” rather than “distributed” effects. VI. CONCLUSION

Fig. 11. Comparison simulation—measurements, route B.

The values considered for the scattering coefficients are 0.25 m and 0.75 m; they do not represent necessarily the best values, since their optimization lies outside the purpose of and values seem this paper. Nevertheless, the likeliest to be of the same order of magnitude as the dimensions of the scattering objects present in the tunnel transversal section. The predicted received power levels related to route B are plotted versus measurements in Fig. 11. The agreement is again rather satisfactory, provided that the same degree of wall roughness is considered and the route curvature is taken into account. Actually, the curvature is rather mild and, therefore, propagation can be expected to be slightly affected by the bend; in order to estimate the actual effect produced by the curve, a simulation assuming a perfectly straight tunnel has been carried out, with a mean error and an error standard deviation equal to 2.55 dB and 5.68 dB, respectively. An overall evaluation of the prediction reliability is concisely and the error provided in Table II, where the mean error standard deviation are computed for each route; the mixed rays—modes approach here proposed provides an accuracy in line with previously published works [4]–[6]. The exact reasons of the prediction errors cannot be easily identified and must be further investigated. In this regard, it should be reminded that (5)–(8) represent an approximated analytical solution even for the ideal, straight rectangular case [17]. Moreover, the equivalent roughness models represent a

Propagation inside tunnels is a quite complex phenomenon whose description requires nontrivial field prediction models. Common ray-tracing tools represent a possible, satisfactory solution in theory, but in practice their actual reliability strongly depends on the accuracy of the environment description; moreover, they are not trivial to handle and often undergo a heavy computational burden. The modal approach is generally simpler and faster, but it appears convenient just in rather ideal, unrealistic cases (straight, empty tunnel with circular/rectangular section). Finally, heuristic models provide a fast but rough knowledge of the channel, which can be perhaps useful for a preliminary, coarse system design, but practically unfit to support the network deployment and optimization phases. The model proposed in this paper aims at catching some benefits from these different approaches, but avoiding their main drawbacks at the same time. It is essentially based on a multimodal description of the propagation, which is nevertheless enriched by some heuristic correctives (based on the geometrical optics theory) in order to extend its prediction capability also to real cases. The model is compared with some other different existing models and with measurements carried out inside an underground line in the neighborhood of Naples. The achieved performance is in line with the published scientific data. APPENDIX Radio propagation inside a rectangular tunnel is physically produced by several radio waves which advance into the tunnel bouncing on its walls; each propagating wavefront can be and ) describing the marked by the couple of angles ( way it impinges on the vertical/horizontal wall, respectively. Every characteristic mode of the tunnel is defined by the set of

FUSCHINI AND FALCIASECCA: MIXED RAYS—MODES APPROACH

Fig. A1. Geometrical description of a mode at the beginning of the curve.

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Fig. A3. Geometrical description of case a2: reflections on both vertical walls.

For higher order modes and for the only “case b,” the first reflection occurs on the internal wall and (necessarily) the second on the external one; a third case “b3” must be then considered.2 “Case a1”–with reference to Fig. A2, the ray representative of the mode covers the curved tunnel by successive reflections on the external wall; the reflection angle which determines the re(Fig. A2) and can be evaluated as fraction loss is therefore explained hereafter. In the reference system O , represented in Fig. A2, the equation of the OP line is (A1) The coordinates of point the following system:

can therefore be computed through

Fig. A2. Geometrical description of case a1: reflections just on the external vertical wall.

rays with the same angles values, which are expressed by (11) and (12) for the rectilinear case. In order to evaluate the effect of the presence of a curve on the mode propagation loss, it is necessary to estimate how the mode characteristic angles change inside the bend. Assuming the curvature can affect only the vertical wall for simplicity reasons, the problem can be therefore approximately simplified to the 2-D case represented in Fig. A1, where the (m,n) mode is simply represented by a couple of rays characterized by an inci. dence angle on the vertical walls equal to , It is easy to show that for the lower order modes ( dependent on the R, w, and values) both of the two with considered rays undergo the first reflection on the external wall of the curved tunnel. In these conditions and depending on the and R, the second reflection may occur again on values of the external wall or instead on the inner one. Therefore, the “case a” in Fig. A1 can be further split into the cases “a1” and “a2” schematically represented in Figs. A2 and A3 (similarly, cases “b1” and “b2” could be achieved from “case b” of Fig. A1).

(A2)

Moreover, the angular coefficient of the line CP is equal to (A3) and, therefore, the angle can be finally computed as

between the lines OP and CP

(A4) 2At

mobile communication frequencies and for usual tunnel dimensions, the value is quite large and, therefore, the modes belonging to class “b3” fade rather quickly along the tunnel, so that their impact on propagation is often quite small.

m

(10)

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The arched distance between two consecutive reflections (i.e., and and evaluated along the tunnel axis) is given between by (A5) With “Case a2”–different reflection angles must be considered, for the external and internal tunnel walls, respectively ( and in Fig. A3). The same procedure already described . Then, for “case a1” can be applied for the evaluation of starts from the expression of the line the computation of (Fig. A3)

(A13) The total number of reflections experienced by the mode inis then estimated as side the curved tunnel of length (A14) In order to achieve the mode refraction loss inside the bend, the values achieved from (A12) and (A14) must be introduced into (31). ACKNOWLEDGMENT

(A6) The coordinates of point following system:

are therefore the solution of the

The authors would like to thank Ansaldo STS (www.ansaldosts.com) for providing the measurements used in the final comparisons, and Wireless Future (www.wirelessfuture.it) for supporting the execution of simulations. REFERENCES

(A7) with being obviously and . It is worth noticing necessarily that is positive in “case a2” since the line intersects the circumference of radius R. Therefore, the sign of discriminates between “case a1” and “case a2” . The angular coefficient of the line CP is then equal to (A8)

of

Equation (A8) immediately leads to the following expression (angle between the lines and CP ): (A9)

The incidence angle associated with the (m,n) mode is assumed to be the average between the two reflection angles (A10) With reference to Fig. A3, the distance between two consecutive reflections can be estimated as

(A11) Similar considerations should be now repeated for “cases b1, b2, and b3”. (Details are here omitted for the sake of brevity). At last, the following expression for the incidence angle and the distance between consecutive reflections are considered for the (m,n) mode: (A12)

[1] C. Fich, “The Copenhagen driverless metro started operation in october 2002 from vision to reality,” presented at the 19th Dresden Conf. Traffic Transport. Sci., Dresda, Germany, Sep. 2003. [2] A. Paier, T. Zemen, J. Karedal, N. Czink, C. Dumard, F. Tufvesson, C. F. Mecklenbrauker, and A. Molisch, “Spatial diversity and spatial correlation evaluation of measured vehicle-to-vehicle radio channels at 5.2 GHz,” in Proc. 13th Digital Signal Process. Workshop and 5th IEEE Signal Process. Educ. Workshop, 2009, pp. 326–330. [3] Y. Hwang, Y. P. Zhang, and R. G. Kouyoumjian, “Ray-optical prediction of radio-wave propagation characteristics in tunnel environments—Part 1: Theory,” IEEE Trans. Antennas Propag., vol. 46, no. 9, pp. 1328–1336, Sep. 1998. [4] Y. Hwang, Y. P. Zhang, and R. G. Kouyoumjian, “Ray-optical prediction of radio-wave propagation characteristics in tunnel environments—Part 2: Analyses and measurements,” IEEE Trans. Antennas Propag., vol. 46, no. 9, pp. 1337–1345, Sep. 1998. [5] D. Didascalou, J. Maurer, and W. Wiesbeck, “Subway tunnel guided electromagnetic wave propagation at mobile communications frequency,” IEEE Trans. Antennas Propag., vol. 49, no. 11, pp. 1590–1596, Nov. 2001. [6] T. Wang and C. Yang, “Simulations and measurements of wave propagations in curved road tunnels for signal from GSM base stations,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2577–2584, Sep. 2006. [7] Y. Kishiki, J. Takada, G. S. Ching, N. Lertsirisopon, M. Kawamura, H. Takao, Y. Sugihara, S. Matsunaga, and F. Uesaka, “Application of reflection on curved surfaces and roughness on surface in ray tracing for tunnel propagation,” in Proc. 4th Conf. Antennas Propag., 2010, pp. 1–5. [8] C. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [9] D. Dudley, M. Lienard, S. F. Mahmoud, and P. Degaque, “Wireless propagation in tunnels,” IEEE Antennas Propag. Mag., vol. 49, no. 2, pp. 11–26, Apr. 2007. [10] Z. Sun and I. F. Akyldiz, “Channel modeling and analysis for wireless networks in underground mines and road tunnels,” IEEE Trans. Commun., vol. 58, no. 6, pp. 1758–1768, Jun. 2010. [11] C. Holloway, D. Hill, R. Dalke, and G. Hufford, “Radio wave propagationcharacteristics in lossy circular waveguides such as tunnels, mine shafts, and boreholes,” IEEE Trans. Antennas Propag., vol. 48, no. 9, pp. 1354–1366, Sep. 2000. [12] A. Emslie, R. Lagace, and P. Strong, “Theory of the propagation of UHF radio waves in coal mine tunnels,” IEEE Trans. Antennas Propag., vol. AP-23, no. 2, pp. 192–205, Mar. 1975. [13] R. Martelly and R. Janaswamy, “Modeling radio transmission loss in curved, branched and rough-walled tunnels with the ADI-PE method,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 2037–2045, Jun. 2010. [14] S. F. Mahmoud and J. R. Wait, “Guided electromagnetic waves in a curved rectangular mine tunnel,” Radio Sci., vol. 9, no. 5, pp. 567–572, 1974.

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[15] Y. P. Zhang and Y. Hwang, “Characterization of UHF radio propagation channel in curved tunnels,” in Proc. 7th Int. Symp. Personal, Indoor Mobile Radio Commun., 1996, vol. 3, pp. 798–802. [16] Y. P. Zhang, “Novel model for propagation loss prediction in tunnels,” IEEE Trans. Vehic. Technol., vol. 52, no. 5, pp. 1308–1314, Sep. 2003. [17] K. Laakman and W. Steier, “Waveguides: Characteristics modes of hollow rectangular dielectric waveguides,” Appl. Opt., vol. 15, no. 5, May 1976. [18] R. E. Collin, Field Theory of Guided Waves. New York: McGrawHill, 1960. [19] V. Degli Esposti, D. Guiducci, A. de’Marsi, P. Azzi, and F. Fuschini, “An advanced field prediction model including diffuse scattering,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1717–1728, Jul. 2004. [20] M. Lienard and P. Degaque, “Natural wave propagation in mine environment,” IEEE Trans. Antennas. Propag., vol. 48, no. 9, pp. 1326–1339, Sep. 2000. [21] Z. Xu and H. Zheng, “The effect of wall roughness on the electromagnetic wave propagation in coal mine underground,” in Proc. IEEE Int. Symp. Knowl. Acq. Model., 2008, pp. 482–485. [22] S. Zhang and B. Y. Li, “The influence of scattering on multipath channel characteristic of tunnel,” in Proc. 6th IEEE Int. Symp. Antennas, Propag. EM Theory, Nov. 2003, pp. 576–578. [23] M. Nilsson, J. Slettenmark, and C. Beckman, “Wave propagation in curved road tunnels,” in Proc. IEEE Antennas Propag. Int. Symp., 1998, vol. 4, pp. 1876–1879. [24] M. Lienard, S. Betrencourt, and P. Degauque, “Theoretical and experimental approach of the propagation at 2.5 GHz and 10 GHz in straight and curved tunnels,” in Proc. IEEE Veh. Tech. Conf., 1999, vol. 4, pp. 2268–2271.

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Franco Fuschini was born in Bologna, Italy, in 1973. He received the Ph.D. degree in electronics and computer science from the University of Bologna, Bologna, Italy, in 2003. From 2004 to 2006, he held a postdoctoral position at the Department of Electronics and Computer Science (DEIS), University of Bologna. From 2007 to 2011, he was with the Marconi Wireless Consortium (Italy). Currently, he is Research Associate at DEIS. His main fields of interest are antenna systems and propagation models for real environments, radio systems network planning, and radio-frequency identification.

Gabriele Falciasecca was born in Bologna in 1945. In 1970, he joined the Laboratories of the Fondazione Bordoni at Pontecchio Marconi, Bologna, Italy, where he was involved in the Italian research programme on millimeter waveguide communication systems. In 1973, he was Assistant Professor and Lecturer in microwave techniques with the Department of Electronics and Computer Science, University of Bologna, where he has been Full Professor of Microwaves since 1980. From 1994 to 2000, he was Director of the Department. Mr. Falciasecca has been the President of the Emilia-Romagna Technological Development Agency since 2001. He has been the President of the Guglielmo Marconi Foundation since 1997 and is a member of the “Advisory Board” of Fondazione Ugo Bordoni. He is Chairman of the Scientific Committee of the Consortium “Elettra2000,” devoted to the study and to the diffusion of scientific results in the field of health issues related to electromagnetic waves. His main fields of research are mobile radio systems, microwaves, optical systems, millimeter waves, radio propagation, radio navigation, and landing aids.

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Optimum Wireless Powering of Sensors Embedded in Concrete Shan Jiang, Student Member, IEEE, and Stavros V. Georgakopoulos, Senior Member, IEEE

Abstract—The optimization of wireless powering of sensors embedded in concrete is studied here. Our analytical results focus on calculating the transmission loss and propagation loss of RF waves penetrating into concrete at different humidity conditions. Specifically, this analysis leads to the identification of an optimum frequency range within 20–80 MHz that is validated through antenna coupling full-wave EM simulations. Also, an optimized rectenna is designed in order to calculate the battery charging time. Finally, the effects of reinforced bars to RF power transfer are analyzed. Index Terms—Concrete, energy transmission, rebar, rectenna, wireless sensors.

I. INTRODUCTION ARIOUS Nondestructive Testing (NDT) technologies for construction and performance monitoring have been studied for decades. In the past few years, health monitoring of infrastructure has been done by active acoustic transducers [1] and inverse Synthetic Aperture Radar (ISAR) [2], which are labor-intensive and expensive techniques. Recently, the rapid evolution of Wireless Sensor Network (WSN) technologies has enabled the development of sensors that can be embedded in concrete to monitor the structural health of infrastructure. Such sensors can be buried inside concrete and they can collect and report valuable volumetric data related to the health of a structure during and/or after construction. For example, embedded sensors can collect data, such as, temperature, displacement, pressure, strain, humidity, and detect cracking and rebar corrosion. The cost of such monitoring systems is significant. The expensive nature of structural monitoring systems is a direct result of the high installation and maintenance costs associated with wired systems. The installation of a monitoring system can represent up to 25% of the total system cost with over 75% of the installation time focused solely on the installation of wires [3]. In outdoor applications, such as bridges, potentially harsh environmental conditions necessitate additional efforts to install cables in weatherproof conduits thereby raising installation costs. A promising solution that can decrease the cost of monitoring systems and reduce their deployment time is based on the use of wireless embedded sensors. Such wireless embedded sensors

V

Manuscript received March 17, 2011; revised July 02, 2011; accepted August 08, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported in part by the Dissertation Year Fellowship that was provided by Florida International University. The authors are with the Department of Electrical and Computer Engineering, Florida International University, Miami, FL 33174 USA (e-mail:[email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173147

need to operate for long time. However, sensor batteries have finite life time. Therefore, in order to enable long operational life of wireless sensors, novel wireless powering methods, which can charge the sensors’ rechargeable batteries wirelessly, need to be developed. Various wireless powering methods have been proposed in the past. Specifically, power for sensors was scavenged from bridge vibrations in [4]. An air core coil was connected to a voltage doubler to collect power in [5]. RFID technology was applied in [6]–[8] to transfer power using inductive coupling between master and sensor magnetic coils. Also, the effects of concrete’s dielectric constant and loss tangent to the radiation pattern and gain of a microstrip patch antenna were studied in [9]. A circularly polarized rectenna for wireless power transfer was designed in [10]. One antenna-rectenna model working at 5.7 GHz was built in [11] to convert RF power to DC power. Finally, planar inverted-F antennas working at 915 MHz were buried into a bridge pier in order to study the feasibility of wireless communications inside concrete [12]. However, RF communications in concrete suffer from high losses. This problem is inherent to all wireless communications with sensors buried in concrete, and it stems from the high attenuation of RF signals in concrete. In this paper, analytical as well as computational methods are used to identify optimal conditions for wireless powering of sensors embedded in concrete. Two types of loss are analyzed in order to find the optimum frequency range: a) the transmission loss caused by the reflection of the electromagnetic wave at the air-concrete interface, and b) the propagation loss generated by the wave’s propagation through the lossy concrete media. The electromagnetic properties of concrete are modeled by the extended Debye model. Then, the performance of the optimum frequency range is tested by antenna coupling simulations. Also, an optimum rectifying circuit is designed to convert RF power to DC power. Finally, the effects of rebars on wireless power transfer are analyzed. II. PLANE WAVE MODEL For a plane wave penetrating concrete, the total power loss is the sum of transmission loss and propagation loss, and it represents the difference between the power transmitted and the power received at certain depth inside concrete. We develop analytical formulations that describe both loss mechanisms and then calculate the total losses for two cases. The first case calculates the losses of a plane wave impinging on an air-concrete interface at normal incidence assuming the thickness of concrete is infinite, as shown in Fig. 1(a). The second case calculates the losses of a plane wave impinging on an air-concrete interface at normal incidence assuming that the thickness of concrete is finite, as shown in Fig. 1(b).

0018-926X/$26.00 © 2011 IEEE

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different humidity conditions (from 0.2% to 12.0%). The humidity condition describes the percentage of water by volume and it is represented here by the symbol, . Furthermore, the transferred power at a propagation depth, , inside concrete can be written using (2) as (5) where the transmission coefficient

is given by [15] (6)

and

is the attenuation constant [15] (7)

Fig. 1. Plane wave penetrating concrete at normal incidence: (a) Concrete halfspace, and (b) Concrete slab.

Based on (5) the transmission loss, describing the power loss caused by the air-concrete interface, can be calculated in dB as follows:

A. Power Attenuation for Normal Incidence (8)

We formulate analytical equations to calculate the transmission loss and propagation loss for the scenario of Fig. 1(a). The incident power is written as [13]

Also, the propagation loss inside the concrete can be written as

(1) is the incident electric field and stands for where the intrinsic impedance of air. Similarly, the transferred power in concrete is written as (2) where

is the transmitted electric field,

is the

stands for intrinsic impedance of concrete, and the relative complex permittivity of concrete [14]. The real part of the relative permittivity is (3) where stands for the difference between the values of the real part of the relative permittivity at low and high frequencies, and is the relaxation time. The real part of the complex relative permittivity represents the ability of the medium to store electrical energy. The imaginary part of the complex relative permittivity represents the energy losses due to dielectric relaxation as follows: (4) where is the DC electrical conductivity of concrete and is the effective conductivity. This model can be considered as an extension of the Debye model (extended Debye model). The Debye model parameter values are available in [14] for six

(9) Therefore, the total loss of the normal incidence is written as

(10) and it depends on the complex permittivity of concrete and the depth of propagation. The power losses are plotted in Fig. 2 for four different humidity conditions of concrete and a propagation depth of 0.25 m. Fig. 2(a) illustrates that the transmission loss decreases dramatically as the frequency increases from 1 MHz to 20 MHz, and then remains almost constant for frequencies higher than 20 MHz. The propagation loss is plotted in Fig. 2(b), and it increases slowly for frequencies up to 100 MHz and then increases dramatically for higher frequencies. Furthermore, transmission loss and propagation loss are added together to obtain the total loss for air-concrete propagation, as shown in Fig. 2(c). As expected, due to the reverse variations of the two losses, an optimum frequency range exists, within which there is significantly smaller power loss. For example, the total loss in the frequency range of 20–80 MHz for wet concrete (12% humidity) is about 5–11 dB less than the total loss at the lowest or highest frequency in our analysis. This observation is particularly useful for wireless powering of sensors embedded in concrete, where we seek to minimize transmission losses in order to deliver maximum power to the sensors. It should be pointed out that the optimum frequency range of 20–80 MHz includes the bands of shortwave radio (3–30 MHz) and VHF TV (54–72 MHz and

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76–88 MHz). Therefore, sensors embedded in concrete operating in the optimum frequency range will be able to efficiently harvest signals in these bands for charging. Our results also illustrate that total loss increases as the humidity of concrete increases. This is expected, since as the moisture content increases from 0.2% to 12%, the conductivity of concrete increases almost 30 times [14] thereby causing more losses. Specifically, we discuss the humidity effects because many applications require monitoring the properties of concrete while it is still wet and curing. For example, high strength concrete may take 6 to 60 days to cure depending on the structure and the weather conditions. Therefore, high strength concrete needs to be closely monitored by sensors while it is curing (it is wet) to ensure that it reaches its full strength before any mechanical load is applied to it. Hence, when designing wireless powering systems, the concrete’s moisture content should be taken into account to maximize the RF power transfer. The total loss for the concrete slab scenario of Fig. 1(b) can be deduced following a similar procedure to the one presented in [16] as (11) where is given in (12), shown at the bottom of the page, is the reflection coefficient given by (13), shown at the bottom of the page, is the attenuation constant in concrete, is the phase constant, and is the phase of . It should be pointed out that the total loss given by (11) for , i.e., for a very thick concrete slab (i.e., concrete halfspace), becomes equal to the total loss given by (10). The analysis of the concrete slab case is performed because wave reflections on both sides of a concrete slab affect the power transfer versus the case of concrete half-space, where reflections at the air-concrete interface occur only once. Also, concrete slabs are widely used in construction, i.e., buildings, bridges, etc. Fig. 3 illustrates the total power loss for four different concrete slab thicknesses and for a concrete humidity condition of 12%. These results indicate that the losses inside the optimum frequency range of 20–80 MHz are 5–14 dB smaller than the losses at the lowest or highest frequencies of our analysis. Also, the oscillatory behavior of the total loss is expected due to the superposition of primary and reflected waves. Therefore, use of the optimum frequency range can lead in significant performance improvement of the power transmission. B. Power Attenuation for Oblique Incidence Fig. 2. Power attenuation for normal incidence and different humidity conditions, h, when d = 0:25 m. (a) Transmission loss. (b) Propagation loss. (c) Total loss.

To examine transmissions at oblique angles of incidence for a general wave polarization, it is convenient to decompose the electric field into perpendicular and parallel

(12) (13)

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Fig. 3. Total attenuation for plane wave penetrating concrete slab at normal incidence when h = 12%, and d = 0:25 m.

components and analyze them separately. The transmission coefficient for parallel and perpendicular polarization and is given by , respectively, where is the angle of incidence and is the angle of transmission defined by Snell’s law of refraction [15], . Following similar procedure to the one outlined for the normal incidence calculation, we use (8) to calculate the transmission loss for various incidence angles. Following similar steps to the normal incidence case of concrete half-space, the average total loss across all incidence angles ranging from 0 to 89 degrees is calculated for frequencies from 1 MHz to 1 GHz. The normal incidence corresponds to 0 degree. Fig. 4 plots the average total loss versus frequency for both polarizations. Based on Fig. 4, it can be concluded that the average total loss for an oblique incident plane wave is larger than the normal incidence case, as expected, but both cases exhibit similar variation. For example, the average total loss for parallel polarization in the 20–80 MHz frequency range is approximately 3 dB larger than the total loss for normal incidence in the case of 12% humidity. Therefore, when the wave is normally incident and the frequency is between 20–80 MHz, an embedded sensor will receive significantly larger power than the other cases we examined. Also, for the case of oblique incidence we see that the optimum frequency range still matches the optimum range that we identified for the normal incidence (20–80 MHz).

III. DIPOLES FOR EMBEDDED SENSORS In the previous section we identified an optimum frequency range of 20–80 MHz that provides minimum transmission losses. In this section, antenna analysis is performed in order to validate and confirm this important finding. All simulations are performed with either Ansoft Designer or Nexxim. Also, all Ansoft Designer simulations use the extended Debye model for concrete (as discussed in Section II).

Fig. 4. Average total loss across all incidence angles from 0 to 89 degrees when d = 0:25 m. (a) Parallel polarization. (b) Perpendicular polarization.

Fig. 5. Setup of two dipoles in the air-concrete model. (L = 4 m, H = =2 at the dipole resonant frequency, D = 0:25 m).

A. Coupling Between Two Dipoles Specifically, the coupling between two resonant half-wavelength dipoles is analyzed using Ansoft Designer. One dipole resides in air with length of , picked based on [17], and the other dipole resides inside the concrete with length that was picked based on our simulations to of match the resonant frequency of the dipole in air, as shown is the permitin Fig. 5; is the free-space wavelength and tivity of concrete. Fig. 6 illustrates the results of our analysis

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Fig. 6. Coupling for different concrete humidity conditions for dipoles resonating at 500 MHz (Fig. 5 setup).

Fig. 7. Dipoles’ coupling for different resonant frequencies when the humidity condition of concrete is 12% (Fig. 5 setup).

where the frequency range is normalized with the resonant frequency of the dipole antennas (the dipole in air and the dipole in concrete). Specifically, Fig. 6 plots the coupling difference between various humidity conditions and the dry concrete (0.2% humidity) for dipoles with a resonant frequency of 500 MHz. It should be pointed out that in our case we need to maximize the coupling (minimize the power losses) between the two dipoles in order to obtain optimized wireless communication between the two antennas. Also, we need to maximize the wireless power transfer between the two dipoles in order to charge the battery of embedded sensors in concrete. As expected, the coupling for lower humidity conditions is larger than the one for higher humidity conditions, as shown in Fig. 6. Specifically, the coupling of the two dipoles for 12% humidity is approximately 14.5 dB smaller than the coupling of two dipoles for dry concrete (0.2% humidity). The plane wave analysis predicts an 11.31 dB difference between the same corresponding humidity conditions, which is shown in Fig. 2. The difference between plane wave analysis and dipoles’ simulation results is due to the pattern and gain of the dipole antennas. Therefore, in order to design a transmitting-receiving power system for the air-concrete interface and maximize the power transfer between the two dipoles, we must take into account the humidity condition of concrete since it has detrimental effect to the coupling between the two dipoles. Also, a pair of dipoles (one residing in air and one residing inside the concrete, as shown in Fig. 5) with matching resonant frequencies is designed for various resonant frequencies, i.e., 70 MHz, 100 MHz, 200 MHz, 500 MHz and 1 GHz. Then, the coupling between the two dipoles is calculated for each case of resonant frequency. Fig 7 plots the difference between the coupling of two dipoles for each resonant frequency and the coupling of two dipoles for a resonant frequency of 70 MHz. This was done because 70 MHz resides in the optimum operation frequency range according to the results of the previous section. Fig. 7 illustrates that the coupling of the dipoles resonating at 70 MHz is larger than the coupling of the dipoles at other resonant frequencies (since the difference is positive). For example,

the coupling of the two dipoles resonating at 1 GHz is about 3.5 dB smaller than the coupling of the dipoles resonating at 70 MHz. This indeed validates our findings from the previous section, and indicates the existence of an optimum frequency range at 20–80 MHz. B. Effects of Rebars So far, we have presented results involving antennas in homogeneous concrete. However, in reality and in most applications reinforcement bars (rebars) are embedded in concrete to improve its mechanical stability. Rebars are metallic and therefore they affect the performance of antennas that are embedded in concrete as well as the wireless powering of sensors. Here, we present results describing the effects of rebars on coupling between a transmitting and a receiving dipole, as shown in Fig. 8. The material of the rebars is set to Steel-1008, whereas the grid size and the period lengths are shown in Fig. 8. First, the rebars are set 10 cm above the half-wavelength dipole that is embedded in concrete; rebars are located between the two dipoles, as shown in Fig. 8(a). Second, the rebars are set 10 cm below the half-wavelength dipole that is embedded in concrete; rebars are located below the two dipoles, as shown in Fig. 8(b). Fig. 9 illustrates the effects of rebars’ position on antenna coupling in the frequency range of 70 MHz to 1 GHz. It is observed that coupling decreased when rebars were inserted between the dipoles. Also, this coupling reduction is more significant at lower frequencies. For example, when the humidity condition of concrete is 12%, the coupling at 70 MHz reduced by approximately 16 dB when rebars were added between the dipoles, whereas the coupling at 1 GHz only reduced by 1 dB, as shown in Fig. 9(a). This is due to the fact that at lower frequencies the grid formed by the rebars creates a more effective electromagnetic shield because the wavelength is larger than the period of the rebar grid. Also, Fig. 9 shows that when rebars are set below the two dipoles, their coupling is larger at lower frequencies. In fact, at low frequencies, coupling for the case with rebars below the dipoles is even larger than the coupling for the

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Fig. 8. Setup for studying the rebar effects (L = 4 m, H = =2 at the dipole resonant frequency, D = 0:25 m, l = 120 mm, d = 100 mm, r = 7:95 mm). (a) Geometry side view of rebars located between the dipoles. (b) Geometry side view of rebars located below the dipoles. (c) Geometry top view.

case of no rebars. This happens because at lower frequencies, the wavelength is larger and the rebars form a more reflective element (compared to higher frequencies) under the dipole embedded in concrete, which in turn increases coupling. The oscillation behavior of the coupling is also expected due to the effects of the rebar layer on the impedance of the dipole inside the concrete [18]. In conclusion, Fig. 9 shows that for the case of rebars below the dipoles coupling peaks at low frequencies whereas in the case of rebars between the dipoles coupling exhibits peaks at higher frequencies (e.g., for the case of 5.5% humidity, maximum coupling occurs in the range of 500 MHz to 700 MHz). Therefore, in order to design a wireless powering system for an air-concrete interface with maximized power transmission, rebars must be taken into account since they significantly impact coupling between antennas. IV. RECTENNA DESIGNS FOR EMBEDDED SENSORS Following our analysis of dipole antennas for embedded sensors we proceed to discuss some practical considerations for embedded sensor antennas. Specifically, in this section, a rectification circuit along with the antenna (rectenna) is used to convert the received electromagnetic power to DC power for charging the sensor’s battery. A. Rectenna Design A pair of square patch antennas is designed for 70 MHz, which is included in our 20–80 MHz optimum frequency range. , [17], and one One patch is in air with length of , which was picked is in concrete with length of based on our simulations to match the resonant frequency of is the the patch in air; is the free-space wavelength and

Fig. 9. Dipoles’ coupling for different rebar conditions. (a) Humidity condition of concrete is 12%. (b) Humidity condition of concrete is 5.5%.

permittivity of the substrate. Each patch is designed on a thick Rogers RO4003 substrate and fed by a coaxial probe [17]. The geometry configuration of the patch antenna is shown in Fig. 10. This type of patch antenna is selected due to its low-mass and compact size [19]. Specifically, we simulate the wireless power transfer system consisting of two patch antennas operating at 70 MHz buried at m inside concrete with 0.2% humidity and a depth m. The patch antenna located for a distance in air is connected to a 1 W power source, and the patch that is embedded in concrete is connected to a rectifier circuit to form the rectenna, as shown in Fig. 11. This system is analyzed using NEXIMM software. The rectifying circuit consists of one HSMS-2850 Schottky detector diode, a 6 nF capacitor and a 100 Ohm load. The power delivered to the load is 56.6 mW, which is equivalent to a wireless power transfer efficiency of 5.66%. In order to optimize the efficiency of the wireless powering system we design matching circuits for the antennas and the rectifier. Fig. 12 illustrates the complete circuit layout where the

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nor voltage regulator need to be incorporated into the circuit for this type of battery [21]. Also, the size of this battery is suitable for nodes embedded in the concrete. Assuming that we use the same nodes as in [23], the rechargeable button cell battery consumes 1.25 mAh per day and can work 64 days without charging while in sleep mode. In working days, this battery consumes 6.9 mAh per day and can work for only 11 days with 5.8% duty cycle. This battery can be fully charged using our wireless powering system of Fig. 12, which has 25.9 mA output current, in 3.1 hours. V. CONCLUSION Fig. 10. Setup of two patches in the air-concrete model. (L at the patch resonant frequency, D = 0:25 m).

= 4 m, H = =2

Fig. 11. Wireless powering system setup: C = 6 nF, D is a HSMS-2850 Schottky detector diode, load = 100 and source is a power source with 1 W output and 50 internal resistor.

The air-concrete plane wave model is used to calculate the transmission loss, propagation loss and total loss for normal incidence and oblique incidence. These losses are related to the electrical properties of concrete, operational frequency, and the incidence angle. For sensors that are embedded in medium depths, maximum power is received when the transmitted wave is normally incident at the air-concrete interface and the operational frequency is between 20–80 MHz. Therefore, sensors embedded in concrete operating in the optimum frequency range will be able to efficiently harvest signals in the bands of shortwave radio and VHF TV. Also, our results illustrate that the moisture content of concrete, as well as the period and location of the reinforced bars should be carefully considered when designing the wireless power transfer system. Therefore, future work will include detailed analysis of the effects of various rebar configurations. REFERENCES

Fig. 12. Wireless powering system setup: L = 100 nH, C = 30 pF, L = 60 nH, C = 25 pF, L = 70nH; C = 45 pF, C = 6 nF, D is a HSMS-2850 Schottky detector diode, Load = 100 and Source is the power source with 1 W output and 50 internal resistor.

two patches are both matched to 50 Ohm using matching networks. The rectifying circuit (diode, capacitor and load) is also matched to 50 Ohms. The matched system increases power delivered to the load to 67.1 mW, which is equivalent to an improved wireless power transfer efficiency of 6.71%. B. Rechargeable Battery A common problem for energy harvesting devices is that the amount of energy harvested must be sufficient to power the electronic device of interest. For our application, approximately a power of 50 mW is required to run commercial sensor nodes [20] and we will use a rechargeable battery as a power source. The 80 mAh Nickel-metal hydride button cell battery [21] is specifically chosen to power the wireless sensor node embedded in concrete since it can withstand more harsh environments and it offers more energy per unit volume than the nickel-cadmium type one [22]. It should be noted that neither a charge controller

[1] J. T. Bernhard, K. Hietpas, E. George, D. Kuchma, and H. Reis, “An interdisciplinary effort to develop a wireless embedded sensor system to monitor and assess corrosion in the tendons of prestressed concrete girders,” in Proc. IEEE Topical Conf. Wireless Commun. Tech., Oct. 2003, pp. 241–243. [2] H. C. Rhim, “Condition monitoring of deteriorating concrete dams using radar,” Cem. Concr. Res., vol. 31, pp. 363–373, 2001. [3] J. P. Lynch, K. H. Law, E. G. Straser, A. S. Kiremidjian, and T. W. Kenny, “The development of a wireless modular health monitoring system for civil structures,” in Proc. MEDAT Workshop, Nov. 2000, pp. 1–4. [4] E. Sazonov, H. Li, D. Curry, and P. Pillay, “Self-powered sensors for monitoring of highway bridges,” IEEE Sens. J., vol. 9, no. 11, pp. 2422–1429, Nov. 2009. [5] B. Carkhhuff and R. Cain, “Corrosion sensors for concrete bridges,” IEEE Instrum. Meas. Mag., vol. 6, no. 2, pp. 19–24, Jun. 2003. [6] B. Jiang, J. R. Smith, M. Philipose, S. Roy, K. Sundara-Rajan, and A. V. Mamishev, “Energy scavenging for inductive coupled passive RFID systems,” IEEE Trans. Instrum. Meas., vol. 56, no. 1, pp. 118–125, Feb. 2007. [7] M. M. Andringa, D. P. Neikirk, N. P. Dickerson, and S. L. Wood, “Unpowered wireless corrosion sensor for steel reinforced concrete,” in Proc. IEEE Sens., Oct. 2005, pp. 155–158. [8] J. J. Casanova, Z. N. Low, and J. Lin, “A loosely coupled planar wireless power system for multiple receivers,” IEEE Trans. Ind. Electron., vol. 56, no. 8, pp. 3060–3068, Aug. 2009. [9] K. M. Z. Shams, M. Ali, and A. M. Miah, “Characteristics of an embedded microstrip patch antenna for wireless infrastructure health monitoring,” in Proc. IEEE AP-S Intel. Symp., 2006, pp. 3643–3646. [10] M. Ali, G. Yang, and R. Dougal, “A new circularly polarized rectenna for wireless power transmission and data communication,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 205–208, Jul. 2005. [11] K. M. Z. Shams and M. Ali, “Wireless power transmission to a buried sensor in concrete,” IEEE Sens. J., vol. 7, no. 12, pp. 1573–1577, Dec. 2007. [12] X. Jin and M. Ali, “Reflection and transmission properties of embedded dipoles and PIFAs inside concrete at 915 MHz,” in Proc. IEEE AP-S Intel. Symp., 2009, pp. 1–4.

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[13] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. Hoboken, NJ: Wiley, 2005. [14] L. Sandrolini, U. Reggiani, and A. Ogunsola, “Modeling the electrical properties of concrete for shielding effectiveness prediction,” J. Phys. D: Appl. Phys., vol. 40, pp. 5366–5372, 2007. [15] D. M. Pozar, Microwave Engineering, 3rd ed. Hoboken, NJ: Wiley, 2005. [16] K. G. Ayappa, H. T. Davis, G. Crapiste, E. A. Davis, and J. Gordon, “Microwave heating: An evaluation of power formulations,” Chem. Eng. Sci., vol. 46, no. 4, pp. 1005–1016, 1991. [17] J. D. Kraus and R. J. Marhefka, Antennas for All Applications, 3rd ed. New York: McGraw-Hill, 2002. [18] P. Raumonen, L. Sydanheimo, L. Ukkonen, M. Keskilammi, and M. Kivikoski, “Folded dipole antenna near metal plate,” in Proc. IEEE AP-S Intel. Symp., 2003, pp. 848–85. [19] J. Huang, Z. A. Hussein, and A. Petros, “A VHF microstrip antenna with wide-bandwidth and dual-polarization for sea ice thickness measurement,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2718–1722, Oct. 2007. [20] N. G. Elvin, N. Lajnef, and A. A. Elvin, “Feasibility of structural monitoring with vibration powered sensors,” Smart Mater. Structu., vol. 15, no. 4, pp. 977–986, 2006. [21] H. A. Sodano, G. E. Simmers, R. Dereux, and D. J. Inman, “Recharging batteries using energy harvested from thermal gradients,” J. Intel. Mat. Syst. Str., vol. 18, no. 1, pp. 3–10, Jan. 2007. [22] P. Ruetschi, F. Meli, and J. Desilvestro, “Nickel-metal hydride batteries. The preferred batteries of the future?,” J. Power Sources, vol. 57, pp. 85–91, 1995. [23] A. Mianwaring, J. Polastre, R. Szewczyk, D. Culler, and J. Anderson, “Wireless sensor networks for habitat monitoring,” in Proc. WSNA, 2002, pp. 1–10.

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Shan Jiang (S’11) received the B.S. and M.S. degrees in electrical engineering from Tianjin Polytechnic University, Tianjin, China, in 2005 and 2008, respectively, and is currently working towards the Ph.D. degree in the Department of Electrical Engineering, Florida International University, Miami. Her research interests include electromagnetic wave propagation, antennas and RF circuits.

Stavros V. Georgakopoulos (S’93–M’02–SM’11) received the Diploma in electrical engineering from the University of Patras, Patras, Greece, in 1996, and the M.S. and Ph.D. degrees in electrical engineering from Arizona State University (ASU), Tempe, in 1998, and 2001, respectively. From 2001–2007 he held a position as Principal Engineer at the Research and Development Department of SV Microwave, Inc., where he worked on the design of high reliability passive microwave components, thin-film circuits, high performance interconnects and calibration standards. Since 2007, he has been with the Department of Electrical and Computer Engineering, Florida International University, Miami, where he is now Assistant Professor. His current research interests relate to applied electromagnetics, antennas, wireless communications and wireless sensors.

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Portable Real-Time Microwave Camera at 24 GHz Mohammad Tayeb Ghasr, Member, IEEE, Mohamed A. Abou-Khousa, Senior Member, IEEE, Sergey Kharkovsky, Fellow, IEEE, R. Zoughi, Fellow, IEEE, and David Pommerenke, Senior Member, IEEE

Abstract—This paper presents a microwave camera built upon a two-dimensional array of switchable slot antennas. The camera borrows from modulated scattering techniques to improve isolation among the array elements. The camera was designed to measure vector electric field distribution, be compact, portable, battery operated, possess high dynamic range, and be capable of producing real-time images at video frame-rate. This imaging system utilizes PIN diode-loaded resonant elliptical slot antennas as its array elements integrated in a simple and relatively low-loss waveguide network thus reducing the complexity, cost and size of the array. The sensitivity and dynamic range of this system is improved by utilizing a custom-designed heterodyne receiver and matched filter for demodulation. The performance of the multiplexing scheme, noise-floor and dynamic range of the receivers are presented as well. Sources of errors such as mutual-coupling and array response dispersion are also investigated. Finally, utilizing this imaging system for various applications such as 2-D electric field mapping, and nondestructive testing is demonstrated. Index Terms—Electric field mapping, microwave camera, nondestructive testing, real-time imaging, switchable slot antenna.

I. INTRODUCTION

T

HE utility of microwave and millimeter wave imaging techniques has been successfully demonstrated for a variety of applications including: nondestructive testing and evaluation (NDT&E) of materials and composite structures [1], [2], medical imaging [3]–[6], and security applications [7]–[9], to name a few. The successes associated with these works is partially due to the fact that electromagnetic waves, at microwave (300 MHz–30 GHz) and millimeter-wave (30 GHz–300 GHz) frequencies, penetrate a wide range of

Manuscript received March 10, 2011; revised July 11, 2011; accepted August 06, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported in part by a grant from NASA Marshall Space Flight Center (MSFC), Huntsville, AL. M. T. Ghasr and R. Zoughi are with the Applied Microwave Nondestructive Testing Laboratory (amntl), Missouri University of Science and Technology, Rolla, MO 65409 USA (e-mail: [email protected]; [email protected]). M. A. Abou-Khousa was with the Applied Microwave Nondestructive Testing Laboratory (amntl), Electrical and Computer Engineering Department, Missouri University of Science and Technology, Rolla, MO 65409 USA. He is now with the Imaging Research Laboratories, Robarts Research Institute, The University of Western Ontario, London, ON N6A 5K8, Canada (e-mail: [email protected]). S. Kharkovsky is with Civionics Research Centre, School of Engineering at the University of Western Sydney, Penrith, NSW 2751, Australia (e-mail: [email protected]). D. Pommerenke is with the Electromagnetic Compatibility Laboratory, Electrical and Computer Engineering Department, Missouri University of Science and Technology, Rolla, MO 65409 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173145

optically-opaque and non-conducting materials, such as various composites, ceramics, concrete, wood, clothing, and interact with their interior structures. Signals at microwave and millimeter-wave frequencies are non-ionizing and are not considered to be sources of hazardous radiation, leading to their ever-increasing utility in a wide variety of diverse imaging applications. Moreover, the relatively large available signal bandwidth at these frequencies, enable the possibility of producing three-dimensional (3-D) images [7], [8]. Microwave imaging is based on collecting the vector (i.e., magnitude and phase) scattered electric field distribution from an object over a known two-dimensional (2-D) plane or ideally over a surface enclosing the object (e.g., a sphere). This may be accomplished by mechanically raster scanning a transceiver over the object [10], or irradiating the object with an incident wave and then collecting the scattered electric field distribution over a known surface (i.e., 2-D plane) [4], [5]. Mechanical raster scanning is an established and accurate method due to the absence of errors which otherwise exists in imaging array configurations such as mutual coupling and variation in the response of the array elements. In addition, such systems do not suffer from the required array-element spacing constraint. However, the drawback of mechanical raster scanning is that it commonly requires a relatively long time to collect the electric field data [10]. For reasons that are rather obvious, the trend in microwave imaging has been to move towards real-time imaging capability in the form of measuring the vector electric field distribution over a receiving antenna array. Microwave and millimeter wave imaging systems that measure the coherent spatial scattered field distribution directly, using a multiplexed array of antennas, commonly incorporate imaging algorithms such as those derived from synthetic aperture radar (SAR) techniques to back-propagate the electric field to the scattering object [7], [8], or use algorithms based on reconstruction techniques to obtain information about the geometrical and/or dielectric distribution of the scattering object [5], [11]. Nowadays, custom-designed microwave and millimeter wave imaging systems are capable of real-time image production while producing images with relatively high spatial resolution. The high resolution feature is obtained in the near-field of the array [7], [8], [12], while the depth-of-focus is aperture-limited, i.e., limited by the size of the array [12]. In the far-field of the array, these systems may operate as phased-arrays to enable narrow angular scanning in their field-of-view. In this case the resolution becomes a function of the array beamwidth, which is also aperture limited. There are many challenges in designing systems that utilizes SAR imaging, stemming from the requirements associated with a or smaller array tightly-spaced measurement grid (e.g., element spacing) dictated by the Nyquist sampling criterion.

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GHASR et al.: PORTABLE REAL-TIME MICROWAVE CAMERA AT 24 GHz

The array elements of imaging arrays are typically multiplexed, at the RF level into a single receiver. There are two conventional methods for this purpose. The first method utilizes switching capabilities in the array (switched-array) to route the signal from an array element to the transceiver [8]. Switchedarray systems utilize a set of RF switches to multiplex the signal, from each array element, with a receiver that performs the task of coherent detection. At the upper end of microwave frequency range RF switching becomes expensive, and bulky. Another challenge is to design small, yet efficient array elements. The dimensions of the array elements must be much smaller than in order to fit in a tightly-spaced 2-D grid. To date, the development of large-scale 2-D imaging arrays have been limited due to these challenges. At high frequencies exceeding 20 GHz, only 1-D successful imaging systems based on switched-array have been developed [8]. Although as mentioned earlier that an RF-multiplexed array of antennas mimics the performance of a mechanically scanned system, practical multiplexing requires high isolation between the array elements through the switching network, and very low signal loss (i.e., insertion loss) between the array elements and the receiver. Any amplitude or phase dispersion over the array elements must be properly measured and calibrated for. The second method utilizes modulated scattering technique (MST) to “spatially tag” the scattered signal at specific locations within a collector array. The use of MST for electric field measurements was first introduced in 1955 [13]. Since then, this method has been widely implemented in 1-D and 2-D configurations using various linear scatterers, such as small dipole antennas, and has been extensively used for antenna pattern and radar cross section measurement in addition to some investigations involving medical applications [14]. Several MST-based 1-D imaging systems have been designed and built with an array of sub-resonant dipoles at frequencies up to 18 GHz [3]–[5], [14]. The use of MST overcomes some of the difficulties associated with switched arrays (i.e., bulkiness and isolation) by modulating the scattered signal with a low-frequency modulating signal. Using MST, the measured signal is uniquely distinguished from any other signal present at the receiver, and is therefore spatially localized to the probe location (which is known) by utilizing proper modulation and demodulation techniques. The main advantage of MST is that multiplexing is performed at a relatively very low frequency, typically 100’s of KHz. Traditional MST suffers from several limitations [14]–[16]. The commonly used small dipole antenna, provides for very small modulation depth, defined as the power ratio of the modulated signal to the incident signal, in the order of 40 dBc to 70 dBc [3], [15]. This significantly limits the sensitivity and dynamic range of the overall system. Another limitation associated with MST is the signal transfer from the scatterer to the receiver. Spatial collection schemes or passive combiners are lossy solutions, lowering the overall system dynamic range [14], [17]. Furthermore, the mutual coupling among the array elements (e.g., dipoles) can significantly reduce the system dynamic range. These problems become even more significant and challenging at higher frequencies such as those in the millimeter-wave region [16], [18].

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This paper describes a novel design for a real-time imaging system. This design overcomes many of the mentioned limitations associated with the current imaging systems. It is a combination of a switched RF multiplexed system and MST measurement approaches, utilizing a PIN diode-loaded switchable resonant slot antenna as its array elements [17], [18] to perform the switching function directly at the array element locations rather than using an RF switching network. Subsequently, the isolation among the array elements, and consequently the dynamic range is markedly improved. The cornerstone of the imaging array used in this design is a low-loss waveguide network which routes the modulated signal from each array element to a set of four receivers. Each receiver down-converts the modulated signals to an IF signal where a sensitive demodulator is used to match filter the modulated signal. This unique design results in a system with a relatively high dynamic range, and enables image production at video frame rate rendering it a real-time imaging system or “camera.” This paper describes the design and development of a novel portable mm mm mm 2-D microwave camera operating at 24 GHz. This camera, which is primarily a 2-D coherent vector electric field mapping device, provides for adequate spatial sampling, reasonable aperture size, relatively large dynamic range, and video frame-rate image production. The camera collector consists of a 24 24 (576 elements) array of switchable slot antennas, spaced by , where is the free-space wavelength. Each element is loaded with a PIN diode facilitating fast switching for the purpose of real-time mapping of the electric field distribution at the aperture of the camera. The various design and construction aspects of this system were optimized and integrated to produce a standalone imaging system. Description, optimization steps, and design features of the various components of this design are presented here. The performance of the multiplexing scheme, noise-floor and the dynamic range characteristics of the receivers are also investigated and summarized. Finally, the utility of this camera for imaging several objects is illustrated. II. MICROWAVE CAMERA DESIGN A general schematic of the microwave camera is shown in Fig. 1, in which many individual antennas are used to collect the electric field distribution on a predetermined 2-D measurement plane. The combiner along with the antenna array perform the task of spatial multiplexing required to spatially sample the electric field distribution and route the corresponding signals to a high dynamic range RF receiver. Subsequently, the RF receiver provides the coherent vector electric field information from each antenna in this array. The RF circuitry also generates the incident electric field necessary for illuminating the object to be imaged. The spatial multiplexing and the RF circuits are synchronized by a processor that collects the data. Additional steps performed by the processor, such as any required calibration or signal processing, make these data representative of the scattering source or the object being imaged. The basic designs associated with the RF circuitry and the processors are well established, yet they must be optimized for this specific application. The array designed for this type of electric field distribution

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Fig. 2. Top and bottom views of the resonant slot antenna as manufactured on a PCB.

Fig. 1. General schematic of the microwave camera.

measurement is commonly referred to as a “retina,” for its similarity to the function of a human eye retina [4]. The proof-of-concept design described in [17] was founded on efficient spatial multiplexing of scattered electric field over the 2-D space of slot array. The analysis showed the possibility of effective and rapid “electrical scanning” of the array. The efficacy of that design was mainly due to employing PIN diode loaded resonant slot antennas instead of the commonly used linear scatterers (e.g., dipole antennas). However, the design associated with that system also exposed several design deficiencies requiring significant improvement or redesign. Consequently, a new approach, to the overall design, was considered to not only address those deficiencies, but to also address other issues related to the extension of the collector design to a more viable imaging system. The utilization of a rectangular waveguide network in the collector, instead of the lossy free-space collection system used in [17] is one of the main features of this imaging system. Rectangular waveguides are widely used low-loss transmission lines, and can therefore improve the efficiency of the overall system by reducing attenuation and radiation losses of the desired electric field picked up by the resonant slot array elements. In addition, the use of rectangular waveguides provides significant control over redirecting, redistributing, and adding signals from a large number of slot antennas, as will be seen later. Integrating the high-frequency transceiver components into the frame of the retina, beside its appealing compactness, eliminates errors caused by long and flexible coaxial transmission lines that are otherwise required. Several aspects of this design were optimized through extensive simulations and experiments. The design features and properties of each component of this imaging system are described in details in the following sections. Hereon, the term retina refers to the unit consisting of the slot antenna array and the collector, while the term collector refers to the waveguide network collecting the signal from the slots and combining them. A. Slot Antenna Fig. 2 shows the outlines of the slot antenna as they appear on the two sides of a PCB. The design and the electromagnetic

properties of this slot antenna are presented in detail in [18] and will not be repeated here. Briefly, this slot antenna operates as a switch when loaded with a PIN diode (MA4AG907) [19] with a very high radiation efficiency ( 97%) when the PIN diode is reverse or zero biased (turned OFF). Conversely, when the PIN diode is forward biased (or ON) the slot becomes reflective with a very small leakage (i.e., slot is closed) [18]. These attributes make the slot a good candidate for the waveguide-fed imaging array considered here. Furthermore, the slot can be easily integrated into a rectangular waveguide wall while maintaining efficient signal coupling. Fig. 3 shows the simulated (using CST-MWS [20]) and measured reflection coefficient, looking into the waveguide for the ON and OFF states of the PIN diode. The slot prototype was simulated and then manufactured on a 0.5 mm-thick Rogers RO4350 laminate [21]. When the slot is open and radiating (i.e., the PIN diode is OFF), the magnitude of the reflection coefficient is less than 15 dB. However, when the slot is closed (i.e., the PIN diode is ON), the reflection coefficient becomes very high ( 0.2 dB). The measured slot response shows a slight shift in the resonant frequency coupled with an increase in reflection coefficient when the slot is open. The slight shift in the resonant frequency is due to slight inaccuracies in electrical dimensions of the manufactured slot. The increase in the measured reflection coefficient when the diode is OFF, and the increased measured leakage when the diode in ON is due to the non-perfect contact between the slot PCB and the testing waveguide flange. This was experimentally verified by increasing or reducing the pressure on the slot PCB. B. Collector Design The retina size of 152 mm 152 mm accommodates an array of 24 rows by 24 columns with an interspacing of 6.25 mm corresponding to at 24 GHz. The proof-of-concept design in [17] utilized a free-space collection scheme where an antenna placed behind the retina was used to collect the modulated scattered signal. For that relatively small retina of 37.5 mm by 31.25 mm in [17], the signal experienced a loss of nearly 25 dB when traveling from a slot at the retina to the collector antenna. If similar design is used for a larger array, this loss will significantly increase, rendering the collection approach inefficient and effectively unusable. A transmission line was deemed necessary to guide the collected scattered signals into the receiver. Rectangular waveguides are attractive for use at microwave and millimeter wave frequencies for their low loss transmission properties. Moreover, a slot antenna can be readily integrated into the sidewall of a rectangular waveguide by matching the polarization of the slot antenna to the vector surface current on

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Fig. 3. Measured and simulated magnitude of reflection coefficient for the resonant PIN diode-loaded slot antenna.

the waveguide wall. A single waveguide can feed or reciprocally collect signal form many slots in a corporate-feed style. For example, a single waveguide can be meandered behind the array to collect the signal from all slots in an array. However, the total loss in the waveguide collector network is proportional to the number of slot antennas on the waveguide wall since each PIN diode-loaded slot antenna, when closed, contributes a small amount of unwanted leakage/radiation loss ( 0.28 dB per slot), resulting in a relatively high level of loss in the waveguide. Consequently, the collector waveguides were limited to one waveguide per row of the retina, resulting in 24 collector waveguides in this design. Finally, the end-ports of these waveguides were combined to route the collected signals to the receiver. Another issue to be considered when placing the slots on the waveguide walls is the space management, given the array element interspacing requirement of , and the variety of options (e.g., broad or narrow wall) for slot placement on the waveguide walls. The broad dimension of the waveguide must be larger than or else the waves do not propagate inside it. Therefore, it is physically impossible to place two waveguides side-by-side with their broad sidewalls in one plane with a center-to-center spacing of . This forces the placement of the slots on the narrow sidewall of the waveguide. A standard K-band waveguide has a narrow dimension of 4.3 mm (slightly larger than the slot height), which leaves an adequate 1.95 mm of wall thickness between each two waveguides when placed at 6.25 mm distance from each other. Fig. 4(a) shows the aluminum block in which the three walls of 24 parallel waveguides are machined. This serves as the signal collection network. The slot array is then fabricated on a PCB constituting the fourth wall of the waveguides. The design of the slot antenna array will be described in details in the next section. Fig. 4(b) shows a picture of the assembled retina, with the slot array PCB mounted, using an aluminum rim on top of the waveguide network. The PCB is connected to the waveguide array using conductive epoxy, ensuring no signal leakage or coupling between adjacent waveguides. The rim on top of the PCB serves two purposes. First, it provides for a

Fig. 4. (a) Aluminum block with 24 parallel waveguides, and (b) assembled retina.

secure mounting of the slot array PCB onto the base. Secondly, it completes the flange on the side of the retina for terminating these waveguides into signal combiners, as shown. C. Slot Antenna Array Mounting the slot on the aperture of a standard waveguide is an effective method for testing and optimizing its electromagnetic and switching performance characteristics. However, in the retina, the slot is placed on the narrow side-wall of a rectangular waveguide. Subsequent to establishing the slot dimensions as explained earlier, the signal coupling and radiation properties of the slot, when placed on the narrow sidewall of a waveguide were simulated using CST-MWS. The structure simulated is shown in Fig. 5 with the waveguide outlines in white. The waveguide has three solid metallic walls and the fourth wall is created by the two-layer PCB containing the slot, as explained earlier. The bias structure was not considered in this simulation since prior extensive simulations showed that it does not adversely affect the electrical properties of the slot. Initially, simulations were performed with a slot placed on a standard K-band (WR-51) rectangular waveguide with dimensions of 10.7 mm by 4.3 mm. The simulated radiated power defined as , is shown in Fig. 6. This definition of radiated power is valid since the radiation efficiency of this slot is high. Fig. 6 shows that for a standard waveguide, the radiated power when the slot is open (i.e., when the diode is OFF) is less than 6 dB, and when the slot is closed (i.e., when the diode is ON) the radiated power (representing leakage) is less than 20 dB. While the leakage level is acceptable, the radiated power when the slot is open is lower than the ideal value of 3 dB. Reducing the broad dimension of the waveguide effectively addresses the issue of low radiated power level when using a standard waveguide. The frequency of operation (24 GHz) is at the higher end of K-band. Therefore, the waveguide

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Fig. 5. Simulated structure for the slot antenna on the narrow sidewall of waveguide.

Fig. 7. Ratio of the current on the narrow wall to current on the broad wall of standard and a modified waveguide.

D. Slot Array Design

Fig. 6. Simulated radiated power from PIN diode loaded slot antenna on the waveguide narrow sidewall.

broad dimension may be reduced to 6.25 mm before reaching the cutoff frequency of the waveguide [22]. Consequently, the waveguide broad dimension was reduced from the standard 10.7 mm to 7.7 mm bringing the cutoff frequency to 19.5 GHz, well below the frequency of operation. Looking at this structure from the radiated power point-of-view, in this modified waveguide case, the leakage when the slot is closed increases slightly. More importantly, the radiated power when the slot is open increases to 3.1 dB, which is twice as much compared to the standard waveguide case. In other words, approximately half of the power reaching the slot from port 1 is radiated. A 3 dB radiated power (50% total efficiency) is considered optimum in the sense that if the slot is used in the receiving mode (as it will be used in the microwave camera), half of the power reaching the slot will travel to each side of the waveguide and no power will be reflected back from the slot. The 3 dB increase in efficiency is due to the increase in the current intensity on the narrow wall (containing the slot) in comparison to the broad wall when the waveguide broad wall dimension in reduced. This phenomenon can be quantized by calculating the ratio of the current intensities on the two different waveguide walls [22], as shown in Fig. 7 and was also verified by simulation.

A close-up view of the slot antenna array is shown in Fig. 8(a). The array was fabricated on a two layer 0.5 mm-thick RO4350B laminate [21]. Since the back layer is not accessible in this design (because of the waveguide collection network), only the top layer is used for placing the biasing network. The 576 resonant slots antennas were placed in a 24 24 grid with center-to-center spacing of 6.25 mm . Matrix addressing scheme, commonly used in electronic displays, was used in this design. In a matrix-addressing scheme for an array, only addresses (or bias lines in this case) are needed. In this scheme, to address a slot at the intersection of a row and a column, its corresponding rows and columns are addressed. To perform the electrical scan of the retina, first a row is enabled by properly biasing the line, and then the various elements in that row are scanned by sequentially addressing the columns. Subsequently by going through all rows, a complete 2-D electrical scan of the retina is performed. By placing LEDs in the bias structure we combined the row and column bias lines at each junction before feeding them to the anode of the PIN diodes, as shown in Fig. 8(b). The LEDs are required since the cathode of the PIN diode is grounded in addition to the fact that the slot must be idly closed (i.e., PIN diode is normally ON). Thus, to address a particular slot, its corresponding row and column lines are pulled low (i.e., turned off). The LEDs perform logic OR operation, that is the PIN diode will only be turned OFF if both of its corresponding row and column lines are pulled low. Low profile (0402 size) surface-mount red and green LEDs were used for the column and row address connections, respectively. The row address lines were etched as 0.127 mm-wide horizontal lines onto the PCB top layer at midpoint between the rows. On the other hand, thin (0.127 mm-diameter) insulated wires were used for the column address lines, which were soldered at each slot location to its corresponding red LED. Slight disturbance on the slot antenna radiation properties are anticipated from the close proximity of the LEDs and the bias lines,

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Fig. 9. Schematic of the switched waveguide combiner.

Fig. 8. (a) Close-up views of the retina PCB showing the bias lines, bias LEDs, and the slots, and (b) bias circuit.

however a compromise is made between this disturbance and the low cost and ease of fabrication of this current design. E. Combiner Waveguide The 24 collector waveguides, shown in Fig. 4, are the first stage of guiding the signal from each slot to the receiver. The signal picked-up by each slot travels to both ends of its corresponding waveguide in the 24 waveguide collector. Several approaches were investigated for combining the output ports of these 24 waveguides into a single or few output ports. The schematic of the chosen combiner is shown in Fig. 9. This approach utilizes a switchable iris at each end of each collector waveguide. The iris is switchable and facilitates the routing of the signal from each collector waveguide to a set of four coaxial ports (two on each side). The coaxial ports are created by a quarter-wave monopole feed at each end of the combiner waveguide. The iris was also a resonant PIN diode-loaded elliptical slot similar to the retina slots, optimized to be placed in the wall between the waveguides. By properly switching the desired iris, the signal picked up by a slot on the array is routed to the four coaxial ports by opening the respective iris on each side on

the collector waveguide. Meanwhile, all other irises are maintained in a closed state to enhance the isolation within the array, and to reduce the loss in the combiner waveguides. Using this method, the maximum attenuation experienced by a signal before reaching the nearest port is estimated to be less than 12 dB. This attenuation is mainly due to the leakage in the array slots and the combiner irises. As mentioned, to ensure maximum signal transfer from the retina to the output ports, only one of the combiner irises should be open at a given time. In ideal situations when the irises have zero leakage, the connection between each collector waveguide and a combiner waveguide, is considered as a switchable E-plane Tee. This Tee configuration was extensively simulated using CST-MWS [20]. The switchable iris is a PIN diode-controlled resonant slot, similar in design to the slot of the retina and optimized for operating inside the Tee structure for the frequency of operation. The simulation results showed that when the switch is open, at the resonant frequency of 24 GHz, the power division in the Tee is 3.9 dB which comparing to the ideal value of 3 dB results in an insertion loss of 0.9 dB, while the return loss was 10 dB. On the other hand, when the switch is closed, the isolation is 20 dB. Moreover, the signal traveling through the collector waveguide experiences a small amount of loss. For the final application of designing a combiner for the microwave camera, 24 irises were manufactured on a single 0.5 mm-thick 2-layered PCB with RO4350B laminate. This board was sandwiched between the combiner waveguide and the 24 collector waveguides, as shown in Fig. 10. F. Signal Source and Receiver The transceiver design was based on a heterodyne scheme, where the first harmonic of a single side-band of the modulated signal is demodulated. This design scheme is illustrated in Fig. 11. The main RF source generates a signal, which is sent through a transmitting antenna to irradiate the target. The received signal, which is square-wave amplitude-modulated by the retina, is down-mixed to the IF stage where it is filtered and amplified. The LO frequency is set such that the fundamental harmonic on the lower sideband is mixed to the center

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Fig. 10. Combiner waveguide assembly.

frequency of the IF filter. The two spectra insets in Fig. 11 show the modulated signal before and after being down-converted to the IF frequency range. The narrowband IF filter passes this harmonic, while at the same time it attenuates the relatively strong carrier along with all other harmonics. Subsequently, the IQ demodulator provides in-phase (I) and quadrature (Q) outputs proportional to the real and imaginary parts of the electric field at the retina aperture. As mentioned in the previous section, due to the combiner design, this imaging system requires four such receiver channels. This scheme was first introduced in [23], in order to improve measurement accuracy by eliminating the quadrature error associated with the high-frequency components. In [23] the mixing was performed down to baseband, such that the IF frequency is the fundamental frequency of the modulating signal. In the design presented here the mixing is performed to a fixed IF frequency of 10.7 MHz irrespective of the modulation frequency. That is to say, given commercial off-the-shelf (COTS) components at standard IF frequencies (e.g., 10.7 MHz) the ability to independently select the modulation frequency provides for significant design flexibility. This design has several other advantages over the design introduced in [23]. First, the fixed IF frequency provides the flexibility of changing the modulation frequency without changing the IF stage hardware. Second, the higher IF frequency of 10.7 MHz used here, compared to the low IF frequency corresponding to the modulation frequency of less than 1 MHz used in [23], translates to lower flicker noise. Third, the higher IF frequency translates to larger frequency difference between the RF sources reducing the design complexity associated with the sources. There is one fundamental difference between this design and the more traditional MST receivers [14]; namely, utilizing the IQ demodulator to perform both the quadrature detection and matched filtering (i.e., demodulation). In this scheme the IQ demodulator serves as a quadrature lock-in amplifier, which performs the task of quadrature coherent detection on a single tone, which is the first harmonic of the square wave modulated signal. The disadvantage of this scheme is that capturing the first harmonic results in approximately 3.9 dB loss in power, 3 dB of which is for ignoring one side-band and another 0.9 dB accounts

Fig. 11. Schematic of the microwave camera including the RF sources and one channel of the receiver.

for ignoring the harmonics of the square wave signal. However, this scheme offers a good compromise between the loss in power and the following benefits. First, a heterodyne receiver provides higher dynamic range by eliminating quadrature errors associated with high frequency homodyne receivers. Second, the design provides flexibility in the choice of modulation frequency. Finally, the highly accurate IQ demodulator at the IF frequency also acts as a matched-filter eliminating the need for lock-in amplifiers for demodulation and thus reducing system complexity. Overall, the signal power loss is compensated for by the resulting higher dynamic range and sensitivity of the receiver and the ability to filter and amplify the IF signal. Another disadvantage of this design as compared to [23] is the requirement of an additional IF source. However at standard commercial IF frequencies, such as the 10.7 MHz used here, there is an abundance of commercially available phase-locked sources. Individually testing the various components of the receiver showed that the RF receiver front-end and the IQ demodulator both exhibited a dynamic range exceeding 90 dB. However, the overall dynamic range of the receiver is limited by signals passing through the IF bandpass filter (100 KHz bandwidth) including the noise generated by the leaked carrier. Consequently, the overall dynamic range of the receiver was experimentally measured. The retina was excluded from this measurement due to the difficulty in controlling the magnitude and phase of the incident electric field on each individual slot. Instead the measurements were performed using a standalone slot to create the

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Fig. 13. Pictures of the (a) front and (b) back of the camera showing the array at the front and the various RF and control components at the back.

III. RESULTS Fig. 13 shows pictures of the front and back of the assembled microwave camera. The major component of this microwave camera is the switchable imaging array. Performing the switching operation at the antennas rather than using bulky RF switches resulted in a thin array design with a final size of mm mm mm. The various RF and digital control circuitries are mounted on the back of the array in a multi-level stacked configuration. This configuration resulted in the total size of the microwave camera being mm mm mm, which makes it handheld and readily portable. This microwave camera provided real-time imaging at a video frame rate of 22 frames per second. This section will present the efficacy of this design in measuring electric field distribution including a few of imaging examples. A. Calibration and Electric Field Pattern Measurements Fig. 12. (a) Dynamic range of the receiver, and (b) phase error within the dynamic range.

modulated signal (i.e., replacing the retina slots). The electric field on the slot was controlled using laboratory equipment such as precision attenuators and phase shifters. Fig. 12 illustrate the receiver dynamic range and the accuracy by which phase of a signal may be measured. Fig. 12(a) shows the dynamic range of the receiver, indicating a linear response range of approximately 70 dB for each of the four channels. Fig. 12(a) shows the corrected signal output power after compensating for this IF gain. Fig. 12(b) shows that the phase measurement error within the dynamic range of the receiver is less than 10 degrees at the low end of its dynamic range and it is less than 2 degree at its upper half of its dynamic range for a range of approximately 50 dB. The receiver saturates when the input power level exceeds 35 dBm. The saturation is due to the high IF stage gain. The cumulative gain of the IF stage is 40 dB. This high gain is necessary to compensate for the losses in the array. Moreover, this high gain is readily achievable since other strong signals, that will otherwise saturate the IF amplifier and the IQ demodulator (e.g., the carrier), are filtered out. The ability to introduce this gain is crucial in compensating for the losses in the array, and hence utilizing the dynamic range of the receiver to the fullest extent.

Array dispersion, defined as the variation in the response of array elements to a uniform plane wave illumination [4], is the largest contributor to errors in mapping electric field distribution. To some extent these variations are due to inaccuracies in manufacturing of the slots, mounting the components on the array, and due to differences between nominal values of the component properties. However, experiments showed that these variations are typically very small. The largest variation in the response of the array elements comes from the signal path connecting each array element to the receiver. This path takes the signal through the collector waveguide, the iris, the combiner waveguide, and finally through a short cable connected to the receiver where in each portion of this path the signal undergoes attenuation and reflection. It was anticipated that signal amplitude will depend linearly (in dB) on the distance between the slot and the output port. Utilizing four ports, one at each corner of the array, alleviates some of this signal loss. In this case the signal experiencing the maximum attenuation will be from the slots in the middle portion of the collector array. Typically a response calibration is performed to correct for any distortions [24]. The response calibration is performed by illuminating the retina with a known electric field. In this case the known electric field was generated by an open-ended K-band rectangular waveguide aperture. Several measurements at various distances, between the camera and the waveguide covering a range of two wavelengths from 387 mm to 411 mm with steps of 2 mm, were performed. At these distances the magnitude

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Fig. 14. Measured versus simulated electric field distribution produced by a K-band open ended waveguide from a distance of 390 mm, (a) simulated magnitude (dB), (b) measured magnitude (dB), (c) simulated phase (deg), (b) measured phase (deg).

variations over the aperture of the retina does not exceed 1 dB. On the other hand the phase variations go through a full cycle. At each distance, the correction coefficients are computed by referencing the measured electric field pattern over the aperture of the camera to the simulated waveguide pattern. Subsequently, the corrected measurements from the four ports are combined using a maximum ratio combiner (MRC) [25]. The results of the various distances are averaged to remove the effect of any multiple reflection or placement errors. Subsequent to applying MRC, the maximum dispersion over the array was measured to be 15 dB. The maximum loss is suffered by the signal picked up by the middle slots, since they must travel the longest distance to reach the four receivers at the four corners of the array. As a result, in the worst case the overall dynamic range of the system is reduced by this loss. Fig. 14 shows the corrected measurement versus the simulated electric field distribution produced by the open-ended waveguide from a distance of 390 mm. This measurement was not part of the calibration set. The magnitude plot shows slight distortion. This distortion is very small and it is only noticed due to the small range of the actual electric field amplitude. The root mean squared error (RMSE) in the magnitude for this measurement is 0.025. The phase distribution on the other hand shows a strong resemblance to the simulated phase pattern. The RMSE was 3.5 degrees. Overall, these accuracies are considered very high for such a high-frequency system. Furthermore, for imaging applications and array processing techniques such as synthetic aperture radar algorithms, the overall phase pattern is much more important than the absolute error. Several other measurements of various electric field patterns were made and the RMSE compared to the simulated fields were computed. Overall, the magnitude measurements showed an RMSE of 0.03 compared to the simulated results, while phase measurement showed a RMSE of 13 degrees. These errors are quite acceptable since the experimental setups were po-

sitioned manually and the 13 degrees corresponds to less than 0.5 mm in position error at 24 GHz. Therefore, for all practical purposes, this retina measures the electric field pattern accurately. Further improvements would require a better calibration of the retina that takes into account and corrects for the limited isolation in the waveguide network and the very small mutual coupling among the slots which was fully studied in [26]. B. Imaging Results The microwave camera is a primarily a coherent electric field mapping device. When measuring the scattered electric field from an object, an image of that object may be reconstructed using techniques such as SAR imaging. This portable microwave camera possibly may be used in various applications involving nondestructive testing, medical imaging, security, and contraband detection. The video frame rate and small form-factor of the microwave camera, makes it a desirable candidate for high throughput applications (e.g., airport passenger security check-points, conveyed product inspection, etc.). The retina of this microwave camera is designed as a receiver requiring an external signal source to irradiate the target. This camera may be operated in two modes namely: reflection and through-transmission modes. In reflection mode, the transmitting antenna and the retina are both on the same side and the resulting image represents reflective properties of the object being imaged. In through-transmission mode, the object is placed between the transmitting antenna and the retina, resulting in an image corresponding to both the reflection and transmission properties of the object (i.e., attenuation and scattering). 1) Reflection Mode: Operating the microwave camera in reflection mode requires that the irradiating antennas (transmitters) and the receiving retina be both on the same side of the object being imaged. The choice and configuration of the illuminating antenna(s) plays a direct role on the fidelity of the

GHASR et al.: PORTABLE REAL-TIME MICROWAVE CAMERA AT 24 GHz

Fig. 15. Experimental reflection mode setup for imaging of a balsa wood sample containing a thin copper tape inclusion: (a) front view with a single transmitter, (b) side view with single transmitter, and (c) front view showing four transmitters.

resulting image. The downside of using a single or few transmitters is that the pattern of this external transmitter will affect the contrast of the resulting image. In an experiment, a mm mm mm balsa wood sample with a thin 6.35 mm 6.35 mm copper film inclusion was imaged in reflection mode, first using a single transmitter as shown in Fig. 15(a) and Fig. 15(b), and then using four transmitters, as shown in Fig. 15(c). The transmitters in this experiment were K-band open-ended rectangular waveguide aperture antennas. The sample was placed 150 mm away from the camera. SAR imaging technique [8] was then used to reconstruct the image of the balsa wood from its mapped scattered field. Fig. 16(a) shows the reconstructed image when using only one transmitting antenna. An indication of the copper film inclusion can be seen in the middle of the image. The dashed line indicates the boundary of the sample. The image also shows a strong specular reflection from its left edge, which is closest edge to the transmitter. The choice of irradiating antenna and its radiation pattern on the target affects the fidelity of the obtained image. Non-uniform illuminations and specular reflections may cause some areas of the target not to be illuminated properly. This problem is alleviated to some extent by using multiple transmitters. Fig. 16(b) shows the reconstructed image the sample when illuminated using four transmitting antennas from the four corners of the microwave camera as shown in Fig. 15(c). In this image, the sample is more uniformly illuminated and thus its image represents its shape and reflection properties to a better extent than the case of single antenna illumination, case in point that the balsa wood is a weak scatterer and the copper inclusion is a strong scatterer. Mono-static mode, in which each element is used as transmitter and receiver, is expected to produce higher quality images since the target is illuminated from various angles which greatly reduced distortions due to specular reflections [8]. 2) Through-Transmission Mode: In through-transmission mode, the camera is placed in front of a transmitting antenna and the object is placed in between the transmitter and receiver. Usually, a small transmitter antenna is used to create a fairly broad pattern, such that the object and the retina will be somewhat uniformly illuminated. A reference image without the presence of the target is needed to correct for the variation in the pattern of the transmitter antenna [17]. Fig. 17(a) shows the camera operation in the through transmission mode. In this case, the transmitter is a K-band open-ended rectangular waveguide antenna, and the target is a two layer balsa wood

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Fig. 16. SAR image of the balsa wood sample with copper inclusion in reflection mode: (a) using a single transmitter, and (b) using 4 transmitters.

Fig. 17. Imaging in through transmission mode of a balsa wood sample with a small rubber inclusion: (a) setup, and (b) SAR image.

composite with a small rubber inclusion. The images on the PC monitor show the raw magnitude, the raw phase and the SAR images obtained in real-time. Fig. 17(b) shows the SAR of this sample, clearly showing the rubber inclusion and the effect of the user’s hand holding the sample from the bottom corner and top left side. In this configuration, the target either attenuates, or re-scatters the microwave energy towards the retina, and for this reason the rubber inclusion and the edges of the sample are clearly seen in the focused image while the majority of the low permittivity balsa wood is seen transparent to microwave energy. IV. SUMMARY This paper presented design and performance of a portable mm microwave camera operating at 24 GHz based on an array of 576 switchable slot antennas and MST measurement techniques. Several aspects of this design resulted in substantially improving the overall system performance. Many optimization challenges were overcome that resulted in an efficient feed for the slots, practical slot addressing scheme, lowloss waveguide collector network, and a custom-designed heterodyne receiver. Many elements of this design were founded on extensive full-wave simulations and experimental verification. The improved array element (resonant slot) and the signal collection scheme, provide for stronger signals at the receiver thus enhancing the overall SNR and correspondingly the dynamic range of the system. As a microwave camera, this system enables real-time imaging at a video frame rate of 22 frames per second and is intended for real-time imaging applications. This frame rate was made possible by the relatively high modulation rate of 1 MHz. The design of the heterodyne receiver allowed for

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optimizing the modulation rate for the specific frame rate and dynamic range requirements. This microwave camera produces SAR images focused at the location of the object due to its accurate vector electric field mapping. In the through-transmission mode, the produced images were of high fidelity despite the relatively long operating wavelength. The design of this camera does not accommodate a desirable true mono-static reflection mode imaging. Utilizing simple transmitters placed around the camera, helped in producing images of simple structures in the reflection mode. However, specular reflection and non-uniform illumination prohibited the formation of meaningful images of complex targets in the reflection mode. A more practical one-sided system would require sufficient number of transmitters, or a suitably designed transmitting antenna, to illuminate the target uniformly from various angles. A short video demonstrating the functionality of this camera is available at [27].

REFERENCES [1] R. Zoughi, Microwave Non-Destructive Testing and Evaluation. The Netherlands: Kluwer, 2000. [2] S. Kharkovsky and R. Zoughi, “Microwave and millimeter wave nondestructive testing and evaluation—Overview and recent advances,” IEEE Instrum. Meas. Mag., vol. 10, no. 2, pp. 26–38, Apr. 2007. [3] J.-C. Bolomey and C. Pichot, “Microwave tomography: From theory to practical imaging systems,” Int. J. Imaging Syst. Technol., vol. 2, pp. 144–156, 1990. [4] A. Franchois, A. Joisel, C. Pichot, and J.-C. Bolomey, “Quantitative microwave imaging with a 2.45-GHz planar microwave camera,” IEEE Trans. Med. Imag., vol. 17, no. 4, pp. 550–561, Aug. 1998. [5] T. Henriksson, N. Joachimowicz, C. Conessa, and J.-C. Bolomey, “Quantitative microwave imaging for breast cancer detection using a planar 2.45 GHz system,” IEEE Trans. Instrum. Mea., vol. 59, no. 10, pp. 2691–2699, Oct. 2010. [6] M. Klemm, J. A. Leendertz, D. Gibbins, I. J. Craddock, A. Preece, and R. Benjamin, “Microwave radar-based differential breast cancer imaging: Imaging in homogeneous breast phantoms and low contrast scenarios,” IEEE Trans. Antennas Propagat., vol. 58, no. 7, pp. 2337–2344, Jul. 2010. [7] J. M. Lopez-Sanchez and J. Fortuny-Guasch, “3-D radar imaging using range migration techniques,” IEEE Trans. Antennas Propagat., vol. 48, no. 5, pp. 728–737, May 2000. [8] D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 9, pp. 1581–1592, Sep. 2001. [9] R. Solimene, F. Soldovieri, and G. Prisco, “A multiarray tomographic approach for through-wall imaging,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 4, pp. 1192–1199, Apr. 2008. [10] M. T. Ghasr, D. Pommerenke, J. T. Case, A. D. McClanahan, A. AflakiBeni, M. Abou-Khousa, S. Kharkovsky, K. Guinn, F. DePaulis, and R. Zoughi, “Rapid rotary scanner and portable coherent wideband Q-band transceiver for high-resolution millimeter wave imaging applications,” IEEE Trans. Instrum. Meas., vol. 60, no. 1, pp. 186–197, Jan. 2011. [11] M. Pastorino, Microwave Imaging. Hoboken, NJ: Wiley., 2010. [12] D. M. Sheen, D. L. McMakin, and T. E. Hall, “Near field imaging at microwave and millimeter wave frequencies,” in Proc. IEEE/MTT-S Int. Microw. Symp., Jun. 3–8, 2007, pp. 1693–1696. [13] J. H. Richmond, “A modulated scattering technique for measurement of field distributions,” IRE Trans. Microw. Theory Tech., vol. 3, no. 4, pp. 13–15, Jul. 1955. [14] J.-C. Bolomey and G. E. Gardiol, Engineering Applications of the Modulated Scatterer Technique. Norwood, MA: Artech House, 2001. [15] J.-C. Bolomey, B. J. Cown, G. Fine, L. Jofre, M. Mostafavi, D. Picard, J. P. Estrada, P. G. Friederich, and F. L. Cain, “Rapid near-field antenna testing via arrays of modulated scattering probes,” IEEE Trans. Antennas Propagat., vol. 36, no. 6, pp. 804–814, Jun. 1988.

[16] M. A. Abou-Khousa and R. Zoughi, “Multiple loaded scatterer method for E-field mapping applications,” IEEE Trans. Antennas Propagat., vol. 58, no. 3, pp. 900–907, Mar. 2010. [17] M. T. Ghasr, M. A. Abou-Khousa, S. Kharkovsky, R. Zoughi, and D. Pommerenke, “A novel 24 GHz one-shot, rapid and portable microwave imaging system,” in Proc. IEEE I2MTC, Victoria, Canada, May 12–15, 2008, pp. 1798–1802. [18] M. A. Abou-Khousa, M. T. Ghasr, S. Kharkovsky, D. Pommerenke, and R. Zoughi, “Modulated elliptical slot antenna for electric field mapping and microwave imaging,” IEEE Trans. Antennas Propagat., vol. 59, no. 3, pp. 733–741, Mar. 2011. [19] M/A-COM Technology Solutions [Online]. Available: http://www.macomtech.com/datasheets/MA4AGP907_FCP910.pdf [20] CST-Computer Simulation Technology [Online]. Available: http://www.cst.com. [21] Rogers Corp., RO4000 Laminates Datasheet [Online]. Available: http://www.rogerscorp.com/documents/726/acm/RO4000-Laminatesdata-sheet-and-fabrication-guidelines-RO4003C-RO4350B.aspx [22] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998. [23] J.-H. Choi, J.-I. Moon, and S.-O. Park, “Measurement of the modulated scattering microwave fields using dual-phase lock-in amplifier,” IEEE Antennas Wireless Propagat. Lett., vol. 3, pp. 340–343, 2004. [24] P. Garreau, K. Van’t Klooster, J. C. Bolomey, and D. Picard, “Modulated scattering techniques calibration procedure for a 2-D array,” in IEEE Int. Symp. Antennas Propagation Soci. (AP-S. 1992) Digest. Held in Conjunction With: URSI Radio Science Meeting and Nuclear EMP Meeting, Jul. 18–25, 1992, vol. 3, pp. 1550–1553. [25] G. L. Stuber, Principles of Mobile Communication, 2nd ed. Norwell, MA: Kluwer Academic Publishers, 2001. [26] M. T. Ghasr, “Real Time and Portable Microwave Imaging System,” Ph.D. dissertation, ECE Dept., MST, , Rolla, MO, 2009. [27] MissouriSandT, Nondestructive Testing [Online]. Available: http://www.youtube.com/user/MissouriSandT#p/u/23/PeS2SFNb6dY

Mohammad Tayeb Ghasr (S’01–M’10) received the B.S. degree in electrical engineering degree (magna cum laude) from the American University of Sharjah (AUS), Sharjah, in 2002, the M.S. degree in electrical engineering from the University of Missouri-Rolla, Rolla, in 2004 and the Ph.D. degree in electrical engineering from Missouri University of Science and Technology (Missouri S&T), MO, in 2009. Currently, he is an Assistant Research Professor with the Applied Microwave Nondestructive Testing Laboratory (amntl), Electrical and Computer Engineering Department, Missouri University of Science and Technology (Missouri S&T). His research interests include microwave and millimeter-wave instrumentation and measurement, RF circuits, antennas, and numerical electromagnetic analysis.

Mohamed A. Abou-Khousa (S’02–M’09–SM’10) was born in Al-Ain, UAE, in 1980. He received the B.S.E.E. degree (magna cum laude) from the American University of Sharjah (AUS), Sharjah, UAE, in 2003, the M.S.E.E. degree from Concordia University, Montreal, QC, Canada in 2004, and the Ph.D. degree in electrical engineering from Missouri University of Science and Technology (Missouri S&T), MO, in 2009. Currently, he is a Research Scientist with the Imaging Research Labaratories at Robarts Research Institute, London, Ontario, Canada. Prior to his current position, he was RF research Engineer at Robarts. His efforts at Robarts are focused on developing RF hardware and system-level solutions to improve the performance of the high-field magnetic resonance imaging (MRI) scanners. His research interests include high count RF coil array design, millimeter wave and microwave instrumentation, numerical electromagnetic analysis, modulated antennas, and subsurface imaging.

GHASR et al.: PORTABLE REAL-TIME MICROWAVE CAMERA AT 24 GHz

Sergey Kharkovsky (M’01–SM’03–F’11) received the Diploma in electronics engineering from Kharkov National University of Radioelectronics, Kharkov, Ukraine, in 1975, and the Ph.D. and D.Sc. degrees in radiophysics from the Kharkov State University, Kharkov, Ukraine, in 1985, and from the Institute of Radio-Physics and Electronics (IRE) of National Academy of Sciences of Ukraine, in 1994, respectively. Currently he is an Associate Professor in the Civionics Research Centre, School of Engineering at the University of Western Sydney (UWS), Australia. Prior to joining UWS in July 2011 he was a Member of the Research Staff at IRE from 1975 to 1998, a Professor in the Electrical and Electronics Engineering Department at the Cukurova University, Adana, Turkey, from December 1998 to February 2003, and a Research Associate Professor in the Applied Microwave Nondestructive Laboratory (amntl), the Electrical and Computer Engineering Department at Missouri University of Science and Technology, formerly University of Missouri-Rolla, from March 2003 to June 2011. His current research interest is microwave and millimeter wave physics and techniques, sensor technologies, imaging, material characterization and nondestructive evaluation of composite structures. Dr. Kharkovsky is a member of the American Society of Nondestructive Testing (ASNT) and the ASNT University Programs Award Committee, and an Associate Editor for the IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT.

R. Zoughi (S’85–M’86–SM’93–F’06) received the B.S.E.E, M.S.E.E, and Ph.D. degrees in electrical engineering (radar remote sensing, radar systems, and microwaves) from the University of Kansas where from 1981 until 1987 he was at the Radar Systems and Remote Sensing Laboratory (RSL). Currently he is the Schlumberger Endowed Professor of Electrical and Computer Engineering at Missouri University of Science and Technology (Missouri S&T), formerly University of Missouri-Rolla (UMR). Prior to joining Missouri S&T in January 2001 and since 1987 he was with the Electrical and Computer Engineering Department at Colorado State University (CSU), where he was

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a professor and established the Applied Microwave Nondestructive Testing Laboratory (amntl). He held the position of Business Challenge Endowed Professor of Electrical and Computer Engineering from 1995 to 1997 while at CSU. He is the author of a textbook entitled Microwave Nondestructive Testing and Evaluation Principles (Kluwer, 2000), and the coauthor with A. Bahr, and N. Qaddoumi of a chapter on Microwave Techniques in an undergraduate introductory textbook entitled Nondestructive Evaluation: Theory, Techniques, and Applications edited by P.J. Shull (Marcel and Dekker, Inc., 2002). He is the coauthor of over 110 journal papers, 260 conference proceedings and presentations and 90 technical reports. He has ten patents to his credit all in the field of microwave nondestructive testing and evaluation. Dr. Zoughi is the Editor-in-Chief of the IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT. He is serving his second term as an at-large AdCom member of the IEEE Instrumentation and Measurement (I&M) Society and also serves as the society’s VP of Education. He is also an IEEE I&M Society Distinguished Lecturer. He has been the recipient of numerous teaching awards both at CSU and Missouri S&T. He is the recipient of the 2007 IEEE Instrumentation and Measurement Society Distinguished Service Award, the 2009 American Society for Nondestructive Testing (ASNT) Research Award for Sustained Excellence Award and the 2011 IEEE Joseph F. Keithley Award in Instrumentation and Measurement. He is also a Fellow of the American Society for Nondestructive Testing.

David Pommerenke (M’98–SM’03) received the Diploma in electrical engineering and the Ph.D. degree in transient fields of electrostatic discharge from the Technical University of Berlin, Berlin, Germany, in 1989 and 1995, respectively. After working at Hewlett Packard for five years he joined the Electromagnetic Compatibility (EMC) Laboratory at the University Missouri-Rolla (currently Missouri University of Science and Technology) in 2001 where he is a Professor in electrical engineering. He has published more than 100 papers and is inventor on 10 patents. He has been distinguished Lecturer for the IEEE EMC Society in 06/07. His main research interests are system level ESD, numerical simulations, EMC measurement methods and instrumentations.

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Is Orbital Angular Momentum (OAM) Based Radio Communication an Unexploited Area? Ove Edfors, Member, IEEE, and Anders J. Johansson, Member, IEEE

Abstract—We compare the technique of using the orbital angular momentum (OAM) of radio waves for generating multiple channels in a radio communication scenario with traditional multiple-in-multiple-out (MIMO) communication methods. We demonstrate that, for certain array configurations in free space, traditional MIMO theory leads to eigen-modes identical to the OAM states. From this we conclude that communicating over the sub-channels given by OAM states is a subset of the solutions offered by MIMO, and therefore does not offer any additional gains in capacity. Index Terms—Antenna arrays, antenna radiation patterns, channel capacity, free-space propagation, MIMO, orbital angular momentum, radio communication.

I. INTRODUCTION

I

T WAS RECENTLY shown that the photon orbital angular momentum (OAM) can be used in the low frequency radio domain and is not restricted to the optical frequency range [1]. These findings and the claimed prospects for opening a new frontier in wireless communications, with “promise for the development of novel information-rich radar and wireless communication concepts and methodologies” [1], motivates the investigation in this paper. Here we focus on the wireless communication aspects of [1] and the follow-up paper [2]. We start by identifying the conditions under which electromagnetic waves with specific OAM characteristics1 are generated in [1], [2] and continue by comparing with properties of traditional communication using multiple-in-multiple-out (MIMO) antenna systems [5]–[7]. We pay special attention to the singular value decomposition (SVD) based derivation of channel capacity for MIMO systems [8], when applied to MIMO systems under free-space propagation conditions. Spatial multiplexing under free-space conditions may seem like a contradiction, but this very concept has been investigated in various forms for almost a decade [9]–[12]. When restricting ourselves to using circular antenna arrays, the SVD-based analysis in combination with properties of circulant

Manuscript received February 24, 2011; revised May 26, 2011; accepted July 15, 2011. Date of publication October 24, 2011; date of current version February 03, 2012. The authors are with the Department of Electrical and Information Technology, Lund University, 221 00 Lund, Sweden (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173142 1These radio waves with specific OAM characteristics are often called “twisted radio beams” [3], [4] in popular science contexts.

Fig. 1. Illustration of the creation of Laguerre-Gaussian laser beams from planar laser beams by using transparent spiral phase plates introducing a linear phase delay over phase delay with azimuthal angle. OAM state implies a one revolution. Three different phase plates are illustrated in gray, for OAM state 0, 1 and 2. The colored surfaces are contour surfaces indicating where the phase of the laser beams is zero.

matrices [13], [14] can deliver the same beam-forming and the same OAM properties as in [1], [2]. The beam forming process for all eigenmodes/OAM states can be performed by a discrete Fourier transform (DFT) [15], which was also observed in [1]. The analysis also reveals that the eigenmodes of the resulting MIMO system are not necessarily unique, making OAM radio communication a sub-class of traditional MIMO communication with circular antenna arrays. Finally, we conclude the analysis by comparing the channel capacity of OAM-based communication, resulting from MIMO with circular antenna arrays, with known limits on the capacity of MIMO communication [7]. This shows that OAM based communication can achieve nearly optimal capacity gain, as predicted by MIMO theory, when the antenna arrays are closely spaced compared to the Rayleigh distance. II. SHORT REVIEW OF RADIO OAM Radio OAM can be seen as a development of techniques used in laser optics, where Laguerre–Gaussian (LG) mode laser beams are created using spiral phase plates [16]. The phase fronts of the created LG beams are helical in the sense that the phase front varies linearly with azimuthal angle, as illustrated in Fig. 1. As a means of creating radio waves with OAM properties the authors of [1] and [2] use antenna arrays consisting of concentric uniform circular arrays (UCAs). The antenna elements in the UCAs are fed with the same input signal, but with a successive delay from element to element such that after a full turn the phase has been incremented by a an integer multiple of . The basic principle of one of these UCAs is shown in Fig. 2.

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Fig. 2. Eight-element UCA with phase rotation proximate OAM state .

on element

to ap-

In [1] they calculate the far-field intensity patterns using NEC2 [17] and conclude that the results are very similar to those obtained in paraxial optics. It is also pointed out that the different OAM states in a beam can be decomposed by integrating the complex field vector weighted with along a circle around the beam axis. In practice there will be a finite number of antennas in an UCA measuring the field and the integration operation is approximated by a discrete Fourier transform (DFT) of the field in the antenna positions (or the antenna outputs). In [1] it is also concluded that with a limited number of antennas, , there is an upper limit on the largest OAM number that can be resolved, namely . Before we investigate these array design strategies for approximating beams with certain OAM states and apply standard MIMO theory to the resulting systems, let us briefly review the basics of narrow-band MIMO systems. Narrow-band MIMO systems have been addressed in numerous publications during the last decade and a standard formulation of the input/output relationship in complex base-band notation is (1) where is the vector of inputs, the vector of outputs, the MIMO channel matrix, and the vector of additive receiver noise. In many cases is assumed to be random, e.g., in wireless MIMO communication scenarios with relative movements in the propagation environment. Here we assume that is both known and has specific properties. The additive noise is assumed to be a vector of independent and identically distributed (i.i.d.) zero-mean, circularly symmetric, complex Gaussian noise components such that , where is the noise variance on each receiver branch and is the identity matrix. The channel capacity of the MIMO system above has been known for a long time, for both known and unknown channel at the transmitter side. We will review a technique here to derive the capacity of the system, first introduced by Telatar [8], which is essential to the analysis in the rest of the paper. We use the singular value decomposition (SVD) of the channel matrix (2)

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where and are unitary matrices containing the left and right singular vectors of , respectively, while is a diagonal matrix with the positive singular values , in decreasing order, on its diagonal, and is the rank of . The left and right singular vectors in and are obtained from the eigen-decompositions of the Hermitian matrices and , respectively, while the singular values along the diagonal in are the square roots of the corresponding eigenvalues ( and share the same set of positive eigenvalues). From a capacity point of view, nothing is changed if we perform pre-processing to obtain our transmitted vector with the unitary matrix , and post-processing of our received vector with the unitary matrix . To describe the pre- and post-processing, we use the notation (3) (4) The pre- and post-processing operations above can also be seen as receiver- and transmitter-side beam-forming, where the left and right singular vectors of the channel matrix are used as steering vectors. Performing these operations on the original MIMO system (1) gives us an equivalent system (5) (6) where we, in the last step, use the SVD in (2) and denote the . Since is unitary, the new noise vector noise by has the same distribution as itself, i.e., . The corresponding capacity for known channel at the transmitter becomes (7) where all the available power is distributed across the channels, according to the water-filling principle, such that (8)

III. CAPACITY OF FREE SPACE MIMO SYSTEMS To be able to compare free space MIMO systems against each other we need the channel matrix in (1) for a given configuration of the antenna arrays and a measure which quantifies the performance of a particular configuration. We will use the channel capacity of the MIMO system, relative to the capacity of a single-in-single-out (SISO) system, operating at the same antenna separation and using the same total transmit power. We call this the capacity gain of the MIMO system over a SISO system. A. Channel Matrices in Free Space Given the distance between a pair of antenna elements in free space, we denote the (narrow band) transfer function from transmit antenna input to receive antenna output as (9)

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where the free space loss is given by , the additional phase rotation due to propagation distance is introduced by the complex exponential term, denotes the wavelength of the used carrier frequency, and contains all relevant constants such as attenuation and phase rotation caused by antennas and their patterns on both sides. For the above model to be relevant in our analysis of antenna arrays, we assume that all antenna elements (in each array) are equal and that the propagation distance is large enough to provide: • a valid far-field assumption between any pair of transmit/ receive antenna elements, and • relative array sizes (diameters) small enough to assume that the antenna diagrams are constant over the directions of departure/directions of arrival involved. The first of these requirements essentially says that the freespace loss formula should be valid, while the second one implies that the antenna diagrams of the used antenna elements should be smooth enough in the direction of the opposing array to allow them to be approximated well by a constant (included in above). These requirements are not strictly necessary for the analysis, but greatly simplifies the expressions. Given that we have an MIMO system, where is the distance between receive antenna element and transmit antenna element , the channel matrix becomes

.. .

.. .

..

.

.. .

(10)

where (11) is given by (9). B. Capacity Gain In the rest of the paper we will use a relative capacity measure to evaluate the gain of applying MIMO instead of SISO communication. We assume that we have equal transmitters and receivers for both systems, with the same antennas and equal receiver noise figures. The SISO system uses a single transmitter-receiver pair, while the MIMO system uses multiple units on each side. As a basis for the capacity gain, we assume that our SISO system needs a certain transmit power to achieve a certain SNR on the receiver side. The required transmit power can be calculated using a simple link budget. By using the propagation loss as given by (9), the required transmit power becomes

Fig. 3. Distance between two antenna elements on concentric circles with and , respectively, placed on a common beam axis at a distance radii from each other. The angle between first elements in the two arrays is denoted .

We now evaluate the capacity of the studied MIMO system for the same transmit power, , and compare their channel capacities. We define the MIMO capacity gain as (14) is the channel capacity for the studied where MIMO system for known channel at the transmitter, i.e., where is given by in (7). Following the analysis in [7], the capacity gain of any MIMO system is limited by the number of antennas at each side as . Having established our metric for comparing the merits of different MIMO systems, we move on to the specific antenna array geometries addressed in this paper. IV. CIRCULAR ARRAYS ON THE SAME BEAM AXIS IN FREE SPACE The basic system configuration described in [1] concerns antenna arrays with one or more concentric UCAs used to create electric fields with different OAM states. Here we focus on the simplest MIMO system using such antenna structures; two UCAs facing each other on the same beam axis at a distance . One is the transmitting array and one is the receiving array, as illustrated in Fig. 3, where the dots indicate antenna element positions. The distance between arbitrary points on the two concentric circles, with radii and , on which the antenna elements are placed, with an angle between the points is (15) evenly distributed antennas on the first circle Assuming and evenly distributed antennas on the second circle, neighboring antenna elements on the two circles are

(12)

(16)

where is the receiver noise variance and is the distance between the transmitting and receiving antenna. This choice of transmit power gives a reference SISO channel capacity of

radians apart, respectively. Without loss of generality, we can assume that the first antenna element on the first circle is placed at zero radians, while the first antenna element on the second circle is placed at an angle . By changing the value on we can obtain all possible

(13)

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relative rotations between the two antenna arrays. Using the angles between two elements on each array as described in (16), while introducing a relative array rotation , the angle between transmit element and receive element becomes (17) where and . If the distance and the array radii and are given, the distance between elements in the two arrays can be expressed as, substituting (17) in (15), (18) An important observation at this stage is that for the same on both sides, the elnumber of antennas ement-to-element distances only depend on the difference . This relation holds for all array separations , array radii and , and angles between first elements. Assuming that the antenna elements of both arrays are co-polarized,2 we use the distances between transmit and receive antenna elements (18) and the free space transfer function (9), to express the elements of the MIMO channel matrix (10) as

where is a diagonal matrix, with unit magnitude complex numbers as diagonal elements, used to rotate the complex eigenvalues into singular values on the positive real axis. If is the th eigenvalue (diagonal element) in , the th diagonal element of is . By observing that both transmitand receive-side beam-formers, (3) and (4), are given by the rearranged DFT in (23), we have verified that the linear phase rotations across UCAs proposed in [1] and [2] to approximate OAM states in radio beams coincide with the eigen-modes derived with standard MIMO theory for our free-space scenario.3 The circulant property also helps us in the calculation of singular values of the channel matrix, which are the magnitude of the DFT of the first column of , sorted in decreasing order. V. WAVE FRONT PROPERTIES AND CHANNEL CAPACITY After verifying that the free-space MIMO model gives the same beam-forming vectors as the ones proposed in [1], we also want to verify the phase properties of the wave front as predicted by this model. Using the steering vectors in (3), with , we can calculate the received signal in a point in space as (27)

For the same number of antennas on both sides the matrix becomes circulant [13], since its elements inherit the property that only depends on the difference , through (19). This implies that the channel matrix is diagonalized by the unitary DFT matrix

where is the channel matrix from the transmitter array elements to a single receiving antenna element in the investigated point in space, using (9), and is a vector with a single one in the position corresponding to the th OAM state. In Fig. 4 we show the resulting phase plots for an 8-element transmit array. The phase plots are calculated for three different distances, 1/4, 4 and 400 times the Rayleigh distance for the entire antenna arrays4

(20)

(28)

(19)

with entries (21) With this notation we can write the eigen-decomposition as (22) where contains the eigenvalues of the channel matrix. To change this into an SVD, where the singular values are real, non-negative, and sorted in decreasing order, we first modify the eigen-decomposition by rearranging the order of the eigen-vectors so that the eigenvalues are sorted according to decreasing magnitude. Denote this sorted eigen-decomposition (23) With this notation, the matrices in the SVD of the channel matrix (2) can be expressed: (24) (25) (26) 2Here

we focus on a single polarization while, in principle, two independent MIMO systems can be achieved if we exploit both polarization states.

is set to the transmit UCA diameter where the aperture . The plot shows that we do not have very clean helical phases below the Rayleigh distance, but the gains of several of the OAM states makes them useful for communication. When the distance increases above the Rayleigh distance, we observe much cleaner helical phases, but the normalized gains of all non-zero OAM states fall rapidly, since the rank of the channel matrix approaches one. This makes all but the zeroth OAM state essentially useless at these distances for communication at realistic SNRs. Several of the OAM states at are so weak that we can clearly observe numerical problems in the plots. We have verified that our simple MIMO model generates the helical wave fronts expected in OAM, but at the same time, the pure OAM states are not necessarily unique in the sense that they provide the only set of eigenmodes for the channel. In the 3It can also be shown that if the number of antenna elements on one side is an integer multiple of the number of antennas on the other side, the resulting channel matrix becomes rectangular circulant [14] and the matrix containing the singular vectors corresponding to the side with fewer elements is a DFT matrix. The singular vectors on the other side, however, do not form a DFT matrix but can be described in closed form with harmonic functions. 4While the antenna arrays may be closer than the (array) Rayleigh distance, any pair of transmit/receive antenna elements of the arrays are considered to be at far field distances from each other.

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Fig. 5. Phase plots from an 8-element UCA with radius of . A numerical SVD is performed of the channel, the right singular vectors are used as beam-formers and phase plots are shown for the distance 400 times the . Phase is coded as shades of gray, from black Rayleigh distance , and the phase plots are sorted in order to white representing the range of gain (singular value). Normalized gains are shown below each phase plot and . the plot area is

Fig. 4. Phase plots at three different distances from an 8-element UCA with . The DFT is used as transmit beam-former and phase radius of plots are shown for distances of 1/4, 4 and 400 times the Rayleigh distance . Phase is coded as shades of gray, from black to white representing the , and the phase plots are sorted in order of gain (singular value). range Normalized gains are shown below each phase plot and the plot area is .

example provided in Fig. 4, there are singular values with multiplicity two, which leads to non-unique singular vectors. The non-uniqueness is illustrated by the phase plots in Fig. 5 for the same setup as in Fig. 4, but a numerical SVD is performed to obtain the eigenmodes rather than using the closed form expressions. Two of the eigenmodes (singular values of multiplicity one) coincide with OAM states 0 and 4 for in Fig. 4, while the other six display quite different characteristics (singular values with multiplicity 2). The set of singular values are the same for both cases, making them equivalent from a communication point of view. When calculating the phase diagrams above, we notice that higher order OAM states become very weak beyond the

Fig. 6. Capacity gain over single antenna (SISO) system at at UCA sizes 4 4, 8 8, and 16 16, at an SNR of 30 dB. Curves are calculated for array radii 100 and array separation distances from 10 times below to 1000 times above the Rayleigh distance (20.0000 ).

Rayleigh distance. To investigate this further, we calculate the channel capacity gain, as defined in (14), for three different configurations with 4 4, 8 8, and 16 16 antenna elements, at a per-receiver branch SNR of 30 dB. The channel capacity for the MIMO configurations is maximized over all relative rotations of the two arrays. The results are shown in Fig. 6. We can see that in all three cases the capacity gain achieved by using OAM-based MIMO communication almost reach the theoretical maximum [7] of 4, 8, and 16 times that of a SISO system below the Rayleigh distance, while performance degrades considerably at larger distances. At 1000 times the Rayleigh distance only one eigen-mode (OAM state 0) is useful for communication and the only gain available is the the array gain. This behavior can also be understood from the radiation patterns displayed in [1] and [2], where all but OAM state 0 have a null in the forward direction.

EDFORS AND JOHANSSON: IS ORBITAL ANGULAR MOMENTUM (OAM) BASED RADIO COMMUNICATION AN UNEXPLOITED AREA?

VI. DISCUSSION AND CONCLUSION In the investigation above, we have shown that OAM-based radio communication, as proposed in [1], can be obtained from standard MIMO theory, under certain conditions. Inspired by the discussion in [1], we made a system design with UCAs facing each other on the same beam axis in free space, which leads to circulant channel matrices, for all numbers of antenna elements (same number on both sides), all antenna radii (can be different at both ends), and all relative rotations of the arrays. Such matrices are diagonalized by the DFT, which means that the OAM states presented in [1] are one, not necessarily unique, set of eigen-modes of these channels. In our evaluation of the expected performance of such systems we showed that well above the Rayleigh distance there is a single dominant eigen-mode/OAM state. This leads to only a small capacity gain over a SISO system, essentially due to the array gain. No multiplexing gain is achieved since only one of the modes will carry information at realistic SNRs. Well below the Rayleigh distance, the investigated systems almost achieve the maximum capacity gains predicted by MIMO theory when using the OAM based eigenmodes of the channels. This means that the system based on UCAs is a relatively good choice in free space, since there is very little extra gain to achieve with other array geometries. The helical phase of the OAM states remain coherent over vast distances, but the amount of energy that can be received beyond the Rayleigh distance with a limited-size array decays rapidly, as compared to free space attenuation, for all but OAM state 0. Since the Rayleigh distance increases with array radius and frequency, the distances at which we have a multiplexing gain can be increased with larger array radii or a move to higher frequencies. One application example would be to use the sub-millimeter range of wavelengths, where planar UCAs of reasonable size can be integrated into, e.g., wallpaper and used to provide short-range high data-rate links based on spatial multiplexing under free-space conditions in indoor environments. Since transmitter and receiver beam-forming can be done with the DFT, OAM based communication between UCAs in free space has a potential to deliver high performance at short distances with a low computational complexity, when fast transforms are used. It is, however, only under very specific conditions that OAM based communication provides an optimal solution. The traditional and more general MIMO communication concept can handle all array geometries and propagation environments, including those where OAM based communication is optimal. Our main conclusion is that exploiting OAM states does not bring anything conceptually new to the area of radio communications. It is well covered by traditional MIMO communication, using channel eigen-modes for transmission, in the sense that OAM states of radio waves will be automatically exploited whenever the array configurations and propagation environments call for it. REFERENCES [1] B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett., vol. 99, no. 8, pp. 087701-1–087701-4, Aug. 2007.

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[2] S. M. Mohammadi, L. K. S. Daldorff, J. E. S. Bergman, R. L. Karlsson, B. Thidé, K. Forozesh, T. D. Carozzi, and B. Isham, “Orbital angular momentum in radio—A system study,” IEEE Trans. Antennas Propag., vol. 58, no. 2, pp. 565–572, Feb. 2010. [3] C. Barras, “Twisted radio beams could untangle the airwaves,” New Scientist Mar. 2009 [Online]. Available: http://www.newscientist.com/article/dn16591-twisted-radio-beams-could-untangle-the-airwaves.html [4] E. Cartlidge, “Adding a twist to radio technology,” Nature News Feb. 2011 [Online]. Available: http://www.nature.com/news/2011/110222/ full/news.2011.114.html [5] J. H. Winters, “On the capacity of radio communication systems with diversity in a rayleigh fading environment,” IEEE J. Sel. Areas Commun., vol. SAC-5, no. 5, pp. 871–878, Jun. 1987. [6] G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J., pp. 41–59, Autumn, 1996. [7] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun., no. 6, pp. 311–335, 1998. [8] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, pp. 585–595, Sep. 1999. [9] J.-S. Jiang and M. A. Ingram, “Distributed source model for shortrange MIMO,” in Proc. IEEE Veh. Tech. Conf., Orlando, FL, Oct. 2003, vol. 1, pp. 357–362. [10] F. Bøhagen, P. Orten, and G. E. Øien, “Construction and capacity analysis of high-rank line-of-sight MIMO channels,” in Proc. IEEE Wireless Communications and Networking Conf. (WCNC), New Orleans, LA, Mar. 2005, vol. 1, pp. 432–437. [11] I. Sarris and A. R. Nix, “Design and performance assessment of maximum capacity MIMO architectures in line-of-sight,” IEE Proc.–Commun., vol. 153, no. 4, pp. 482–488, Aug. 2006. [12] F. Bøhagen, P. Orten, and G. E. Øien, “On spherical vs. plane wave modeling of line-of-sight MIMO channels,” IEEE Trans. Commun., vol. 57, no. 3, pp. 841–849, Mar. 2009. [13] P. J. Davies, Circulant Matrices, ser. Wiley-Interscience. New York: Wiley, 1979. [14] J. Fan, G. E. Stewart, and G. A. Dumont, “Two-dimensional frequency analysis for unconstrained model predictive control of cross-directional processes,” Automatica, vol. 40, pp. 1891–1903, 2004. [15] A. V. Oppenheim and R. W. Shafer, Discrete-Time Signal Processing, 2nd ed. Englewood Cliffs, NJ: Prentice Hall, 1999. [16] M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun., vol. 112, pp. 321–327, Dec. 1994. [17] Numerical Electromagnetics Code. [Online]. Available: http://www. nec2.org/ Ove Edfors (M’00) received the M.Sc. degree in computer science and electrical engineering in 1990 and the Ph.D. degree in signal processing in 1996, both from Luleå University of Technology, Sweden. In the spring of 1997, he worked as a researcher at the Division of Signal Processing at the same university, and in July 1997, he joined the staff at the Department of Electrical and Information Technology, Lund University, Sweden, where he has been a Professor of radio systems since 2002. His research interests include radio systems, statistical signal processing and low-complexity algorithms with applications in telecommunication. Anders J. Johansson (M’03) received the Masters, Lic.Eng., and Ph.D. degrees in electrical engineering from Lund University, Lund, Sweden, in 1993, 2000, and 2004, respectively. From 1994 to 1997, he was with Ericsson Mobile Communications AB developing transceivers and antennas for mobile phones. Since 2005, he has been an Associate Professor in the Department of Electrical and Information Technology, Lund University, Sweden. His research interests include antennas and wave propagation for medical implants as well as antenna systems and propagation modeling for MIMO systems. He is involved as one of the researchers in the NeuroNano Research Center at Lund University, which is a interdisciplinary research initiative, and where he is responsible for the development of the telemetry part of the project.

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Communications A Circularly Polarized Ring-Antenna Fed by a Serially Coupled Square Slot-Ring The-Nan Chang, Jyun-Ming Lin, and Y. G. Chen

Abstract—An aperture-coupled ring-antenna is presented in this communication. The antenna is fed by a microstrip line through a unique aperture configuration. The aperture contains a square slot ring with four short branch slots protruding toward the center of the ring. It is studied that axial-ratio and return-loss bandwidths of 8.7% centered at 2.53 GHz can be achieved. Within 2.42 GHz–2.64 GHz, the gains are all greater than 7 dBic. Index Terms—Circular polarization, slot. Fig. 1. An overview of the ring-antenna excited by a coupling ring-slot.

I. INTRODUCTION An antenna is inseparable from a transmission line. If an antenna matches well with a transmission line, it can radiate more efficiently. Various methods have been devised to physically connect or electrically couple a transmission line to an antenna. The microstrip patch and the ring-antenna are two frequently used antennas. To excite a patch, we can use a coaxial cable or a microstrip line to directly connect to the microstrip patch antenna. The microstrip line can also be coupled to the patch antenna through a slot [1] or a cross-slot [2]–[4] depending on whether a linearly polarized or a circularly polarized wave will be generated. In [2], a parallel feed configuration which contains several Wilkinson’s power dividers to accurately control the amplitude and phase of the coupling slots is demonstrated. In [3], a serial feed configuration which uses an open-ended microstrip line to sequentially excite the coupling slots is presented. In [4], an antenna array with cross-slot coupled elements is reported by Chang etc. To excite a ring-antenna, the direct connecting method is seldom used as it is hard to match to a 50 system [5]. In [6], an open-ended microstripline is bent in U-shape to proximity couple to a square ring-antenna. The achieved axial-ratio bandwidth is about 1.3%. In [7], the microstripline is coupled to a square ring-antenna with an embedded small square patch. The achieved axial ratio bandwidth is only 0.03%. In this communication, a simple idea to design CP antenna using coupled slot-ring to a ring-antenna is presented. One advantage of the ring-antenna is that it occupies less space in comparison with a patch when both resonate at the same frequency. Though different methods to excite the ring have been investigated in [5], none of them can be directly applied to generate circularly polarized waves. On the other hand, the method proposed in [6] and [7] can generate only very narrow axial-ratio bandwidth for circularly polarized waves. It is also of interest to note that the usual cross-slot aperture-coupled method applied to the patch antenna was not considered to excite the ring-antenna in [5] or in any other papers. In [8], a square ring-slot aperture-coupled method has been presented. But, it was used to realize Manuscript received January 31, 2011; revised June 11, 2011; accepted August 17, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. The authors are with the Electrical Engineering Department, Tatung University, Taipei, Taiwan 104 (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2173138

a circularly polarized 2 2 2 patch array. In [9], a ring-antenna is aperture-coupled through three slots by Chang etc. However, the ring-antenna aperture-coupled by a ring-slot has not been thoroughly investigated. In this communication, we integrate the ring-antenna with a modified square ring-slot to generate circularly polarized waves. The modification is through adding four branch slots to the square ring-slot. The four branch slots are placed inside the square ring-slot. Therefore, both the ring-slot and the ring-antenna can be kept compact. II. ANTENNA CONFIGURATION AND ANALYSIS Fig. 1 depicts the geometry of a ring-antenna. The metal ring-antenna is attached on the topside of two distinct stacked layers. The first layer is 13 mm foam which has a relative dielectric constant close to 1. The second layer is a 0.8 mm thickness FR4 substrate. It should be pointed out that the use of a thicker air layer may enlarge the size of the square ring compared with the use of a thinner FR4 substrate. However, considerable enhancement in gain can be achieved through the use of an air layer. It also enhances the return-loss and axial-ratio bandwidths. The feed network consists of a square slot-ring with four inward branch slots on the top surface of the FR4 substrate. Below it, there is a microstrip line. Since a metal ring instead of a patch is used as the antenna. The antenna size can be reduced. In [3], a patch with physical dimensions 45 mm by 45 mm is required to radiate at a center frequency of 2.4 GHz. The patch is coupled by an on-FR4 microstrip line through a cross slot. Each side length of the cross slot is 28.55 mm. In our structure, the ring-antenna is with an outer side length of Ro = 35 mm and with an inner side length of Ri = 24 mm. We will show that the ring-antenna along with a novel coupling configuration can not only reduce the size of the antenna but also largely enhance the axial-ratio and return-loss bandwidths of the antenna. Detailed coupling configuration is shown in Fig. 2. The design starts with replacing the patch of [3] by a ring-antenna. In [3], the microstrip line is associated with a cross slot. The cross slot in [3] is placed close to the central area of a patch. Since there is no metal in most central area in a ring-antenna, the cross slot is better replaced by a ring-slot. To strengthen our idea, the ring-slot is stacked to the ring-antenna. In Fig. 2, the outline of the ring-slot is completely within the ring-antenna while viewing downward along the z axis. By this way, the size of the rectangular ring-slot is compatible to the size of the

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Fig. 2. Details to show (a) the relative positions between the patch, the slot and the feed line (b) the relative positions between the slot and the feed line.

W = 33 mm

ring-antenna. The outer side-length of the ring-slot is s , which is 1 mm smaller than the outer side-length of the ring-antenna. In Fig. 2, the optimized widths of the square slot-ring and the four branch : . The optimized four branch slots slots are all equal to s . are of equal length of s Like the conventional structure of [3], a microstrip line running under the new slot configuration is required. The four extended short slots are arranged to be sequentially excited by the microstrip line. It is investigated that the new coupling method can also effectively generate rotation of current along the surface of the metal ring-antenna resulting in a left-hand circularly polarized wave. If the microstrip line traces counterclockwise inside the slot-ring, i.e. symmetrically with respect to the center vertical line in Fig. 2(b), the antenna would produce a right-hand CP radiation. The width of the microstrip line : . The microstrip line is open ended, where an open is w stub length is denoted by os . The length of the open stub is crucial to affect the antenna’s performances. Fig. 3 shows the tuning effect by varying Los while other parameters are listed in Table I. When , it is seen that the simulated 10 dB return-loss : os bandwidth covers from 2.27 to 2.61 GHz, or 13% centered at 2.44 GHz. Within this bandwidth, the simulated 3 dB axial-ratio bandwidth is limited from 2.42 to 2.64 GHz, or 8.7% centered at 2.53 GHz. It is also seen that the simulated gains are all greater than 7 dBic within the overlapped bandwidth. If the foam is replaced by 1.6 mm thickness FR4 substrate, the gain is dropped to 2 dBi at 2.45 GHz; the CP characteristic is completely destroyed. The effect of tuning length and width of the branch slots (include the width of the square-ring) on antenna’s return-loss and axial-ratio value is respectively shown in Fig. 4 and Fig. 5. In this simulation, all parameters are taken from Table I except that s is fixed at 0.9 mm while s is varied and s is fixed at 12 mm while s is varied. It is shown that both parameters are crucial to promise for wide returnloss and axial-ratio bandwidths. If we focus on only the return-loss bandwidth, we can decrease either s or s from its optimum value. However, decrease of s will narrow down the axial-ratio bandwidth

T = 0 9 mm L = 12 mm

F = 1 5 mm

L

L = 15 47 mm

L

T

L

L

L T

T

Fig. 3. The effect of the stub length on the (a) return-loss, (b) axial-ratio, and (c) gain of the antenna.

TABLE I MAIN GEOMETRICAL PARAMETERS USED IN THE PROPOSED ANTENNA

and decrease of whole region.

T

s

may even worsen the axial-ratio value in the

III. EXPERIMENTS The test sample is fabricated with the main geometrical parameters in Table I. It has a ground plane dimensioned with 90 2 90 mm. The simulated and measured return-loss, axial-ratio, and gain responses of

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Fig. 4. The effect of the length (L ) of the branch slots on (a) return-loss and (b) axial-ratio of the antenna.

Fig. 6. The simulated and measured (a) return-loss, (b) axial-ratio, and (c) gain responses of the antenna.

Fig. 5. The effect of the width (T ) of the branch slots on (a) return-loss and (b) axial-ratio of the antenna.

referred to an ideal isotropic circularly polarized antenna. The CP gain can be measured by a partial gain method [4]. Since the return-loss bandwidth is larger than the axial-ratio bandwidth, it is concluded that the ring-antenna can effectively be excited by the present method to yield an overlapped bandwidth around 8.7%. The measured XZ-and YZ-plane radiation patterns at 2.5 GHz are shown in Fig. 7. The patterns are measured by a spinning linear (or rotating source) method. In this method, the axial-ratio value is the difference in dB between adjacent ripples. It is shown in Fig. 7 that the ripples are less than 3 dB within 630 from the boresight direction in either plane. IV. CONCLUSION

the antenna are shown in Fig. 6. The simulation is obtained by electromagnetic software of IE3D. The gain is denoted in unit of dBic which is

Aperture coupling through a cross-slot is conventionally used to separate the microstrip line from the patch antenna. In this communication,

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A Pseudo-Normal-Mode Helical Antenna for Use With Deeply Implanted Wireless Sensors Olive H. Murphy, Christopher N. McLeod, Manoraj Navaratnarajah, Magdi Yacoub, and Christofer Toumazou

Abstract—A pseudo-normal-mode helical antenna as part of a deeply implanted wireless sensor was designed. Justification for using this type of antenna along with simulation, in vitro and in vivo experimental results are presented in this communication. The circumference of the helical coil is 0 43 (wavelength in-body) and its height is 0 23 which includes substantial insulation. While losses from such a deeply implanted antenna are inevitable, the work presented here shows accurate frequency tuning can be achieved prior to implantation. The relative size, safety of use and results presented here make this pseudo-normal-mode helical antenna an excellent candidate for use with deeply implanted wireless sensors. Fig. 7. Measured radiation patterns of the antenna at 2.5 GHz.

Index Terms—Biomedical applications of electromagnetic (EM) radiation, biomedical telemetry, helical antennas, implantable biomedical devices.

the method is extended to excite a ring-antenna. A ring-slot instead of a cross-slot as the coupling aperture is employed to cope with the shape of the ring-antenna. It is investigated that the axial-ratio bandwidth is 8.7% which is wider than 4.6% in [3] of a conventional design. The size of the proposed antenna is also smaller than the contrast one.

REFERENCES [1] D. M. Pozar, “Microstrip patch antenna aperture-coupled to a microstripline,” Electron. Lett., vol. 21, no. 2, pp. 49–50, 1985. [2] D. M. Pozar and S. M. Duffy, “A dual-band circularly polarized aperture-coupled stacked microstrip antenna for global positioning satellite,” IEEE Trans. Antennas Propag., vol. 45, no. 11, pp. 1618–1625, 1997. [3] H. Kim, B. M. Lee, and Y. J. Yoon, “A single-feeding circularly polarized microstrip antenna with the effect of hybrid feeding,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 74–77, Apr. 2003. [4] K. H. Lu and T. N. Chang, “Circularly polarized array antenna,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3288–3292, Oct. 2005. [5] P. M. Bafrooei and L. Shafai, “Characteristics of single-and doublelayer microstrip square antennas,” IEEE Trans. Antennas Propag., vol. 47, no. 10, pp. 1633–1639, Oct. 1999. [6] K.-F. Tong and J. Huang, “New proximity coupled feeding method for reconfigurable circularly polarized microstrip ring antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 7, pp. 1860–1866, Jul. 2008. [7] H.-M. Chen, Y.-K. Wang, Y.-F. Lin, C.-Y. Lin, and S.-C. Pan, “Microstrip-fed circularly polarized square-ring patch antenna for GPS applications,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1264–1267, Apr. 2009. [8] R. Caso, A. Buffi, M. R. Pino, P. Nepa, and G. Manara, “A novel dual-feed slot-coupling feeding technique for circularly polarized patch arrays,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 183–186, Apr. 2010. [9] T. N. Chang and J.-M. Lin, “Serial aperture-coupled dual-band circularly polarized antenna,” IEEE Trans. Antennas Propag., vol. 59, no. 6, pp. 2419–2423, Jun. 2011.

I. INTRODUCTION The medical community has embraced the use of intelligent implants and are utilizing the data that is gathered to promote early detection, effective treatment and long term healthy living [1], [2]. There have been many advances in various different aspects of health-care: cardiac implants for the continuous monitoring of blood pressure [3], [4]; ocular implants for the restoration of sight [5] and treatment of disease [6]; cochlear implants which have provided partial hearing [7] and the use of swallowable devices for disease detection, imaging and drug delivery [8]–[10]. The methods of communicating with implanted devices vary depending on: the architecture of the implant; the depth of implantation; and the frequency of operation. Inductive telemetry has been used for many years for powering pacemakers [11] and for data transmission to and from implants [12], [13] but its use is limited by the size and quality of the implanted coil and even at low frequencies (1 MHz) where the near-field is large the depth of the implanted coil is limited to just a centimeter [13]. Antennas for use with higher frequency telemetry have also been developed. Planar loops, inverted F antennas, monopoles, dipoles, spirals and meanders have all been investigated for use with deeply implanted devices due to their low profile [14]–[18] and while planar antennas are easy to tune, the achievable power transmission and radiation efficiency are poor and few in vivo results have been provided [17]. It has been shown that making a planar antenna multi-dimensional, by adding volume, leads to higher gain [19]. A novel approach using pre-existing vascular stents as antennas has been examined, but unfortunately in vivo results are either not presented or those presented show relatively good power transmission is achievable but from an non-vessel, shallow implanted test site [20], [21]. While the stent-antenna uses pre-approved bio-compatible materials it is still too long for use in certain vessels and cannot be used in the chambers of the heart [21]. Manuscript received July 21, 2010; revised March 11, 2011; accepted July 02, 2011. Date of publication October 24, 2011; date of current version February 03, 2012. This work was supported by a Wellcome Trust Technology Transfer Translation Award. O. H. Murphy, C. N. McLeod and C. Toumazou are with the Centre for BioInspired Technology, Institute of Biomedical Engineering and the Department of Electrical and Electronic Engineering, Imperial College, London SW7 2AZ, U.K. (e-mail: [email protected]). M. Navaratnarajah and M. Yacoub are with the Heart Science Centre, Harefield Hospital, Harefield, Middlesex UB9 6JH, U.K. Digital Object Identifier 10.1109/TAP.2011.2173106

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This communication investigates the constraints when designing an implanted antenna and suggests the use of a pseudo-normal-mode helical antenna for use with a passive transponder in cavities deep within the human body. The results concentrate on in vitro and in vivo measurements which show the frequency response and transmission losses incurred within the body, showing the importance of understanding the modes of operation of the antenna within a complex multi-boundary media such as the human body. Section II will present the challenges of designing an antenna for use within the human body and the reasons for choosing a normal-mode helical antenna design in terms of its electrical size, its modes of operation and the power absorption associated with it. Section III will present the geometric design of the pseudo-normal mode helical antenna, while Sections IV–VI will present the electromagnetic simulations, in vitro and in vivo measurements, respectively. II. DEEPLY IMPLANTED ANTENNAS When designing an antenna for use deep within the human body extra constraints exist in comparison to designing an antenna for use in free space. Most important are (A) the limitations on the geometric size of the antenna and (B) the strict regulations regarding the amount of power absorption within the human body which is considered a lossy multi-boundary media. A. Electrically Small Antennas—Normal-Mode Helical To fit an antenna within a vessel or cavity, whilst operating at high frequencies results in electrically small antennas. Historically, work has been carried out in a similarly lossy environment, the sea, when designing antennas for submarine use [22]. It was found that electrically small antennas in the form of a magnetic dipole within a lossless radome performed better than an electric dipole, due to the fact that the magnetic properties of air and water were similar while the electric conductivity differed largely, this is also the case within the human body [23]. For simpler magnetic dipoles it has been shown there was a direct proportionality between the radiation power factor and the volume occupied by the antenna; therefore, extending a single loop to multiple loops adds volume and increases the efficiency of the antenna [24], [25]. Such a multiple loop antenna can also be referred to as a helical antenna but it can no longer be considered a pure magnetic dipole. Wheeler first described the normal-mode helical antenna (NMHA) as a superposition of electric and magnetic dipoles to radiate a wave with circular polarization [26]. Kraus later adopts and expands this theory to provide the well documented foundations for NMHA operation [27], [28]. It is this NMHA which is of particular interest as an implanted antenna as it meets the criterion of size and is suitable for use within an environment such as the human body. B. Safety and Performance of a NMHA Antenna in the Multi-Boundary Human Body By analyzing the mechanisms of both the near and far field of electrically small antennas it is possible to determine the performance and safety of an implanted antenna. Normal-mode antennas do not have any radiating near-field, just a reactive near-field which can occupy just a few millimeters of the surrounding tissue [29]. The far-field radiation extends through the remainder of the body and into free space [30]. Applying this analysis to implanted antennas is non-trivial but fortunately it has been shown that the well documented specific absorption rate (SAR) can be applied to both the near and far field [31]–[33]. In the body there is no magnetic field component as the permeability of the tissue is similar to that of air; therefore all interactions occur through mechanisms described by the electric field, including the current induced by the magnetic field. With this in mind the electric field induced by a particular antenna design is of significant interest [34]. For a NMHA, it can be seen that the short electric dipole has a normal

TABLE I FREE-SPACE AND IN-BODY PARAMETERS

TABLE II PSEUDO-NORMAL-MODE HELICAL ANTENNA GEOMETRY

electric field while the magnetic dipole, in the form of a small loop, has a tangential electric field. This angle of incidence is of considerable interest when the antenna is implanted and surrounded by insulation and tissue of different permittivity [35], [36]. The human body consists of many interfaces between tissues of different conductivity and permittivities and if 1 1 = 2 2 for two different media, the interface is charged at a rate which is proportional to the difference in the current densities in the media. If the electric field is tangential to the interface, no interfacial dispersion is observed [37]. Therefore to reduce the amount of absorbed power the implanted antenna should approach the characteristics of a magnetic dipole; however as shown, the better performing NMHA with circular polarization consists of both electric and magnetic dipoles and it is clear that a compromise between radiation efficiency and absorption loss is necessary. It has already been proved using theoretically implanted helical antennas (in an air capsule) at multiple frequencies that power levels lower than 36 mW and 11 mW meet ICNIRP and IEEE safety standards, respectively [33], [38], [39]. This NMHA will be implanted with a transducer and the received and transmitted power will be of the order of fractions of milliwatts and therefore falls well below this range. For implanted antennas, the previously mentioned radomes are replicated using biocompatible insulation material, which has already been shown to improve the radiation efficiency [35], [40]. While some work has been carried out on the radiation characteristics of implanted helical antennas [33], [41], there has been relatively little work examining the transmission losses between a deeply implanted NMHA and an external antenna [42]; therefore, this communication will concentrate on the frequency response of a deeply implanted NMHA and the transmission losses incurred.

6

III. DESIGN OF A NORMAL-MODE HELICAL ANTENNA This NMHA is designed to work in the high power short-range-devices European frequency band which exists between 863–870 MHz [43]. For a particular frequency within this band Table I shows the corresponding wavelength (), relative permittivity () and conductivity () for free-space (f s) and in-body (ib), respectively [29]. Table II shows the geometry of the implanted antenna.  which as can be seen The criteria for a NMHA is that nL fs ; however, is not valid from Table II is valid in free space as nL within the human body as nL > ib . If the wavelength within the body is approximately 8 times smaller than that in free-space all of the corresponding geometries should also be 8 times smaller to satisfy nL ib . In reality this would result in prohibitively small antenna dimensions. It is extremely difficult to tune such a small antenna without additional passive components whilst having so few turns. Also, to maintain such a narrow spacing requires a wire diameter which is currently not available using Nitinol, the preferred material due to its proven bio-compatibility and memory shaping. Increasing the number of turns

 



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Fig. 1. (a) Pseudo-normal-mode helical antenna surrounded by insulation and simulated in a homogenous human body environment, and (b) Polarization patand E , tern of implanted pseudo-normal-mode helical antenna, E ,   .

= 90

= 90

to achieve tuning would result in an antenna approaching an electricdipole; therefore this pseudo-normal-mode helical antenna (pNMHA) will be implemented in the body based on the dimensions satisfying normal-mode operation in free-space. In addition, the axial ratio (2S= 2 D2 ) for ib is 0.5, which means there will not be the perfectly circular polarization as predicted when the axial ratio is unity, but the polarization approaches that of a magnetic dipole with the major axis of the polarization ellipse horizontal. For free-space the axial ratio approaches that of an electric dipole with the major axis of the polarization ellipse vertical. This geometry is designed to resonate in the body and will have a resonance at a much higher frequency in free-space, reducing f s significantly and matching the theoretical axial ratio to the physical geometrical axial ratio which approaches unity. For ease of implantation it is necessary to keep the length of the antenna to a minimum but this is governed by the tuning of the antenna. The insulation thickness and the position of the antenna within the body also determine the performance of the antenna; therefore, extensive 3D electromagnetic simulations are necessary to help design this pNMHA. IV. 3D ELECTROMAGNETIC SIMULATION CST Microwave Studio™ has been used to perform a full 3D electromagnetic simulation of the pNMHA within the human body, using a simple homogenous human body model [44]. By varying the number of turns and consequently the height of the pNMHA it is possible to tune the antenna. The insulation thickness is chosen to be 6 mm in diameter to maximize the radiation efficiency, whilst remaining within the geometric constraints of the heart cavity, the entire antenna and insulation is surrounded by a homogenous box of heart tissue whose permittivity and conductivity is known, as seen in Table I [29]. The antenna is envisaged to sit vertically in a cavity of the heart. Fig. 1(a) shows the antenna and insulation within the homogenous environment and Fig. 2 shows the tuned response of the antenna (S11 -CST). It is known, that while the antenna itself is the radiating element, once implanted in the body, the antenna, insulation and surrounding tissue becomes the radiating element [40]. Fig. 1(b) shows the simulated polarization pattern and it is seen that almost perfectly circular polarization still exists, as expected when implanted within a homogenous body model. A maximum radiation efficiency of 027 dB and directivity of 2.65 dBi was achieved during simulation. V. IN VITRO MEASUREMENTS The antenna is formed using 0.33 mm Nitinol and a suitable mandrel, it is then attached to a co-axial cable. The higher frequency response in free space is recorded and then the insulation is added to arrive at the desired frequency response. Any substantial tuning needs to be carried

Fig. 2. Simulated (CST), in vitro (bio-phantom) and in vivo (implant) frequency response of a pseudo-normal-mode helical antenna. TABLE III INGREDIENTS AND QUANTITIES FOR BIO-PHANTOM MATERIAL

out by changing the number of turns which is not possible once the insulation is added so the higher frequency free space response is used as a calibration point. The insulation is added using a mould and is then tested in a bio-phantom material to emulate the characteristics of the body. The bio-phantom material used is a mixture of sodium-chloride, deionized water, polyethylene powder and a mixing agent, TX-151 [45] and the permittivity and conductivity is tested using an Agilent 85070E Dielectric Probe Kit. The correct ratio of the bio-phantom ingredients is vital and Table III shows the amounts required to get a permittivity of 60.10 and a conductivity of 1:21Sm01 at the frequency of interest. This quantity of bio-phantom is approximately equivalent to the size of the human heart. Due to environmental factors during mixing and the fact that the bio-phantom mixture is a highly viscous suspension and is not fully homogenous, an average response within 5% is considered successful. The bio-phantom has a life of approximately 48 hours but was retested for any drift before each experiment. The antenna is shown in Fig. 3(a) and is approximately 1 cm in length (0:23ib ) of which approximately 70% is insulation. The diameter, including insulation is equivalent to 0:43ib . Fig. 2 shows the reflection coefficient (S11 —bio-phantom) for the insulated antenna on the end of a long co-axial cable which is connected to a Rhode and Schwarz ZVL Vector Network Analyzer. For both the in vitro and in vivo experiments the long cables form part of the network analyzer’s calibration. At this stage any frequency de-tuning can be rectified by increasing the amount of insulation. While the emphasis is on tuning the antenna, S21 was also recorded using a simple tuned dipole also with circular polarization to reduce polarization mismatch. Fig. 2 shows S21 for the bio-phantom. This was measured at a distance of approximately 5 cm. VI. IN VIVO RESULTS The pseudo-normal-mode helical antenna was placed within the right ventricle of a live but anesthetized porcine test subject (30 kg landrace) and connected to the Rhode and Schwarz ZVL Vector Network Analyzer. The test subject’s chest was stitched closed so that accurate transmission measurements could be taken. Fig. 3(b) shows the cable protruding from the chest of the live test subject while Fig. 3(c) shows the angiographic image of the implanted antenna within the heart with

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Fig. 3. (a) Miniature pseudo-normal-mode antenna with insulation, scale in cm, (b) Porcine test subject, co-axial cable visible protruding from its chest, and (c) Fluoroscopic image showing co-axial cables and the implanted antenna and two external medical clamps used as markers.

external medical clamps used as markers to verify the position for the transmission measurements. The test subject is kept alive for the duration of the experiment as upon death the physiological differences which occur immediately within the heart such as thrombosis can effect the results. Fig. 2 shows the reflection coefficient (S11 -implant) for the implanted antenna. The reflection coefficient (S22 -implant) of the external dipole is also shown in Fig. 2. (S21 -implant) between the implanted antenna and the external dipole was recorded at a distance of 5 cm from the surface of the porcine test subject’s chest, directly above the implanted antenna. An inert spacer was used to maintain the correct distance and angle between the dipole and the skin of the porcine test subject. As seen in Fig. 2 the frequency response of the simulated antenna, while giving similar frequency values for S11 minima to the in vitro and in vivo experiments, are substantially different in their broadband response. The electromagnetic simulation predicts a relatively narrowband response, indicative of the abrupt variation of impedance versus frequency for this antenna topology. In reality, as shown in the in vitro and in vivo measurements, which are very closely matched, the impedance response, while of a higher value, does not vary to the same extent, giving a broadband response. On closer inspection this is related to the real impedance or, more precisely, the resistive losses as the radiation resistance is based on the geometry which is the same for both simulation and measurements. The resistive losses are substantially higher in reality than those theoretically reported for Nitinol which can be attributed to the impact of annealing, the resultant oxide and the poor electrical contact due to the difficulty in soldering to Nitinol. The discrepancy in S21 between the in vitro and in vivo experiments, while large, is a factor of the depth of implantation, indicating that larger bio-phantom volumes should be used for more accurate results. VII. CONCLUSION The results presented in this communication show the feasibility and suitability of placing an insulated pseudo-normal-mode helical antenna within the cavities of the heart from a functional, geometric and safety point of view. The minute design was easily simulated and then fine tuned using an in vitro procedure involving a bio-phantom mixture to mimic the human body. It has been shown that there is a very close correlation for the tuned frequency response between the in vitro and in vivo measurements and that the simulation provides a good starting point, but that care must be taken when simulating and using materials such as Nitinol, in particular when the bandwidth response is important. The bio-phantom mixture is vital for realizing accurate frequency responses but it is most beneficial for relatively broadband responses which is the case with this pseudo-normal-mode helical antenna. While S21 is not as closely predicted the tuning of the antenna is foremost in facilitating communication and the promising results presented here will lead to future work involving even smaller pNMHA with less insulation.

ACKNOWLEDGMENT The authors would like to thank the staff of Elpen Pharmaceuticals, Pikermi Attikis, Greece, in particular Dr. A. Papalois for providing experimental facilities and also Dr. T. Sakelaridis, Dr. V. Pangiotakopoulos and Dr. M. Argiriou of Evangelismos General Hospital, Athens, Greece, for their time and expertise during the experimental procedures.

REFERENCES [1] R. A. M. Receveur, F. W. Lindemans, and N. F. de Rooij, “Microsystem technologies for implantable applications,” J. Micromech. Microeng., vol. 17, no. 5, pp. 50–80, Apr. 2007. [2] W. Mokwa, “Medical implants based on microsystems,” Meas. Sci. Technol., vol. 18, no. 5, pp. 47–57, May 2007. [3] Y. Rozenman, R. S. Schwartz, H. Shah, and K. H. Parikh, “Wireless acoustic communication with a miniature pressure sensor in the pulmonary artery for disease surveillance and therapy of patients with congestive heart failure,” J. Am. Coll. Cardiol., vol. 49, no. 7, pp. 784–789, Feb. 2007. [4] J. A. Potkay, “Long term, implantable blood pressure monitoring systems,” Biomed. Microdevices, vol. 10, no. 3, pp. 379–392, Jun. 2008. [5] J. D. Weiland, W. Liu, and M. S. Humayun, “Retinal prosthesis,” Annu. Rev. Biomed. Eng., vol. 7, pp. 361–401, Aug. 2005. [6] P.-J. Chen, D. C. Rodger, S. Saati, M. S. Humanyun, and Y.-C. Tai, “Microfabricated implantable parylene-based wireless passive intraocular pressure sensors,” J. Microelectromech. Syst., vol. 17, no. 6, pp. 1342–1351, Dec. 2008. [7] F.-G. Zeng, S. Rebscher, W. Harrison, X. Sun, and H. Feng, “Cochlear implants: System design, integration, and evaluation,” IEEE Rev. Biomed. Engrg., vol. 1, pp. 115–142, Dec. 2008. [8] C. McCaffrey, O. Chevalerias, C. O’Mathuna, and K. Twomey, “Swallowable-capsule technology,” IEEE Perv. Comput., vol. 7, no. 1, pp. 23–29, Jan. 2008. [9] P. Hanninen, V. Liimatainen, V. Sariola, Q. Zhou, and H. N. Koivo, “Swallowable biotelemetry device for analysis of irritable bowel syndrome,” in Proc. Int. Micro-NanoMechatronics and Human Science Symp., Nov. 2008, pp. 513–518. [10] A. Moglia, A. Menciass, M. O. Schurr, and P. Dario, “Wireless capsule endoscopy: From diagnostic devices to multipurpose robotic systems,” Biomed. Microdevices, vol. 1, pp. 115–142, Apr. 2008. [11] M. Theodoridis and S. Mollov, “Distant energy transfer for artificial human implants,” IEEE Trans. Biomed. Eng., vol. 52, no. 11, pp. 1931–1938, Nov. 2005. [12] M. Catrysse, B. Hermans, and R. Puers, “An inductive power system with integrated bi-directional data-transmission,” Sens. Act. A: Phys., vol. 115, no. 2–3, pp. 221–229, 2004. [13] U.-M. Jow and M. Ghovanloo, “Design and optimization of printed spiral coils for efficient transcutaneous inductive power transmission,” IEEE Trans. Biomed. Circuits Syst., vol. 1, no. 3, Sep. 2007. [14] Z. N. Chen, G. C. Liu, and T. See, “Transmission of RF signals between MICS loop antennas in free space and implanted in the human head,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1850–1854, Jun. 2009. [15] J. Kim and Y. Rahmat-Samii, “Implanted antennas inside a human body: Simulations, designs, and characterizations,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1934–1943, Aug. 2004.

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[16] K. Gosalia, M. Humayun, and G. Lazzi, “Impedance matching and implementation of planar space-filling dipoles as intraocular implanted antennas in a retinal prosthesis,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2365–2373, Aug. 2005. [17] E. Chow, C.-L. Yang, A. Chlebowski, S. Moon, W. Chappell, and P. Irazoqui, “Implantable wireless telemetry boards for in vivo transocular transmission,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 3200–3208, Dec. 2008. [18] P. Soontornpipit, C. Furse, and Y. C. Chung, “Design of implantable microstrip antenna for communication with medical implants,” IEEE Trans. Microw. Theory Tech, vol. 52, no. 8, pp. 1944–1951, Aug. 2004. [19] S. Soora, K. Gosalia, M. Humayun, and G. Lazzi, “A comparison of two and three dimensional dipole antennas for an implantable retinal prosthesis,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 622–629, Mar. 2008. [20] K. Takahata, Y. Gianchandani, and K. Wise, “Micromachined antenna stents and cuffs for monitoring intraluminal pressure and flow,” J. Microelectromech. Syst., vol. 15, no. 5, pp. 1289–1298, Oct. 2006. [21] E. Chow, Y. Ouyang, B. Beier, W. Chappell, and P. Irazoqui, “Evaluation of cardiovascular stents as antennas for implantable wireless applications,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 10, pp. 2523–2532, Oct. 2009. [22] H. A. Wheeler, “Fundamental limitations of a small VLF antenna for submarines,” IRE Trans. Antennas Propag., vol. 6, no. 1, pp. 123–125, 1958. [23] R. F. Fano, L. J. Chu, and R. B. Adler, Electromagnetic Fields, Forces and Energy, 1st ed. New York: Wiley, 1960, ch. 6. [24] H. A. Wheeler, “Fundamental limitations of small antennas,” Proc. I.R.E., vol. 35, no. 12, pp. 1479–1484, Dec. 1947. [25] H. A. Wheeler, Small Antennas, vol. 23, no. 4, pp. 462–469, Jul. 1975. [26] H. A. Wheeler, “A helical antenna for circular polarization,” Proc. I.R.E., vol. 35, no. 12, pp. 1484–1488, Dec. 1947. [27] J. D. Kraus, “The helical antenna,” Proc. I.R.E., vol. 37, no. 3, pp. 263–272, Mar. 1949. [28] J. D. Kraus and R. J. Marhefka, Antennas for All Applicaions, 3rd ed. New York: McGraw Hill, 2002, ch. 8. [29] Dielectric Properties of Tissues [Online]. Available: http://niremf.ifac. cnr.it/tissprop/ [30] P. Nikitin, K. Rao, and S. Lazar, “An overview of near field UHF RFID,” in Proc. IEEE Int. Conf. on RFID, Mar. 2007, pp. 167–174. [31] IEEE Recommended Practice for Measurements and Computations of Radio Frequency Electromagnetic Fields With Respect to Human Exposure to Such Fields, 100 kHz to 300 GHz, EEE Std C95.3-2002 (R2008), Jun. 2008. [32] P. Riu and K. Foster, “Heating of tissue by near-field exposure to a dipole: A model analysis,” IEEE Trans. Biomed. Eng., vol. 46, no. 8, pp. 911–917, Aug. 1999. [33] L. Xu, M.-H. Meng, H. Ren, and Y. Chan, “Radiation characteristics of ingestible wireless devices in human intestine following radio frequency exposure at 430, 800, 1200, and 2400 MHz,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2418–2428, Aug. 2009. [34] Environmental Health Criteria 137 (1993): Electromagnetic Fields (300 Hz–300 GHz) WHO, Geneva, Switzerland, 1993 [Online]. Available: http://www.inchem.org/documents/ehc/ehc/ehc137.htm [35] K. Yekeh and R. Kohno, “Wireless communications for body implanted medical device,” in Proc. Asia-Pacific Microwave Conf. Asia-Pacific, Dec. 2007, pp. 1–4. [36] R. W. P. King and G. S. Smith, Antennas in Matter—Fundamentals, Theory and Applications, 1st ed. Cambridge, MA: MIT Press, 1981, ch. 8. [37] Bioengineering and Biophysical Aspects of Electromagnetic Fields, F. S. Barnes and B. Greenebaum, Eds., 3rd ed. Baca Raton, FL: CRC Press, 2007, ch. 3. [38] “Guidelines for Limiting Exposure to Time-Varying Electric, Magnetic, and Electromagnetic Fields (up to 300 GHz),” International Committee for Non-ionizing Radiation Protection (ICNIRP), 1998 [Online]. Available: http://www.icnirp.de/documents/emfgdl.pdf [39] IEEE Standard for Safety Levels with Respect to Human Exposure to Radio Frequency Electromagnetic Fields, 3 kHz to 300 GHz, IEEE Std C95.1-2005, Apr. 2006. [40] A. J. Johansson, “Performance measures of implant antennas,” in Proc. 1st Eur. Conf. on Antennas Propag., Nov. 2006, pp. 1–4. [41] L. Chirwa, P. Hammond, S. Roy, and D. Cumming, “Electromagnetic radiation from ingested sources in the human intestine between 150 MHz and 1.2 GHz,” IEEE Trans. Biomed. Eng., vol. 50, no. 4, pp. 484–492, Apr. 2003.

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[42] S. I. Kwak, K. Chang, and Y. J. Yoon, “The helical antenna for the capsule endoscope,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 2005, vol. 2B, pp. 804–807. [43] Electromagnetic Compatibility and Radio Spectrum Matters (ERM); Short Range Devices (SRD); Radio equipment to be Used in the 25 MHz to 1000 MHz Frequency Range With Power Levels Ranging up to 500 mW; Part 1: Technical Characteristics and Test Methods, ESTI Standard, EN 300 220-1 V2.3.1, 2010. [44] CST, Computer Simulation Software [Online]. Available: http://www. cst.com/Content/Applications/Markets/MWandRF.aspx [45] K. Ito, K. Furuya, Y. Okano, and L. Hamada, “Development and characteristics of a biological tissue-equivalent phantom for microwaves,” Electron. Commun. Jpn. (Part I: Commun.), vol. 84, no. 4, pp. 67–77, Dec. 2000.

A Novel Folded UWB Antenna for Wireless Body Area Network Cheng-Hung Kang, Sung-Jung Wu, and Jenn-Hwan Tarng

Abstract—A novel folded ultrawideband antenna for Wireless Body Area Network (WBAN) is proposed, which can effectively reduce the backward radiation and proximity effects of human bodies. The proposed antenna has a low-profile 3D structure that consists of a bevel-edge feed structure and a metal plate with folded strip. The bevel edge feed structure achieves broadband impedance matching and the metal plate acts as the main radiator. Moreover, the folded strip not only extends the lower frequency band but also provides additional resonant frequency around 6 GHz. The final bandwidth covers from 3.1 GHz to 12 GHz. The proposed antenna shows the directional patterns with low backward radiation due to the patchlike structure and the ground plane also prevents from the proximity effects of human bodies. Furthermore, the simulated SAR values of the proposed antenna are lower than the values of omnidirectional disc planar monopole. These features demonstrate that the proposed antenna is suitable for WBAN application. Index Terms—Body-area network, 3-D antenna, ultrawideband.

I. INTRODUCTION Wireless body area network (WBAN), a communication system which transmits large amount of information/data near the human body, is attracting more and more attention in wireless communications [1]. WBAN integrated with proper sensors can observe and transmit vital signs of patients, police or fire personnel without cables. With increasing attention directed toward WBAN, the ultrawideband (UWB) technology becomes an active solution for these applications because of its low transmission power and high data rates. The antenna and propagation measurements for WBAN are discussed in [2], [3]. The results show that the path loss and the rms delay spread are highly related to antenna structure. Several studies provide various antenna designs used in WBAN [4]–[7]. In [2], a 3-D monopole antenna placed perpendicular to the human body is designed for WBAN. However, the 3-D monopole antenna is too high so it Manuscript received May 21, 2010; revised May 15, 2011; accepted July 25, 2011. Date of publication October 24, 2011; date of current version February 03, 2012. This work was supported by the National Science Council, R.O.C., under Grants NSC-99-2219-E-009-001 and NSC-99-2221-E-009-028-MY2. The authors are with National Chiao Tung University, Hsinchu, Taiwan 300, ROC (e-mail: [email protected]). Color versions of one or more of the figures in this comunication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173101

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Fig. 3. Simulated return loss of various lengths of L and L .

Fig. 1. (a) Configuration of the proposed ultrawideband antenna. (b) The antenna unbent into a planar structure.

Fig. 4. Simulated return loss of various lengths of H .

Fig. 2. Measured and simulated return loss of the proposed antenna.

will obstruct the human daily activities when applied to the human body. Therefore, planar monopole antennas with low antenna height are widely developed in the UWB community. However, when these antennas lie on the human body, the operating frequency, bandwidth and radiation efficiency are easily interfered with by the human body. Except the aforementioned influences, specific absorption rate (SAR) is another important issue in WBAN. In [5], it has been shown simulated results that the antenna with omnidirectional pattern exhibits low radiation efficiency and high SAR values compared to directional antenna when placed on human model. Low backward radiation, low height with compact form, and low mutual effect between the antenna and the human body are three major requirements [5]–[7] for WBAN antennas. These features increase the difficulty of antenna design. To solve this issue, some studies propose using a reflector in antenna design to reduce the backward radiation and enhance directionality. In [5], a reflector was added to a 3 GHz–6 GHz slot antenna and this additional reflector enhanced the directionality and radiation efficiency. But the additional reflector affects the antenna bandwidth which becomes 4 GHz to 6.5 GHz and is not wide enough for the UWB system. In this communication, a novel directional UWB antenna is proposed for the WBAN application. The proposed antenna consists of a bevel

Fig. 5. Simulated return loss of various ground size.

edge feed structure and a truncated metal plate with folded strip. The size of the proposed antenna is 25 2 22 2 10 mm3 with ground plane 50 2 50 mm2 and the bandwidth covers from 3.1 GHz to 12 GHz. Section II presents the geometry and design concept of the proposed antenna. The design parameters and simulated SAR values are also introduced in Section II. Radiation patterns are shown in Section III. Finally, Section IV draws some conclusions.

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Fig. 6. Current distribution of the proposed antenna at (a) 4 GHz, (b) 7 GHz, (c) 10 GHz.

II. ANTENNA DESIGN CONCEPT AND PERFORMANCE Fig. 1 shows the geometry of the proposed antenna. The proposed antenna consists of a bevel edge feed structure and a truncated metal plate with folded strip. To enlarge bandwidth, the bevel edge structure achieves slow impedance variation by using traveling wave concept. The metal plate is designed as a main radiator. The truncated edge and the folded strip of the metal plate extend the current path to lower the operating frequency. In addition, because of the patch-like structure, the wave radiates toward the z direction and the ground plane reduces the backward radiation. The whole antenna size is 25 mm 2 22 mm 2 10 mm with a 50 mm 2 50 mm ground plane. The final parameters of the proposed antenna are H1 = 4 mm, H2 = 6 mm, H3 = 3 mm, W1 = 20 mm, W2 =W 3 = 5 mm, L1 = 22 mm, L2 = 8 mm, L3 = 7 mm, L4 = 10 mm. Fig. 2 shows the simulated and measured return losses. The simulation was performed using a commercial simulator while the measurements were taken by an E8364B network analyzer. The measured bandwidth covers from 3.1 GHz to 12 GHz and agrees with the simulated results. The minor discrepancies of simulation and measurement may be attributed to the connector, which is not considered in the simulation. Fig. 2 also shows the simulated result of the proposed antenna without the folded strip. It is evidenced that the folded strip not only determines the lower operating frequency from 3 to 4 GHz but also creates an additional resonant frequency around 6 GHz. The folded strip is the key factor of antenna design. The effect of folded strip is presented in Fig. 3. The folded strip extends the current path and creates lower resonant frequencies. According to Fig. 3, the length of L3 affects the lowest frequency and when the length of L4 increases, the impedance in the middle frequency becomes mismatched. By suitably adjusting the length of the strip, we can make the whole frequency band under the 10 dB return loss condition. The effect of the feed structure is shown in Fig. 4. The tapered profile of the feed structure achieves the slow impedance variation for obtaining the ultra-wide bandwidth. The slope of the bevel edge should be carefully designed to achieve wideband matching. In our experiments, the H1 should be 4 mm to obtain better impedance matching for the whole operating frequency. Fig. 5 shows the parametric simulations with regard to the different ground plane size. For the proposed antenna which radiates as a patch antenna, the ground plane should be large enough to resonant the desired frequency. In the simulating results, it can be observed that the impedance match is interfered and the bandwidth becomes narrow when the ground plane becomes 30 2 30 mm2 which is close to the main radiator. Therefore, the ground plane size of the proposed antenna should be larger than 40 2 40 mm2 to generate the wanted resonance mode. In the final design, 50 2 50 mm is chosen for the ground plane by the dimension and better impedance match. Furthermore, the current distribution of the proposed antenna in 4 GHz, 7 GHz and 10 GHz is exhibited in Fig. 6. In the low frequency, the current distributes along the edge of the truncated plate and the folded strip, which is like a patch antenna. Therefore, the truncated

Fig. 7. Comparison measured return loss between in free space and on the body.

part and the length of the folded strip determines the lowest frequency. In addition, a resonance mode can be observed at the folded strip in middle band. For high band, the tapered profile fed structure travels the energy to the plate and radiate as combination of general monopole and patch antenna. Moreover, the shape of the tapered profile crucially affects the impedance match within the whole band. The proposed antenna combines the patch-like radiator and traveling wave concept to achieve the ultra-wide bandwidth and directional patterns. In order to verify the proximity effect of human body, return loss of the proposed antenna in free space and on the body are measured, as shown in Fig. 7. The spacing between the proposed antenna and the human skin is 2 mm. The proximity effect of the human body slightly affects the impedance matching of the proposed antenna because that directivity of the proposed antenna is outward from the human body. Furthermore, a truncated body model is considered in simulation to estimate the specific absorption rate (SAR) value and radiation efficiency by software SEMCAD X. Two different body model is considered in the simulations, one is single layer muscle model and another is three layers body model with skin, fat and muscle according to the [5]. Full dimensions of the models are skin: 120 2 110 2 1 mm3 with "r = 38,  = 2:7 [S/m], fat: 120 2 110 2 3 mm3 with "r = 5:1;  = 0:18 [S/m] and muscle: 120 2 110 2 40 mm3 with "r = 50:8;  = 3 [S/m] according to [5]. Moreover, in order to reveal the SAR values in relative way, a planar disc monopole antenna with omnidirection pattern, as shown in Fig. 8, is also involved in the simulation for comparison. For keeping the same distance from the model to the top of the antenna (antenna height: 10 mm), the proposed antenna is 1 mm away from the body model and the planar disc monopole antenna (antenna height: 1.6 mm) is 10 mm away from the body model. Table I shows the simulated results of peak SAR values and the total radiation efficiency included the muscle and body models. As expected, the SAR values of the proposed antenna are lower than one of the planar UWB monopole antenna due to the patch-like structure and directional patterns of the proposed antenna.

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Fig. 8. Geometry of the planar disc monopole antenna and its simulated return loss. TABLE I PEAK SAR VALUES AND RADIATION EFFICIENCY (NORMALIZE TO 1 W)

Fig. 9. Simulated and measured radiation patterns (a) at 4 GHz XZplane, (b) at 4 GHz YZplane, (c) at 7 GHz XZplane, (d) at 7 GHz YZplane, (e) at 10 GHz XZplane, (f) at 10 GHz YZplane.

III. RADIATION PATTERNS The antenna power gain radiation patterns in free space are measured in an anechoic chamber with an Agilent E362B network analyzer and NSI2000 far-field measurement software. The xz- and yz-plane radiations at 4 GHz, 7 GHz and 10 GHz are illustrated in Fig. 9. The measured patterns agree with the simulated patterns while some minor discrepancies of simulation and measurement in xy-plane and yz-plane can be attributed to the interference of the coaxial cable and the absorber. The radiations perform directional patterns in xz-plane with peak gains of 5.8 dBi, 4 dBi, and 3 dBi for each frequency, respectively. In the yz-plane, the radiation patterns on the whole operating frequency are nearly directional patterns. The power levels of backward radiation are less than 05 dBi. It is evidenced that the proposed antenna is desirable in WBANs applications to reduce the backward radiation. IV. CONCLUSION A novel folded UWB antenna for WBAN applications has been proposed. The proposed antenna utilizes the bevel edge feed structure and the truncated metal plate with folded strip to achieve ultrawide bandwidth from 3.1 GHz to 12 GHz. The effects of the feed structure and the truncated metal plane are discussed in order to provide brief guidelines. The measured results show that the antenna is only slightly affected by the proximity effect of human body. Moreover, the simulated SAR

values are lower than the omnidirectional disc planar monopole. The patch-like structure can reduce the backward radiation and enhance the directionality. These features demonstrate that the proposed antenna is suitable for WBAN applications.

REFERENCES [1] T. Zasowski, F. Althaus, M. Stager, A. Wittneben, and G. Troster, “UWB for noninvasive wireless body area networks: Channel measurements and results,” in Proc. IEEE Conf. Ultra Wideband Systems and Technologies, Reston, VA, Nov. 2003, pp. 285–289. [2] A. Alomainy, Y. Hao, C. G. Parini, and P. S. Hall, “Comparison between two different antennas for UWB on-body propagation measurements,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 31–34. [3] A. Alomainy, A. Sani, A. Rahman, J. G. Santas, and H. Yang, “Transient characteristics of wearable antennas and radio propagation channels for ultrawideband body-centric wireless communications,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 875–884, 2009. [4] M. Klemm and G. Troester, “Textile UWB antennas for wireless body area networks,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3192–3197, 2006. [5] M. Klemm, I. Z. Kovcs, G. F. Pedersen, and G. Troster, “Novel smallsize directional antenna for UWB WBAN/WPAN applications,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 3884–3896, 2005. [6] Z. Shaozhen and R. Langley, “Dual-band wearable textile antenna on an EBG substrate,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 926–935, 2009. [7] N. Haga, K. Saito, M. Takahashi, and K. Ito, “Characteristics of cavity slot antenna for body-area networks,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 837–843, 2009. [8] G. A. Conway and W. G. Scanlon, “Antennas for over-body-surface communication at 2.45 GHz,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 844–855, 2009.

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Hybrid Mode Wideband Patch Antenna Loaded With a Planar Metamaterial Unit Cell Jaegeun Ha, Kyeol Kwon, Youngki Lee, and Jaehoon Choi

Abstract—A wideband patch antenna loaded with a planar metamaterial unit cell is proposed. The metamaterial unit cell is composed of an interdigital capacitor and a complementary split-ring resonator (CSRR) slot. A dispersion analysis of the metamaterial unit cell reveals that an increase in series capacitance can decrease the half-wavelength resonance frequency, thus reducing the electrical size of the proposed antenna. In addition, circulating current distributions around the CSRR slot with inmode radiation, creased interdigital finger length bring about the mode. Furthermore, the while the normal radiation mode is the mode can be combined with the mode without a pattern distortion. The hybridization of the two modes yields a wideband property (6.8%) and a unique radiation pattern that is comparable with two independent dipole antennas positioned orthogonally. Also, the proposed antenna achieves high efficiency (96%) and reasonable gain (3.85 dBi), even though . the electrical size of the antenna is only

TM

TM TM

0 24

TM

0 24

Fig. 1. Configuration of the proposed antenna. A patch with an interdigital capacitor is on the top side, and a CSRR slot is etched on the bottom side as a ground plane. The size of the ground plane is 40 mm 35 mm.

2

0 02

Index Terms—Hybrid mode, metamaterials, patch antennas, wideband antennas.

I. INTRODUCTION While the extraordinary electromagnetic features of the negative index medium were predicted by Veselago in 1968 [1], it took over thirty years to experimentally verify the feasibility of an artificial negative index medium [2]. Since then, it has been expected that this state-of-the-art technology will uncover enormous possibilities in electromagnetics and optics [3]. This achievement has encouraged many researchers to study the exotic properties of metamaterials, leading to the development of diverse applications [4]–[8]. The transmission line (TL) approach of metamaterials was established in 2002 [9]–[11]. Metamaterial TLs are called composite right/left-handed (CRLH) TLs because they have both right- and left-handed properties. In other words, a CRLH TL supports not only a positive phase constant, but also a negative phase constant in a specific frequency region and a zero phase constant at a nonzero frequency. The introduction of CRLH TL theory led to the rapid development of metamaterials for TL applications, including filters, mixers, and couplers [12]–[14]. At the same time, extensive studies have been performed on CRLH resonant antennas. Owing to the unique dispersion curves of CRLH TLs, compact CRLH resonant antennas can be realized [15]–[20], although they have inherent drawbacks of narrow bandwidth. To achieve a wideband characteristic, left-handed metamaterial was used as a compact radiating element [21]. However, the proposed antenna in [21] requires an additional matching circuit, which increases the overall antenna size. Also, coplanar waveguide-fed monopole antennas with metamaterial loading were reported in [22], [23]. Even though the broadband [22] or multi-band properties [23] were achieved by loading metamaterial, the radiation patterns were largely dependent upon the operating frequency. Manuscript received October 07, 2010; revised July 05, 2011; accepted July 20, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported by the IT R&D program of KCA (Korea Communications Agency) [KI002071, Study of technologies for improving the RF spectrum characteristics by using the meta-electromagnetic structure]. The authors are with the Department of Electronics and Computer Engineering, Hanyang University, Seoul, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173114

Fig. 2. Equivalent circuit model of the CRLH unit cell of the proposed antenna.

In this communication, a small and wideband microstrip patch antenna loaded with a planar CRLH unit cell is presented. In order to impose CRLH properties on a patch antenna, the antenna includes an interdigital capacitor for series capacitance and a complementary splitring resonator (CSRR) slot for shunt inductance. CSRR slots can be coupled with a TL or a waveguide in order to achieve CRLH characteristics [12], [24]. Owing to the CSRR and the interdigital capacitor, a CRLH unit cell is implemented in fully planar technology, and its dispersion characteristics are analyzed for small antenna application. In addition, the current distributions circulating around the CSRR slot induce a unique radiation mode that is orthogonal to the normal radiation mode. Moreover, combining two radiation modes provides a wideband property and a unique radiation pattern with high antenna efficiency, which is verified both numerically and experimentally. II. ANTENNA DESIGN Fig. 1 shows the configuration of the proposed antenna. In order to construct a single planar CRLH unit cell in the antenna, an interdigital capacitor is inserted into the patch for series capacitance, and a CSRR slot is etched on the ground plane for shunt admittance. The equivalent circuit model of the CRLH unit cell is shown in Fig. 2. The patch with an interdigital capacitor is represented as a series LC circuit (LR and CI ), while the CSRR slot is represented as a shunt LC resonant tank (LC and CC ). In addition, the capacitance between the patch and the ground plane (CR ) connects the shunt resonant tank to the patch [24]. From the equivalent circuit model, the dispersion relation can be written as:

d = cos01 1 +

ZY 2

!2 =!C2 0 1 !2 =!R2 0 CR =CI = cos01 1 + 2 (1 0 !2 =!z2) p p !R = 1= LR CR ; !C = 1= LC CC ; and !z = 1= LC (CR + CC ):

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Fig. 3. The return loss characteristics of the proposed antenna for various interdigital finger lengths (L3).

The resonance frequencies of the proposed CRLH unit cell can be derived from the dispersion relation in (1). The zeroth-order resonance (ZOR), where d = 0, occurs at !C . At the ZOR, the phase constant ( ) becomes zero and infinite wavelength propagation is allowed. The negative first-order resonance, where d = 0 , also occurs at !Z . The negative resonances support backward wave propagation with the same field distribution of positive order resonances. In addition, the positive order resonances arise where d = +n , but the equations for those resonances are not given here for the sake of simplicity. We note that the slope of the dispersion curve can be controlled by adjusting the series capacitance (CI ). For example, an increase in series capacitance (CI ) will give rise to the increase in the slope of the dispersion curve so that the positive resonances occur at the lower frequencies. Fig. 3 depicts the numerical return loss characteristics of the proposed antenna for various interdigital finger lengths (1 mm to 5 mm). The numerical simulations were performed using Ansys HFSS v. 12 [25]. The first-order resonance frequency (n = 1), or the positive halfwavelength resonance frequency, decreases from 5.5 GHz to 4.3 GHz as interdigital finger length (L3) increases from 1 mm to 5 mm. This is because the increase of L3 provides the increased series capacitance. Therefore, the electrical size of the proposed antenna can be reduced by increasing the interdigital finger length. However, the ZOR (n = 0) frequency is almost fixed at 2.18 GHz regardless of the value of L3, since it is only related to the capacitance and inductance of the CSRR slot (CC and LC in Fig. 2). Also, the ZOR is suppressed as the interdigital finger length becomes longer. Since the bandwidth of the ZOR is extremely narrow, it is not practical to use in an antenna application. In addition, it is interesting to note that the negative order resonance does not occur in the proposed antenna. In order to understand this phenomenon, let us consider the equivalent circuit model depicted in Fig. 2 and the full-wave simulated frequency response of the CRLH TL shown in Fig. 4. At the ZOR frequency of 2.18 GHz (!C ), the shunt admittance of the CRLH unit cell (Y) becomes zero because the resonant tank of the CSRR is open-circuited. Therefore, the reflection zero occurs at this frequency as shown in Fig. 4. In regard to the CRLH antenna, the impedance matching can be achieved at this frequency if the resistance is properly matched to the port impedance. However, the shunt admittance at the negative first-order resonance (n = 01) frequency of 1.86 GHz (!Z ) is infinity and short-circuited, causing a transmission zero to occur at this frequency as shown in Fig. 4. For an antenna with short-circuited shunt admittance, the impedance matching cannot be attained, regardless of the value of resistance. Therefore, unlike other CRLH resonant antennas, the proposed antenna does not have a negative order resonance.

Fig. 4. Simulated s-parameters of the transmission line with the CRLH unit cell for the proposed antenna (when L3 = 1 mm).

Fig. 5. Return loss characteristics of the proposed antenna when the interdigital finger lengths are between 6 mm and 8 mm.

Fig. 6. HFSS-simulated current distributions of the proposed antenna (L3 = mode (3.83 GHz) (a) on the patch and (b) on the ground plane.

8 mm) at the TM

For a conventional patch antenna, the current distributions at the fundamental mode form a half-wavelength resonance, and the fundamental radiation mode of the patch antenna oriented along the x-direction is the TM10 mode [26]. For the proposed antenna, an increase in the interdigital finger length can be used not only to reduce the size of the antenna, but also to excite another novel radiation mode. Fig. 5 shows the return loss properties of the proposed antenna for various interdigital finger lengths (6 mm to 8 mm). The first-order resonance frequency (n = 1) can be decreased further from 4.1 GHz to 3.6 GHz as the interdigital finger length increases. Moreover, when L3 is larger than 7 mm, another radiation mode is induced on the proposed antenna. Fig. 6(a) and (b)

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Fig. 7. HFSS-simulated electric field distributions of the proposed antenna (L3 = 8 mm) (a) at 3.61 GHz for the TM mode and (b) at 3.83 GHz for the TM mode.

show the current distributions on the patch and the ground plane, respectively at 3.83 GHz when L3 is 8 mm. The circulating current distributions around the CSRR slot on the ground plane induce y-oriented currents on the patch, which allows the TM01 mode to be supported in the proposed antenna, even though the feeding currents are directed along the x-direction. The resonance frequency of the TM01 mode depends on the width of the patch, rather than the length. The electric field distributions of the proposed antenna (L3 = 8 mm) at the normal TM10 mode (3.61 GHz) and at the TM01 mode (3.83 GHz) are illustrated in Fig. 7(a) and (b), respectively. At the TM10 mode, the electric field distributions on the x-oriented two edges are 180 out-of-phase as shown in Fig. 7(a). In contrast, at the TM01 mode, the electric field distributions on the y-oriented two edges are 180 out-of-phase as shown in Fig. 7(b). Therefore the two radiation modes that are orthogonal to each other can be attained in the proposed antenna. Furthermore, the TM01 mode can be combined with the TM10 mode in order to achieve a wideband characteristic. Owing to the adjustability of the TM10 mode (n = 1) frequency described in Fig. 3, the operating frequency of the TM10 mode can approach and combine with that of the TM01 mode when L3 is 7 mm as shown in Fig. 5. Thus, we can achieve a wide bandwidth with hybrid mode for the proposed antenna. Although the operating frequency bands of the TM10 and TM01 modes overlap, they do not interfere with each other due to the phase difference. Fig. 8 shows the electric field distributions of the proposed antenna (L3 = 7 mm) at 3.80 GHz for different input signal phases (0 , 90 , 180 , 270 ). In Fig. 8(a) and (c), when the input signal phases are 0 and 180 , the TM01 mode dominates the antenna radiation. On the contrary, when the input signal phases are 90 and 270 , the TM10 mode dominates as shown in Fig. 8(b) and (d). Therefore, the two radiation modes can operate independently because they have a phase difference of 90 . III. EXPERIMENTAL RESULTS Fig. 9 shows the prototype of the proposed antenna fabricated on a Teflon substrate ("r = 2:1, tan  = 0:001) with a thickness of 1.57 mm. L3 was optimized to 7.3 mm to achieve a wide bandwidth without changing the other design parameters. The simulated and measured return loss characteristics of the proposed antenna are shown in Fig. 10.

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Fig. 8. HFSS-simulated electric field distributions of the proposed antenna (L3 = 7 mm) at 3.80 GHz when the input signal phases are (a) 0 , (b) 90 , (c) 180 , and (d) 270 .

Fig. 9. The prototype of the proposed antenna. L3 was optimized to 7.3 mm for wideband property. (a) Top view. (b) Bottom view.

Fig. 10. Simulated and measured return loss characteristics of the proposed antenna.

The measured data were in good agreement with those of the simulation. The measured 10 dB return loss bandwidth was 260 MHz (6.8%) extended from 3.67 GHz to 3.93 GHz. The bandwidth of the proposed antenna was more than three times wider than that of a conventional patch antenna operating at the same frequency band (2.1%) because the two radiation modes are combined. The electrical size of the patch was 0:240 2 0:240 2 0:020 (19 mm 2 19 mm 2 1:57 mm) at the center frequency of 3.80 GHz. Owing to the CRLH structure and

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Fig. 12. HFSS-simulated 3D radiation patterns at 3.80 GHz for (a) a horizontal polarization and (b) a vertical polarization.

Fig. 13. Simulated and measured antenna efficiency and peak gain. Solid line: simulation. Dotted line: measurement.

Fig. 11. Radiation patterns of the proposed antenna at 3.80 GHz obtained from the full-wave simulation and the measurement in (a) the xz- and (b) the yz-planes.

the increased interdigital finger length, a 55% reduction in antenna size has been achieved, as compared to a conventional patch antenna size (30 mm 2 26 mm 2 1:57 mm). A patch antenna directed along the x-direction at the fundamental radiation mode (TM10 mode) has a monopolar pattern in the xz-plane for a vertical polarization and an omnidirectional pattern in the yz-plane for a horizontal polarization. Fig. 11 shows the numerical and experimental radiation patterns of the proposed antenna at 3.80 GHz. The proposed antenna exhibits a dipolar pattern for a vertical polarization and a near-omnidirectional pattern for a horizontal polarization in both the xz- and yz-planes. The proposed antenna supports back radiation due to the CSRR slot, and this back radiation is not significantly affected by the size of the ground plane. In fact, the increase in the size of the ground plane yields only a higher broadside radiation. For the proposed antenna, the TM10 mode illustrated in Fig. 8(b) and (d) brings about a dipolar pattern in the xz-plane and a near-omnidirectional pattern in the yz-plane, whereas the TM01 mode illustrated in Fig. 8(a) and (c) causes a dipolar pattern in the yz-plane and a near-omnidirectional pattern in the xz-plane. Therefore, the radiation pattern of the proposed antenna is like that of an antenna consisting of two x- and y-directed dipoles without destructive interference because of the existence of the hybrid

mode with 90 phase difference. In other words, the radiation pattern of the proposed antenna is near-isotropic with a maximum deviation of 11 dB for a horizontal polarization, and it is dipolar in all planes for a vertical polarization as shown in Fig. 12(a) and (b). Furthermore, the radiation pattern has the frequency-independent property over the frequency band of interest. Fig. 13 shows the simulated and measured antenna efficiency and peak gain of the proposed antenna. A 3D spherical far-field measurement system was used for the measurement. The measured antenna efficiency was higher than 80% over the 10 dB return loss bandwidth with the peak value of 96%. Also, the measured peak gain of the TM10 mode was 3.85 dBi and that of the TM01 mode was 2.36 dBi. The TM10 mode gain represents the vertical-polarization gain in the xz-plane or the horizontal-polarization gain in the yz-plane, and vice versa for the TM01 mode. Although the physical size of the proposed antenna is much smaller than a conventional patch antenna, the gain and the efficiency are not much degraded. IV. CONCLUSION A novel wideband patch antenna with a hybrid of TM10 and TM01 modes is proposed. The proposed antenna contains a single planar CRLH unit cell composed of a CSRR slot and an interdigital capacitor. By increasing the interdigital finger length, the electrical size of the antenna was decreased due to the increased series capacitance. The proposed antenna achieves a 55% reduction in patch size compared to a conventional patch antenna. Additionally, the increased interdigital finger length along with the CSRR slot generates the TM01 mode radiation, which can be combined with the normal TM10 mode. The combination of these two modes provides a wideband property (6.8%) and unique radiation pattern that are near-isotropic for the horizontal polarization and dipolar for the vertical polarization. Regardless of

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the small size of the proposed antenna, very high efficiency (96%) and moderate gain (3.85 dBi) are attained. The gain of the proposed antenna is only 2.2 dB lower than that of a conventional one operating at the same frequency band with the same ground size. Based on the antenna performances mentioned in the previous sections such as small size, high efficiency, and near-isotropic radiation pattern, one can conclude that the proposed antenna is applicable for a mobile RFID reader system requiring isotropic coverage.

REFERENCES [1] V. G. Veselago, “The electrodynamics of materials with simultaneously negative values of " and ,” Soviet Phys., vol. 10, no. 4, pp. 509–514, Jan. 1968. [2] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [3] M. Kaku, Physics of the Impossible. Jacksonville, FL: Anchor Books, 2008. [4] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science, vol. 314, pp. 977–980, Nov. 2006. [5] R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science, vol. 323, pp. 366–369, Jan. 2009. [6] N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science, vol. 308, pp. 534–537, Apr. 2005. [7] J. Y. Chin, J. N. Gollub, J. J. Mock, R. Liu, C. Harrison, D. R. Smith, and T. J. Cui, “An efficient broadband metamaterial wave retarder,” Opt. Expr., vol. 17, no. 9, pp. 7640–7647, Apr. 2009. [8] H. F. Ma, X. Chen, H. S. Xu, X. M. Yang, W. X. Jiang, and T. J. Cui, “Experiments on high-performance beam-scanning antennas made of gradient-index metamaterials,” Appl. Phys. Lett., vol. 95, pp. 094107–0941073, Sep. 2009. [9] C. Caloz and T. Itoh, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip “LH line”,” in Proc. IEEE Antennas Propag. Society Int. Symp. (AP-S), San Antonio, TX, Jun. 16–21, 2002, pp. 412–415. [10] A. K. Iyer and G. V. Eleftheriades, “Negative refractive index metamaterials supporting 2-D waves,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA, Jun. 2–7, 2002, vol. 2, pp. 1067–1070. [11] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [12] D. Yuan Dan, Y. Tao, and T. Itoh, “Substrate integrated waveguide loaded by complementary split-ring resonators and its applications to miniaturized waveguide filters,” IEEE Trans. Microw. Theory Tech, vol. 57, no. 9, pp. 2211–2223, Sep. 2009. [13] I. H. Lin, K. M. K. H. Leong, C. Caloz, and T. Itoh, “Dual-band subharmonic quadrature mixer using composite right/left-handed transmission lines,” IEE Proc. Microw. Antennas Propag., vol. 153, no. 4, pp. 365–375, Aug. 2006. [14] I. H. Lin, M. DeVincentis, C. Caloz, and T. Itoh, “Arbitrary dual-band components using composite right/left-handed transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 52, pp. 1142–1149, Apr. 2004. [15] Y. Dong and T. Itoh, “Miniaturized substrate integrated waveguide slot antennas based on negative order resonance,” IEEE Trans. Antennas Propag., vol. 58, no. 12, pp. 3856–3864, Dec. 2010. [16] A. Erentok and R. W. Ziolkowski, “Metamaterial-inspired efficient electrically small antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 691–707, Mar. 2008. [17] M. A. Antoniades and G. V. Eleftheriades, “A folded-monopole model for electrically small NRI-TL metamaterial antennas,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 425–428, 2008. [18] J. Kim, G. Kim, W. Seong, and J. Choi, “A tunable internal antenna with an epsilon negative zeroth order resonator for DVB-H service,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 4014–4017, Dec. 2009. [19] M. Schubler, J. Freese, and R. Jakoby, “Design of compact planar antennas using LH-transmission lines,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., Fort Worth, TX, Jun. 6–11, 2004, vol. 1, pp. 209–212.

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[20] S. Yoo and S. Kahng, “CRLH zor antenna of a circular microstrip patch capacitively coupled to a circular shorted ring,” Progr. Electromagn. Res. C, vol. 25, pp. 15–26, 2012. [21] M. Palandoken, A. Grede, and H. Henke, “Broadband microstrip antenna with left-handed metamaterials,” IEEE Trans. Antennas Propag., vol. 57, no. 2, pp. 331–338, Feb. 2009. [22] M. A. Antoniades and G. V. Eleftheriades, “A broadband dual-mode monopole antenna using NRI-TL metamaterial loading,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 258–261, 2009. [23] J. Zhu, M. A. Antoniades, and G. V. Eleftheriades, “A compact tri-band monopole antenna with single-cell metamaterial loading,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1031–1038, Apr. 2010. [24] J. Baena, J. Bonache, F. Martin, R. Sillero, F. Falcone, T. Lopetegi, M. Laso, J. Garcia-Garcia, I. Gil, and M. Portillo, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 1451–1461, Apr. 2005. [25] HFSS ver. 12 ANSYS Inc. [26] G. Kumar and K. Ray, Broadband Microstrip Antennas. London, U.K.: Artech House, 2003, ch. 2.

Explicit Relation Between Volume and Lower Bound for Q for Small Dipole Topologies Guy A. E. Vandenbosch

Abstract—A rigorous relation is derived between any local change of the volume of an electrically small radiating device and the lower bound of its radiation Q factor. The relation concerns the actual volume and not the circumscribing volume. This means that also (incremental) changes in volume can be studied where the circumscribing volume remains the same. The relation clearly proves that any arbitrary increase in volume decreases the Q, and any arbitrary reduction in volume increases the Q. When directly applied to volumes embedded within a sphere, it is almost trivial to rigorously prove the well-known fact that the full sphere provides the absolute minimal Q. The communication ends with a simple analytical proof of the limit as introduced by Thal for dipole type spherical TM modes. To the knowledge of the author, these explicit relations between Q factor and occupied volume have not been described in literature yet. Index Terms—Radiation Q factor, small antennas.

I. INTRODUCTION Since the beginning of the study of Q factors for electrically small radiating devices, the spherical shape always has played a crucial role. In the early days, Chu [1] derived an approximating minimum by expanding the field around a vertically polarized antenna into a series of spherical TMn0 wave functions. Collin and Rothschild [2] calculated the exact value of , also based on spherical modes. Later, McLean [3] calculated the minimum directly from the fields outside a sphere with radius completely enclosing the device. Thal [4] upgraded McLean’s result for spherical wire antennas, incorporating the internal reactive

Q

Q

a

Q

Manuscript received December 06, 2010; revised May 25, 2011; accepted August 08, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. The author is with the Department of Electrical Engineering, Division ESAT-TELEMIC (Telecommunications and Microwaves), Katholieke Universiteit Leuven, B-3001 Leuven, Belgium (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173127

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energy. Very recently, Hansen and Collin proposed simple formulas to reach the same goal as Thal [5]. Best has published several papers describing designs of antennas with Q factors as close as possible to the fundamental limits [6], [7]. Best’s papers use the spherical topology as a basis. The use of magnetic materials inside the sphere seems also to be promising in this respect [8]. Since the early days in the study of Q factors, it was well-known that smaller spheres yield higher Q factors. However, if we don’t change the radius of the sphere containing the radiator, but modify the volume internally, what will happen then to the Q? In [9] rigorous expressions were presented to calculate the reactive energies stored in the electromagnetic field surrounding an electromagnetic device. These expressions were used there to find the Thal limit, but starting from an a priori current flowing on an a priori defined sphere. In [10], a procedure was given to calculate the lower bound of Q for any arbitrary volume carrying any possible current. This resulted again in Thal’s limit for an a priori defined sphere. The current a priori assumed in [9] was now found as part of the solution. In this communication, a fundamental relation between the lower bound for the Q factor of a device and a local change of the volume occupied by this device is derived. Its interpretation is extremely simple: any increase in volume, in any direction, yields a smaller lower bound for Q. Based on this, it is almost trivial to prove the well-known fact that the sphere is indeed the topology yielding the lowest Q factor. It has to be emphasized that we do not start from the spherical shape a priori, but we find the spherical shape a posteriori. At the end of the communication, an alternative and very simple analytical proof is given for the limit as set by Thal [4] for spheres carrying a dipole type current. II. THE MINIMUM Q OF AN ARBITRARY TOPOLOGY Consider an electrically small volume V . For cases where no magnetic materials are present, in [10] a technique is given to calculate the Q factor Qr in terms of the current J flowing inside V . The Qr is actually determined by the charge distribution  generated by J. Using the same nomenclature as in [10]

Qr =

V

0

V V k 3 V V

S

V

 0

3

R2 dV 0 dV

1+ R 3

k03 Qr

R2 dV 0 = 0 inside V:

1 + k03 Qr R2 R 3

dS 0 = 8(0 ) = 0 on S:

1

2k03 Qr 3

dV 0 =

R

0 r0 dV 0 = V

III. INFINITESIMAL CHANGE OF A TOPOLOGY

1 + k03 (Qr + dQr ) R2 R 3

0 + d0 S

0 dS mod

=0

(5)

on an infinitesimally modified surface S mod . Straightforward working out and neglecting infinitesimally small contributions of the second order leads to

0 S

+

1 + k03 Qr R2 R 3 d0

S

+

0

0 dS mod

1 + k03 Qr R2 R 3

k03 dQr

3

0 R2 dS mod

0 dS mod

= 0 on Smod :

(6)

The last two contributions in (6), integrals over S mod , differ from the corresponding integrals over S infinitesimally to the second order. This means that S mod can be replaced in these integrals by S . The distance R in the first contribution can be written as

R(S mod ) = r + ndn 0 r0 0 n0 dn0

 r0r + 0

(2)

 R(S ) +

1 jrr 00 rr j 1 rR0(Sr) 0

ndn 0 n dn 0

0

0

0

0

ndn 0 n dn

0

(7)

where r and r0 are located on S , n and n0 are the unit vectors normal to S at these locations, and dn and dn0 are the distances between S and S mod at these locations. This means that

) 1 (r 0 r )  1 1 0 (ndn 0 nRdn R(S mod ) R(S ) (S )2 R(S mod )2  R(S )2 + 2 ndn 0 n dn 1 (r 0 r ): 1

0

0

Note that the operator 8 is “singular” on S . It projects a non-zero function into the zero function on S . Also note that (3) is actually valid inside the entire volume V , i.e., at the inside of S . Since (2) is valid at all points inside V , its gradient also has to be zero inside V . Working out

(4)

Assume an infinitesimal change dV in the volume V . The condition for the new lower bound for the Q factor Qr + dQr can be expressed as

S

(3)

2k03Qr P 3

with P the electric dipole moment of the charge distribution. Equation (4) shows that (2) is equivalent to solving the topology statically for a constant incident electric field, simultaneously requiring that the resulting electric dipole moment is parallel to this field. Note that this is only possible for certain directions of the incident field. Expression (4) forms the link between the practical numerical calculation procedure introduced in [10] and the theoretical approach as explained in [11]. Note that, although the result is obviously the same, the procedure to obtain this result is quite different. This can be seen as a mutual validation. Our result is fundamentally based on the equations for the reactive energies as stated in [9]. [11] is based on considering the topology as a scatterer.

(1)

It was obtained for the volume V filled with vacuum and assuming lossless conductors. The solution of (2) with the lowest Qr yields the largest intrinsic bandwidth of the topology. Since both for perfect conductors and homogeneous dielectrics the charge distribution is actually located only at the surface, we only need to consider the integration of a surface charge distribution  over the surface S of the volume V , yielding

0

0 r

0 3 R1 dV 0 dV

with R = jr 0 r0 j = (x 0 x0 )2 + (y 0 y 0 )2 + (z 0 z 0 )2 the distance between source point r0 and observation point r. The prime indicates source coordinates. Also in [10] a rigorous technique is given to derive the lower bound for Qr for any possible current. The necessary condition is given by

0

the gradient of the squared distance function and taking into account that the integrated charge within V is zero yields

0

0

0

0

(8)

Using the principal radii of curvature in each point of S , we can write 0 dS mod

= 112

1 + dn0 2 + dn0 dS 0

 1 + 1 + 1 0

1

0

2

dn0 dS 0 :

(9)

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Using (8) and (9) in the first term of (6), and taking into account (3), this term can be written as

1

0

 6

3

3

0 (2@ (r)n 1 ndn) 0 S



+

1

0

R

S

0 r(

 S

S

1

R

ndn

0

0 n dn 0

02@ (r)n 1 n dn 0

1

2

01

+

0

0

k03 Qr

R2 )(

3

1

01

+

dn dS 0

02

1

0

1

S

1

k03 Qr

R

1

0 n dn

ndn

0

+ S

d (

+

0

S

0

+

+

3

1

)dn0 dS 0 :

R

+

3

k03 dQr 3

S

R )dS

S S

S S

6 4

1

R

+

k03 Qr 3

R2

1

S

0

3 0

+ S S

1

R

k03 Qr 3

R2

R

SS

0

=

1

1 0

+

1

2 0

+

R

k03 Qr

+

R

3

k03 Qr

+ 1

0

k03 Qr 3

1

3

R

S

S

k03 Qr 3

1 n dn dS dS

R2

0

0

0

R2 dS 0

1

R2

01

k03 Qr

+

3

+

1

3 d0 ( S S

1

R

k03 Qr

+

3

1

R2 dS

(13)

+

01

1

02

dn0 dS 0 = 0: (14)

R2 )dS 0 dS

d0 ( 3 (

=

S

S

1

R

+

k03 Qr 3

R2 )dS )dS 0 = 0: (15)

Working out the fifth term, taking into account the fact that the total charge has to remain zero, delivers

3 0 S S

k03 dQr 3

k03 dQr 3

R2 dS 0 dS 3 0 r2 + r02

S S

02k03 dQr = 3

S

3 rdS 1

0 2r 1 r

0

dS 0 dS

0 r0 dS 0 S (16)

with P the electric dipole moment of the charge distribution. The equation becomes

S

dn0 dS 0 dS

1 ndn dS:

dn0 dS 0 dS

02

3

dS 0 dS

0

1 ndn dS dS

R2

3

+

0

Working out the fourth term, and taking into account (3), we obtain

2 +

1

02k03 dQr jPj2 =

3 dndS S

1

3 0

(11)

0

R

S

=

0 n dn

1

1 ndn dS dS

Working out the third term, and taking into account (3), we obtain

0

ndn

0

3 r

=2

Multiplying with  3 and integrating over S delivers then

3 0 r

dS 0 dS 1 k3 Qr + 0 R2 R 3

0

3 0 r

=2

R2 dS 0 = 0 on S:

(12)

R2

3

3 0 r0

S S

(10)

0

2

3

S S

R2

k03 Qr

0

=

dS 0 6 4dn 1 1 1 k3 Qr + dn0 dS 0 + 0 R2 R 3 01 02 0

0 n dn  r

ndn

k03 Qr

+

R

+

02

R2 dS 0 dS

3

R2 dS 0 dS = 0:

3

1

3 0 r

dS 0

The occurrence of the two Dirac impulses in the right hand side of (10) deserves a more detailed explanation. It is well known that the 01=R3 term in the integrand of the first term at the right hand side in the limit (r 0 r0 ) ! 0 produces a Dirac impulse. The sign of the Dirac impulse depends on how r approaches r0 . If this approach follows a path with positive dn the sign is negative, if it follows a path with negative dn the sign is positive. It is this effect that produces the sudden jump of the normal electric field across a surface with a charge distribution. However, in the integral considered, this effect does not occur, due to the fact that the local charge, producing the effect, is actually not located on S . This local charge has been moved to the new surface S mod by modifying the volume. This means that the Dirac impulses inherently incorporated in the 01=R3 term have to be canceled out explicitly by the Dirac impulses introduced. For a positive dn this leads to a canceling positive sign (top), and for a negative dn this leads to a canceling negative sign (bottom). For the n0 dn0 term the path (i.e., how r approaches r0 ) is always at the other side compared to the ndn term. Together with the additional minus sign in the expression this leads to the same sign as for the ndn term. Equation (6) now becomes

0 r

k03 dQr

3 0

k03 Qr

Since r = 0r0 the first term can be made symmetrical in source and observation coordinates

dS 0

S S

1

+

R

S S

R2 ) 1 (ndn 0 n0 dn0 )dS 0 6 4dn

3

+

R

3

k03 Qr

+

R

0 (

+

1

k03 Qr

+

S S

+

0r 1

r

1

3 d0

+

0 R2 dS mod

0 R13 + k03Qr 2

0 S

+

R

S

k03 Qr

1149

3 r

0 ( S

1

R

+

k03 Qr 3

64

S

R2 )dS 0

1 ndn dS

3 dndS 0

2k03 dQr jPj2 = 0 (17) 3

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where the top sign has to be chosen for positive dn, and the bottom sign has to be chosen for negative dn. Since (3) is valid inside the whole volume V , its gradient inside V is also zero. This can be exploited by choosing dn negative, which yields 3

04 jj2 dndS 0 2k03dQr jPj2 = 0:

(18)

S

Solving for dQr delivers

6 dQr = 0 3 S k0

jj2 dndS jj2 dndS 6 S jPj2 = 0 k03 rdS 2 :

(19)

S

Note that (19) is also valid for positive dn. This is because the sum of the first two terms in (17) is continuous at the surface S , as a consequence of the introduction of the explicit Dirac impulses in (10) to remove the discontinuity. Expression (19) is the key novelty in this communication. Its main feature is that it allows to study in a very simple way the effect of local changes in the volume on the Q factor bound in terms of the charge distribution. The change in Qr for a small modification of the surface at a specific location is proportional to the square of the amplitude of the charge density there. The change is always negative for any increase of the volume, and thus positive for any reduction of the volume. This means that any increase of the volume of a device always leads to a smaller lower bound for Q, even if the increase is towards the inside, thus not enlarging the circumscribing sphere, and any decrease of the volume of a device always leads to a larger lower bound for Q. To the knowledge of the author, this is the first time that this very simple and practical property is rigorously proven. Since in most cases the currents are flowing on surfaces rather than within volumes, we need to derive a similar statement for surface currents. Since a surface can always be considered as a volume for a thickness going to zero, it can also be concluded that any increase of the current carrying area of a device always leads to a smaller lower bound for Q, even if the increase does not enlarge the circumscribing sphere, and any decrease of the current carrying area of a device always leads to a larger lower bound for Q. Also an open surface can be considered as the limit of a volume with the thickness going to zero. An infinitesimal change of the shape of the open surface can be decomposed into first changing the volume at one side of the infinitesimal small thickness, then followed by the other side. Applying (19) then delivers for surfaces

limt!0 ( S 6 dQr = 0 3 k0

j+ j2 dndS 0 j0 j2 dndS ) jPj2

S

:

(20)

The plus and minus sign stand for the side of the surface in the positive and negative n direction. The charge density on the surface is distributed over the two sides of the surface in such a way that the total field inside is zero. This total field takes into account the “constant incident electric field” due to the right hand side of (4). The expression (20) says that in order to lower the Q factor, any change of the surface has to be in the direction of the largest charge density, or in other words, the largest electric field component normal to the surface. IV. VALIDATION AND APPLICATION In this section first expression (19) is proven analytically for two canonical topologies: the spherical dipole and the cylindrical dipole with normal-to-axis polarization. Also for the thin plate with normal polarization, the proof can be given analytically. For these topologies, the current delivering the minimum Q factor is analytically known. The

Fig. 1. (a) The spherical dipole (b) the cylindrical dipole with normal-to-axis polarization, and (c) the cap antenna.

first topology is bounded in three dimensions, the second one in two dimensions (solution expressed per meter length), and the last one in one dimension only (solution expressed per square meter). The spherical dipole (Fig. 1(a)) consists of a current flowing on a spherical surface with radius a, in a spherical coordinate system given by = sin   . The charge distribution is  = 02=(j!a) cos  . In [9] and [10] it was already shown numerically that, in the limit k0 a ! 0, for this structure

J

i

1:5 2 lim Qr = = 3 k a 0 (k0 a)3 k0 V

!

(21)

which perfectly agrees with Thal [4]. For a uniform positive dn over the whole surface of the sphere, i.e., make the sphere a little bit bigger by choosing a new radius a + da, the integral in (19) is easily calculated as

S

jj2 dndS = 316!2 da:

(22)

The electric dipole moment is easily calculated as 2

P = 83a iz : j!

(23)

Inserting in (19) delivers

dQr = 0

6 316! da 9 = 0 3 4 da k03 649! a 2k0 a

(24)

which is indeed the derivative of expression (21). The cylindrical dipole (Fig. 1(b)) with normal-to-axis polarization consists of a current flowing on a cylindrical surface with radius a and length l >> a, in a cylindrical coordinate system given by = cos ' ' . The charge distribution is  = 01=(j!a) sin '. Using expression (1), while neglecting any end effects and formulating the integrals per meter length, the Q factor can be calculated analytically. The result is

J

i

3 3 lim Qr = 3 2 = 3 : k a 0 k0 a l k0 V

!

(25)

For a uniform positive dn over the whole surface of the cylinder, i.e., make the cylinder a little bit bigger by choosing a new radius a + da, the integral in (19) per unit length becomes 2 jj2 dnad' = !21a2 sin2 'ad'da = !2 a da: (26) ' 0

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1151

Assume a rectangular plate with dimensions Lx and Ly , in the x polarization regime. The thickness t of the plate is small compared to its x and y dimensions. Since the current has to be smooth, and it has to be zero at the ends, the simplest current approximation is an x directed current of the form

J = 1 0 L2xx

2

ix :

(30)

Deriving the charge, equally dividing it over both sides, and inserting it in (29), the calculation of the change of the lower bound for Q is straightforward. All integrals can be calculated analytically

dQr = 0 Fig. 2. The lower bound for the Q factor of the cap antenna as a function of  , for three values of  . For  the structure is actually a cylinder closed at the top and bottom by the caps.

=0

Inserting in (19) delivers

6 dQr = 0 k03

 ! a  a !

= 0 k36a3 l da 2

(28)

0

which is indeed the derivative of expression (25). The cap antenna (Fig. 1(c)) consists of a central cylinder of diameter D and height h feeding two spherical caps. In a spherical coordinate system the top cap extends from the connection between the cylinder and the cap at cyl up to the opening angle of the cap cap . The bottom cap is the image of the top cap. The lower bound for the Q factor obtained with the technique of [10] as a function of cap is presented in Fig. 2 for three different values of cyl . For this structure the lower bound for Q shows a maximum, located at cap = cyl (where the cap disappears). At these points the Q factor is of course the same as the Q factor of the corresponding cylinder dipole antenna, depicted in the top curve. These curves illustrate clearly that starting from the cylinder (with zero cap), enlarging the surface, i.e., by enlarging the cap, in either direction, decreases the lower bound, as predicted by (19). Note that for small cyl , the lower bound approaches the absolute lower bound of a sphere for cap = 90 . Expression (19) can be used also to study the lower bound for Q as a function of the thickness t of the conductor used. To this goal, starting from an infinitely thin conductor, we move both sides of the conductor equally and in opposite directions over half the thickness dt. Realizing that in (20) a uniform change in conductor thickness to the first order does not have effect on the charge distributions at both sides of the conductor, we obtain easily

dQr = 0

6 k03

S

j+ j2 + j0 j2 jPj2

Qr = Qr (t = 0) 0

(27)

lda l

Q becomes

dS

dt 2

:

9t : k03 Lx3 Ly

(31)

(32)

Such an approximate calculation, based on assumed currents can be useful to estimate the order of magnitude of the effect of the thickness.

The electric dipole moment per unit length is calculated as

P = 0 a i: j! y

9dt : k03 Lx3 Ly

(29)

This shows that the change in Q is just proportional to the change in conductor thickness, as long as the thickness of the conductor is much smaller than its transversal dimensions. Also, it is easily seen that any increase in the thickness lowers the lower bound for Q. A practical case can be considered, the rectangular plate. The current delivering the minimum Q is not analytically known, but an approximation is used, based on physical insight. The approximation can always be improved by using the charge distribution obtained from numerical tools.

V. THE SPHERICAL SHAPE In this section, first it is rigorously proven that the sphere is the topology with the smallest lower bound for Q for a given radius. Then, this smallest value is derived analytically. Evidently, this yields the well-known value known in literature already, from numerical calculations [9], [10], or based on the proof in [4]. It has been long known that the spherical dipole has the absolute minimal Q factor for any topology with a maximum dimension of 2a. With (19) this is almost trivial to prove. Consider any topology with maximum distance 2a between any two points. This means that the topology can be completely embedded within a sphere of radius a. According to (19) any increase in the volume leads to a decrease of the lower bound for Q. By increasing the volume up to the complete sphere, we reach the absolute minimum lower bound for Q. Any further increase of the volume would result in a larger sphere to contain the topology. We conclude this theoretical communication by proving (21) analytically. Since (3) is valid inside the entire volume V , all its derivatives are also zero inside this volume. Taking the gradient and evaluating it at the origin, located in the center of the sphere, yields

0 S

Since for a sphere r0

1 02 a3 3

= a, we obtain

k03 Qr

P

1 0 2k03 Qr r0 dS 0 = 0: r03 3

0 r0 dS 0 = S

1 0 2k03Qr P = 0: a3 3

(33)

(34)

Since differs from zero, the term between brackets has to be zero, which directly yields (21). VI. CONCLUSIONS A simple relation is found between a change in volume and the corresponding change in lower bound for the Q factor of a small radiating device. From his relation, it is seen that a larger volume always results in a lower Q, independent of the way in which the volume is enlarged. The relation is validated for two canonical topologies, for which analytical solutions are available. It is illustrated through a numerical example. The effect of conductor thickness can be studied using the relation. The well-known value of the lower bound for Q for a sphere, the topology yielding the absolute minimum lower bound for Q, is derived analytically.

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REFERENCES [1] L. J. Chu, “Physical limitations on omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, Dec. 1948. [2] R. E. Collin and S. Rothschild, “Evaluation of antenna Q,” IEEE Trans. Antennas Propag., vol. AP-12, pp. 23–27, Jan. 1964. [3] J. S. McLean, “A re-examination of the fundamental limits on the radiation Q of electrically small antennas,” IEEE Trans. Antennas Propag., vol. 44, pp. 672–676, May 1996. [4] H. L. Thal, “New radiation Q limits for spherical wire antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2757–2763, Oct. 2006. [5] R. C. Hansen and R. E. Collin, “A new Chu formula for Q,” IEEE Antennas Propag. Mag., vol. 51, no. 5, pp. 38–41, Oct. 2009. [6] S. R. Best, “Low Q electrically small linear and elliptical polarized spherical dipole antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1047–1053, Mar. 2005. [7] S. R. Best, “A low Q electrically small magnetic (TE mode) dipole,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 572–575, 2009. [8] O. S. Kim, O. Breinbjerg, and A. D. Yaghjian, “Electrically small magnetic dipole antennas with quality factors approaching the Chu lower bound,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 1898–1906, Jun. 2010. [9] G. A. E. Vandenbosch, “Reactive energies, impedance, and Q factor of radiating structures,” IEEE Trans. Antennas Propag., vol. 58, no. 4, Apr. 2010. [10] G. A. E. Vandenbosch, “Simple procedure to derive lower bounds for radiation Q of electrically small devices of arbitrary topology, accepted,” IEEE Trans. Antennas Propag., vol. 59, no. 6, pp. 2217–2225, Jun. 2011. [11] A. D. Yaghjian and H. R. Stuart, “Lower bounds on the Q of electrically small dipole antennas,” IEEE Trans. Antennas Propag., vol. 58, no. 10, Oct. 2010.

On the Generalization of Taylor and Bayliss n-bar Array Distributions Srinivasa Rao Zinka and Jeong Phill Kim

Abstract—Taylor’s asymptotic analysis theory is used to design the generalized Taylor and Bayliss patterns. Such a design technique allows generating array factors with arbitrary sidelobe level and envelope taper. For both the Taylor and Bayliss patterns, array excitation is obtained by the Elliott’s pattern zero matching technique. A few examples are provided to validate the presented theory. Also, variation of different array characteristics with respect to the sidelobe tapering is explained through graphical data. Index Terms—Array synthesis, Bayliss, Chebyshev, optimum difference pattern, optimum sum pattern, sidelobe taper, Taylor, Zolotarev.

d < =2, Riblet provided a modified formulation to achieve the optimal distribution for odd numbered arrays only. In a recent paper [3], McNamara broadened the Riblet’s formulation to even numbered arrays with d < =2 by using the more general Zolotarev polynomials. When it comes to the difference patterns, McNamara-Zolotarev difference1 pattern [4] is optimum in the Dolph-Chebyshev sense [5], [6]. It has the narrowest first null beamwidth and largest normalized difference slope on boresight for a specified sidelobe level. Even though all the sum and difference patterns mentioned till now are ideal from the perspective of the SLR, these distributions suffer from poor array taper efficiency (ATE) [7] and edge brightening (undesirable upswing in the amplitude of the excitations near the array edges). In addition, equal far-out sidelobes tend to pickup undesired interference and clutter. In order to overcome these limitations, a dilation technique can be used, in which a sidelobe taper is introduced after the  array factor zeros [8]–[10]. All these n  distributions are continfirst n uous sources. An equivalent discrete distribution can be obtained either by sampling the continuous distribution or by using a different method proposed by Elliott [11]. In [12], [13], authors inherently used the Elliott’s method to obtain discrete Taylor and Bayliss array analogues. The patterns corresponding to the continuous (or discrete) Taylor and Bayliss distributions possess sidelobe taper of the order 1=kx , which corresponds to the special case2 of  = 0 [8]. If a higher order sidelobe taper (i.e., 1=kx+1 ) is required, then the  values greater than 0 should be used as done by Rhodes [14]. However, the Rhodes distribution is a continuous source. The present authors have previously provided a formulation for obtaining a discrete equivalent of the Rhodes distribution [15]. In this communication, authors extend their technique to generalize the discrete Bayliss distribution too. Also, authors would like to point out that the array factor zeros used in this communication are different from those used in [13], [16] by McNamara. The reason for choosing different sets of array factor zeros is explained in Section II from the view point of Taylor’s asymptotic analysis. Finally, array excitation coefficients are computed from the chosen array factor zeros by using the Elliott’s technique [11]. II. SUMMARY OF THE TAYLOR’S ASYMPTOTIC ANALYSIS Due to the drawbacks of equal sidelobe distributions mentioned in Section I, a sidelobe taper is required in most of the practical situations. In [8], Taylor provided a comprehensive analysis regarding the effect of the array excitation edge tapering on the pattern’s asymptotic behavior. For the sake of continuity, Taylor’s theory is briefly reproduced here. Let the line source shown as embedded in Fig. 1, which has a total length 2a, have a distribution function A(x). Then the array factor is given as a

AF (kx ) =

I. INTRODUCTION Dolph-Chebyshev linear arrays [1] are ideal in the sense that they provide the narrowest first null beamwidth possible for a given sidelobe ratio (SLR). In a discussion [2], Riblet noted that the formulation by Dolph is optimal only for inter-element spacing d  =2. For Manuscript received March 11, 2011; revised June 06, 2011; accepted August 03, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. The authors are with School of Electrical and Electronic Engineering, Chung-Ang University, Seoul 156-756, Korea (e-mail: srinivas.zinka@gmail. com; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173146

0a

A(x)ejk x dx:

(1)

From the view point of asymptotic analysis, the degree of abruptness with which the source distribution begins and ends at x = 6 a will be a matter of importance. Then suppose that

A(x)  K 0 (x + a) as x ! 0a A(x)  K + (a 0 x) as x ! +a

(2)

Thus if  = 0, A(x) assumes the fixed non-zero values K 0 and K + at the edges and the distribution is said to have a pedestal. If  = 1, 1Authors used the word difference in order to avoid confusion with another distribution by the same author [3]. 2For physically realizable continuous distributions, the parameter any real number greater than 0.

0018-926X/$26.00 © 2011 IEEE

 can be

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 2, FEBRUARY 2012

1153

Fig. 2. Schematic diagram of a linear array of M elements having an interelement spacing of d.

!0

9

Fig. 1. The distribution function embedded in the complex -plane.  is ar. Furthermore, it is assumed that bitrarily small but non-vanishing and  A is holomorphic in the race court shaded region.

(9)

the distribution falls to zero linearly at its ends (e.g., cosine distribution). All values of  greater than 01 were considered. However, it should be noted that for  values in the domain 01 <  < 0, the distribution function will be non-uniformly-bounded (i.e., physically non realizable). Evidently, it is possible to write the distribution function A(x) as

A(x) = B (x)(a2 0 x2 )

(3)

where B (x) does not vanish at x = 6a. Now, let the variables x and kx be embedded in the complex domains 9 and respectively, such that 9 = x + jxI and = kx + jkxI . The functions A(x) and AF (kx ) are then profiled on the axes of reals of the functions A(9) and AF ( ), respectively. Furthermore, it is assumed that B (9) is different from zero at 9 = 6a and regular in the racetrack shaped region of Fig. 1. Then, asymptotic forms of integrals along the deformed paths C1 and C3 (Fig. 1) are given respectively as

1) 0j a+j I1  B (0a)(2a) 0(+ e

+1 I3  B (a)(2a) 0( + 1) ej a0j

+1

(4)

as j j ! 1. Also, it can be shown that for large j j, jI2 j will always be negligible compared to jI1 j or jI3 j (I2 is integral along the path C2 ). In [8], Taylor chose B (9) as an even function because he was concerned primarily with the sum patterns. In order to extend Taylor’s results to the difference patterns, the present authors choose B (9) as an odd function. So, by combining I1 and I3 , asymptotic forms of the integral (1) for both sum (6) and difference (1) patterns are as given below

( + 1) 1) AF 6  2B (a)(2a) 0(+ cos kx a 0 2 kx +1  ( + 1) 0( + 1) 1  AF  j 2B (a)(2a) sin kx a 0 : 2 kx+1

(5)

From (5), it can be intuitively perceived that the array factor zeros tend to 6 kxn !

and 1 ! kxn

6 n + 2 a 6 n +  +2 1 a

as n ! 1 for sum and difference3 patterns, respectively, where n = 1; 2; 3; 1 1 1. A comprehensive proof for the above equation was given in [8, Theorem IV]. III. SYNTHESIS OF THE GENERALIZED DISCRETE TAYLOR BAYLISS ARRAY DISTRIBUTIONS

Generalized discrete Taylor and Bayliss array distributions were already addressed by McNamara in [16] and [13], respectively. However, the array factor zeros used by McNamara were different from those given by (6). The reasons for choosing different sets of zeros compared to (6) were not mentioned in those papers. Also, choosing such zeros will not provide enough information regarding the array factor sidelobe tapering rate. So, in this communication, authors provide a  array different formulation for synthesizing the generalized discrete n distributions. A. Generalized Discrete Taylor Distribution A symmetric linear array of M elements with uniform inter-element spacing d as shown in Fig. 2 is considered. For the time being, it is assumed that d  0:5. For a given sidelobe ratio R, M 0 1 array factor zeros of the Dolph-Chebyshev array pattern are given as

n 0 1) 6 d2 cos01 1c cos (2 2(M 0 1) n = 1; 2; 3; 1 1 1 ; ceil [(M 0 2)=2]

DC kxn =

(7)

where c = cos h(cos h01 R=(M 0 1)). These zeros will be referred to as parent array factor zeros. Similarly, zeros given by (6) will be named as generic array factor zeros [16]. In designing simple Taylor distribution, far end array factor zeros (n  n  ) are equated to those of the uniform array of M elements [8]. In order to extend the original Taylor distribution for higher order sidelobe tapering, the authors choose far end zeros as (from (6))

 2 6 kxn = 6 n+ 2 Md n = n ; n + 1; . . . ; ceil [(M 0 2)=2)]

(8)

where  is a real number greater than 0. When  = 0, the above zeros simply become the zeros of the uniform array. So, after the dilation procedure, array factor zeros corresponding to the generalized Taylor distribution are given as T DC kxn = 6 kxn ; n  n : n  n kxn ; 0

(6)

AND

3For the difference patterns, the default array factor zero k cluded in (6).

(9)

= 0 is not in-

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where dilation factor T is defined as

kx6n : T = DC kxn

(10)

In comparison, the array factor zeros chosen by McNamara are [4]:

DC ; MT kxn n  n DC n 0 kDC ; n  n  kxn + ( + 1) 2Nd xn MT where, dilation factor  is defined as 00 kxn =

(11)

kxn MT = DC : (12) kxn 00 From (11), it is evident that for n  n DCthe array factor zeros kxn depend upon the Dolph-Chebyshev zeros kxn , which is not the case with the 00 formulation (9). When  = 01, the modified zeros kxn reduce to the 00 DC Dolph-Chebyshev zeros kxn . On the other hand, when  = 0, kxn are equal to those of the simple Taylor distribution. When  > 0, the sidelobe tapering is more rapid than 1=kx . But, the exact amount of 00

tapering cannot be estimated from the formulation (11). This is not a problem with the formulation (9) because of the readily available asymptotic relation between  and AF (5). From (5), it is understood that the sidelobe tapering is of the order (1=kx )+1 . For d < 0:5, the same dilation technique but with the McNamara sum pattern zeros as the parent zeros should be used [3]. B. Generalized Discrete Bayliss Distribution

The Bayliss array distribution is the difference pattern counterpart of the Taylor distribution [10]. But, Bayliss was not aware of the existence of the optimum difference pattern. So, he started with a polynomial which was obtained by taking the derivative of the optimum sum pattern. Then an iterative procedure was used to make all the sidelobes equal. Later, McNamara devised a technique to obtain the optimum difference patterns using the Zolotarev polynomials [4]. The same author also provided a method to synthesize discrete Bayliss distributions [13]. However, for higher order tapering, the array factor zeros used by McNamara do not coincide with those given by (6). So, a different formulation is provided here to obtain the generalized discrete Bayliss distribution. Since the Zolotarev polynomials are less familiar compared to the Chebyshev polynomials, a simple graph is provided with relevant parametric equations (Fig. 5). For a complete treatment, refer [4], [6]. K (m) is the complete elliptic integral of the first kind, to the parameter m. H (#; m) is the Jacobi eta function. The sn(#; m), cn(#; m) and dn(#; m) are the Jacobi elliptic functions, while zn(#; m) is the Jacobi zeta function [17]. Open source routines for implementing these functions using arbitrary-precision floating-point arithmetic are available, e.g., [18]. Similar to the Dolph-Chebyshev array, array factor of the McNamara-Zolotarev difference pattern array is given in terms of the (M 0 1)th order Zolotarev polynomial as [4]

AF (kx ) = ZM 01 c sin where

c = csc c=1

k d

2

kx d

2

;m

; if d  =2 : if d  =2

(13)

(14)

The parameter m is decided by the amount of SLR required. For the computational aspects, refer to [4, Appendix]. From the above formulation and Fig. 5, array factor zeros are given as

MZd = 6 kxn where n

2 sin

d

1 xn c

0

= 1; 2; 3; . . . ; ceil[(M 0 2)=2)].

(15)

Fig. 3. Generalized discrete Taylor distributions with different  values.

Analogous to the Taylor distribution, to obtain the generalized Bayliss distribution, far end zeros are replaced by (again, from (6))

2 6 n +  +2 1 Md n = n ; n + 1; . . . ; ceil [(M 0 2)=2)] :

1 = kxn

(16)

When  = 0, the above zeros approach the zeros of the maximum slope difference pattern (which corresponds to the linear odd array excitation) as n ! 1. So, after the dilation procedure, array factor zeros corresponding to the generalized Bayliss distribution are given as 0 kxn =

MZd ; n  n B kxn : 1 n  n kxn ;

(17)

where dilation factor  B is defined as

kx1n B = MZd : kxn

(18)

In comparison, the array factor zeros chosen by McNamara are [13]: 00 kxn =

MZd ; MB kxn n  n MZd msd 0 kMZd ); n  n kxn +  (kxn xn

(19)

where, dilation factor  MB is defined as

kx00n MB = MZd : kxn

(20)

msd represents the zeros of the maximum slope difference In (19), kxn pattern. Both Taylor and Bayliss distributions for different  values are plotted in Figs. 3 and 6, respectively. As can be seen, for all  values, the array excitation does not change much in the central region. However, the array excitation changes considerably at the edges by varying the . When  = 0, which is the case for the simple Taylor and Bayliss patterns, array excitation terminates abruptly in a pedestal at its edges. For all other  values, the array excitation drops to zero at the edges. The decreasing rate of the array excitation at the edges is a function of  which can be observed from Figs. 3 and 6. Also, array factors corresponding to Figs. 3 and 6 are plotted in Figs. 4 and 7, respectively. From these figures, it can be seen that all the sidelobes closer to  = 0 are almost at the same level. But, the tapering rate of  zeros) corresponding to each  value the far end sidelobes (i.e., after n is significantly different from those of others. Also, the authors would like to note that the far end sidelobes corresponding to  < 0 actually increase with the  value, as expected from (5).

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Fig. 4. Array factors corresponding to the excitations shown in Fig. 3.

Fig. 5. Representation of a 9th order Zolotarev polynomial and it’s zeros.

Fig. 6. Generalized discrete Bayliss distributions with different  values.

IV. EFFECT OF THE SIDELOBE TAPERING ON VARIOUS ARRAY CHARACTERISTICS Various characteristics of the n  array distributions are examined in  array distributions with respect this section. Also, comparison of the n to their ideal counterparts is included. These charts are helpful when there is a need for tradeoff among different array performance criteria. The ATE versus SLR for arrays with different  values are plotted in Fig. 8. The ATE initially increases, reaches a peak and monotonically decreases with the SLR. Also, it is observed that the ATE peaks shift to the left side as the  value increases. This is because for smaller SLR,

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Fig. 7. Array factors corresponding to the excitations shown in Fig. 6.

Fig. 8. ATE versus SLR for Taylor array distributions.

Fig. 9. ATE versus n  for Taylor array distributions.

the power in the main-lobe increases rapidly with the  value which in turn increases the ATE. However, as the SLR increases the power in the main-lobe saturates and beamwidth starts controlling the ATE [7]. So, for larger SLR values, all arrays exhibit approximately equal efficiency. As a matter of fact, the Dolph-Chebyshev array performs slightly better than the other arrays due to its smaller beamwidth.  is shown for different  values. When n  = 1, In Fig. 9, ATE versus n all the arrays attain their maximum ATE values. The ATE initially decreases, reaches a minimum and eventually saturates to the ATE of the corresponding Dolph-Chebyshev array. Next, beam broadening with  can be observed in Figs. 10 and 11. Similar respect to the SLR and n

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Fig. 13. Boresight slope versus n  for Bayliss array distributions.

Fig. 10. Beamwidth versus SLR for Taylor array distributions.

Fig. 14. Beamwidth versus SLR for Bayliss array distributions. Fig. 11. Beamwidth versus n  for Taylor array distributions.

Fig. 15. Beamwidth versus n  for Bayliss array distributions. Fig. 12. Boresight slope versus SLR for Bayliss array distributions.

then the boresight slope of the gain pattern is given as [19] explanations can be provided for all the other Bayliss pattern related diagrams (Figs. 12–15). Before concluding this section, the definition of the boresight slope 0 array used for plotting Figs. 12 and 13 is provided. If the squinted complex gain pattern is defined as [7]

K

Garray :(kx ) = Ge (kx)

A

jk k A

x

M scan m=1 f m exp [ ( x 0 x ) m ]g (21) M 2 m=1 j m j

Karray = @k@ x (Garray ) k =k G k

 e ( xscan )

jA x A

M m=1 ( m m ) M 2 m=1 j m j

:

(22)

In deriving the above equation it is assumed that the variation of the element gain pattern ( e ( x )) with respect to the x is negligible. This approximation is indeed acceptable as long as the boresight direction

G k

k

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is close to the broadside direction. For the exact definitions of Am and Ge (kx ), refer to [7]. From (22), it is evident that the effect of the array coefficients on the slope is solely determined by the term inside the 0 which is used to plot the square brackets. So, the parameter Karray diagrams is defined as

Karray = 0

M (Am xm ) m=1 M Am 2 : m=1 j

(23)

j

0 Also, it can be observed that Karray is independent of the scan direction. The effect of the scan direction is entirely incorporated into the term Ge (kxscan ) of (22).

V. CONCLUSION Theory related to the design of the generalized Taylor and Bayliss patterns has been presented. With this design technique, one can achieve arbitrary sidelobe level and envelope taper. Even though [16] and [13] dealt generalization of the n-bar patterns, the theory presented in those papers does not comply with the Taylor’s asymptotic analysis. The theory presented in this communication is exact and eliminates this drawback. In addition, this theory eliminates the necessity of Bayliss’ cumbersome (but not accurate enough) coefficient and parameter tables [10]. Also, various comparison charts are presented which are important when it comes to tradeoff among different array characteristics.

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[16] D. A. McNamara, “Generalised Villeneuve n  distribution,” IEE Proc. H Microw. Antennas Propag., vol. 136, no. 3, pp. 245–249, 1989. [17] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, 5th ed. New York: Dover, 1972. [18] F. Johansson et al., 2011, mpmath: A Python Library for Arbitrary-Precision Floating-Point Arithmetic (Version 0.17) [Online]. Available: http://code.google.com/p/mpmath/ [19] G. Kirkpatrick, “A relationship between slope functions for array and aperture monopulse antennas,” IRE Trans. Antennas Propag., vol. 10, no. 3, p. 350, May 1962.

Amplitude-Only Low Sidelobe Synthesis for Large Thinned Circular Array Antennas Will P. M. N. Keizer

Abstract—This communication presents results for the low sidelobe synthesis using amplitude-only tapering applied on the turned ON elements of large circular thinned arrays. The low sidelobe synthesis was carried out with the same iterative Fourier transform method that was developed earlier to restore the original sidelobe performance in case of defective array elements. The presented low sidelobe results refer to highly thinned circular arrays with diameters ranging from 25 to 133.3 wavelengths and involved both sum and difference patterns. In addition, sector nulling in combination with low sidelobes was numerical investigated. Index Terms—Array antennas, low sidelobes, pattern synthesis, thinned arrays.

REFERENCES [1] C. L. Dolph, “A current distribution for broadside arrays which optimizes the relationship between beam width and side-lobe level,” Proc. IRE, vol. 34, no. 6, pp. 335–348, 1946. [2] H. J. Riblet and C. L. Dolph, “Discussion on “a current distribution for broadside arrays which optimizes the relationship between beam width and side-lobe level”,” Proc. IRE, vol. 35, no. 5, pp. 489–492, 1947. [3] D. A. McNamara, “An exact formulation for the synthesis of broadside Chebyshev arrays of 2n elements with interelement spacing d < =2,” Microw. Opt. Technol. Lett., vol. 48, pp. 457–463, 2006. [4] D. A. McNamara, “Direct synthesis of optimum difference patterns for discrete linear arrays using Zolotarev distributions,” IEE Proc. H Microw. Antennas Propag., vol. 140, no. 6, pp. 495–500, 1993. [5] O. R. Price and R. F. Hyneman, “Distribution functions for monopulse antenna difference patterns,” IRE Trans. Antennas Propag., vol. 8, no. 6, pp. 567–576, 1960. [6] R. Levy, “Generalized rational function approximation in finite intervals using Zolotarev functions,” IEEE Trans. Microw. Theory Tech., vol. 18, no. 12, pp. 1052–1064, 1970. [7] A. K. Bhattacharyya, Phased Array Antennas, Floquet Analysis, Synthesis, BFNs, and Active Array Systems. Hoboken, NJ: Wiley, 2006. [8] T. T. Taylor, “Design of line-source antennas for narrow beamwidth and low sidelobes,” IRE Trans. Antennas Propag., vol. AP-3, pp. 16–28, 1955. [9] T. T. Taylor, “Design of circular apertures for narrow beamwidth and low sidelobes,” IRE Trans. Antennas Propag., vol. 8, no. 1, pp. 17–22, 1960. [10] E. T. Bayliss, “Design of monopulse difference patterns with low sidelobes,” Bell Syst. Tech. J., vol. 47, pp. 623–650, 1968. [11] R. S. Elliott, “On discretizing continuous aperture distributions,” IEEE Trans. Antennas Propag., vol. 25, no. 5, pp. 617–621, 1977. [12] A. T. Villeneuve, “Taylor patterns for discrete arrays,” IEEE Trans. Antennas Propag., vol. 32, no. 10, pp. 1089–1093, 1984. [13] D. A. McNamara, “Performance of Zolotarev and modified-Zolotarev difference pattern array distributions,” IEE Proc. Microw. Antennas Propag., vol. 141, no. 1, pp. 37–44, 1994. [14] D. R. Rhodes, “On the Taylor distribution,” IEEE Trans. Antennas Propag., vol. 20, no. 2, pp. 143–145, 1972. [15] S. R. Zinka, I. B. Jeong, W. K. Min, J. H. Chun, and J. P. Kim, “On the generalized Villeneuve distribution,” in Proc. Asia Pacific Microwave Conf., 2009, pp. 17–20.

I. INTRODUCTION Array thinning involves the removal (turning off) of radiating elements from an array antenna. The main motivation to use thinning is the reduction in cost and weight. This technique allows getting nearly the same narrow beamwidth as for a filled array of equal size. Another advantage is that when the turned ON elements operate with equal amplitude, lower sidelobes can be obtained as for the same filled array illuminated with uniform weighting. For these reasons thinned array antennas are over more than 40 years in use with high performance US military phased array radar systems such as Pave Paws [1] (operating at UHF, 22 m diameter aperture, 2677 elements with 1792 active), HAPDAR [2] (L-band, 69 m diameter aperture, 4300 elements with 2165 active), Cobra Dane [1] (L-band, 29 m diameter aperture, 34 768 elements with 15 360 active) and SeaBased X-Band (SBX) Radar [3] having a 248 m2 aperture populated by > 45000 transmit/receive (T/R) modules. SBX is presently the largest solid-state phased array radar in the world in terms of element positions and the number of T/R modules. All these radars provide large instantaneous wideband operation, feature monopulse target tracking capability and are intended for the detection, tracking and recognition of intercontinental ballistic missiles. The use of thinned arrays in the Manuscript received November 01, 2010; revised June 14, 2011; accepted August 09, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. The author, retired, was with the TNO Physics and Electronics Laboratory, 2597 AK, The Hague, Netherlands. He is now at 2343 JH Oegstgeest, The Netherlands (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173119

0018-926X/$26.00 © 2011 IEEE

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above mentioned high performance radar systems, makes obvious the importance of this category of array antennas. In order to improve the sidelobe performance of thinned arrays Mailloux and Cohen [4] proposed to use a three-level discrete amplitude weighting scheme to approximate a 050 dB sidelobe Taylor amplitude distribution in combination with statistical thinning of the elements. The thinning was arranged to smooth the average amplitude illumination in order to reduce the peak sidelobes caused by the amplitude quantization. In [5] an almost similar multi-level amplitude weighting scheme was described for thinned arrays. This communication addresses the synthesis of “continuous” amplitude tapers for the illumination of the turned ON elements to lower the maximum peak sidelobe level of the receive patterns, sum and difference, of four large thinned circular array antennas. It is worth to mention that the use of difference patterns with thinned planar arrays was not dealt before in open literature. The considered circular array antennas have diameters ranging from 25 to 133.3 wavelengths and fill factors between 27% and 50%. The fill factor is defined as the ratio of the number of turned ON elements versus the total number (turned ON and OFF) of elements. Each of the four considered arrays has an already defined turned ON element distribution that did not change during the synthesis of the amplitude taper. The motivation to perform the low sidelobe synthesis on existing planar thinned arrays, was to get a substantial lower maximum peak sidelobe performance then offered by analytical tapers. Low sidelobe circular Taylor distributions applied to the considered four arrays reduced the peak sidelobe level with only about 1.5 dB compared to equal amplitude illumination. The used technique to synthesize an amplitude taper is the iterative Fourier transform (IFT) method [6] that is able to restore the original far-field pattern of a planar array antenna in case of defective elements across the aperture. By considering defective elements as turned OFF elements, the IFT method of [6] is directly, without any modification, applicable for the synthesis of low sidelobe tapers with thinned planar arrays. The published pattern synthesis results include also sector nulling combined with low sidelobes obtained with amplitude-only tapering.

II. LOW SIDELOBE SYNTHESIS APPROACH Since the IFT method [6] has the ability to deal with a large number of defective elements randomly dispersed across a planar aperture, this method is very well suited for the synthesis of amplitude-only low sidelobe tapering of thinned circular arrays. Furthermore, the IFT method allows to take user defined element excitation constraints into account. For the sum patterns without sector nulling, a 10 dB dynamic range requirement for the amplitudes of the turned ON elements, defined as the ratio of the maximum and minimum amplitude, was applied, while with sector nulling the dynamic range requirement was raised to 15 dB. For the difference patterns the dynamic range of the amplitudes of the turned ON elements was set to 20 dB, for both with and without sector nulling. For all considered arrays the elements were positioned in a square lattice spaced 0.5 wavelengths apart. Due to this element spacing the array factor of each of the considered arrays covers in u 0 v space the area defined by f01  u  1; 01  v  1g where u = sin  cos '; v = sin  sin ';  and ' the angular far-field positions. A substantial part of the far-field directions of those array factors are therefore sited in invisible u 0 v space defined by u2 + v 2 > 1. The low sidelobe synthesis was considered fully successful when all far-field directions of the sidelobe region, including those of invisible u-v space, did not exceed the user defined peak sidelobe level (PSL) requirements. The far-field directions located into the invisible part of u-v

Fig. 1. Computed far-field sum pattern of the thinned 100  circular array with 30% fill factor when illuminated by the synthesized amplitude taper for 40 dB sidelobes. (a) Principal u-cut. (b) Main beam and PSL distribution of whole visible u-v space of the far-field.

0

space were incorporated in the synthesis to assure that the array sidelobe performance will not be degraded when the main beam is scanned away from broadside. An extended description of the used IFT method is included in [6]. A MATLAB program listing of exactly the same IFT method suited for the pattern recovery of linear arrays troubled by defective elements is given in [7]. This MATLAB program takes the amplitude range constraint for the active elements into account and is implemented in such a way that any synthesized taper always fulfills the used dynamic range requirement; see the MATLAB listing in [7]. The same remark applies to the IFT method described in [6]. III. NUMERICAL RESULTS Three of the four array configurations considered in this communication are the same thinned array antennas as described in [8]. These arrays have a circular aperture shape and the turned ON and OFF elements are positioned in a square grid at 0.5 wavelength spacing. The fourth thinned array has a diameter of 133.3 wavelengths and uses the same square element grid. The applied fill factors yield for equal illumination of the turned ON elements about the lowest maximum peak sidelobes for the considered array diameters. The elements of the considered arrays feature an isotropic embedded element pattern. Fig. 1(a) shows for the thinned circular array, having a 100-wavelength diameter and a 30% fill factor, the u-cut of its far-field going through the main beam peak for the amplitude tapering arranged by the IFT method. The synthesis was carried out with a 040 dB maximum PSL requirement. The obtained far-field, representing the sum pattern,

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TABLE I FAR-FIELD RESULTS FOR FOUR THINNED CIRCULAR ARRAYS WHEN A SUM AMPLITUDE TAPER IS SYNTHESIZED USING THE IFT METHOD

TABLE II FAR-FIELD RESULTS FOR SAME FOUR THINNED CIRCULAR ARRAYS OF TABLE I WHEN A N AZIMUTH DIFFERENCE AMPLITUDE TAPER IS SYNTHESIZED USING THE IFT METHOD

Fig. 2. Computed far-field azimuth difference pattern of the thinned 100  circular array with 30% fill factor when illuminated by the synthesized amplitude taper for 34 dB sidelobes. (a) Principal u-cut. (b) Main beams and PSL distribution of the whole visible u-v space of the far-field.

0

matches the 040 dB maximum PSL requirement as can be noted from the peak sidelobe distribution depicted in Fig. 1(b) that covers all sidelobes (including the main lobe) located in visible u 0 v space. The synthesis terminated after 51 iterations when the design requirements were completely fulfilled. For a plot of the far-field for the equal amplitude distribution of the turned ON elements featuring a 035:4 dB maximum PSL, the associated peak sidelobe distribution and the thinned element distribution across the aperture, the reader is referred to [8]. The u-cut of the azimuth difference pattern of the same array synthesized for 034 dB peak sidelobes using amplitude-only tapering is shown in Fig. 2(a). The corresponding histogram of the far-field peak sidelobes and two main lobes, is depicted in Fig. 2(b). This result required 47 iterations and matches completely the maximum PSL design requirement of 034 dB including the dynamic range requirement of 20 dB for the amplitude taper. Table I summarizes the low sidelobe synthesis results using amplitude–only tapering obtained for the sum patterns of four circular arrays with diameters ranging from to 25 wavelengths to 133.3 wavelengths. On examination Table I one can see that by applying amplitude tapering for the turned ON elements, the maximum PSL improves with more than 4 dB compared to equal amplitude illumination of the turned ON elements. The reduction in antenna directivity due to amplitude tapering is small and varies between 0.09 dB and 0.3 dB. The increase in 3 dB beamwidth due to amplitude tapering is modest. Table II summarizes the far-field results when a low sidelobe difference amplitude taper is synthesized using the IFT method for the same four arrays as shown in Table I. The listed results apply to the azimuth difference pattern and represent the lowest peak sidelobe values

Fig. 3. Computed far-field sum pattern of the thinned 100  circular array with 30% fill factor when illuminated by the synthesized amplitude taper for 40 dB sidelobes and a 65 dB sector null.

0

0

that could be obtained with the IFT method. The listed parameter Kr denotes the relative angle sensitivity. Comparing the PSL results of Table I with those of Table II it can be noted that for each of the four considered arrays the maximum PSL value of the difference pattern is 6 dB higher than the maximum PSL of the corresponding sum pattern. Fig. 3 displays the far-field of the sum pattern of same thinned circular array of Fig. 1 synthesized for a 040 dB maximum PSL in combination with a 065 dB rectangular null sector located in the far-field at f0:4  u  0:5; 0:2  v  0:3g. The second null region present in the far-field at f00:5  u  00:4; 00:3  v  00:2g is a mirror

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TABLE III FAR-FIELD RESULTS FOR FOUR THINNED CIRCULAR ARRAYS WHEN A SUM AMPLITUDE TAPER IN COMBINATION WITH SECTOR NULLING IS SYNTHESIZED USING THE IFT METHOD

TABLE IV FAR-FIELD RESULTS FOR FOUR THINNED CIRCULAR ARRAYS WHEN AN AZIMUTH DIFFERENCE AMPLITUDE TAPER IN COMBINATION WITH SECTOR NULLING IS SYNTHESIZED USING THE IFT METHOD

in combination with a 060 dB rectangular null sector located at f0:4   0:5; 0:2  v  0:3g in u 0 v space. This result required 3605 iterations to match fully the design requirements. Table IV summarizes the sector nulling results of the considered four thinned arrays when operating with an azimuth difference pattern. For each array the locating of the nulling sector was specified at f0:4  u  0:5; 0:2  v  0:3g. For the 25 wavelength diameter array the depth of the null was limited to 055 dB. For the three larger arrays a null depth of 060 dB could be realized. All the presented far-field patterns match fully the specified far-field design requirements while none of the associated tapers did violate the dynamic range constraint for the turned ON element amplitudes. When the considered arrays will be applied in phased radars, the best way to implement the amplitude taper is using active weighting [6]. With active weighting the taper is realized by appropriate settings of the amplitude control devices inside the T/R modules instead of using fixed tapering arranged by a passive beamforming network. Pattern calculations for quantized tapering performed for amplitude control devices with 15 dB control range for the sum patterns and 20 dB control range for the difference patterns, revealed that quantized tapering with six bit resolution degraded the peak level of the sidelobes with less than 0.2 dB for all considered arrays. The increase of the PSL in the nulling sector did not exceed 3 dB. A comparative assessment with earlier published results was not possible due the very limited information about the maximum PSL results in [4], [5] of the simulated far-field patterns. Furthermore no numerical values for the degree of thinning were disclosed in [4] for the considered circular arrays. As far as is known, no other papers on amplitude tapering, discrete of continuous, applied to thinned planar arrays have been published. Even publications for large filled planar arrays featuring the type of results presented in this communication, are scarce.

u

Fig. 4. Computed far-field azimuth difference pattern of the thinned 100  circular array with 30% fill factor when illuminated by the synthesized amplitude taper for 34 dB sidelobes and a 60 dB sector null.

0

0

image of the original null sector that is typical for sector nulling accomplished by amplitude—only tapering of the aperture. This result was obtained after 4449 iterations when the design objectives were fully matched. The synthesis of a sector null in combination with lower sidelobes was also performed for the three other considered arrays. Table III summarizes the sector nulling results for the sum pattern of the four thinned circular arrays. In each case the location of the nulling sector was specified at f0:4  u  0:5; 0:2  v  0:3g. For the array with a diameter of 25 wavelengths the depth of the was limited to 060 dB. On comparing the directivity and 3 dB beamwidth results of Tables I and III, one can notice that the presence of the nulling sector in the far-field hardly degrades the directivity, less than 0.23 dB difference, and the change in 3 dB beamwidth is almost negligible. The high number of iterations, 4449, needed to the realize the nulling sector of Fig. 3, is due to the 065 dB depth requirement for this sector, a very tough one for massively thinned planar arrays. The result of an almost similar synthesis applied to a filled 15 2 15 element square array [9] supports this conclusion. The far-field pattern of this array was designed for 020 dB sidelobes including two rectangular nulling sectors, one with a depth of 030 dB and the other with a 060 dB depth. The 060 dB nulling depth requirement was only partially met despite the not very demanding 020 dB PSL requirement for all sidelobes outside the two nulling sectors and the use of complex weighting. Furthermore the design of [9] experienced a huge 1.4 dB loss in directivity due to the presence of the two nulling sectors. Fig. 4 shows the azimuth difference far-field pattern of the same thinned aperture as of Fig. 2 synthesized for a 034 dB maximum PSL

IV. CONCLUSION The presented results show for the first time that for large thinned circular arrays amplitude weighting is quite feasible for both sum and difference patterns and can improve the maximum peak sidelobe level of the sum pattern with at least 4 dB compared to equal amplitude illumination of the active elements. It is also demonstrated that sector nulling for both sum and difference patterns is possible to a depth of at least 055 dB even for massively thinned arrays and with very minor degradation of directivity and 3 dB beamwidth.

REFERENCES [1] E. Brookner, Aspects of modern radar, 1st ed. Norwood, MA: Artech House, 1988. [2] P. J. Kahrilas, “HAPDAR—an operational phased array radar,” Proc. IEEE, vol. 56, no. 11, pp. 1967–1975, 1968.

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[3] “Raytheon datasheet,” Sea-Based X-Band Radar (SBX) for Missile Defense, Raytheon Datasheet [Online]. Available: www. raytheon.com/capabilities/rtnwcm/groups/rms/documents/content/rtn_rms_ps_sbx_datasheet.pdf [4] R. J. Mailloux and E. Cohen, “Statistically thinned arrays with quantized element weights,” IEEE Trans. Antennas Propag., vol. 39, no. 4, pp. 436–447, Apr. 1991. [5] T. Numazali, S. Mano, T. Katagi, and M. Mizusawa, “An improved method for density tapering of planar array antennas,” IEEE Trans. Antennas Propag., vol. 35, no. 9, pp. 1066–1070, Sep. 1987. [6] W. P. M. N. Keizer, “Element failure correction for a large monopulse phased array antenna with active amplitude weighting,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2211–2218, Aug. 2007. [7] W. P. M. N. Keizer, “Low sidelobe pattern synthesis using iterative Fourier techniques coded in MATLAB,” IEEE Antennas Propag. Mag., vol. 51, no. 2, pp. 137–150, Apr. 2009. [8] W. P. M. N. Keizer, “Large planar array thinning using iterative FFT techniques,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3359–3362, Oct. 2009. [9] O. M. Bucci, L. Caccavale, and T. Isernia, “Optimal far-field focusing of uniformly spaced arrays subject to arbitrary upper bounds in nontarget directions,” IEEE Trans. Antennas Propag., vol. 50, no. 11, pp. 1539–1553, Nov. 2002.

Power Synthesis for Reconfigurable Arrays by Phase-Only Control With Simultaneous Dynamic Range Ratio and Near-Field Reduction Giulia Buttazzoni and Roberto Vescovo

Abstract—An iterative method of power synthesis for reconfigurable arrays of arbitrary geometry is presented, which is based on the method of successive projections. The algorithm allows to synthesize a number of desired patterns, each reconfigurable into any of the others by phase-only control. The excitation amplitudes are optimized, and their dynamic range ratio (DRR) is reduced below a given threshold. Furthermore, the radiated field can be reduced below a prescribed level in a given region close to the antenna. As a particular important case, the method allows to perform a “discrete” phase controlled beam-scanning. Index Terms—Dynamic range ratio reduction, near-field reduction, phase control, power synthesis, reconfigurable arrays, scanning, successive projections.

I. INTRODUCTION In many practical applications, such as for example air traffic control radars, satellites and wireless communications, antennas are required to generate different patterns, each reconfigurable into any of the others. With antenna arrays of many elements, the reconfigurability is often obtained by modifying only the excitation phases, thus allowing the use of simpler feeding networks. Hence the excitation amplitude of each array element holds constant, even if it may be different from the excitation amplitudes of the other elements. The amplitude variations can be reduced by reducing the dynamic range ratio (DRR). Moreover, the environment surrounding the antenna can include electronic devices that can be disturbed by the radiated field, or mounting Manuscript received June 08, 2010; revised January 26, 2011; accepted August 08, 2011. Date of publication October 24, 2011; date of current version February 03, 2012. The authors are with the University of Trieste, 34127 Trieste, Italy. (e-mail: [email protected], [email protected]). Digital Object Identifier 10.1109/TAP.2011.2173103

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equipments that produce interference. A convenient way to reduce such interferences is to reduce the radiated electric field in the region of interest. Several synthesis techniques are available in the literature, most of which are based on powerful tools such as stochastic algorithms [1]–[3] and the method of projections [4], [5]. In [6], a least square solution is described for the phase synthesis of planar and conformal arrays. In [7] a constrained least square optimization is used to study the phase-only control in spherical conformal arrays, with comparison to the combined phase and amplitude control. In [8] an algorithm based on the discrete Fourier transform is used to solve a beam-scanning problem for circular arrays, in presence of null constraints. In [9] a linear array beam-scanning is performed by phase control, with pattern reduction in constant directions and DRR control. In [10] a synthesis procedure for reconfigurable arrays is proposed, based on the method of projections, which reduces the DRR when linear or rectangular arrays are involved. In [11] a power synthesis problem for phase controlled reconfigurable conformal arrays is solved by a generalized projection algorithm, which however does not allow DRR reduction. In [12] a simple method is proposed, based on the method of projections, to solve a power synthesis problem for phase-only reconfigurable arrays of arbitrary geometry, in presence of an upper bound on the DRR. Several techniques have also been devised to reduce the near-field. In [13] a pattern synthesis technique is proposed that forms nulls in given points of the near-field region, based on a constrained least-mean-square approximation. This approach is generalized in [14], where the near-field radiated power is minimized only in the boundary of an obstacle, thus allowing to isolate large objects. Also in [15], [16] electric field nulls are imposed in assigned points of the near-field region, but the radiation pattern is synthesized imposing that it belong to a prescribed mask, and using the method of projections. However, the techniques in [13]–[16] are not suitable for reconfigurable arrays. In [17] the method of projections is used in conjunction with a version of the Broyden-Fletcher-Golfarb-Shanno (BFGS) iterative method, to solve a power synthesis problem for reconfigurable conformal arrays in presence of an upper bound on the near-field in a region close to the antenna. The approach also allows to control the cross-polar component, but does not control the DRR. In this communication, we propose a simple and accurate method for the power synthesis of reconfigurable arrays of arbitrary geometry. The method allows to generate a given number of patterns, each reconfigurable into any of the others by phase-only control, simultaneously reducing the DRR below a given threshold and, in addition, the near-field amplitude below a prescribed threshold in a given region close to the antenna. II. FORMULATION OF THE PROBLEM

N

Let us consider an array of radiating elements, referred to a Carte). The radiation pattern in the direction of the sian system ( 0 plane and the electric field in a point r are given by

x y

O x; y; z

F (a; ') = ( ; )=

E a r

'

N

n=1 N n=1

an Fn (')

(1)

an En (r)

(2)

where a = [ 1 . . . N ]T is the column vector of the complex excitations, while n ( ) and En (r) are the pattern and the electric field, respectively, of the excitation vector vn = [0 . . . 1 . . . 0]T having unity in the -th position. The dynamic range ratio of a is DRR(a) = maxn fj n jg minn fj n jg.

a ; ;a F ' n a = a

0018-926X/$26.00 © 2011 IEEE

; ; ; ;

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We want to determine a number S of array patterns belonging to S suitable masks Ms = ffs ('): ms1 (')  jfs (')j  ms2 (')g ; s = 1; . . . ; S , where ms1 (') and ms2 (') are the lower and the upper bounds, respectively, of the s-th mask, in such a way that each pattern can be transformed into any of the others by modifying only the excitation phases. We also require that the DRR of the excitation amplitudes do not overcome a prescribed threshold, and that, in a region Vr close to the antenna, the electric field amplitude do not overcome an assigned value. This problem can be formulated as follows: find S excitation vectors a1 ; . . . ; aS , as = [a1s ; . . . ; aNs ]T , in such a way that

F (as ; ') 2 Ms ; s = 1; . . . ; S jan1 j = 1 1 1 = janS j = n ; n = 1; . . . ; N; DRR()  D0 ; where  = [1 ; . . . ; N ]T E(as ; r)  E0 ; s = 1; . . . ; S; r 2 Vr :

(3a) (3b) (3c) (3d)

Condition (3a) imposes that each pattern belong to a mask. Condition (3b) imposes that the excitation amplitude of each array element be constant during the reconfiguration process, so as to ensure phase-only control. The amplitudes n in (3b) may vary from one element to the others, but (3c) imposes that their DRR do not overcome a prescribed threshold D0 . Condition (3d) imposes an upper bound E0 on the field amplitude in the region Vr close to the antenna. III. THE SOLVING PROCEDURE The proposed algorithm is an evolution of the method in [12], where the solution is searched in a set of vectors having, as components, S scalar functions and S column vectors aimed to provide the far-field patterns and the optimal excitations, respectively. We here modify the research space in [12] to include vectors containing, as additional components, S vector functions aimed to approximate the constrained nearfields. Thus, we introduce the set W of the 3S -tuples

~ = (g1 ('); . . . ; gS ('); k1 (r); . . . ; kS (r); w1 ; . . . ; wS ) w where

gs (') is a complex scalar function defined in I'

(4)

= [0; ], =

ks (r) a complex vector function defined in Vr , and ws

[w1s ; . . . ; wNs ]T a complex column vector. In W we introduce the subset U of the 3S -tuples ~ = (f1 ('); . . . ; fS ('); e1 (r); . . . ; eS (r); u1 ; . . . ; uS ) u

(5)

where fs (') 2 Ms , jes (r)j  E0 for r 2 Vr , us = [u1s ; . . . ; uNs ]T with jun1 j = 1 1 1 = junS j = n , and DRR()  D0 ( = [1 ; . . . ; N ]T ). We also introduce the subset V of W of the 3S -tuples

v~ =(F (v1 ; '); . . . ; F (vS ; '); E(v1 ; r); . . . ; E(vS ; r); v1 ; . . . ; vS ) (6) where vs is an arbitrary excitation vector, while F (vs ; ') and E(vs ; r) are the corresponding radiation pattern and electric field, respectively. Following [12] we adopt, as a solution, a point of U minimizing a suitable distance from V . To reach such a point, starting from a ~ 0 2 U we perform the iterative procedure u~ n+1 = suitable point u PU PV u~ n ; n = 0; 1; 2; . . ., where PU and PV are the projection operators onto U and V , respectively, implemented in the Appendix. The ~ n to V is non-increasing [12], sequence fn g of the distances from u ~ng thus it converges. Therefore, this iteration generates a sequence fu ~k 2 U of points of U closer and closer to V . We stop the iteration at u such that k < " or (k01 0 k ) =k <  , where " and  are suitable

Fig. 1. Array geometry and near-field region for examples 1 and 2.

~ k are considthresholds. The last S components u1 ; . . . ; uS of u ered as the S optimal excitation vectors. Then, the S optimal radiation (k) (k) patterns are F (us ; '), and the S optimal near-fields are E(us ; r), (k) ~ k 2 U , the vectors us satisfy (3b) and (3c) rigrespectively. Since u orously (so phase control and DRR reduction are always ensured), 6 fs(k) (') fs(k) (') 2 Ms and je(sk) (r)j  E0 . But, F (us(k) ; ') = (k) (k) 6 es (r), so (3a) and (3d) are not satisfied exactly, and E(us ; r) = ~ k to in general. However, in the examined cases the distance from u V resulted to be small, and our solution satisfied (3a) and (3d) with quite satisfactory approximation. Moreover, if both sets U and V were convex, the iterative procedure would converge toward the global minimum of the distance [18]. In our problem, V is a linear subspace of W , and therefore is convex. Instead, U is non-convex. Therefore, the ~ n g may provide a local minimum of the distance. In this sequence fu regard it is to be noted that the constraint (3d) is convex in the space of the excitations, so it does not contribute to produce traps [19]. To re~ 0 should be close duce the risk of falling into traps, the starting point u ~ 0 is seto the solution, which however is unknown. In our examples u (0) (0) lected in such a way that, for each s, fs (') 2 Ms , es (r) = 0, and u(0) s = 0. This choice gave very good results. However, to analyze the effects of different choices, in Section IV the algorithm is tested using a great number of random starting points. (k)

(k)

IV. NUMERICAL EXAMPLES

Let us refer to the array of Fig. 1, consisting of N = 70 elementary Huygens radiators having the maximum radiation intensity in the radial direction and equally spaced on two circular arcs centered on the z –axis of the Cartesian system O(x; y; z ). The arcs lie on the two planes jz j = =4, have radius 10:83 and aperture angle from  = 045 to  = 45 . The near-field constraint is imposed in Vr = f(x; y; z ): jxj; jy j; jz jg. We set " = 1005 and  = 1006 for the stop test. Each integral on I' is approximated with a summation by dividing I' into 360 equal subintervals, and taking the values of the integrand function at the center of each subinterval. Similarly, each integral in Vr is approximated with a summation by dividing Vr into equal cubes of side =8. We used Matlab 7.9.0 on a PC with eight Intel Xeon CPUs (E5430 @ 2.66 GHz) and 16 GB of RAM. A. First Example: A Reconfigurability Problem With reference to the above array, let us consider the S = 4 masks in Fig. 2. The maximum values of the four masks are equal, and are set to unity. In other words, at a given (large) distance from the array, the four far-field patterns are required to have the same maximum value. We first solved the problem in presence of only the constraints (3a) and (3b). This will be referred to as the “reduced” problem. After 395 iterations (0.4 sec) the synthesized patterns resulted to belong to the masks, with maximum side lobe level SLL = 020:00 dB in the worst case, and (3b) was satisfied exactly. It resulted DRR = 35:58, and we also r (r means “reduced” evaluated the maximum near-field amplitude Emax problem) in Vr . In order to reduce both the DRR and the maximum field amplitude in Vr , we then solved the “complete” problem setting

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Fig. 4. Contour plots of the electric field amplitude in the worst case (a),(b) and in the best case (c,)(d). (a),(c): “reduced” problem; (b),(d): “complete” problem. The square line in each figure represents the intersection between and the considered plane.

V

Fig. 2. Assigned masks (solid lines) and synthesized patterns for the “complete” problem, with the presented algorithm (dashed line) and with the algo( ) and ( ) are rithm in [17] (dotted lines). The bounds of the masks, piecewise linear, with ( 180 ) = (180 ) = 0,

m 0

m '

m

m '

TABLE I EFFECT OF THE RANDOM STARTING POINTS (FIRST EXAMPLE)

m (010 ) =m (10 ) = m (15 ) = m (35 ) = m (024 ) = m (24 ) = m (021 ) = m (31 ) = 0; m (02:5 ) =m (2:5 ) = m (22:5 ) = m (27:5 ) = m (012 ) = m (12 ) = m (19 ) = 0:89; m (09 ) = 0:43; m (0180 ) = m (180 ) = 0:1, m (012 ) = m (12 ) = m (13 ) = m (37 ) = m (025 ) = m (25 ) = m (020:5 ) = m (32 ) = 0:1; m (03:5 ) = m (3:5 ) = m (21:5 ) = m (28:5 ) = m (012 ) = m (12 ) = m (20 ) = 1; m (011 ) = 0:48.

For comparison purposes we solved the “complete” problem also with the method in [17]. Since the latter does not control the DRR and the presented method does not control the cross-polar pattern, the comparison was performed in absence of constraint (3c). The starting vectors c0 and p0 in [17] were set to zero. Furthermore, to carry out the projections and to approximate the integrals in [17, Eq. (6)] we kept our partitioning of ' and r . Finally, we adopted our convergence test. The algorithm in [17] required 1128 iterations (4872 sec). Fig. 2 shows the synthesized patterns, which exhibit SLL = 018 73 dB in the worst case. The DRR was 337. The reduction of the maximum field amplitude in r was 14.01 dB in the constraint points, and 13.66 dB using the thicker mesh. 1) Effect of the Starting Point: With reference to the above problem, the proposed algorithm was tested in correspondence of 1000 random (0) starting points: for each , the modulus of s ( ) was uniformly distributed between the lower and the upper bounds of the mask, and the phase was uniformly distributed in [0 2 ]; for each r 2 r , the mod(0) ulus of es (r) was uniformly distributed in [0 0 ] and the phase of each component was uniformly distributed in [0 2 ]; finally, each com(0) ponent of each vector us was chosen with phase uniformly distributed in [0 2 ] and random amplitude independent of and uniformly distributed in [1 0 1]. On average, 14 561 iterations were required to achieve a solution. Conditions (3b) and (3c) were always satisfied exactly. For each trial, we calculated the highest SLL among the solutions and the minimum near-field reduction. Then, we calculated the mean value of such SLL values and field reductions. We also calculated the highest among these SLL values and the minimum among these field reductions. We considered each trial as good if each of the synthesized patterns belongs to the corresponding mask or exceeds the upper or the lower bound by maximum 1 dB, and, simultaneously, if each of the radiated fields satisfies (3d) or its amplitude exceeds 0 by maximum 1 dB. The obtained values and the percentages of good trials are listed in Table I, together with the percentage of good trials

'

I

V

:

V

f '

'

; 

Fig. 3. Distance



from set

V of the n-th point u~

, for the first example.

r D0 = 5 in (3c) and selecting E0 in (3d) such that Emax =E0 = 15 dB (i.e., we required a 15 dB reduction of the maximum electric field amplitude in Vr ). Our algorithm required 14 452 iterations (361 sec), and Fig. 3 shows the distance from the set V as a function of the iteration

number. The synthesized patterns are shown in Fig. 2. Conditions (3b) and (3c) were satisfied exactly. The reduction of the maximum field r , was 14.93 dB in the worst case using the in r , with respect to max partitioning of r into cubes of side 8, and 13.89 dB using a thicker partitioning into cubes of side 16 ( = 4, = 00 97 ). The maximum field reduction (best case) was obtained for = 3 in the plane = 00 78 . Fig. 4 shows the contour plots of the electric field amplitude in these planes.

V

z

V

: 

E

= = s

z

s

: 

; 

=D ;

;E ;  s

V

S

S

S

E

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 2, FEBRUARY 2012

Fig. 6. Third example: synthesized patterns for the “complete” problem. (a) Method in [16]; (b) presented algorithm.

= 46 (07 5 ) = (7 5 ) = 0 (01 3 ) = (1 3 ) = 0 89 (08 ) = (8 ) = 0 1 (02 ) = (2 ) = 1

Fig. 5. Second example: overlapping of all S synthesized patterns that solve the “complete” beam-scanning problem. The reference mask, M , is such that m : m : ,m : m : : , m : ,m m and m .

when the tolerance is 2 dB instead of 1 dB. In conclusion, conditions (3b) and (3c) were always exactly satisfied, while (3a) and (3d) were almost always satisfied with a tolerance of 2 dB. However, when requiring 1 dB tolerance, the percentage of good trials reduces to nearly 91%. B. Second Example: A Beam-Scanning Problem With reference to the above array, we introduced the masks in Fig. 5, = 0 , selected starting from a reference mask 0 , centered at   and scanning it over the angular sector [045 45 ], with angular step = 46 masks close to each other, with anof 2 . This produced gular separation of 2 , so as to obtain a discrete beam-scanning. All masks include the null intervals [080 070 ] and [70 80 ], where the pattern amplitude has an upper bound of 040 dB. The “reduced” problem required 1159 iterations (5 sec). Condition (3b) was satisfied exactly, in the worst case we obtained SLL = 020 00 dB and the maximum pattern level in the null region NL = 039 99 dB, and DRR = 2 46. We then solved the “complete” problem setting 0 = 1 in (3c), and requiring a 15 dB reduction of the maximum field amplitude in r . The algorithm required 29 128 iterations (2497 sec). Fig. 5 shows an overlapping of all the synthesized patterns, which shows that good results were obtained for all patterns. In particular, we found SLL = 020 00 dB and NL = 040 00 dB, in the worst case. Conditions (3b) and (3c) were satisfied exactly. Also (3d) was essentially satr c isfied, as the minimum field reduction was max max = 14 92 dB, r c where max and max are the maximum field amplitudes in the grid points (step = 8) in the “reduced” and in the “complete” problems, 16) respectively. The field reduction on the thicker mesh (step = r c was max max = 13 35 dB.

M

S

V

'

;

;

;

:

:

:

E

E =E

:

E =

:

E =E

D

:

=

:

C. Third Example We here compare the presented method to that described in [16], which refers to the case = 1, is based on the method of projections and gives very high near-field reductions. The array, the desired pattern and the geometry of the problem are those of the first example in [16]. We first solved the “reduced” problem (771 iterations, 0.5 sec) and subsequently the “complete” problem (508 868 iterations, 290 sec) with the method in [16]. The synthesized pattern solving the “complete” problem is shown in Fig. 6(a) (also the pattern solving the “reduced” problem, not shown here, satisfies the mask constraint). We obtained DRR = 42 63 (“reduced” problem) and DRR = 618 61 (“complete” problem), and the maximum field amplitude reduction (evaluated on r c a mesh of 2 spaced points) was max max = 48 89 dB. Then the problem was solved using the proposed algorithm. The “reduced” problem required 2173 iterations (1 sec), giving DRR = 87 38 and r in r . Then we solved the “complete” the maximum field level max

S

: =

E =E

E

V

:

:

:

D

problem with the proposed algorithm, setting 0 = 10 and, to obtain the same maximum field amplitude already obtained with the method in c c , where max is the maximum field am[16], we imposed 0 = max plitude previously obtained in the “complete” problem, corresponding to a near-field reduction of only 3.53 dB. Our algorithm required 27 206 iterations (54 sec) and gave DRR = 10, thus (3c) was satisfied exactly. Condition (3a) was well approximated, as is shown in Fig. 6(b) (also the pattern solving the “reduced” problem, not shown here, satisfies (3a)). Constraint (3d) was satisfied with good accuracy, resulting c  max = 0 . Thus, we obtained the same final maximum field amplitude in r as with the method in [16], but with a strongly reduced DRR value. We also solved the “complete” problem (with the proposed algorithm) imposing a 20 dB reduction of the maximum field amplitude r in r ( max 0 = 20 dB) without DRR increase with respect to the reduced case ( 0 = 87 38). The algorithm required 11 740 iterations (23 sec), and (3c) was satisfied exactly; (3a) and (3d) were satisfied with good accuracy, resulting SLL = 034 99 dB, NL = 049 92 dB c and max 0 = 019 97 dB. Thus the final maximum field amplitude in r resulted to be 16.44 dB lower than that obtained with the method in [16]. In addition, the proposed method allows also the phase-only reconfigurability and the DRR reduction. It is to be noted that a DRR reduction can also be obtained by simply switching off the array elements having the minimum excitation amplitudes. Such an approach, however, in general does not allow to obtain the desired DRR values, and furthermore may strongly deteriorate the radiation pattern. Moreover, in [16] the near-field reduction is obtained by nulling the radiated field in suitably chosen points located in the near-field region, thus reducing the field in a neighborhood of such points. Instead, the presented algorithm allows an accurate near-field control by simply imposing the maximum near-field level. As a final remark we note that, although the algorithm requires several iterations, each iteration involves simple operations. Thus, the global CPU time is quite acceptable.

E E

E

V

E

V E =E D

V

E

E =E

:

:

:

:

V. CONCLUSION We proposed an algorithm of power synthesis for reconfigurable arrays of arbitrary geometry, which is an evolution of the recent approach described in [12]. The method allows to synthesize a number of patterns having a desired shape and possibly including null regions aimed to reject interferences. The switch from one beam to another is performed by phase-only control. Furthermore, an upper bound on the dynamic range ratio can be imposed, to control the variations of the excitation amplitude from one array element to the others, and a threshold can be imposed on the maximum field amplitude in a region close to the antenna. Very good results have been obtained with the proposed choice of the starting point. However, satisfactory results can often be obtained also with random starting points. The solving procedure is easy to implement and the examples show its effectiveness. As a particular important application, the algorithm allows to perform the phase-only beam-scanning on large scan zones, even in presence of stringent constraints on the DRR, on the near-field and on the desired radiation patterns.

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APPENDIX Extending [12], we define in

w; w~ i =

h~

S

0

Imposing, for

W the scalar product

s=1

0

0

g ;g

ks ; ks iV 0

h

3

0

where h s s0 i is defined in [12, Eq. (8b)], the superscript transpose and conjugate, and

H means

ks (r) 1 ks (r) dV 0 3

=

V

(7)

w  w; w

with the asterisk denoting conjugate. This yields the norm k ~ k = 0 0 h ~ ~ i, thus a distance between ~ and ~ is given by ( ~ ~ ) = 0 k ~ 0 ~ k. The Projector U : Given ~ 2 , we want to find the point ~ = U ~ 2 minimizing the squared distance

w; w w w Pw U

w

P

2(w~ ;u~ ) =

S

w

w W

u

gs 0fs k2 + ks0es

k

s=1

2

2 V + k s 0 s kE

w u

(8)

where k1k and k1kV are the norms given bypthe scalar products [12, Eq. H . We must minimize (8b)] and (7), respectively, and k kE = (8) with respect to s , s and s with ~ 2 : s and s are given in [12], while s ( ) is given by

e r

f e

u

u u u U f

u

ks (r) es (r) = E0 k (r) jk (r)j

if if

u

ks (r)  E0 ks (r) > E0 :

~ 2 ~ 2 W , we want to find v ~ = PV w The Projector PV : Given w V minimizing the squared distance

2 (w~ ; v~) =

S

s=1

gs 0F (vs )k2 + ks0E(vs) 2V + kws 0vs kE2 :

k

(9)

Substituting (1) and (2) into (9) yields

2 (w~ ; v~) =

N

S

vnsvms(Fmn + Emn) 3

s=1 n;m=1 N 3 3 3 ( ns ( ns + ns ) + ns ( ns + ns )) 0 n=1

v H K

+

gs gs d' +

v H K

ks 1 ks dV 3

3

I

V

ws 0 vs )H (ws 0 vs )

0(

where

Fmn = Emn =

Fn(') Fm(') d'; Hms = I gs (') Fm(') d'; En (r) 1 Em (r)dV; Kms = ks (r) 1 Em (r)dV: V 3

I

3

3

V

3

p = 1; . . . ;N and s = 1; . . . ;S , the conditions

@2 = N (Fpn + Epn + pn ) vns 0 (Hps + Kps + wps) = 0 @vps n=1

gs;gsi + hks; ks iV + wsH ws

h

1165



S matrix equations J vs = zs ; s = 1; . . . ;S

where pn is the Kronecker delta, yields the

(10)

where J = F + E + IN and zs = hs + ks + ws , with F = [Fmn ], E = [Emn ], IN the identity matrix of rank N , and hs , ks the s-th columns of the matrices H = [Hms ] and K = [Kms ], respectively. Solving (10) to minimize (9), we obtain vs = J 1 zs , where J 1 is the inverse of J. If J is singular or badly conditioned, J 1 can be replaced by the pseudo-inverse matrix of J. 0

0

0

REFERENCES [1] J. A. Ferreira and F. Ares, “Pattern synthesis of conformal arrays by the simulated annealing technique,” Electron. Lett., vol. 33, no. 14, pp. 1187–1189, Jul. 1997. [2] F. J. Ares-Pena, J. A. Rodriguez-Gonzalez, E. Villanueva-Lopez, and S. R. Rengarajan, “Genetic algorithms in the design and optimization of antenna array patterns,” IEEE Trans. Antennas Propag., vol. 47, no. 3, pp. 506–510, Mar. 1999. [3] J. R. P. Lopez and J. B. Verdeja, “Synthesis of linear arrays using particle swarm optimisation,” in Proc. EuCAP, 2006, pp. 1–6. [4] H. Elmikati and A. Elsohly, “Extension of projection method to nonuniformly linear antenna arrays,” IEEE Trans. Antennas Propag., vol. 32, no. 5, pp. 507–512, May 1984. [5] G. M. O. M. Bucci, G. D’Elia, and G. Panariello, “Antenna pattern synthesis: A new general approach,” Proc. IEEE, vol. 82, no. 3, pp. 358–371, Mar. 1994. [6] L. Vaskelainen, “Phase synthesis of conformal array antennas,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 987–991, Jun. 2000. [7] L. Vaskelainen, “New optimising techniques in conformal array antenna design,” in Proc. EuCAP, 2007, pp. 1–6. [8] R. Vescovo, “Scanning circular array with null constraints on the radiation pattern,” Electron. Lett., vol. 32, no. 9, pp. 790–791, Apr. 1996. [9] R. Vescovo, “Constrained beam-scanning for linear arrays of antennas,” Electron. Lett., vol. 40, no. 20, pp. 1242–1243, Sep. 2004. [10] O. M. Bucci, G. Mazzarella, and G. Panariello, “Reconfigurable arrays by phase-only control,” IEEE Trans. Antennas Propag., vol. 39, no. 7, pp. 919–925, Jul. 1991. [11] O. M. Bucci and G. D’Elia, “Power synthesis of reconfigurable conformal arrays with phase-only control,” IEE Proc. Microw. Antennas Propag., vol. 145, no. 1, pp. 131–136, Feb. 1998. [12] R. Vescovo, “Reconfigurability and beam-scanning with phase-only control for antenna arrays,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1555–1565, Jun. 2008. [13] H. Steyskal, “Synthesis of antenna patterns with imposed near-field nulls,” Electron. Lett., vol. 30, no. 24, pp. 2000–2001, Nov. 1994. [14] L. Landesa, F. Obelleiro, J. Rodriguez, J. A. Rodriguez, F. Ares, and A. G. Pino, “Pattern synthesis of array antennas with additional isolation of near field arbitrary objects,” Electron. Lett., vol. 34, no. 16, pp. 1540–1542, Aug. 1998. [15] O. M. Bucci, F. D’Agostino, C. Gennarelli, G. Riccio, and C. Savarese, “Array pattern synthesis with null field constraints in the near-field region,” in Proc. IEEE Antennas Propagat. Soc. Int. Symp., 2001, vol. 3, pp. 716–719. [16] R. Vescovo, “Power pattern synthesis for antenna arrays with null constraints in the near-field region,” Microw. Opt. Technol. Lett., vol. 44, no. 6, pp. 542–545, Mar. 2005. [17] O. M. Bucci, A. Capozzoli, and G. D’Elia, “Power pattern synthesis of reconfigurable conformal arrays with near-field constraints,” IEEE Trans. Antennas Propag., vol. 52, no. 1, pp. 132–141, Jan. 2004. [18] L. G. Gubin, B. T. Polyak, and E. V. Raik, “The method of projections for finding the common point of convex sets,” USSR Comput. Math. Math. Phys., vol. 7, pp. 1–24, 1967. [19] T. Isernia and G. Panariello, “Optimal focusing of scalar fields subject to arbitrary upper bounds,” Electron. Lett., vol. 34, no. 2, pp. 162–164, Jan. 1998.

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Design and Experiment of a Single-Feed Quad-Beam Reflectarray Antenna Payam Nayeri, Fan Yang, and Atef Z. Elsherbeni

Abstract—Reflectarray antennas show momentous promise as a cost-effective high-gain antenna, capable of generating multiple simultaneous beams. A systematic study on various design methods of single-feed multi-beam reflectarray antennas is presented in this communication. Two direct design methods for multi-beam reflectarrays, geometrical method and superposition method, are investigated first. It is demonstrated that although both methods could generate a multi-beam radiation pattern, neither approach provides satisfactory performance, mainly due to high side-lobe levels and gain loss in these designs. The alternating projection method is then implemented to optimize the phase distribution on the reflectarray surface for multi-beam performance. Mask definition and convergence condition of the optimization are studied for multi-beam reflectarray designs. Finally a Ka-band reflectarray prototype is fabricated and tested which shows a good quad-beam performance. Index Terms—Alternating projection method, intersection approach, multi-beam, optimization, reflectarray.

I. INTRODUCTION Reflectarray antennas combine the advantages of both printed arrays and parabolic reflectors and create a high gain antenna with a low-profile, low-mass and low-cost [1], [2]. They have received considerable attention over the years and are quickly finding applications in satellite communications, cloud/precipitation radars, and commercial usages [3], [4]. In addition to these advantages that are mainly due to the use of printed circuit technology, the reflectarray allows for an individual control of the phase shift of each element in the array. As a result, the reflectarray can achieve contoured beam performance without any additional cost [5]. Similarly, multi-beam performance can also be realized by designing the phase shift of the elements appropriately. Multi-beam antennas have numerous applications, such as electronic countermeasures, satellite communications, and multiple-target radar systems [6]. These multi-beam antennas are typically based on reflectors with feed-horn clusters [7] or large phased arrays [8]. Horn array feeds for reflector antennas on communication satellites can provide multiple beams with tailored earth coverage patterns. For phased array antennas, multiple simultaneous beams can be generated by connecting the array to a beamforming network with multiple ports. Considering the complexity of fabricating these antennas and deployment for space applications, these multiple beam designs are relatively high cost. The numerous advantages of reflectarrays, in particular the low-mass and low-cost features, makes the multiple beam reflectarray a suitable antenna candidate. Manuscript received November 29, 2010; revised June 14, 2011; accepted August 26, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported in part by the NASA EPSCoR program under contract number NNX09AP18A. P. Nayeri and A. Z. Elsherbeni are with the Electrical Engineering Department, The University of Mississippi, University, MS 38677 USA (e-mail: [email protected]; [email protected]). F. Yang is with the Electrical Engineering Department, The University of Mississippi, University, MS 38677 USA and also with the Electronic Engineering Department, Tsinghua University, Beijing, China (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173126

Reflectarrays can generate single or multiple beams with single or multiple feeds. A two-beam reflectarray prototype using a single feed was demonstrated in [9] while [10], [11] present a single-feed reflectarray generating four simultaneous beams. Multi-feed multi-beam reflectarrays with shaped patterns were also studied in [12]. In addition, multi-feed single-beam reflectarray antennas were investigated in [13]. In these papers, different design approaches have been introduced to achieve the multi-beam performance. The main objective of this communication is to provide a comprehensive and systematic comparison of various multi-beam design approaches, including both direct design methods and iterative optimization techniques, through a case study of a single-feed quad-beam reflectarray. II. DIRECT DESIGN METHODS FOR MULTI-BEAM REFLECTARRAYS A. Basics of Two Direct Design Methods Two direct design methods are available for multiple beam reflectarray antennas. The basic idea behind the first approach, geometrical sub-arrays method, is simply to divide the reflectarray surface into where each sub-array can then radiate a beam in the required direction [10]. Although the array division and beam allocation can be arbitrary, it is feasible to define them based on the directions of the beams they are designed to generate. It should be noted that with this approach each of the power from the feed horn while using 1 zone receives 1 of the aperture surface. Another approach for multi-beam reflectarray designs is by using the superposition of the aperture fields associated with each beam on the reflectarray aperture [2]. To generate beams with a single feed, the tangential field on the reflectarray surface can simply be written as

N

=N

=N

N

ER (xi ; yi ) =

N n=1

An;i (xi ; yi )ej

(x ;y ) :

(1)

Here n;i and n;i are the required amplitude and phase of the th element which will radiate the th beam. In reflectarrays the amplitude of each element is fixed by the feed position and element location, which are independent of the beam direction, therefore

8

A

i

n

ER (xi ; yi ) = AFeed i (xi ; yi )

N 1

n=1

ej

(x ;y ) :

(2)

The summation of the complex field distributions in (2) will give the overall required amplitude and phase distributions. A basic problem exists here which is due to the fixed amplitude distribution imposed by the feed in reflectarray antennas. Although the required phase in (2) can be satisfied by proper element designs, the amplitude requirement cannot be satisfied in reflectarray antennas. The reason is that in (2)

N n=1

ej

(x ;y ) = 6 1:

(3)

As a result of this difference in the amplitude distribution on the aperture, reflectarrays designed using the superposition approach may show a deteriorated performance. B. Comparison of Direct Design Methods To demonstrate the multi-beam design capabilities of these approaches, we study a quad-beam reflectarray antenna. The antenna is designed for the operating frequency of 32 GHz and has a circular aperture with a diameter of 17 at the design frequency. The element periodicity is 2 and ideal phasing elements are used here to

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TABLE I CALCULATED RADIATION CHARACTERISTICS OF THE SINGLE-BEAM AND MULTI-BEAM REFLECTARRAYS

Fig. 1. Normalized radiation patterns of the single- and quad-beam designs.

compare different design approaches. A centered prime focus left hand circularly polarized (LHCP) feed is used for this design and positioned with an F=D ratio of 0.735 that gives an edge taper of 012.39 dB. The reflectarray elements are dual-linear, which reflect the LHCP incident wave to a right hand circularly polarized (RHCP) wave. The radiation patterns of the reflectarrays are calculated using spectral transformation of the aperture fields as described in [2]. This quad-beam reflectarray is designed to generate four beams in the directions of (1;2;3;4 = 30 , '1 = 0 , '2 = 90 , '3 = 180 , '4 = 270 ). Here  is the angle between the axe normal to the reflectarray plane and the beam direction. To compare the performances of these two reflectarrays, we also design a single-beam reflectarray antenna as a reference whose beam is in the direction of ( = 30 , ' = 0 ). In Fig. 1 the normalized radiation pattern of this single beam design is compared with the multi-beam reflectarrays. Since the reference beam is in the ' = 0 plane, only this plane is given here, and a similar pattern was observed in the orthogonal plane. From these results it can be seen that both design methods can realize a quad-beam performance with a single-feed horn. In both designs the four beams are generated in the required directions. Comparison of the radiation patterns shows small beam deviations in the multi-beam reflectarray designs. The radiation patterns also show that the side-lobe level is below 011 dB for the geometrical design and below 017 dB for the superposition design. The side-lobe levels (SLL) are much higher than the reference single-beam design. Furthermore, the quad-beam antenna designed using the superposition method shows a beamwidth identical to the reference single-beam design, while in the geometrical design the beamwidth is much wider. The antenna directivity is an important measure to compare the radiation performance of these multi-beam design methods. For multi-beam reflectors using a single-feed, theoretically the power of each beam will be reduced by 1=N . Therefore ideally it is expected that generating four beams will reduce the antenna directivity by 6 dB. However, the geometrical design has a directivity reduction of 11.73 dB and the superposition design exhibits a directivity reduction of 7.02 dB. Some important results of these three reflectarrays are summarized in Table I. For the multi-beam reflectarray designed with the geometrical approach, the amplitude distribution in each zone is maximum at the corner of that zone (near the array center) and minimum at the outer edge, which results in a significant increase in the side-lobe level. The wider beamwidth and lower directivity in the geometrical designs however requires further attention. The reduction in antenna directivity is the result of using one fourth of the array surface and one fourth of the power from the feed horn to generate each beam. This reduction of array surface is also the reason for the increase in beamwidth. For the

multi-beam reflectarray designed using the superposition method, the high side-lobes are due to the amplitude error in (2) which alters the required illumination taper on the aperture. Also in comparison of the calculated antenna directivity with the reference, this design approach shows a directivity loss about 1 dB higher than the ideal directivity reduction (6 dB), which is mainly due to the high side-lobe levels of this design. In summary the shortcomings of both these direct design approaches are tabulated here. Geometrical design method: 1) High side-lobe due to illumination taper; 2) Gain loss and beam broadening due to dividing the array surface into sub-arrays; 3) Small beam deviation. Superposition design method: 1) High side-lobe due to amplitude error; 2) Gain loss due to the increase in side-lobe level; 3) Small beam deviation. As a result of the above problems associated with the direct methods of multi-beam reflectarray design, it is necessary to implement some form of optimization routine to achieve desirable performance. III. ALTERNATING PROJECTION METHOD FOR MULTI-BEAM REFLECTARRAY DESIGN A. The Alternating Projection Method Another approach in multi-beam reflectarray design is to view this as a general array synthesis problem. In reflectarrays however the synthesis of radiation patterns is restricted by the fact that the amplitude of each reflectarray element is fixed by the feed properties and element location. As a result, design of multi-beam reflectarrays requires a phase only synthesis approach. The alternating projection method (APM) also known as the intersection approach [14] has been applied successfully to the phase synthesis of antenna arrays. A simple example of a two-beam reflectarray has been demonstrated using this method [9]. This method is basically an iterative process that searches for the intersection between two sets, i.e. the set of possible radiation patterns that can be obtained with the reflectarray antenna and the set of radiation patterns that satisfy the mask requirements (set M ). Comparing to other phase synthesis methods developed for array antennas, the main advantage of the alternating projection method is the significantly reduced computational time for convergence of the solution [14], which makes it suitable for large reflectarray antennas. The pattern requirements for the design are usually defined by a mask, i.e., two sets of bound values, between which the pattern must lie. The general form of the radiation patterns that satisfy the mask requirements is

set M  fF (u; v) : M (u; v)  jF (u; v)j  M (u; v)g ; u = sin  cos '; v = sin  sin ' L

U

(4)

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Fig. 3. Radiation pattern of the optimized design at 32 GHz.

Fig. 2. 2D view of the mask model for the quad-beam reflectarray.

where F is the far-field radiation pattern of the array and (u; v ) are the angular coordinates. MU and ML set the upper and lower bound values of the desired pattern in the entire angular range. With set M defined, the alternating projection method can be implemented to obtain the desired radiation pattern. Implementing the alternating projection method requires definition of two projection operators: the mask projector (PM ) and the inverse projector (PI ) [15]. The mask projector uses the upper and lower bounds of the mask to correct the radiation pattern. The inverse projection (PI ) consists of a series of functions which projects the pattern back to the array excitation coefficients. It calculates new phase values for the reflectarray elements while the elements amplitude remain unchanged. B. Implementing APM for Multi-Beam Reflectarray Design The first step is to define the mask for multi-beam operation. Typical masks for different contour beams can be found in the literature [16]; however for multi-beam designs the mask definition is different. The required masks for multi-beam radiation patterns are typically circular contours defined in the direction of each beam. Since in this quad-beam design we don’t want to change the beamwidth, which is directly related to the aperture size and illumination, the mask upper and lower bounds in the beam area were defined according to the reference single-beam design. This upper and lower bounds are defined as

If If

(u; v )

(u; v )

2 main beam : MU (u; v) = 0 dB 2 0 3 dB beamwidth : ML (u; v) = 03 dB:

(5)

The main objective of this optimization is to minimize the side-lobe level. While it is possible to control the side-lobe level by defining an upper bound (MU ) at certain values, in order to further minimize the side-lobe level, both upper and lower bounds in the side-lobe area were set to zero. A 2D figure of this beam mask model for the quad-beam reflectarray is plotted in Fig. 2 using dashed lines. It should be noted that in practice it was found that for this quad-beam reflectarray design, defining mask levels to zero or to an achievable level showed almost similar results. In all optimization routines, it is necessary to define a cost function that should be minimized and can also control the number of iterations required for the convergence of the solution. Since in this optimization the requirements in the main beam will be satisfied by the projection operators with the bounds set in (5), the cost function need only to take into account the side-lobe performance of the array [17]. Thus, the cost

is evaluated over every point in the (u; v ) space which does not belong to the main beams using the following equation: If

jF (u; v)j > MU (u; v) 2 (jF (u; v )j 0 MU (u; v )) : Cost = u +v 1

(6)

With the mask and cost function defined, the optimized phase distribution of the reflectarray elements can be obtained with an iterative procedure. It should be noted here that the optimization is considered to be converged when the cost function becomes stable. In most cases the optimization converges with only a few iterations; however a suitable starting point can reduce the number of iterations. In this design the phase distribution obtained by the superposition method in Section II is used as the starting point for the optimization. The alternating projection method is then implemented to improve the reflectarray performance by optimizing the phase distribution on the reflectarray aperture. A far-field pattern of 400 2 400 points evenly spaced in the angular coordinates was computed for each candidate reflectarray at each cost evaluation. For this quad-beam design, the solution converges after 23 iterations. Although the number of iterations required for the optimization generally depend on the problem at hand, in most cases the APM will converge with just a few iterations [14]–[16]. The radiation pattern of the optimized design is given in Fig. 3. A quad-beam performance is obtained for the reflectarray with side-lobes below 026 dB. It can be seen that implementing the optimization here has corrected the amplitude problems associated with the initial superposition design ((3)). As a result, the side-lobe level has been reduced by about 9 dB and all four beams are exactly scanned to 30 off broadside. Also the calculated directivity for this antenna is 26.95 dB, which is about 1 dB higher than the initial superposition design (Table I). These results clearly demonstrate the effectiveness of the phase optimization process. The computational time for the APM optimization with 30 iterations was 456 seconds on a 2.2 GHz Intel core Duo CPU with 4 GB RAM. This is to be compared with 16.5 seconds for both direct design methods. It is important to point out that although in some cases the APM optimization might converge to local minima, for this symmetric quadbeam design the optimization did not get trapped and the solution converged smoothly. Similar radiation pattern results were observed when the phase distribution obtained by the geometrical approach was used for the starting point. For non-symmetric multi-beam designs however the problem with local minima is more challenging and some approaches that can circumvent the local minima problem such as [9] may be required.

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Fig. 4. (a) Optimized phase distribution of the reflectarray elements, (b) fabricated quad-beam reflectarray.

IV. KA-BAND QUAD-BEAM REFLECTARRAY PROTOTYPE A. Prototype Fabrication The optimized quad-beam prototype is fabricated on a 20 mil Rogers 5880 substrate. The reflectarray has a circular aperture with a diameter of 15.94 cm. The phasing elements are variable size square patches with a unit-cell periodicity of =2 at the design frequency of 32 GHz. The unit-cell simulations are carried out using the commercial software Ansoft Designer [18], where the fabrication limit of our LPKF ProtoMat S62 milling machine is also taken into account by enforcing the minimum gap size between the elements and the achievable fabrication tolerance. It is worthwhile to point out that in general the reflection characteristics of the phasing elements are angle dependent and oblique incidence needs to be considered. Our simulations showed that for these elements normal incidence can present good approximations for oblique incidence angles up to 35 ; thus the prototype was designed based on the simulated reflection coefficients obtained with normal incidence. The optimized phase distribution for the reflectarray elements and the photograph of the fabricated array with 848 square patch elements are shown in Fig. 4. The centered prime focus LHCP feed horn is mounted on a mechanical alignment system and positioned with an F/D ratio of 0.735. To avoid blockage from the supporting strut of the feed horn, the array is rotated 45 in the reflectarray plane so the main beams are in the directions of (1;2;3;4 = 30 , '1 = 45 , '2 = 135 , '3 = 225 , '4 = 315 ). Since dual-linear square patch elements are used in this design, the reflected co-polarized radiation of the reflectarray system is RHCP.

Fig. 5. Measured and simulated co-polarized radiation patterns of the reflecplane, (b) ' plane. tarray antenna: (a) '

= 45

= 135

B. Measured Radiation Patterns The radiation pattern is measured using our planar near-field measurement system. Comparisons of the simulated and measured co-polarized radiation patterns at 32 GHz are shown in Fig. 5. The simulated radiation pattern of the single beam reference design in Section II is also plotted here for comparison. For the quad-beam design, the simulated radiation patterns here also include the aperture blockage caused by the horn and the alignment system, which is calculated using the approach given in [19]. Note that four beams are generated in the required directions which are correctly scanned to 30 and the side-lobe levels are below 018 dB. The measured and simulated results show good agreements in the main lobes where the measured 03 dB beamwidth is 4.35 for both vertical and horizontal planes. Some discrepancies exist in the side-lobe regions, which are mainly due to fabrication errors and element design approximations. The beam level reduction is primarily due to the alignment errors of the measurement setup and the azimuth non-symmetry of the feed horn radiation pattern; however this reduction is less than 2.15 dB

Fig. 6. Measured and simulated cross-polarized radiation patterns of the reflectarray antenna in ' plane.

= 45

for any beams. Both simulated and measured cross-polarized radiation patterns of the reflectarray are given in Fig. 6 for the ' = 45 plane. Almost similar results were observed in the orthogonal plane (' = 135 ). The relatively high cross-polarization level of the reflectarray here is due to the high cross polarization of the feed horn. To reduce the cross polarization, one approach is to use a better feed

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Fig. 7. Measured gain and efficiency of the quad-beam reflectarray antenna.

horn with lower cross-polarization level. This was confirmed by repeating the simulations with an idealized horn antenna model [2]. For this case the cross-polarization level of the reflectarray was reduced to 023.1 dB. Another approach is to use the element rotation technique [2] for phase compensation, since it will only compensate the phase of the co-polarized CP components to form a focused beam. C. Gain and Efficiency The measured gain and aperture efficiency vs. frequency are given in Fig. 7. It should be noted that although all beams showed a similar gain performance vs. frequency, the gain results presented here are for the beam in the direction of  = 30 , ' = 45 . At 32 GHz, the measured gain is 25.3 dB, and the 1 dB gain bandwidth is 8.6%. For multi-beam antennas, the classical definition of aperture efficiency might not be appropriate; therefore a modified definition is used here to calculate the aperture efficiency, i.e. a

=

N

G

2 i i=1 A

4

Fig. 8. Measured radiation patterns of the reflectarray antenna across the 1 dB plane. gain band in the '

= 45

The radiation patterns show quad-beam patterns across the entire band with a slight increase in side-lobe level at the extreme frequencies. The beam squint across the 1 dB gain bandwidth of the antenna is about 2.5 . Considering that the beamwidth of the antenna is more than 4 , the effect of this beam squint is acceptable for this quad-beam prototype. It is interesting to point out that in comparison between the multi-beam design methods, the direct geometrical approach shows the smallest beam squint. While the quad-beam prototype here did not show a very large beam squint across the band, it should be pointed out that in general for multi-beam reflectarrays, beam squint could limit the operating band of the antenna. Thus, for multi-beam designs where a minimum beam squint requirement is specified, additional constraints over the frequency bandwidth needs to be imposed when optimizing the phase distribution of the elements. V. CONCLUSION

(7)

where N represents the number of beams and A is the aperture area. This definition takes into account the measured gains of all four beams, and the aperture efficiency is calculated to be 35.26%. Besides the spillover and illumination effects, the loss in the aperture efficiency comes from the cross-polarization effect, the element loss, and the feed blockage. D. Beam Squint The bandwidth of a reflectarray antenna is usually defined by the 1 dB gain bandwidth [20], [21]. For multi-beam reflectarrays however, the practical bandwidth of the antenna is also limited by the fact that the beams shift with frequency. This is due to the fact that the main beam direction depends on the progressive total phase on the aperture (including all time delay effects from feed radiation and total reflection phase of the elements). It was shown in [22] that beam squint can be minimized in reflectarray antennas by enforcing the condition o = i , where i is defined as the angle from the phase center of the feed to the center of the array and o is the main beam direction. It is clear that this condition cannot be satisfied for multi-beam designs and beam squint would become a limiting factor. The measured radiation patterns at the center and extreme frequencies are given in Fig. 8 for the ' = 45 plane. Similar results were observed in the orthogonal plane.

A systematic analysis of single-feed multi-beam reflectarray antennas is presented in this communication through a case study of a quad-beam design. Two direct design approaches for multi-beam reflectarrays are investigated first, and their performance suggests that an optimization procedure on the elements phase distribution is necessary to achieve a satisfactory performance for complicated multi-beam reflectarrays. The alternating projection method is then implemented to optimize the performance of the multi-beam reflectarray antennas. Required masks, cost definition, and convergence conditions are discussed for multi-beam reflectarrays. Based on the optimization results, a Ka-band reflectarray antenna is designed and measured, which shows a good quad-beam performance using a single feed. ACKNOWLEDGMENT The authors want to thank the reviewers whose thoughtful comments added to the quality and efficacy of this communication. The authors also acknowledge Rogers Corporation for providing free dielectric substrates.

REFERENCES [1] D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarrays,” IEEE Trans. Antennas Propag., vol. 45, no. 2, pp. 287–296, Feb. 1997.

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[2] J. Huang and J. A. Encinar, Reflectarray Antennas. Hoboken, NJ: Wiley, 2008, Institute of Electrical and Electronics Engineers. [3] R. E. Munson and H. Haddad, “Microstrip Reflectarray for Satellite Communication and RCS Enhancement and Reduction,” U.S. patent 4,684,952, Aug. 1987, Washington DC. [4] H. Shih-Hsun, H. Chulmin, J. Huang, and K. Chang, “An offset lineararray-fed Ku/Ka dual-band reflectarray for planet cloud/precipitation radar,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3144–3122, Nov. 2007. [5] D. M. Pozar, S. D. Targonski, and R. Pokuls, “A shaped-beam microstrip patch reflectarray,” IEEE Trans. Antennas Propag., vol. 47, pp. 1167–1173, Jul. 1999. [6] R. C. Hansen, Phased Array Antennas, Wiley Series in Microwave and Optical Engineering. New York: Wiley, 1998. [7] P. Balling, K. Tjonneland, L. Yi, and A. Lindley, “Multiple contoured beam reflector antenna systems,” presented at the IEEE Antennas and Propagation Society Int. Symp., Michigan, Jun. 28–Jul. 2 1993. [8] L. Schulwitz and A. Mortazawi, “A compact dual-polarized multibeam phased-array architecture for millimeter-wave radar,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3588–3594, Nov. 2005. [9] J. A. Encinar and J. A. Zornoza, “Three-layer printed reflectarrays for contoured beam space applications,” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1138–1148, May 2004. [10] J. Lanteri, C. Migliaccio, J. Ala-Laurinaho, M. Vaaja, J. Mallat, and A. V. Raisanen, “Four-beam reflectarray antenna for Mm-waves: Design and tests in far-field and near-field ranges,” in Proc. EuCAP, Berlin, Germany, Mar. 2009, pp. 2532–2535. [11] P. Nayeri, F. Yang, and A. Z. Elsherbeni, “Single-feed multi-beam reflectarray antennas,” presented at the IEEE Antennas and Propagation Society Int. Symp., Toronto, Canada, Jul. 2010. [12] M. Arrebola, J. A. Encinar, and M. Barba, “Multifed printed reflectarray with three simultaneous shaped beams for LMDS central station antenna,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1518–1527, Jun. 2008. [13] F. Arpin, J. Shaker, and D. A. McNamara, “Multi-feed single-beam power-combining reflectarray antenna,” Electron. Lett., vol. 40, no. 17, pp. 1035–1037, Aug. 2004. [14] R. Vescovo, “Reconfigurability and beam scanning with phase-only control for antenna arrays,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1555–1565, Jun. 2008. [15] O. M. Bucci, G. Franceschetti, G. Mazzarella, and G. Panariello, “Intersection approach to array pattern synthesis,” IEE Proc., vol. 137, no. 6, pp. 349–357, Dec. 1990. [16] O. M. Bucci, G. Mazzarella, and G. Panariello, “Reconfigurable arrays by phase-only control,” IEEE Trans. Antennas Propag., vol. 39, no. 7, pp. 919–925, Jul. 1991. [17] D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 771–779, Mar. 2004. [18] Ansoft Designer v 4.1 Ansoft Corporation, 2009. [19] L. Diaz and T. Milligan, Antenna Engineering Using Physical Optics. Norwood, MA: Artech House, 1996. [20] J. Huang, “Bandwidth study of microstrip reflectarray and a novel phased reflectarray concept,” presented at the IEEE Antennas and Propagation Society Int. Symp., CA, Jun. 1995. [21] D. M. Pozar, “Bandwidth of reflectarrays,” Electron. Lett., vol. 39, no. 21, Oct. 2003. [22] S. D. Targonski and D. M. Pozar, “Minimization of beam squint in microstrip reflectarrays using an offset feed,” presented at the IEEE Antennas and Propagation Society Int. Symp., MD, Jul. 1996.

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Oblique Diffraction of Arbitrarily Polarized Waves by an Array of Coplanar Slots Loaded by Dielectric Semi-Cylinders John L. Tsalamengas and Ioannis O. Vardiambasis

Abstract—We study oblique diffraction of arbitrarily polarized planewaves by a finite array of slots of infinite length on a common ground plane, backed by an array of dielectric semi-cylinders. The formulation is based on a combined eigenfunctions expansion and integral equation approach. For the diffracted field, series expansions in cylindrical wave functions are used to which several singular integral terms are superimposed that fully account for the presence of each of the slots. The relevant system of singular integral equations is discretized by an exponentially convergent Nyström method. Noticeably, all matrix elements take simple closed-form expressions. Numerical examples and case studies illustrate the convergence of the algorithm and bring to light the influence of the dielectric loads on the characteristics of the structure. Index Terms—Dielectric cylinders, electromagnetic diffraction, integral equations, Nyström method, slot arrays.

I. INTRODUCTION Fig. 1 shows an array of S slots of infinite length on the ground plane y = 0, loaded by dielectric semi-cylinders ("i ; i ) of radii Ri whose axes are located at y = 0, x = hi , i = 1; 2; . . . ; S . Without loss of generality it is taken that h1 = 0. Region 0 (y > 0) has the properties of vacuum ("0 ; 0 ), and the background medium (region a), taken to be lossless, has parameters ("a ; a ). Dielectric losses in the cylinders may be accounted for via complex constitutive parameters. The primary excitation is an arbitrarily polarized plane wave obliquely incident from region 0. This structure may be used as a polarization selector, a radiation suppressor over a given frequency bandwidth, and a multi-slot/multi-dielectric open waveguide. Previous related studies [1]–[8] only concern certain special (or limiting), definitely simpler, two-dimensional cases wherein a single slot is illuminated by a z -invariant primary excitation in presence of either dielectric or perfectly conducting cylindrical loads. In such cases TEz and TMz waves decouple and can be treated separately [9]. The case S = 1 of the present quasi three-dimensional problem has been treated in [10] via a combined Green’s function and singular integral equation approach. Nevertheless, when S (S > 1) semi-cylinders coexist in region y < 0, as in Fig. 1, the method of [10] breaks down due to insuperable difficulties in obtaining the Green’s function of the structure. To overcome such difficulties, in the present communication a new hybrid formulation technique, flexible and easily implemented, is proposed (Section II). The main idea is to use series expansions in cylindrical wave functions, in addition to properly selected singular integral Manuscript received December 18, 2010; revised April 08, 2011; accepted August 31, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported in part by the “THALES” Project ANEMOS, funded by the ESPA Program of the Ministry of Education of Greece. J. L. Tsalamengas is with the School of Electrical and Computer Engineering, National Technical University of Athens, GR-15773 Zografou, Athens, Greece (e-mail: [email protected]). I. O. Vardiambasis is with the Department of Electronics, Technological Educational Institute (TEI) of Crete, 73133 Chania Crete, Greece (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173129

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 exc; H exc) be the known field excited in region 0 when Let (E  0; H 0 ) = all slots are absent (slots short-circuited). Then (E tot  tot exc  exc   (E ; H ) 0 (E ; H ) defines the scattered field in region 0. The transmitted (total) field in region i for i = 1; 2; . . . ; S or  i ; H i ). All fields and current densities i = a will be denoted by (E jk z   , just like the incident field, where will vary as X ( r) = X ( )e kz

= k0 cos 0 :

(3)

B. Field Representations For i Fig. 1. Geometry of the structure and details of the incident field.

= 0; 1; 2; . . . ; S , let Gei ( ; 0 ) =

terms, for the field inside each cylinder. The important point is that the proposed field representations automatically satisfy the boundary conditions for the tangential electric field on the entire y = 0 plane. Following the method of scattering superposition, expansions in cylindrical wave functions are, also, used for the field transmitted in the outer region a. Finally, for the field in region 0 integral representations in terms of equivalent surface magnetic current densities across the slots are used as in [10]. The continuity of the tangential magnetic field on each slot, and the continuity of the tangential electric and magnetic fields on the boundary of each cylinder, yield a system of coupled singular integral equations (Section III). This system is amenable to a highly accurate solution by the sophisticated Nyström method of [11] as outlined in Section IV. Numerical examples presented in Section V reveal the exponential convergence of the developed algorithm and bring to light the influence of the dielectric loads on the characteristics of the structure. As a useful by-product, the proposed algorithm may be directly used to obtain the characteristics (e.g., propagation constants and modal fields) of the hybrid modes supported by the configuration of Fig. 1. The extension to the more general problem where each slot is covered by two dielectric semi-cylinders (one in region y < 0, as in Fig. 1, and another in region y > 0, with axes at x = hi ; i = 1; 2; . . . ; S , and possibly different radii), which reside on the y = 0 plane, is straightforward and only requires trivial modifications of the analysis. In what follows, the assumed ej!t time-dependence is suppressed.

2

0 4 !"

(2)

i

2

i

i

i

i

i

i

i

i

(4) (5)

=

z

i

(6)

be auxiliary quantities to be used shortly.  q (x) = E tot(x; 0)2 y^ = Mxq (x)^x + Mzq (x)^z 1) Region 0: Let M denote the equivalent surface magnetic current density across the q 0 th slot. Then [1], [10], [12]

Ez0 ( ) = Hz0 ( ) =

S

0 0 j!"

2 0

S

q =1 C

Mxq (x0 )

@Ge0 (x0 ; 0+ ; ) 0 dx @y 0

(7)

Mzq (x0 )G0h (x0 ; 0+ ; )dx0

q =1 C

0 j k2

S

z

0

q =1 C

Mxq (x0 )

@Gh0 (x0 ; 0+ ; ) 0 dx @x0

(8)

where Cq = fx : hq 0 wq  x0  hq + wq g denotes the x-axis interval occupied by the q 0 th slot. 2) Region i, i = 1; 2; . . . ; S : For the field in region i we make use of the representations

1 n=1

ain Jn ( i i )sin(n'i )

+ j!"

2

Mxi (x0 )

i

A. Preliminaries

i

 inc (r) = E0 ue0jk k 1r , H inc (r) = The incident field E inc inc ^ 2 E (r) (Y0 = 1=Z0 = "0 =0 ; k0 = !p"0 0 ), Y0 k originating from region 0, propagates in the direction of ^

 as in Fig. 1. The polarization is described by the unit vector u u (0; '0 ; ) = ux x^ + uy y^ + uz z^, where

i

=

Ezi ( ) =

= 0^x sin 0 cos '0 0 y^ sin 0 sin '0 0 z^ cos 0

( R0 ) 0 H0(2) ( R+ )

(2) (2) H0 ( R0 ) + H0 ( R+ ) 0 4! 1 2 ; R6 = (x 0 x0 )2 + (y 6 y 0 )2   2 2 1 2

= (k 0 k ) ; 0  arg < 2 2

Ghi ( ; 0 ) =

II. FORMULATION

^inc k

H0

H ( ) = i z

1 n=0

0

(1)

C

(9)

b Jn ( i i )cos(n'i )0 i n

Mzi (x0 )Gih (x0 ; 00 ; )dx0

C

=

@Gei (x0 ; 00 ; ) 0 dx @y 0

+ jk

2

Mxi (x0 )

z

i

C

@Ghi (x0 ; 00 ; ) 0 dx @x0

(10)

= sin '0 cos 0 cos 0 cos '0 sin where ( ; ' ) are the polar coordinates of the observation point  in u = 0 cos '0 cos 0 cos 0 sin '0 sin the local coordinate system associated with the i 0 th semicylinder. (2) u = sin 0 sin The integral terms in (9) and (10) represent the z -components of the field that would have been excited in the semi-infinite region y < ^ , with the particular direction specified by the 0 if the surface magnetic current density 0M had been impressed which is normal to k polarization angle ; the special cases = 0;  and = =2; 3=2 across C assuming that a) all slots are absent (short-circuited) and b) the entire region y < 0 is filled with the medium (" ;  ). correspond to the TE and TM polarizations, respectively. ux

i

i

y z

i

inc

z

z

i

i

i

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 2, FEBRUARY 2012

Fig. 2. Translation of the reference system of the p

0 th cylinder.

Fig. 3. Addition theorem for the Hankel function.

3) Region a: Finally, for the field in region a we make use of the expansions

Eza ( ) = Hza () =

1

S

q=1 n=1 S

1

q=1 n=0

Aqn Hn(2) ( a q ) sin(n'q )

(11)

Bnq Hn(2) ( a q ) cos(n'q ):

(12)

Equations (11) and (12) will be supplemented with the following addition-translation formula (parameters are specified in Fig. 2):

(2) (  p )ejm' Hm

1 (2) = Hm0n (  Dpq )Jn (  Rq )ejn' ej (m0n)' n=01

= =

0 kz rt Ez 0 !z^ 2 rt Hz 0 kz rt Hz + !"z^ 2 rt Ez :

(13)

(14) (15)

Remark 2: It is important here to stress out that, in the context of the proposed field representations, both Ez (x; 0) and Ex (x; 0) 1) vanish on the metallic parts of the ground plane and 2) are continuous on the slots, i.e., the boundary conditions for the tangential electric field are satisfied on the entire y = 0 plane.

A. The First Set By satisfying for i

= 1; 2; . . . ; S the continuity i (x; 0 ) = H 0 (x; 0+ ) + H exc (x; 0+ ), x Hz;x i, z;x z;x

0

conditions 2 C after some manipulations omitted here for brevity we obtain the following (2S)2(2S) set of coupled singular IEs: 1

2!0 +

q=1

1

dx

+

i2

2!i

S

2!0

q=1

1 2!i

jkz jkz

1

d

q (x) z

dx

d (zi) (x) dx

0 hi

0

(2) q Mz;x (x )H0 ( 0

(18)

jx 0 x0 j)dx0

(19)

i 0 (2) 0 0 Mz;x (x )H0 ( i jx 0 x j)dx :

(i) (x) = z;x

C

0

(2)

6

(20)

(2)

6

We substitute (9) and (11) into the continuity condition a = Ez (Ri ; 'i ), 0  'i  0, and make use of (4), (13), and the addition theorem for the Hankel functions [13]

Ezi (Ri ; 'i )

H0(2) ( i R) =

1 (2) Hn ( i Ri )Jn ( i xi )ejn' n=01

(21)

(the parameters involved in (21) are specified in Fig. 3). Next, we multiply both sides of the resulting equation by sin(m'i ) for m = 1; 2; . . . ; 1, and integrate from 'i = 0 to 'i =  . After some lengthy but otherwise straightforward manipulations, omitted here for brevity, we end up with the following set of IEs:

(2) ( i Ri )J + (m; i) aim Jm ( i Ri ) 0 j i Hm x

0 Aim Hm(2) ( a Ri ) 0 Jm ( a Ri )

01

S

q=1 n=1

0 (q; i) = 0 Aqn Fmn

m = 1; 2; . . . ; 1; i = 1; 2; . . . ; S where

x0i

0 0 hi

=x

1

2

0

Mxi (x )

C

(22)

Jm01 ( i xi0 ) 6 Jm+1 ( i x0i )

dx0

6 (01)m Hn(2)+m ( a Dqi )e0j(m+n)'

(24)

(25)

with

'qi

= 0;

if q > i; otherwise 'qi

=

:

(26)

C. The Third Set

(16)

In a similar way, from Hzi (Ri ; 'i ) = Hza (Ri ; 'i ),   'i  0, for i = 1; 2; . . . ; S and for m = 0; 1; . . . ; 1, one obtains the IEs:

+

q x (x)

" i (2) ( i Ri ) i Jz (m; i) 0 jkz J 0 (m; i) bm Jm ( i Ri )+ i m Hm x 2!i

+

d2 ki2 + 2 dx

(i) (x)

0 Bmi Hm(2) ( a Ri ) 0 "2m Jm ( a Ri )

1

(23)

6 (q; i) = Hn(2)0m ( a Dqi )ej(n0m)' Fmn

d2 k02 + 2 dx

x

jk j!"i n ai Jn ( i xi ) + z bi Jn 0 ( i xi ) +

i2 n=1 n xi

i n=0 n exc + = Hx (x; 0 ); x 2 Ci ; i = 1; 2; . . . ; S

6 6

In (17), the relations (@=@y )H0 ( i R ) = 6(@=@y)H0 ( i R ) (2) and (@x2 + @y2 0 kz2 + ki 2 )H0 ( i R ) = 0 have been used, with R given in (6).

The prime in the summation over q in (22) is used to remind that the term with q = i is excluded from the sum.

(i) z (x)

jkz d (xi) (x) bni Jn ( i xi ) = Hzexc (x; 0+ ) + 2!i dx n=0 x 2 Ci ; i = 1; 2; . . . ; S

1

+

d q z (x) + jkz

02

q (x) x

xi = x q z;x (x) = C

Jx6 (m; i) =

III. INTEGRAL EQUATIONS

S

where

B. The Second Set

where 'pq = 0, if p > q , otherwise 'pq =  , which enables one to express the elementary cylindrical wave referred to the local coordinate system of the p 0 th cylinder as a series of cylindrical wave functions referred to the local coordinate system of the q 0 th cylinder. Remark 1: In terms of Ez () and Hz (), the transverse (to z -axis)  t ( t ( ) and H ) can be obtained everywhere from components E

j (! 2 " 0 kz2 )Et t j (! 2 " 0 kz2 )H

1173

01

q=1 n=0

+ (q; i) = 0 Bnq Fmn (27)

where "m = 2 0 m0 and

Mzi (x0 )Jm ( i xi0 )dx0 :

Jz (m; i) =

(17)

S

C

(28)

1174

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 2, FEBRUARY 2012

(L has to be selected as high as needed to assure any prescribed accuracy, as specified in Section V). After carrying out the relevant integrations we satisfy the resulting two equations, respectively, at the i , where collocation points x = xim and x = m

D. The Fourth Set The remaining boundary conditions i a '^i 1 E i(Ri ; 'i ) = '^i 1 E a(Ri ; 'i ) H (Ri ; 'i ) H (Ri ; 'i )

; 0  'i  0

i xim = hi + wi tm ; m = hi + wi m :

treated along the same lines end up with the following set of IEs:

Aim m0 i Bm

H

U mi H

0

V mi

"m U J +

0

i Jz 2! i m m0 i m

j i Jx+ (m; i) (m; i) 0 jkz Jx (m; i))

1

0

0 a  0 V mi =  b 0 m = 0; 1; . . . ; 1; i = 1; 2; . . . ; S: J

Here,

n=0

(29)

mk

R

+ n=1

0

J

0

J

H

In view of the edge conditions, Mzi (x0 ) and Mxi (x0 ) will be sought in the form i i Mzi (x0 ) = Fz ( ) ; Mxi (x0 ) = 1 0 ( i )2 Fxi ( i ) (32) 1 0 ( i )2

L

+

=

x

0

wi

hi :

1

p

1 0  2 f ( )d =

1

0

1

 (1 0 n2 )f (n ) L + 1 n=1

(34)

f ( ) d =  L f (t ) L n=1 n 1 0 2

(35)

p

1

0

6

Jx

(m; i) =

Ez

( ) =

Hz

( ) =

S

0

(36)

(m; i) =

q =1

wq

1

L

L n=1

i wi n )

1(

0

Jm+1 ( i wi n ) Fxi (n )

(40)

wi  L J ( w t )F i (t ) L n=1 m i i n z n

(41)

respectively. Equation (40) and (41) result from (24) and (28) by applying the rules (34) and (35), respectively. C. Far-Scattered Field In polar coordinates (; ') the z -components of the far scattered field in region 0 and of the total far-field in region a have the following asymptotic expressions. 1) Region 0: Ez0 (; ') = 2j 0 e0j  sin 'Ez ( 0 cos ') (42)

0

0



2!0

2j 0



e

0j



Hz

( 0 cos ')

(43)

where, for  = 0 cos ', see (44) and (45) at the bottom of the page.

L

1 wq (1 0 n2 )Fxq (n )ej 2(L + 1) q=1 n=1

S

i = Hxexc (m ): (39)

wi  L (1 0 2 ) J m n 2 L + 1 n=1

Hz0 (; ') =

tn = cos (2n 0 1) ; n = cos n 2L L+1

iq Fxq (n )Qmn

n=1

The discrete counterparts of these IEs can be obtained by simply substituting in each of them Jx6 and Jz by

Jz

L

L

B. Discretization of (22), (27) and (29)

(33)

Substitute from (32) into (19) and (20). Then, working in the framework of the Nyström method of [14], any singular integrals encountered can be analytically treated as in [11] (see, also, [15]), and any regular integrals can be computed via the Gauss-Chebyshev rules [13]

(38)

iq iq iq iq , Lmn , Pmn The matrix elements Kmn , and Qmn have simple closed form expressions, omitted here for brevity.

A. Discretization of (16) and (17)

where

iq Fzq (tn )Pmn +

6

i = x i wi

i = Hzexc (xm )

iq Fxq (n )Lmn

q =1 n=1

H

0

n=1

1

S

IV. SOLUTION BY THE NYSTRÖM METHOD

0

q =1

iq Fzq (tn )Kmn

0

(31)

whereas U mi and V mi result from U mi and V mi , respectively, after re(2) (2)0 placing Hm (1) and Hm (1) in the right side of (30) and (31) by Jm (1) (2)0 0 and by Jm (1). The primes in Hm (1) and Jm 0 (1) denote derivatives with respect to the argument.

where

L

1

(30)

0

0

S

jkz nain J ( w  ) bni Jn ( i wi m ) + j!"2 i

i n=0

i n=1 wi m n i i m

0

0

bni Jn ( i wi tm ) + L

Hm(2) ( a Ri ) !a Hm(2) ( a Ri ) (2) mk 0!"a Hm ( a Ri ) Hm(2) ( a Ri )

R mk H Hm(2) ( i Ri ) !i Hm(2) ( i Ri ) V mi = j R (2)

i 0!"i Hm ( i Ri ) mk Hm(2) ( i Ri ) R H U mi = j

a

(37)

This yields the following discrete counterparts of (16) and (17) for m = 1; 2; . . . ; L and for i = 1; 2; . . . ; S :

1

q=1

0

(

"

1 0

mi

2

Aqn Fmn (q; i) n=1 + (q; i) Bnq Fmn n=0

S

Fzq (xn )ejx

+

kz

L+1

cos '

(44) L

(1 0 n2 )Fxq (n )ej

n=1

:

(45)

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 2, FEBRUARY 2012

of the relative error of M and M versus L n , w  = , R  = , h :  , " ,  ' = , GHz.

Fig. 4. log when S=2,

f

1175

(0)

=

4

=

2

(0)

= 1 5

= 2

=

=

=

=

4

= 10

mm w :  R

=

Eza (; ') =  2j  e0j a 1

1

n') A1n +

Hza (; ') =  2j  e0j a 1

1

where



sin(

n=1

1

S

Aqm G0q (m; n)

q=2 m=1

!t = when S ,  :  " , ' = , =

2

= 3

= 10

=

=

=

2

(46)



n') Bn1 + "2n

cos(

n=0

: h

Fig. 5. Snapshot of the near magnetic field at , 2 = 1 5 , = 0 8 , = 1 75 , and = 2. 3

2) Region a:

G6q (m; n) = Jn0m ( a hq )

6

1

S

Bmq G+q (m; n)

q=2 m=0 m

(01)

Jm+n ( a hq )

(47)

(48)

The transverse (to z -axis) field components in regions 0 and a can be obtained via (14) and (15). V. NUMERICAL RESULTS AND DISCUSSION In the following it is assumed, unless otherwise specified, that E0 = V=m, ("a ; a ) = ("0 ; 0 ), and that the semi-cylinders have identical geometrical and physical parameters: "i = "r "0 , i = 0 , wi = w , Ri = R, i = 1; 2; . . . ; S . Only the symmetric case, hi = (i 1)h (i = 1; 2; . . . ; S ), wherein the distance between any two consecutive slots is the same, equal to h, will be considered. In obtaining numerical results, 1

0

the size of the infinite linear algebraic system derived in the preceding section is truncated to the finite value 2LS + 4nmax S by retaining nmax terms in each of the infinite series in (9)–(12) (and, similarly, in (38), (39)). The correctness of the results has been tested by following the criterion of energy balance. Moreover, in the special case S = 1 our results were found to coincide with those of [10]. For increasing L (the number of points of the Gauss-Chebyshev rules used in evaluating the matrix elements) and for nmax = L, Fig. 4 shows the logarithm (base 10) of the relative errors jMp1 (0) 0 Mp1;asym (0)j=jMp1;asym(0)j (p = x; z) at the center x = 0 of the first slot when S = 2. Here Mz1;asym (0) and Mx1;asym (0) are the values to which Mz1 (0) and Mx1 (0) settle down for sufficiently large L (and nmax ). Apparently, the convergence is exponential. (Note: Our computations in this example show that jMz1 (0)j and jMx1 (0)j settle down to their final values 0.4712579942343935 and 0.7371905396420925, respectively, for L = nmax  22; these asymptotic values can be treated as exact values.) The fast convergence of the algorithm enables one to very accurately evaluate both the near-field and the far-field. As an example, for S = 3, Fig. 5 shows a typical snapshot of the total magnetic field at !t = =2 both inside and around the dielectric semi-cylinders when 0 = 3mm,

Fig. 6.  versus R when S , = , = , and " 0j : . = 1

2

=

2

= 10

0 5

= 3

mm, w

:  , '

= 0 25

=

=

w = 1:50 , R = 0:80 , h = 1:750 , "r = 10, 0 = '0 = =2, and

= =2 (TMz case). As seen, the field is strongly localized inside the 2

dielectric resonators. The absorption efficiency abs will be defined by 1 Pabs abs = 2wS inc S

(49)

where 2wS is the total width of the slots, Pabs is the (per unit length of z-axis) power absorbed by the lossy material inside the dielectric semi-cylinders, and S inc = E02 =2Z0 is the power associated with the incident wave. Typical results indicating the connection of abs to several physical and geometrical parameters of the structure are presented in Figs. 6–8. Fig. 6 shows abs versus R when S = 1, 0 = 3 mm, w = 0:250 , 0 = '0 = =2, = =2, and "r = 10 0 j 0:5. The observed standing wave pattern is due to reflections at the boundary of the cylinder. Both the amplitude of the wave and the power which eventually reaches the boundary decrease with increasing R owing to attenuation (skin effect). This means that, for sufficiently large R, the power which arrives at the boundary becomes negligibly small, i.e., the entire power transmitted inside the cylinder is being absorbed and, thus, abs reaches a core value. For S = 2, Fig. 7 shows how abs varies with the distance h between the two slots. The dashed straight line about which abs oscillates pertains to the case S = 1 of a single slot. As expected, the amplitude of the observed standing-wave-like pattern is large for small h, due to strong interactions between the cylinders, but it gradually diminishes as h increases. For several values of S , Fig. 8 shows how the absorption efficiency is affected by possible uncertainty of the complex permittivity value. As seen, abs can be effectively controlled by properly selecting S and "r .

1176

Fig. 7.  :  ,

05

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 2, FEBRUARY 2012

versus h=R when S = 2,  = 3mm, w = 0:25 , R = = ' = =2, = =2, and " = 10 0 j 0:5.

[11] J. L. Tsalamengas, “Exponentially converging Nyström methods for systems of singular integral equations with applications to open/closed strip- or slot-loaded 2D structures,” IEEE Trans. Antennas Propag., vol. 54, no. 5, May 2006. [12] J. L. Tsalamengas and E. C. Pitsavos, “Diffraction of plane waves by a finite array of dielectric-loaded cavity-backed slots on a common ground plane for oblique incidence and arbitrary polarization,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 1070–1079, Apr. 2004. [13] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1972. [14] R. Kress, Linear integral equations. Berlin: Springer-Verlang, 1989. [15] A. A. Nosich and Y. V. Gandel, “Numerical analysis of quasioptical multireflector antennas in 2-D with the method of discrete singularities: E-wave case,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 399–406, Feb. 2007.

Analysis of Radiation Characteristics of Conformal Microstrip Arrays Using Adaptive Integral Method Wei-Jiang Zhao, Le-Wei Li, Er-Ping Li, and Ke Xiao

 versus Re(" ) for S = 1; 2; 5; 10 when  = 3 mm, w = 0:25 , R = 0:5 , h = 1:5 ,  = ' = =2, = 0, and Im(" ) = 00:5.

Fig. 8.

VI. CONCLUSION A computationally efficient Nyström method for analyzing oblique scattering of arbitrarily polarized waves by an array of slots loaded by dielectric semi-cylinders has been presented. Filling the system matrix requires no numerical integration. The algorithm converges exponentially and, thus, extremely accurate results may be obtained both for the near and far fields.

Abstract—A new surface integral equation formulation is presented for characterizing electromagnetic radiation by conformal microstrip arrays on finite curved bodies of arbitrary shapes. The surface equivalence principle is used to reduce the original problem to two equivalent problems, one for the external medium and another for the internal medium. Electric field integral equations are applied to the conducting surfaces, and weighted sums of the field integral equations corresponding to the external and internal dielectric regions with appropriate weighted coefficients are applied to the dielectric interface. The integral equations are solved via the method of moments (MoM) procedure, to which the memory requirement and computational complexity pertinent is reduced by employing the adaptive integral method (AIM). Numerical results are presented to demonstrate the validity and accuracy of the method. Index Terms—Antenna arrays, antenna radiation patterns, conformal antenna, moment method, patch arrays.

REFERENCES [1] J. L. Tsalamengas, “TE/TM scattering by a slot on a ground plane and in the presence of a semi-cylindrical load,” J. Electromag. Waves Applicat., vol. 8, no. 5, pp. 613–646, 1994. [2] R. A. Hurd and B. K. Sachdeva, “Scattering by a dielectric-loaded slit in a conducting plane,” Radio Sci., vol. 10, no. 5, pp. 565–572, May 1975. [3] A. Z. Elsherbeni and H. A. Auda, “Electromagnetic diffraction by two perfectly conducting wedges with dented edges loaded with a dielectric cylinder,” Proc. Inst. Elect. Eng., vol. 136, no. 3, pt. H, pp. 225–234, Jun. 1989. [4] T. J. Park, H. J. Eom, W.-M. Boerner, and Y. Yamaguchi, “TM scattering from a dielectric-loaded semi-circular trough in a conducting plane,” IEICE Trans. Commun., vol. E75-B, no. 2, pp. 87–91, Feb. 1992. [5] T. J. Park, H. J. Eom, Y. Yamaguchi, W.-M. Boerner, and S. Kozaki, “TE plane wave scattering from a dielectric-loaded semi-circular trough in a conducting plane,” J. Electrom. Waves Applicat., vol. 7, no. 2, pp. 235–245, 1993. [6] M. A. Kolbehdari, H. A. Auda, and A. Z. Elsherbeni, “Scattering from a dielectric cylinder partially embedded in a perfectly conducting ground plane,” J. Electrom. Waves Applicat., vol. 3, no. 6, pp. 531–554, 1989. [7] M. K. Hinders and A. D. Yaghjian, “Dual-series solution to scattering from a semicircular channel in a ground plane,” IEEE Microwave Guided Wave Lett., vol. 1, pp. 239–242, Sep. 1991. [8] B. K. Sachdeva and R. A. Hurd, “Scattering by a dielectric-loaded trough in a conducting plane,” J. Appl. Phys., vol. 48, no. 4, pp. 1473–1476, Apr. 1977. [9] A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering. Englewood Cliffs, NJ: Prentice Hall, 1991, pp. 317–318. [10] I. O. Vardiambasis, J. L. Tsalamengas, and J. G. Fikioris, “Plane wave scattering by slots on a ground plane loaded with semicircular dielectric cylinders in case of oblique incidence and arbitrary polarization,” IEEE Trans. Antennas Propag., vol. 46, no. 10, pp. 1571–1579, 1998.

I. INTRODUCTION In many practical applications, antenna platforms are nonplanar and antennas are required to conform to the surfaces of curved platforms for structural reasons [1]. In general, an array is regarded as a conformal array only if it is comparable or large with respect to the radius of curvature of the mounting surface [2], otherwise it behaves nearly like Manuscript received January 20, 2011; revised May 03, 2011; accepted July 15, 2011. Date of publication October 24, 2011; date of current version February 03, 2012. This work was supported in part by the Agency for Science Technology and Research (A*STAR) via a SERC Aerospace Project (No. 0921550102) and in part by the National Natural Science Foundation of China under Grants 61072020 and 61171046. W.-J. Zhao is with the Department of Electronics and Photonics, Institute of High Performance Computing, A*STAR, Singapore 138632, Singapore (e-mail: [email protected]). L.-W. Li is with Institute of Electromagnetics and School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China (e-mail: [email protected]). E.-P Li is with the Department of Electronics and Photonics, Institute of High Performance Computing, A*STAR, Singapore 138632, Singapore and also with Zhejiang University, Hangzhou 310058, China (e-mail: [email protected]). K. Xiao is with the School of Electronic Science and Engineering, National University of Defense Technology, Hunan, China. Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173135

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a planar one and its characteristics can be approximately determined from its corresponding planar array. Among the most suitable antennas for conformal applications are microstrip patch antennas due to their low profile. The integral equation technique based on the Green’s functions of an infinite ground dielectric slab [3] is one of the popular analysis techniques for microstrip patch antennas, which has an assumption that the dielectric substrate and the ground plane are infinite in extent. It can yield, in most cases, reasonably accurate results for impedance behaviors since the impedance is primarily determined by the patch. However, it may fail to provide accurate results for radiation characteristics because the radiation behavior may be severely affected by the finite size and shapes of the substrate. Some integral equation techniques via three-dimensional (3-D) modeling of the dielectric substrates were proposed for rigorous treatment of finite microstrip structures, where the dielectric substrate can be modeled by either the volume integral equation (VIE) [4], [5] or the surface integral equation (SIE) [4], [6], [7]. In general, the SIE approach is more efficient than the VIE approach for homogeneous dielectric problems [8] because it uses a smaller number of unknowns. An EFIE-PMCHWT formulation originally proposed in [9] for conductors with partial dielectric coatings was successfully applied to the analysis of electromagnetic (EM) radiation from circular microstrip antennas excited by a dipole source [7], EM scattering by combined conducting and dielectric structures of arbitrary shapes [10], and EM radiation from arbitrary-shaped microstrip antennas excited by a localized voltage source [11], where the integral equations are solved by the method of moments (MoM) [12]. However, the formulation used in [11] has poor convergence performance which makes it difficult to be used for solving the problems involving electrically large structures. In [11], only electrically small structures were considered, hence the Gaussian elimination process can be employed to solve the resulting matrix system, and this is the reason that the poor convergence of the formulation did not cause concern. In this communication, an innovative formulation based on SIE is presented for accurately predicting the radiation characteristics of conformal patch arrays. Weighted sums of the field integral equations in the formulation were calculated, corresponding to the external and internal dielectric regions with appropriate weighting coefficients on the dielectric interface. The convergence achieved in this communication is found to be better than that of the EFIE-PMCHWT formulation used in [7], [10], [11]. It has a smaller number of unknowns than that in [4], [6], it avoids singularity due to the overlapping of conducting and dielectric surfaces, and it is free of the interior resonance. The integral equations are solved using the MoM, and the adaptive integral method (AIM) [13], [14] is employed to accelerate numerical solutions in the MoM. II. SIE FORMULATION FOR CONFORMAL PATCH ARRAYS Consider a conformal microstrip patch array on a curved dielectric substrate of arbitrary shapes, which is modeled by a hybrid structure of conducting and dielectric materials as shown in Fig. 1(a). The regions exterior and interior to the dielectric substrate are characterized by medium parameters ("1 ; 1 ) and ("2 ; 2 ), respectively. The surfaces Sc1 , Sc2 and Sd represent, respectively, the interfaces between the conductor and free space, the conductor and the dielectric material, and the dielectric region and free space. The excitation considered here is from either a plane wave for scattering problems or a localized voltage source on the conducting patch for radiation problems. The sources of the excitation are provided by the impressed electric and magnetic currents, J i1 or M i1 in free space and J i2 and M i2 inside the dielectric region. According to the equivalence principle, the problem shown in Fig. 1(a) can be solved by considering its two equivalent problems, i.e., exterior and interior equivalences which are shown in Fig. 1(b) and (c), respectively.

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Fig. 1. Problem definition and its equivalence. (a) Original problem; (b) external equivalence; (c) internal equivalence.

The equivalent electric currents J c1 and 0J c2 are assumed on the mathematical surface Sc1 and Sc2 , respectively, while the equivalent electric current J d and magnetic current M are considered on the mathematical surface Sd . Since tangential components of electric and magnetic fields on a dielectric interface are continuous, the electric and magnetic currents on the opposite side of Sd are hence known as 0J d and 0M , respectively. By enforcing the continuity of the tangential electric and magnetic field components on surfaces just internal to Sc1 and Sd and just external to Sc2 and Sd , the following fundamental integral equations are obtained:

j 0E ci11 (J i1 ; M i1 ) tan ; on Sc1 (1a) 0 E ds1 (J c1 ; J d ; M )jtan = 0E di1 (J i1 ; M i1 ) on Sd (1b) tan 0 H s1 (J c1 ; J d ; M )jtan = 0H i1 (J i1 ; M i1 ) on Sd (1c) tan E cs22 (0J c2 ; 0J d ; 0M )jtan = 0E ci22 (J i2 ; M i2 ) ; on Sc2 (2a) tan E ds2 (0J c2 ; 0J d ; 0M )jtan = 0; on Sd+ (2b) H s2 (0J c2 ; 0J d ; 0M ) tan = 0; on Sd+ (2c) E sc11 (J c1 ; J d ; M ) tan =

where the subscript “tan” stands for the tangential component, and Sd0 and Sd+ refer to approaching the dielectric surface Sd from outside and the interior region, respectively. The terms on the right-hand side of the above equations are the incident electric and magnetic fields. The terms on the left-hand side of the above equations denote the scattering electric and magnetic fields. The scattered electric field E s and magnetic field H s due to the electric current J and magnetic current M are given by E s (J ) =

0 jk

1

S

E s (M ) =

J (r 0 )+ 2 k

H (M ) = s

H (J ) =

G(r ; r 0 )dS 0

6 21 n^ 2 M + 00M (r0 ) 2 rG(r ; r 0 )dS 0 S

s

rr0s 1 J (r 0 )

0

jk 

M (r 0 )+ 2 k 1

S

rr0s 1 M (r 0 )

(3b)

G(r ; r 0 )dS 0 (3c)

7 2 n^ 2 J 0 00J (r 0 ) 2 rG(r ; r 0 )dS 0 1

(3a)

(3d)

S

where k and  denote the wave number and wave impedance of the medium, respectively, which are either k1 and 1 or

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 2, FEBRUARY 2012

k2 and 2 depending on the medium surrounding the currents, G(r ; r 0 ) = exp(0jkR)=(4R) represents the 3-D scalar Green’s function for the medium, R = jr 0 r 0 j identifies the distance from a ^ represents the unit vector normal source point r 0 to a field point r , n to the surfaces and pointing toward the free space, the bar integral symbol is used to represent the principal value, the positive and negative signs in (3b) are used when E s1 (M ) and E s2 (M ) are computed, respectively, and the negative and positive signs in (3d) are used when H s1 (J ) and H s2 (J ) are computed, respectively. Multiplying (2b) by 1 =2 and substituting the new equation into (1b), and at the meantime multiplying (1c) and (2c) respectively by 1 and 2 and combining the resultant equations, we establish the following two new equations on Sd

E ds1 (J c1 ; J d ; M ) +

1 s2 E (J c2 ; J d ; M ) 2 d =

tern results. By applying the Galerkin MoM procedure, a set of partitioned matrix equations can be written as (6) shown at the bottom of the page, where the first superscript of the impedance matrix elements represents the medium in which the source radiates, the second and third superscripts denote respectively the surfaces on which the testing field points and equivalent sources locate, with c associated with the surfaces Sc1 and Sc2 , and d associated with the surface Sd . The elements of impedance sub-matrices in (6) can be expressed as follows: 1;2ab Zmn

T

+ 1;2ab Ymn

tan

0E id1 (J i1 ; M i1 ) tan

=

01H i1 (J

;M

)

+

: (4b)

J d (r ) = M (r ) =

n =1 N n=1 N n=1

Inc f cn (r )

2dc 2 Dmn

0

0

0

0

dS

(7c)

T

1;2ab 0 Cmn :

(7d)

A 1n;2b (r ) =

f bn (r 0 )G1;2 (r ; r 0 )dS 0

(8a)

T

rs 1 f bn (r )

1;2b 8n (r ) =

0

0

G1;2 (r ; r 0 )dS 0

(8b)

T

with Tm 2 Sa , and Tm 2 Sb . a and b each represents either d or c. The elements of excitation matrices in (6) can be given by

Vmc

=

f cm (r ) 1 E ci11 dS

(9a)

Vmc

=

f cm (r ) 1 E ci22 dS

(9b)

f dm (r ) 1 E di1 dS

(9c)

f dm (r ) 1 H i1 dS:

(9d)

T

(5c)

Vmd

= T

1 Mn f dn (r ):

1dc 1 Dmn

6 21 n^ 2 f bn (r )

00 f bn (r ) 2 r G1;2 (r ; r )dS

(5b)

Ind f dn (r )

0 1dc Zmn

f am (r )

T

Inc f cn (r )

0 2cc Zm n  2dc Z mn 

(7b)

T

+

(5a)

(5d)

It should be noted that the edges at the interface between Sc1 and Sd and the interface between Sc2 and Sd have not been considered. The edges were also discarded in [11]. From the results presented in [11], the discarding of the edges will not affect the RCS and radiation pat-

1cc Zm n

r 1 f am (r )] 8n1;2b(r )dS

[

where

The set of integral equations, (1a), (2a), (4a) and (4b), is first discretized in the MoM procedure. The equivalent surface current distributions, J c1 , J c2 , J d and M , are expanded in terms of the Rao-WiltonGlisson (RWG) basis functions f n [15]. Let Nc1 , Nc2 and Nd represent the total numbers of interior edges of the triangles approximating the surfaces Sc1 , Sc2 and Sd , respectively. Then, we have

J c2 (r ) =

j!1;2

T

1;2ab Dmn =

A. Discretization of Integral Equations

n =1 N

f am (r) 1 A1n;2b (r)dS

1

1;2ab Cmn =

III. NUMERICAL SOLUTION PROCEDURE

J c1 (r ) =

(7a)

T

T

Equations (1a), (2a), (4a) and (4b) form a new formulation set for determining the above unknown currents. When the linear equation system is cast into a matrix form, the diagonal blocks will have well-balanced scales.

N

[

j!"1;2

(4a)

tan

r 1 f am (r )] 8n1;2b(r)dS

1

= j!"1;2

1 H s1 (J c1 ; J d ; M ) + 2 H s2 (J c2 ; J d ; M )

tan i1 i1

f am (r ) 1 A 1n;2b (r )dS

= j!1;2

1cd Zm n 2cd Zm n

1dd Zmn +

 

2dd Zmn

Hm

= T

1dd [Zmn +

2dd 1dd 2dd 1 =2 Zmn ] and 1 [1 Ymn + 2 Ymn ] in (6) are equal, hence well-balanced scales of the diagonal blocks in the equation are achieved.

1

1cd 1 Cm n 2cd 1 Cm n  1dd 2dd Cmn +  Cmn

 1dd 2dd 1dd 2dd 1 Dmn +  Dmn 1 1 Ymn + 2 Ymn m1 m2 = 1; 2; 1 1 1 ; Nc1 ; = 1; 2; 1 1 1 ; Nc2 ; n1 n2

1

c [In ] c [In ] Ind [Mn ]

m n

=

c [Vm ] c [Vm ] Vmd

[1 Hm ] = 1; 2;

1 1 1 ; Nd

(6)

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 2, FEBRUARY 2012

Fig. 2. Bistatic radar cross section of a conducting cylinder capped by a dielectric cone, at   -polarization.

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Fig. 3. Comparison of the convergence behavior of different formulations.

B. Application of AIM The memory requirement and computational complexity for solving integral equations using an iterative MoM solver are O (N 2 ) and O(N 2 ) per iteration, respectively, where N is the number of unknowns. Such a memory requirement and computational complexity is too expensive for designing and characterizing electrically large conformal microstrip arrays. Hence the AIM is employed in this communication to make the solutions more efficient. The implementation of AIM is quite similar to that presented in [14]. For dielectric involved problems, the computational complexity and memory requirement of AIM for surface scatterers are less than O(N 1:5 log N ) and O(N 1:5 ), respectively [14]. C. Input Impedance Transmission lines are used to feed the antenna patches. The feeding lines are modeled with only one RWG edge element per line width, and a voltage source is applied across the delta gap (a gap with small width) in the feeding edge (assumed as the j th edge on Sc2 ). Let V denote the voltage across the gap. The element of excitation matrices in (6) is equal to lj V for m2 = j and 0 otherwise. The input impedance is the ratio of the feeding voltage V to the total current normal to the feeding edge given by lj Ijc , where lj is the length of the feeding edge. Once Ijc is solved, the antenna input impedance can be obtained directly by

Z

in

=

V lI : j

c j

(10)

IV. NUMERICAL RESULTS Numerical examples are considered in this Section to demonstrate the accuracy and capability of the proposed method. A testing case which was used in [6] is revisited firstly. The geometry is a dielectric cone placed on the top of a conducting cylinder, as shown in Fig. 2. The cylinder and cone have the same radius of 0:3 and the same height of 0:6, where  is the wavelength in free space. The relative permittivity of the dielectric cone is assumed to be 2. The structure is illuminated by a plane wave which is propagating from the tip of the cone toward the cylinder with the incident electric field along the +x-axis direction. The bistatic radar cross section (RCS) in the XOZ plane for   -polarization versus the polar angle  is calculated using the presented method and its values are shown in Fig. 2. The cylinder and cone without its top face are modeled by 442 and 234 triangular patches, respectively. The result obtained with a different formulation [6] is also plotted in the figure for comparison. A very good agreement is observed

Fig. 4. Bistatic RCS for the structure of a dielectric hemisphere capped by a conducting disk,  polarization.

between our result and the published data in [6]. The number of unknowns needed by the present method is 992, and that needed by other integral equation method introduced in [6] is 1630. To compare the convergence behavior of the EFIE, EFIE-PMCHWT, EFIE-PMCHWTm formulations and the present formulation, EM scattering by a dielectric cylinder capped by a conducting disk is considered, where the EFIE is constituted of (1a), (1b), (2a) and (2b), the EFIE-PMCHWT is formed by (1a), (2a), (1b) + (2b) and (1c) + (2c), and the EFIE-PMCHWTm represents the modified EFIE-PMCHWT by multiplying the magnetic field equation by 1 and using M =1 as unknowns instead of M . The dielectric cylinder has a radius of 0:3 and a height of 0:8, where  is the wavelength in free space. The material with relative permittivity of "r = 6 is considered. The structure is illuminated by a normally incident plane wave with the incident electric field along the +x-axis direction. The normalized residual norm in the generalized conjugate residual (GCR) method as functions of the number of iterations for the four different formulations is plotted in Fig. 3. It is found that the EFIE and EFIE-PMCHWT do not converge, and the present formulation converges faster than EFIE-PMCHWTm. To validate the AIM algorithm, we consider a structure of dielectric hemisphere capped by a conducting circular disk shown in Fig. 4. The hemisphere and disk are of a same radius of 1, where  is the wavelength in free space, and the dielectric material has a relative permittivity of "r = 2. The structure is illuminated by a plane wave incident along the 0z -axis direction with the incident electric field in the +x-axis direction. Two methods, i.e., the MoM and AIM are used for calculating the bistatic RCS of the structure. In the calculation by

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Fig. 6.

H -plane pattern for the 4 2 4 cylindrical microstrip array.

Fig. 5. Geometry of a microstrip patch array mounted on a cylindrical body.

AIM, the expansion order is chosen as 3, the Cartesian grid, extending slightly beyond the structure boundary, has 20 2 20 2 12 nodes, and the near field range is chosen as 0:3. The comparison of the results in the X OZ plane is shown in Fig. 4. It can be seen that the AIM result agrees very well with the MoM result. In the following examples, conformal microstrip arrays on curved bodies are considered. First, radiation patterns are calculated for a cylindrical microstrip patch array of 4 2 4 elements printed on a cylindrical ground body of radius a = 7:6 cm as shown in Fig. 5. The cylindrical substrate is of thickness t = 0:254 mm and relative permittivity "r = 2:94. Each array element has the same dimension L 2 W , where L = 7:2 mm and W = 5:0 mm. The inter-element spacing between the centers of two adjacent elements in the - and z -directions are S and Sz , respectively, and here they are both chosen as 15 mm. The ground plane is a part of cylindrical surface with a length of Lg = 90 cm and width of ag , where g = =2. Each element is connected with a feed line around the center of its bottom edge. Each feed line, with a length of 3.6 mm and width of 1 mm, has a feed edge on it which is parallel to and at a distance of 0.9 mm away from its bottom edge. A 1-V source is placed across a delta gap in each feed edge. In such case, the array is an axial array. The patches with ground plane and the dielectric surface are modeled by 5232 and 4522 triangular patches, respectively. In the calculation by AIM, the expansion order is chosen as 3, the Cartesian grid has 14 2 50 2 42 nodes, and the near field range is chosen as 0:4 which is slightly larger than that used for scattering problems. Fig. 6 shows the calculated radiation pattern in X OY plane at 16.2 GHz. Also shown in the figure is the measurement result for a very similar model previously considered in [16], where the ground plane is a complete cylindrical surface. The different simulation and measurement structures will lead to quite different radiation patterns in the backward direction (90 –270 ), but may yield similar radiation patterns in the forward direction. In Fig. 6, the radiation patterns in the forward direction are compared. The agreement is reasonable considering the measurement error.

Fig. 7. Directive gain patterns for the 4 - and -planes.

H

E

2 4 microstrip array at 16.2 GHz in

After validating the method, curvature effects on the radiation patterns and directivities of conformal microstrip arrays on curved bodies are studied. Cylindrical microstrip arrays with three different radii of 4.6 cm, 7.6 cm, and infinite (for the planar case) are considered. The geometries of the arrays and most geometrical parameters except for the radius are the same as those of the previous examples. H -plane (X OY plane) and E -plane (X OZ plane) directive gain patterns for the arrays at 16.2 GHz are calculated, and their results are shown in Fig. 7(a) and (b), respectively. It is observed that the directivities in

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 2, FEBRUARY 2012

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REFERENCES

Fig. 8. Vertical directive gain patterns (in the E -plane) of the 16-element microstrip patch array on conical and hemispherical bodies at 8 GHz.

both planes are affected by the curvature and they will decrease as the curvature increases. The three radiation patterns are almost the same in the E -plane, but apparently different in the H plane. This indicates that the curvature effects on radiation patterns of cylindrical arrays vary significantly on different cutting planes. Finally, two conformal microstrip arrays on conical and hemispherical bodies are considered. For some practical applications of conformal arrays on bodies of revolution, an omnidirectional pattern in the roll plane is often required. This type of radiation patterns can be produced by either wrapping a microstripline around the circumference and feeding it at a number of points evenly distributed along the circumference or using a number of discrete radiators arrayed along the circumference and excited with the same amplitude. The arrays considered have 16 patches distributed along the circumference with an even inter-element spacing of 22.5 and excited uniformly. Such arrays can provide an omnidirectional pattern in the roll plane. The central angles corresponding to the widths of the patches are 13.86 for conical array and 12.09 for the hemispherical array. The z -coordinates for the bottom and top edges of the patches are respectively 2.6414 cm and 3.8538 cm for the conical array, while they are respectively 2.6203 cm and 3.8387 cm for the hemispherical array. Both the conical surface and the hemispherical surface have the same radius of 7.5 cm. Their substrates are of thickness t = 1:4 mm and relative permittivity "r = 2:94. Vertical directive gain patterns (in the E -plane) are calculated for the two arrays at 8 GHz, as shown in Fig. 8. The two patterns, appearing as difference patterns, have similar main lobes but quite different side lobes. V. CONCLUSIONS An efficient approach has been developed in this communication for accurate characterization of conformal patch arrays on finite curved bodies of arbitrary shapes. The numerical results of far-field radiation patterns are obtained and compared with published experimental data, and a good agreement is observed and it serves as a good validation of the presently derived formulation and in-house developed codes. The present formulation exhibits better convergence behavior compared to the well-known EFIE-PMCHWT formulation. In the application of the AIM to antenna problems, to guarantee the solution accuracy, the near field range must be chosen as at least 0:4 for radiation problems, while it can be as small as 0:3 for scattering problems. The application of the present approach to the mutual coupling analysis of conformal patch arrays on singly and doubly curved surfaces will be considered in the future work.

[1] Conformal Antenna Array Design Handbook, R. C. Hansen, Ed. Alexandria, VA: NTIS, 1981, AP-A110091. [2] J. Ashkenazy, S. Shtrikman, and D. Treves, “Conformal microstrip arrays on cylinder,” IEE Proc., vol. 135, pt. H, pp. 132–134, Apr. 1988. [3] J. R. Mosig, R. C. Hall, and F. E. Gardiol, “Numerical analysis of microstrip patch antennas,” in Handbook of Microstrip Antennas, J. R. James and P. S. Hall, Eds. London: Peter Peregrinus, 1989, ch. 8. [4] T. K. Sarkar, S. M. Rao, and A. R. Djordjevic, “Electromagnetic scattering and radiation from finite microstrip structures,” IEEE Trans. Microwave Theory Tech., vol. 38, no. 11, pp. 1568–1575, Nov. 1990. [5] K. Y. See and E. M. Freeman, “Rigorous approach to modelling electromagnetic radiation from finite printed circuit structures,” IEE Proc.Microw. Antennas Propag., vol. 146, no. 1, pp. 29–34, Feb. 1999. [6] S. M. Rao, T. K. Sarkar, P. Mydia, and A. R. Djordjevic, “Electromagnetic radiation and scattering from finite conducting dielectric structures: Surface/surface formulation,” IEEE Trans. Antennas Propag., vol. 39, pp. 1034–1037, Jul. 1991. [7] A. A. Kishk and L. Shafal, “Different formulations for numerical solution of single or multibodies of revolution with mixed boundary conditions,” IEEE Trans. Antennas Propag., vol. 34, pp. 666–673, May 1986. [8] I. Chiang and W. C. Chew, “A coupled PEC-TDS surface integral equation approach for electromagnetic scattering and radiation from composite metallic and thin dielectric objects,” IEEE Trans. Antennas Propag., vol. 54, pp. 3511–3516, Jul. 2006. [9] J. R. Mautz and R. F. Harrington, “Boundary formulation for aperture coupling problem,” Arch. Elek. Übertragung, vol. 34, pp. 377–384, Apr. 1980. [10] J. Shin, A. W. Glisson, and A. A. Kishk, “Analysis of combined conducting and dielectric structures of arbitrary shapes using an E-PMCHW integral equation formulation,” in Proc. IEEE AP-S Int. Symp., 2000, vol. 3, pp. 2282–2285. [11] W. J. Zhao, L. W. Li, and K. Xiao, “Analysis of electromagnetic scattering and radiation from finite microstrip structures using an EFIEPMCHWT formulation,” IEEE Trans. Antennas Propag., vol. 58, pp. 2468–2473, Jul. 2010. [12] R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968. [13] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci., vol. 31, pp. 1225–1251, Sep.–Oct. 1996. [14] W. J. Zhao, L. W. Li, and Y. B. Gan, “Efficient analysis of antenna radiation in the presence of airborne dielectric radomes of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 53, pp. 442–449, Jan. 2005. [15] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, pp. 409–418, May 1982. [16] K. L. Wong and G. B. Hsieh, “Curvature effects on the radiation patterns of cylindrical microstrip arrays,” Microw. Opt. Tech. Lett., vol. 18, pp. 206–209, Mar. 1998.

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Generalized Multilevel Physical Optics (MLPO) for Comprehensive Analysis of Reflector Antennas Christine Letrou and Amir Boag

Abstract—Recent developments of the multilevel physical optics (MLPO) algorithm aiming at the comprehensive analysis of complex reflector antenna systems are presented. The physical theory of diffraction (PTD) line integral along the rim of a reflector is combined with the physical optics (PO) surface integral within the multilevel algorithm. The multilevel scheme is also generalized to combine fields radiated by various components of different sizes, as encountered in complex antenna systems with multiple feeds and/or reflectors. Comparison with published results demonstrates the ability of the MLPO algorithm to cope accurately and efficiently with realistic reflector antenna problems. Index Terms—Fast algorithms, physical optics, physical theory of diffraction, radiation pattern, reflector antennas.

I. INTRODUCTION The physical optics (PO) approximation provides an attractive computational tool for the analysis of large reflector antennas [1]. The PO combined with the PTD (e.g., in the form of incremental length diffraction coefficients [2], [3]) often strikes a balance between the computational burden and accuracy requirements for the reflector antenna analysis. The PO-PTD combination facilitates uniformly accurate evaluation of the co- and cross-polarized radiation patterns, including the far sidelobe regions. Numerically rigorous techniques such as the method of moments, though more accurate, are considerably more computationally demanding and, therefore, mostly employed for small and moderately sized antennas. Combining or hybridizing a method of moments technique with a PO-type surface integration is often the preferred method to address reflector antenna problems involving evaluation of the current on the reflector surface [4]. In contrast, computationally inexpensive geometrical theory of diffraction (GTD), often used to compute fields in far-out sidelobes, is subject to well-known limitations due to the presence of caustics [5], [6] and to the possibly large number of diffraction points in the case of shaped reflectors or reflectors with irregular edges. Also, GTD is valid only when edges are illuminated from the far field, which is not always the case in multireflector antennas. Hence GTD computations of cross-polarized patterns and far sidelobe levels are not considered accurate enough in the case of high performance reflector antennas [7]. A modified PO formulation recently proposed in [8] and [9] promises to describe the diffraction effects while performing only PO type surface integrals with normal vector directions modified based on the observation direction. This approach however is yet to be fully developed for arbitrary three-dimensional geometries. Manuscript received October 29, 2010; revised May 27, 2011; accepted July 20, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported in part by a grant from the Ministry of Science and Technology, Israel, and from the Ministry of Research, France. C. Letrou is with the Institut Télécom, Télécom SudParis, CNRS Lab. SAMOVAR, Evry, France (e-mail: [email protected]). A. Boag is with the School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173118

The combined PO/PTD approach does not suffer from the above limitations, but for large reflectors and wide angle patterns, the straightforward evaluation of the pertinent integrals for a wide range of observation directions is inefficient due to its high computational complexity. This computational burden can pose a significant limitation in situations such as reflector shaping and optimization [10], as well as multibeam multifrequency systems, where repeated evaluation of antenna characteristics is required. The fast Fourier transform (FFT) facilitates numerically efficient evaluation of radiation integrals, but only for planar apertures. The MLPO was introduced in [11] in order to reduce the complexity of evaluating the PO integrals over arbitrary shaped surfaces to a level comparable to that of the FFT-based techniques. The efficacy of the MLPO approach for antenna analysis has already been demonstrated in the case of simple PO analysis (surface integrals only) of lens and reflector antennas. In this communication, we show how the MLPO algorithm can be generalized for the efficient computation of wide angle radiation patterns, accommodating both diffraction and spill over effects. For the sake of simplicity, we present the new algorithmic developments for the case of a single reflector antenna system. In Section II, we formulate the problem under study reducing it to the evaluation of PO and PTD integrals. The presentation of the generalized MLPO algorithm in Section III starts with an outline of the basic multilevel approach and proceeds with the computation of elemental sub-patterns including the PTD contribution followed by a hierarchical aggregation of reflector sub-patterns and additional contributions, such as the feed radiation, into the final pattern. A numerical example is worked out in Section IV to demonstrate the main features of the proposed approach. II. PROBLEM SPECIFICATION Consider a PO-based computation of the radiation pattern of an idealized reflector antenna comprising a primary feed and a single reflector r ) in direction r^ as: surface. We define an antenna far field pattern U (^

U (^r ) = 4r ejkr E (r)

r!1

(1)

where k is the wavenumber and E (r ) is the far electric field radiated by the antenna at observation point r = rr^. In order to compute the reflector antenna wide angle pattern, the PO surface integral contributions have to be augmented with those of the PTD line integral along the reflector rim and further combined with the primary feed pattern. The resulting expression for the far field pattern radiated by the single reflector antenna system is then of the form:

U (^r ) =

A(^r ; r s )ejkr^ r ds + 1

S

D (^r ; r c )ejkr^ r dzl 1

C

U r ejkr^ r

+ f (^)

1

(2)

where S and C denote the reflector surface and rim contour, respectively. The elemental surface contribution A(^ r ; r s ) is related to the r ; r c ) stands for the increequivalent currents on the surface and D (^ r ) is mental length diffraction coefficients (ILDCs). Also in (2), U f (^ the feed pattern in the feed centered coordinate system. It can be either known analytically or obtained via measurement, or from a separate numerical analysis of the feed system. Also, r s denotes a point on the reflector surface S , r c a point on the reflector rim C , and r f the position of the primary feed. These position vectors are defined in the same coordinate system, called the “observation” coordinate system. The fields over surface S are assumed to be known thanks, e.g., to a known incident field and local impedance boundary conditions. In the following, we assume that S is a perfect conductor. Then:

A(^r ; r s ) = j 2kr^ 2 [^r 2 (^n(r s ) 2 H f (r s ))]

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where  = 120 is the intrinsic wave impedance, while H f (r s ) and n ^(r s ) denote, respectively, the incident magnetic field produced by the feed and the outward unit vector normal to S , both at point r s on the surface. r ; r c ) is the far field pattern of the ILDCs given Along the rim, D (^ in [2], [3], normalized to the elemental length dzl :

D (^ r ; r c ) = D (TM) + D (TE) with

2 sin sin l = Ez sin  l cos  + sin = 0 sgn( 0  l ) Hz sin1 l 1 2  cos  + sin sin

Fig. 1. Reflector surface decomposition along polar coordinates in the projection plane.



D (TM) D

(TE)

2

f

2

0

2

0



^l

the surface integrals of the equivalent currents (function A(^ r ; r s ) defined in (3)) and the line integrals of ILDC contributions for patches situated along the rim C , as defined in (4). This integration phase is thus performed for each patch (n = 1; . . . ; NsM ) in its self-centered M coordinate system (i.e., with the origin at point rn for patch SnM ), yielding:

2

f

0

2

2

 2

2 sin l ^l 0 cos l cos l + 1 + 2 sin l cot( l ) ^l

2 cos 2 l sin 2 0

0

U n (; ) = e0jkr 1r M

(4)

where zl is the curvilinear coordinate along the rim. A local coordinate ^l ; y^l ; z^l ) is defined with Ol (r c ) on the rim, z^l tangent to system (Ol ; x ^l orthogonal to z^l in the plane tangent to the reflector the rim, and x surface at Ol . Ezf and Hzf are the zl components of the incident (feed produced) electric and magnetic fields, respectively, in this local coordinate system. The local spherical angular coordinates of the observation direction are denoted (l ; l ) and those of an incident (locally) plane wave are (0l ; 0l ). Also, in (4),

 = cos01

sin l sin  l cos l 0

;

0  l < 2:

In general, full characterization of the far field pattern requires computation of O(Na2 ) of its samples. Here, Na = kRa provides a measure of the antenna electrical size with Ra being the radius of the smallest sphere enclosing the whole antenna system. Complexity of the direct evaluation of (2) for O(Na2 ) observation directions is of O(Na4 ). This estimate is dominated by the cost of calculating the PO surface integral. However, the complexity of directly computing the PTD contour integral is of O(Na3 ) and, therefore, must also be addressed. Our goal is to reduce this overall computational cost to O(Na2 log Na ), which is comparable to that of the FFT-based techniques that are used for planar apertures. III. GENERALIZED MLPO ALGORITHM The generalized PO-PTD based numerical scheme is developed by extending the original MLPO approach. In a preprocessing phase, the reflector surface is hierarchically subdivided into subdomains: surface subdomains (also called “patches”) at level L are denoted by SnL , with n = 1; . . . ; NsL . Also, we denote rnL and RnL the center and the radius of the smallest sphere circumscribing patch SnL , while RL = maxn fRnL g. If a binary subdivision scheme is used along each of the two coordinates spanning the surface, a parent patch of level L is subdivided into four patches of level L +1, called its “children.” An example of a binary subdivision in polar coordinates of the reflector projection plane is shown in Fig. 1. The subdivision process is stopped at level L = M when RM is of the order of the wavelength. The multilevel algorithm starts with the PO surface integration over each of the elemental patches of level M . These elemental subpatterns involve both

^ 

2

A(^ r ; r s )ejkr 1r ds + ^

S

D (^ r ; r c )ejkr 1r dzl : (5) ^

S

\C

Due to the size of these patches (kRM is of O(1)), these radiation patterns are fully described by sampling the directions of observation very coarsely, according to Property 3 in [12]. The integrals are thus evaluated for a very sparse grid comprising O(1), i.e., a small fixed number of directions. The number of quadrature points needed for surface and line integrals over each level M patch is also of O(1). The computational complexity of evaluating the integrals for each level M patch is then of O(1), and the total computational complexity of evaluating the surface and line integrals via (5) for all level M patches is of O(Nr2 ), where Nr = kRr denotes the electrical size of the reflector. Here, Rr denotes the radius of the smallest sphere circumscribing the whole reflector surface. It is noteworthy that the additional cost due to the evaluation of the PTD line integrals scales as O(Nr ), and is expected to be quite small compared to that of the PO surface integrals. The remainder of the algorithm involves multilevel aggregation of subdomain radiation patterns [11]. At each level from M to 1 during the aggregation phase, the “children” patterns must be interpolated prior to aggregation, due to the need to increase the grid density with increasing subdomain size from level L to level L 0 1. Thereafter they must be expressed in their “parent” patch coordinate system: the coordinate sysL points are translated to the coorditems SnL with their origins at the rn L01 L01 nate system Sm with origin rm . Such translations are performed through phase changes in far field patterns. The obtained patterns can be summed and the resulting pattern is amenable to interpolation at the next aggregation step. The computational complexity of such multilevel interpolation and aggregation process has been shown in [11] to be of O(Nr2 log Nr ). Assuming that Nr is comparable with Na , the computational cost of this process is expected to asymptotically dominate the total cost of the radiation pattern evaluation for electrically very large antennas. On the other hand, for moderately sized reflectors, the level M integration time tends to be the dominant computational burden, due to a large constant hidden in its O(Nr2 ) complexity estimate. If an antenna system comprises multiple radiating components such as the main and sub-reflectors, illuminated by single or multiple feed elements, the minimum sufficient sampling rates of individual radiation patterns can be applied to each object in its self-centered coordinate system. Interpolation and origin translations are then used to aggregate partial patterns into the global one, in the same way as for reflector subdomains. The density of angular grids is increased proportionally to

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Fig. 2. Vertical cross section of the offset single reflector antenna system and spheres circumscribing the radiating components and the whole system. ( ) are the center and radius of the smallest sphere circumscribing the feed ( = f ), the reflector ( = r), and the whole antenna ( = a), respectively.

O ;R





the ratio of electrical sizes of individual components, leading to a final interpolation and aggregation step similar to the two-level fast physical optics algorithm [12]. In the case of an offset single reflector antenna system presented in Fig. 2, the reflector and the feed can be considered as radiating objects with electrical sizes r and f , and radiation patterns r and f , respectively. Here, r and f are assumed to be computed in coordinate systems Sr and Sf with their origins at the centers of the smallest spheres circumscribing the reflector and the feed, respectively. Both the reflector and the feed patterns must be interpolated to the sampling rate associated with the radius a of the smallest enclosing sphere of the whole antenna. These interpolated patterns can then be translated to the global coordinate system with its origin at the center of the antenna. Finally, the reflector and feed patterns can be summed to obtain the pattern a of the whole antenna system. This final aggregation of the antenna pattern components is characterized by the complexity of 2 ( a ).

U

N

U

N

U

U

R

U

  =

Fig. 3. 3D patterns computed with MLPO, including PTD contribution. is varying from 0 to along the radial coordinate. The black circle is for = 2, i.e., the boundary between the front and rear patterns. (a) Copolarized pattern. (b) Crosspolarized pattern.



ON

IV. NUMERICAL RESULTS We apply the generalized MLPO algorithm to a single offset parabolic reflector antenna system used in the GRASP9 Technical Description document as an illustrative example of the computation of wide angle patterns (see [5, p. 241]). The antenna configuration is defined by the following initial data (cf Fig. 2): parabolic reflector with diameter r = 40 , magnification factor r = 0 8, height of the reflector “center” with respect to the paraboloid axis = 30 , half angle subtended by the reflector from the source point at the reflector focus: = 29 1 . A Gaussian feed is taken as the primary source, with its radiated fields computed by complex source point formulas (see [5, p. 99]) with the complex shift parameter equal to 1 66 (12 dB taper at the edge of the reflector). From these data, the MLPO code computes the radii of the spheres circumscribing the reflector and the whole antenna system, respectively. Using notations previously introduced for the reflector, the antenna system, and the feed, we obtain for the case under study: r = 22 09 and a = 25 98 (cf Fig. 2). f is taken equal to . The for the multilevel computation of number of decomposition levels the reflector pattern is then computed so as to obtain sufficiently small is 5, patches at level . For the above case, the computed value for patch leading to M = 1 37 for the maximum radius of the level circumscribing spheres. A Gauss quadrature with 8 integrand points along each projected variable describing the reflector surface is used for surface integration over level patches. The PTD integral is per-

D





F=D

:

h



:

b

R

: 

R

: 

M

M

R

: 

R b

M M

: 

M

Fig. 4. Co-polarized pattern cut in the symmetry plane: comparison between GRASP9, MLPO, and direct PO results.

formed with 8 integrand points along the reflector rim for each edge patch at level . Fig. 3 presents the full 3D co- and crosspolarized patterns of the antenna system, including the PTD integral contribution and the feed spillover. These patterns were computed by our MLPO code in 253 seconds on a single processor, and comprise 2945 cuts with constant value, and 1473 values in each of these cuts. To compute the same patterns with approximately the same accuracy (1 dB down to 080 dB on amplitude patterns) GRASP9 requires about 30 min on the same single processor machine. Fig. 4 illustrates the accuracy of the MLPO results by comparison with the patterns obtained with our home made direct PO code, and with GRASP9 reference results obtained with a large number of integration points on the reflector surface. The cut shown on this figure is

M





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TABLE I COMPUTATION TIMES IN SECONDS OF THE MLPO AND DIRECT PO ALGORITHMS, FOR THE SCALE 1 AND SCALE 4 PROBLEMS, WITH AND WITHOUT THE PTD LINE INTEGRAL

Fig. 5. Comparison of the patterns computed by the PO surface integral only, and by the PO + PTD integrals. The fields radiated by the feed are added in both cases. (a) Copolarized pattern in the symmetry plane. (b) Crosspolarized pattern in the  = =4 plane.

in the plane of symmetry of the patterns (xOz plane:  = 0;  ), where the requirements on the surface and edge integrals are particularly high to reach the prescribed accuracy of 1 dB down to 080 dB. For the considered antenna, the shadow boundaries in the symmetry plane occur in the t = 104:2 (top of the reflector) and b = 162:2 (bottom of the reflector) directions, shown on Fig. 2 (black dashed lines). The feed radiation is shadowed in the [t ; b ] interval of  values. As a consequence, the pattern shown in Fig. 4 exhibits an increase of the back radiation around these directions. It should be noted that the respective fields computed by surface integration and edge integration in PO-PTD simulations are quite different from the fields obtained in GO-GTD based simulations from specular reflection of space rays and from edge diffracted rays, respectively. In the latter type of simulation, the rays originating from the feed are blocked by the reflector, yielding a GO field discontinuity along the shadow boundary. On the contrary, the far field obtained by PO surface integration is not null in the angular region “shadowed” by the reflector. Combination of the field obtained by PO integration with the direct radiation from the feed yields the shape of the “shadow” pattern, except in angular regions where the PTD contribution is dominant. For instance, the peak which is observed for   130 in Fig. 4, corresponds both to the direction of maximum radiation of the feed and of equiphase combination of contributions from elliptical slices of the reflector surface parallel to its edge; in GO-GTD simulations, it appears as a caustic for GTD rays, while GO rays are blocked in that direction [5]. The influence of the PTD integral is illustrated in Fig. 5 on two cuts of full 3D patterns obtained with and without the PTD integral contribution. This influence is visible, both on the copolarized pattern in the symmetry plane ( = 0) and on the crosspolarized pattern in the  = =4 plane. The most critical region with respect to the integration accuracy is in the symmetry plane, in the interval of  values ranging between 0140 and 090 , i.e., in the back radiation pattern not only of the reflector but also of the feed (cf Fig. 2). The other region where the PTD contribution is clearly dominant on this pattern, approximately between 40 and 70 , shows the importance of taking into account the PTD fields in far sidelobe regions, even in the front radiation pattern.

Finally, we illustrate the robustness of the MLPO algorithm, by increasing the size of the problem, showing that parameters tuned for a small wide angle pattern problem lead to prescribed accuracy for large sized problems. Increasing the problem size leads to an increased number of decomposition levels, which is determined by the code itself. Oversampling values and numbers of quadrature points are kept the same for all problem sizes, as they only depend on the patch size at the highest level of decomposition. To validate this approach, large antenna problems are constructed by scaling the previously described one, considered as the “scale 1” configuration. Scaling by a factor q is performed by multiplying the frequency by q . Applying the MLPO algorithm to the scale 4 configuration, with the scale 1 oversampling values and numbers of quadrature points, leads to an observed accuracy of 1 dB down to 090 dB on the amplitude 3D patterns. Table I shows representative values for MLPO and direct PO computation times in seconds, obtained for the scale 1 and scale 4 problems with a single Intel Xeon [email protected] GHz processor on a multiuser server for pattern computations without PTD integral (“only PO” column) and with PTD integral (“with PTD” column). The increase in the computation time when accounting for the PTD contribution is presented in the last column. Computation times are subject to variations from one run to another, but these values are “representative” in the sense that they were obtained repeatedly with less than 1% or a few seconds of error. The direct PO computation times for sufficiently sampled wide angle patterns of the scale 4 antenna system were obtained by multiplying by a proper factor the computation times measured when computing undersampled subsets of the full patterns (with no adaptive quadrature rule being used, CPU times are essentially proportional to the number of directions of the computed patterns for large antenna systems). The “ratio” row in the Table is obtained by dividing computation times of scales 4 and 1: in the case of computations involving only surface integrals, this ratio is expected to be close to 16 for the MLPO, and of the order of 256 for the direct PO. The measured values are clearly in good agreement with these predictions. The data of Table I support the claim that the PTD contribution must also be computed by a multilevel algorithm, in order to keep its computation time negligible with respect to the MLPO surface integral computation. The ratio of computation times devoted to the PTD integral should theoretically vary as Nr with MLPO and Nr3 with regular PO (leading to ratios of 4 and 64 respectively). The orders of magnitude of the experimental data presented here (4.3 and 92., respectively) are coherent with these predictions. Computation of the PTD contribution by the direct numerical integration at scales 1 and 4 requires minutes and hours, respectively, in contrast to the computation times of a few seconds (less than 20 s at scale 4) observed with the MLPO.

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V. CONCLUSION The MLPO algorithm augmented with the PTD integral has been developed and applied to a single reflector antenna. Furthermore, the PO and PTD contributions of the reflector have been combined with the feed radiation to obtain the total far field pattern of the antenna over the whole angular range. The algorithm accuracy and computational efficiency have been tested on the case of full 3D pattern computations by comparison to reference results. It has been shown to easily satisfy the accuracy required for wide angle patterns in the demanding case of a single offset reflector antenna. ACKNOWLEDGMENT The authors would like to thank Dr. E. Jorgensen of TICRA for providing reference results obtained with the GRASP9 software.

REFERENCES [1] Y. Rahmat-Samii, “Reflector antennas,” in Antenna Handbook, Y. Lo and S. Lee, Eds. New York: Van Nostrand Reinhold, 1993, vol. II, ch. 15, pp. 1–124. [2] R. Shore and A. Yaghjian, “Application of incremental length diffraction coefficients to calculate the pattern effects of the rim and surface cracks of a reflector antenna,” IEEE Trans. Antennas Propag., vol. 41, pp. 1–11, Jan. 1993. [3] R. Shore and A. Yaghjian, “Correction to application of incremental length diffraction coefficients to calculate the pattern effects of the rim and surface cracks of a reflector antenna,” IEEE Trans. Antennas Propag., vol. 45, p. 917, May 1997. [4] A. Miura and Y. Rahmat-Samii, “Spaceborne mesh reflector antennas with complex weaves: Extended PO/periodic-MoM analysis,” IEEE Trans. Antennas Propag., vol. 55, pp. 1022–1029, Apr. 2007. [5] K. Pontoppidan, Ed., “Technical Description of GRASP9,” TICRA, 2005 [Online]. Available: www.ticra.com [6] V. Galindo-Israel, T. Veruttipong, S. Rengarajan, and W. Imbriale, “Inflection point caustic problems and solutions for high-gain dual-shaped reflectors,” IEEE Trans. Antennas Propag., pp. 202–211, Feb. 1990. [7] D.-W. Duan and Y. Rahmat-Samii, “A generalized diffraction synthesis technique for high performance reflector antennas,” IEEE Trans. Antennas Propag., vol. 43, pp. 27–40, Jan. 1995. [8] Y. Z. Umul, “Modified theory of physical optics solution of impedance half plane,” IEEE Trans. Antennas Propag., vol. 54, pp. 2048–2053, Jul. 2006. [9] T. Shijo, L. Rodriguez, and M. Ando, “The modified surface-normal vectors in the physical optics,” IEEE Trans. Antennas Propag., vol. 56, pp. 3714–3722, Dec. 2008. [10] R. Hoferer and Y. Rahmat-Samii, “Subreflector shaping for antenna distortion compensation: An efficient Fourier-Jacobi expansion with GO/PO analysis,” IEEE Trans. Antennas Propag., vol. 50, pp. 1676–1687, Dec. 2002. [11] A. Boag and C. Letrou, “Multilevel fast physical optics algorithm for radiation from non-planar apertures,” IEEE Trans. Antennas Propag., vol. 53, pp. 2064–2072, Jun. 2005. [12] A. Boag and C. Letrou, “Fast physical optics algorithm for lens and reflector antennas,” IEEE Trans. Antennas Propag., vol. 51, pp. 1063–1068, May 2003.

Fast Dipole Method for Electromagnetic Scattering From Perfect Electric Conducting Targets Xinlei Chen, Changqing Gu, Zhenyi Niu, and Zhuo Li

Abstract—A new fast dipole method (FDM) is proposed for the electromagnetic scattering from arbitrarily shaped three-dimensional (3D), electrically large, perfect electric conducting (PEC) targets in free space based on the concept of equivalent dipole-moment method (EDM) and the fast multipole method (FMM). The electric-field, magnetic-field and combined-field integral equations (CFIE) for this algorithm have been developed and implemented. Although the basic acceleration idea in the FDM has been borrowed from the FMM, the specific implementation of these two algorithms is completely different. In the FDM, a simple Taylor’s series expansion of the distance between two interacting equivalent dipoles is used, which transforms the impedance element into another aggregation-translation-disaggregation form naturally. Furthermore, this algorithm is very simple for numerical implementation for it does not involve the calculation of a number of Bessel functions, Legendre functions for the addition theorem and complex integral operators. The FDM can computational complexity and memory requirement, achieve is the number of unknowns. Numerical results are presented to where validate the efficiency and accuracy of this method through comparison with other rigorous solutions.

O(

)

Index Terms—Combined filed integral equation (CFIE), electromagnetic scattering, equivalent dipole-moment method (EDM), fast dipole method (FDM), fast multipole method (FMM).

I. INTRODUCTION The method of moments (MoM), which converts the integral equation to a dense matrix equation, has been widely studied. However, for large scale electromagnetic (EM) problems the cost to directly solve this dense matrix equation is very expensive and formidable, which requires O(N 2 ) memory consumption and computational complexity for iterative solvers, where N is the number of unknowns. Fortunately, some effective methods have been proposed and applied in recent years, among which the fast multipole method (FMM) [1], [2] and multilevel fast multipole algorithm (MLFMA) [3], [4] are the most representative ones. The FMM reduces both the complexity of a matrix-vector product (MVP) and memory requirement from O(N 2 ) to O(N 1:5 ). And the MLMFA can achieve O(N log N ) complexity and memory requirement using translation, interpolation, anterpolation (adjoint interpolation) and a grid-tree data structure. Recently, MLFMA has been demonstrated to solve an equivalent dense matrix system resulting from MoM with close to 620 million unknowns [5]. Although the FMM and MLFMA are sufficiently efficiency, the formulas are very complicated and hard for coding considering that a large number of integral operators, Bessel functions and Legendre functions are involved. A fast far-field approximation (FAFFA) [6], [7] has simple formula and can be viewed as a smooth transition from FMM, but it can be used only when the two groups are Manuscript received August 19, 2010; revised March 16, 2011; accepted July 23, 2011. Date of publication September 15, 2011; date of current version February 03, 2012. This work was supported in part by the National Nature Science Foundation of China under Grant 61071019 and in part by the Jiangsu Innovation Program for Graduate Education under Grant CXZZ11_0229. The authors are with the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2167906

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well separated. Therefore, it is often mixed with other methods to get good precision, such as the FAFFA-FMM [8], the FAFFA-MLFMA [9] and the RPFMA-FAFFA-MLFMA [10]. More recently, the equivalent dipole-moment method (EDM) [11], [12] has been developed to simplify and accelerate the impedance matrix element filling procedure for surface integral equations (SIE). The EDM is based on the commonly used Rao-Wilton-Glisson (RWG) [13] basis function, in which each RWG triangle pair is viewed as a dipole model with an equivalent dipole moment. Later the EDM is extended and applied to deal with the electric isotropic media and electric anisotropic media based on Schaubert-Wilton-Glisson (SWG) [14] basis function [15], [16]. The main advantage of EDM is that the impedance matrix element can be expressed in an extremely simplified form, which avoids the integral operators, whereas the memory requirement does not change. In this communication, a fast dipole method (FDM) is proposed for solving the EM scattering from arbitrarily shaped three-dimensional (3D), electrically large, PEC targets in free space based on the concept of the EDM and FMM. All the RWG basis functions on the target surface are modeled as equivalent dipole models and divided into several uniform cubes. Each cube is called a group, and contains a few equivalent dipole models. If two groups are in each others’ nearfield, the corresponding impedance matrix elements are computed by conventional MoM/EDM. Otherwise, through a simple Taylor’s series expansion of the distance R between the interacting equivalent dipoles, we can transform the impedance element into an aggregation-translation-disaggregation form naturally. Moreover, the CPU time as well as the memory requirement can also be reduced to O(N 1:5 ). Furthermore, compared with the FMM, the Bessel functions, Legendre functions and most of integral operators do not participate in the whole computation procedure, which makes the formula derivation and coding procedure much easier. In addition, the complexity and memory requirement for computing translation operators are just only O(N ) and the disaggregation process does not require extra memory when the CFIE is used. Compared with the FAFFA method, the computation complexity of the FDM for the far group pairs is close to its when the FAFFA doesn’t use interpolating. The remainder of the communication is organized as follows. In Section II, the EDM is briefly presented for the integrity of the new method. Then in Section III, the derivation and implementation of the FDM is illustrated in detail. The computation complexity and memory requirement are analyzed in Section IV. In Section V, some numerical results about bistatic radar cross section (RCS) of several canonical targets are given to verify the efficiency and accuracy of the method. Finally, conclusions and suggestions for future work are discussed in Section VI. II. BASIC PRINCIPLES OF THE EDM In this section, the basic principles of the EDM are briefly presented. Considering the problem of EM wave scattering by an ^ . The arbitrarily shaped PEC surface whose normal is denoted by n formulations for the electric-field integral equation (EFIE) is derived from the electric-field boundary conditions on the conducting surface

jkn^ 2

G(r; r )J(r ) + k12 rG(r; r )r 1 J(r ) dr ^ 2 E (r) =n 0



0

0

0

0

0

i

(1)

where J denotes the induced surface current, k and  denote the free-space wave number and impedance, and G(r; r0 ) = e0jkjr0r j =(4jr 0 r0 j) denotes the free-space Green’s function, Ei stands the incident field.

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Fig. 1. Geometry of the nth and mth RWG elements and their equivalent dipole models.

Then using a set of RWG basis functions fn (n = 1; . . . ; N ) to I f (r0 ), expand the induced surface current density J(r0 ) = N n=1 n n where N is the number of degrees of freedom and In is the unknown ^ 2 fm (r) as the testing function to discurrent coefficient, and using n cretize (1), we obtain a matrix equation N n=1

Z I

E mn n

=

V ; m = 1; 2; . . . ; N

(2)

m

where

Z 1

E mn

jk

=

fm (r) 1 fn (r ) 0 0

f

f

1 k2 r1 f (r)r 1 f (r ) G(r; r )dr dr 0

m

0

n

0

0

(3) are the elements of the impedance matrix, and vector element given by

V

m

=

V

m

is the excitation

fm (r) 1 Ei (r)dr:

f

(4)

E represents the interaction between the mth As shown in Fig. 1, Zmn and the nth RWG triangle pair, which consist of two adjacent triangles Tm6 (Tn6 ) with the common edge of length lm (ln ) separately. The equivalent dipole moment for the nth RWG element can be expressed as [11]–[13]

mn =

fn (r )dr 0

T

0

 l (r 0 r + ); n

c0 n

c n

(5)

where rcn6 are the position vectors of the centroid of Tn6 defined in the global coordinate and ln is the length of the nth common edge of Tn6 . It can be seen from the above equation that the equivalent dipole moment of a RWG basis function can be simply represented by its geometric parameters. E usually obtained by In EDM method [11], [12], the terms Zmn Gaussian quadrature can be regarded approximately as an interaction between two infinitely small dipoles with equivalent moments instead of two common RWG basis functions.

Z

E mn

e

0jkR

=



4

mm 1 mn

jk + C R

0 m 1 R^ R^ 1 m m

n

jk + 3C R

(6)

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R r r r R R R r

r

in which = mn = m 0 n is the vector from the center point n of the nth equivalent dipole to the center point m of the mth equivalent dipole. R = j j, ^ = =R. n = ( cn+ + cn0 )=2 and m = ( mc+ + cm0 )=2 are denoted in Fig. 1 and

r

r

r

1 1+ 1 jkR

C= 2 R

r

r

r

:

(7)

Similarly, for the magnetic field integral equation (MFIE), the mutual impedance element can be expressed as

M = Zmn

n

n

jkCe0jkR 4 (mm 2 n^ m ) 1 (R 2 mn )

n

n

(8)

n n n

where ^ m = (^ + m + ^ 0m )=j^ +m + ^+0m j is0 the average normal vector of the mth equivalent dipole, and ^ m , ^ m represent the unit normal + and Tm0 respectively. vectors of Tm We note that the distance between the center points of the source and field equivalent dipoles must be greater than 0:15, as elucidated in [12]. In other words, if two RWG triangle pairs are very near and the distance between them is less 0:15, this equivalence is no longer valid and the conventional MoM must be used. It can be seen from the above equations for the EDM the calculations of mutual terms of the impedance matrix for both EFIE and MFIE do not contain integral operators, thus greatly simplifying the matrix filling process and saving much computation time. III. BASIC PRINCIPLES AND IMPLEMENTATION OF THE FDM A. Basic Principles Comparison Between the FDM, FMM and FAFFA In the FMM, the addition theorem is fully utilized for relating far group pairs by expanding the three dimensional Green’s function with a series of the product of spherical Bessel function, spherical Hankel function and Legendre polynomial. Physical interpretation of this expansion is that a spherical wave in free space can be expanded by the sum of an infinite number of plane waves, which transforms the interaction between the source and field points in non-nearby groups into aggregation, translation and disaggregation operators. A fast far field approximation (FAFFA) with simple formulations was developed to estimate RCS of conducting scatterers in [6]. The FAFFA method can be viewed as the natural far field approximation of the FMM and it simplifies the integral operation over every sampling directions on the unit sphere surface to one sampling computation in the direction between two far group centers [6], [9], [10]. Such approximation is valid only when two groups are far enough, otherwise a bigger error will be introduced. The FDM in this work is very similar to the FMM and the FAFFA in the sense that all these three methods are developed to accelerate the MVP in an iterative solver without needing to store many of the matrix elements. For the FDM, the EFIE and MFIE matrix elements are represented by (6) and (8) respectively. For nearfield pairs, the impedance matrix elements are calculated by the traditional MoM and the EDM, while for the far group interaction, a simple Taylor’s series expansion of the distance R between the source and field points transforms the impedance element into another aggregation-translation-disaggregation form naturally, also leading to memory reduction and speed acceleration in the MVP. In the following formulas we can find that the main difference between the FDM and the FAFFA [9], [10] is that the second order term of R is introduced in the phase approximation. In addition, the formulas involved in FDM are much more simpler than those of the FMM, thus significantly simplifying the formula derivation and coding procedure.

Fig. 2. The

mth and nth RWG elements and the groups they belong to.

B. Implementation of the FDM In this work, for the FDM each group’s size is chosen longer than

0:15 and the relative position of two Group i (Gi ) and Group j (Gj )

is classified into two types: • Near group pair: Gi and Gj are overlapping or adjacent; • Far group pair: Gi and Gj are nonoverlapping or nonadjacent. So a MVP can be divided into two parts as follow:

N

n=1

Zmn In =

G 2N n2G

Zmn In +

G 2F n2G

Zmn In ; m 2 Gj (9)

in which G 2N n2G Zmn In denotes the near group interaction and G 2F n2G Zmn In denotes the far group interaction respectively. For the near group interaction computation in (9), the traditional MoM and EDM are employed. For the far group interaction, we consider two RWG elements m (m 2 Gj ) and n (n 2 Gi ) and suppose Gi belongs to the far group of Gj (Gi 2Fj ) shown in Fig. 2. These two elements can be viewed as two equivalent dipoles considering that the distance R = j mn j = j m 0 n j between the centers of the two RWG basis function is definitely larger than 0:15 according to the approximation rule of the EDM and the impedance element Zmn can be represented as (6) and (8) for the EFIE and MFIE respectively. The vector mn can be rewritten as (see Fig. 2)

r

r

r

r

R = rmn = rji + rmj 0 rni ; (10) in which rji = ro 0 ro , rmj = rm 0 ro , rni = rn 0 ro . ro and ro are the center positions of Gi and Gj , and rm and rn are the center

positions of the mth and nth equivalent dipoles. Now, we consider R carefully and expand it using the Taylor series as

R = jRj = jrji + rmj 0 rni j 2 2  rji + ^rji 1 rmj + rmj 0 (^2rrjiji 1 rmj ) 2 2 + 0^rji 1 rni + rni 0 (^2rrjiji 1 rni ) :

(11)

For the amplitude and phase approximation of (6) and (8), R can be simplified as

R  rji

(12)

for amplitude variations and

e0jkR  e0jkr e0jk(^r 1r +(r 0(r^ 1r ) )=2r ) e0jk(0r^ 1r +(r 0(r^ 1r ) )=2r ) (13) for phase variations.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 2, FEBRUARY 2012

Substituting (12) and (13) into (6), the impedance matrix element for EFIE can be rewritten as

E  m e0jk(^r 1r +(r 0(^r 1r ) )=2r ) Zmn m 0jkr e 1 4 I rjkji + C 0 ^rji^rji rjkji + 3C 1 mn e0jk(0r^ 1r +(r 0(^r 1r ) )=2r )

rr

1 jkrji :

(15)

CFIE = EFIE + (1 0 )MFIE

mn and the phase term

In (14), the product of the dipole moment r 1r ) )=2r ) +(r 0(^

e0jk(^r 1r

Mn(rij ) = mn e0jk

1r

(^ r

r 0(r^ 1r ) )=2r

+(

)

(16)

can be considered as a aggregation function which aggregate the signal from the nth equivalent dipole to the group center Oi it belongs to. The term in the brace of (14)

e0jkr E ji ( ji ) = 4

T r

jk rji

I



jk + C 0 ^rji ^rji rji

+ 3C

(17)

can be regarded as an electric field translator which transform the signal from the source group center Oi to the field group center Oj . Finally, the product of the dipole moment m and the phase term e0jk(^r 1r +(r 0(r^ 1r ) )=2r )

m

Mm(rji) = mme0jk

(^ r

1r

r 0(r^ 1r ) )=2r

+(

)

(18)

can be viewed as a disaggregation function which disaggregate the signal from the group center Oj to the mth equivalent dipole. Thus the impedance element for the EFIE in the (14) can be rewritten concisely as

E  Mm (rji ) 1 T E Zmn ji (rji ) 1 Mn (rij ):

(19)

Then, the MVP (9) including both near and far group interactions for the EFIE can be written as

N n=1

E In = Zmn

G 2N n2G

E In + Zmn

Mm(rji) 1 T Eji(rji) G 2F 1 In Mn (rij ): (20) n2G

M r

N n=1

MI = Zmn n +

G 2F

G 2N n2G

MI Zmn n

Mm(rji) 2 n^m )1 TMji (rji) 2

(

T r

n2G

In Mn (rij )

(21)

(23)

where  2 [0; 1] is the combination parameter of the CFIE. It is obvious that the aggregation, translation and disaggregation processes don’t require the calculation of higher order spherical Hankel functions, the Legendre polynomial and spherical integration. The computational complexity and memory requirement of the FDM will be analyzed in the next section.

IV. COMPLEXITY ANALYSIS If we assume there are N unknowns for a problem, we divide them into G nonempty groups. Each group has M = N=G unknowns in average and P near-region groups (including itself) in average. First we consider the computational complexity of a MVP. The total amount of computation consists of four parts: near-region, aggregation, translation, disaggregation. • Near-region: Because each group has P groups in its near-region, the total number of groups needed to be calculated and associated with near-region interaction is GP . And there are M 2 impedance elements needed to be calculated in each group. So the CPU cost relevant to near-region interaction T1 = 1 GP M 2 = 1 P N 2 =G; • Aggregation: Each group has about (G 0 P ) groups in its farregion. So there are G(G 0 P ) far-group pairs, and each pair needs M operations. Therefore the time cost of the aggregation process T2 = 2 G(G 0 P )M  2 GN ; • Translation: Every far-group pairs need the transfer operations, so the overall cost of the transfer process T3 = 3 G(G 0 P )  3 G2 ; • Disaggregation: The disaggregation process is very similar to the aggregation process. Thereby the cost for computing all disaggregations T4 = 4 G(G 0 P )M  4 GN . Then the total computational cost of a MVP for the FDM is

T

It can be seen from the above equation that the calculation process of the MVP for far group interaction can be naturally divided into three E 1 aggregation 2 translation  ji ( ji ); steps: n2G In n ( ij );

3 disaggregation m ( ji ). The aggregation and disaggregation functions are exactly the same, so only the aggregation function and translator are needed and stored for the MVP, which expedite the calculation of the MVP. In a similar way, the MVP for the MFIE can be derived as

M r

(22)

(14)

I

1+

TMji (rji) is the magnetic field translator 0jkr TMji (rji) = jkCe4 rji:

For the MFIE, we can find that the translator is a vector and the disaggregation function differs from the aggregation function with a normal unit vector. However, these aggregation and disaggregation functions in the MVP for the MFIE do not need any additional memory. For the CFIE case, we can obtain the MVP easily through the following equation

in which ^ji ^ji is a dyad,  is the unit dyad and

C = r12 ji

in which

1189

4

=

i=1

Ti  1 P NG

2

 c NG 1

2

+ c2 GN:

+ (2 + 4 )GN + 3 G2 (24)

The total cost is minimized by choosing G = c1 N=c2 , and the Tmin = 2pc1 c2 N 1:5 . Here, 1  4 , c1 and c2 are constants which

are platform and software dependent. It can be seen from the above formula that the FDM can reduce the total computational complexity of the MVP to O(N 1:5 ). The memory requirements of the FDM also consists of four parts: near-region, aggregation, translation, disaggregation. • Near-region: Memory consumption S1 = GP M 2 = P N 2 =G; • Aggregation: Memory consumption S2 = G(G 0 P )M  GN , only the phase terms of aggregation functions are stored;

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2

2

Fig. 3. Bistatic RCSs in  polarization of a 2 m 2 m 2 m PEC cube illuminated by a uniform plane wave with the incident direction of (; ) = (0 ; 0 ).

Fig. 4. Bistatic RCSs in  polarization of a PEC sphere of radius 3 m illuminated by a uniform plane wave with the incident direction of (; ) = (0 ; 0 ).

TABLE I TOTAL CPU TIME AND MEMORY COST OF THE TRADITIONAL MOM, EDM AND FDM FOR THE SCATTERING PROBLEM OF A PEC CUBE WITH SIDE LENGTH OF 2 m

• Translation: Memory consumption S3 = 3 G(G 0 P )  3 G2 , in which 3 is a constant; • Disaggregation: Memory consumption S4 = 0. Disaggregation process does not require extra memory for the EFIE, MFIE and CFIE considering that the disaggregation and aggregation functions are the same in the EFIE and the disaggregation function is just the cross product of the aggregation function m ( ji ) and the average normal vector ^ m for the MFIE. So the total memory requirement is S =p 4i=1 Si P N 2 =G + GN . It can be seen that if we choose G  N , the memory requirements of the FDM can achieve O(N 1:5 ).

n

M r

V. NUMERICAL RESULTS AND DISCUSSIONS In this section, we present some numerical results to test the efficiency and validity of the FDM compared with the traditional MoM, EDM, Mie series solution and our FMM code [17]. We remark that in the following examples, the CFIE ( = 0:5) is chosen and all the simulations are performed on a personal computer with the Pentium(R) Dual CPU E5500 with 2.80 GHz (only one core is used) and 2.0 GB RAM. The single-precision code of FDM and the GMRES iterative solver are employed to obtain an identical residual error  0:001 and no preconditioning is used in all the simulations. First we consider the scattering problem of a PEC cube with side length of 2 m. The cube is divided into triangular patches with an average edge length of 0:1, and the total number of unknowns is 7200. All the unknowns are divided into 152 nonempty groups and the size of each group is 0:4. The bistatic RCS for  polarization calculated by the FDM is agree excellently with the conventional MoM and the EDM as shown in the Fig. 3. Table I summarizes the CPU time and memory requirements. It can be seen that the FDM saves much more time and memory than the conventional MoM and the EDM.

Fig. 5. Bistatic RCSs in  polarization of a PEC pencil target illuminated by a uniform plane wave with the incident direction of (; ) = (0 ; 0 ).

Then the bistatic RCS of a PEC sphere of radius 3 m is considered. The total number of unknowns is 40851. The geometry is grouped into 397 nonempty groups and the size of each group is set to 0:65. The total computation time is 234 s and 658 MB memory is used. The bistatic RCSs in  polarization are shown in Fig. 4. The result obtained by the FDM agrees well with the Mie series solution, which is exact and used as a reference. Furthermore, we consider a PEC pencil target mentioned in [18]. This target was formed with a 3 m capped cylinder with a 0.1 m radius, and a tip extending 0.173 m pointing towards 0 azimuth and elevation. This pencil is meshed into 11342 triangular patches and there are total 17013 unknowns, which are grouped into 315 nonempty groups with the size of 0:4. The total computation time is 72 s and 144 MB memory is used by the FDM. The bistatic RCS in  polarization obtained by the FDM agrees very well with the FMM as shown in Fig. 5. And the FMM costs 79 s and 168 MB memory using the same grouping. Finally, the bistatic RCSs of a 252.3744-mm PEC NASA almond is calculated. The almond is divided into 24150 triangular patches and the number of unknowns is 36225. Totally 430 nonempty groups with the size of 0:55 are obtained. The CPU time is 216 s and the memory cost

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Fig. 6. Bistatic RCSs in  polarization of a 252.3744-mm PEC NASA almond illuminated by a uniform plane wave with the incident direction of (; ) = (0 ; 0 ).

is 505 MB. The bistatic RCS in  polarization is compared with the FMM shown in Fig. 6. And the FMM costs 238 s and 579 MB memory using the same grouping. From the pencil and NASA almond simulations results, it’s can been seen that the FDM and the FMM have the same order cost in CPU time and memory requirement. We remark that the single point Gaussian quadrature is used to calculate the aggregation functions, disaggregation functions and the near group interaction of the FMM. VI. CONCLUSIONS In this communication, the FDM is proposed and implemented to accelerate solving the CFIE in the EM scattering of 3D PEC targets. The new method can reduce both the CPU time and memory requirement to O (N 1:5 ). Also the FDM is base on the concept of the EDM and it is very simple for implementation compared with the FMM. Numerical results show that the FDM method can obtain acceptable results for practical engineering applications. In the near future, the FDM will be extended to solve volume integral equation (VIE) and combined with the characteristic basis function method (CBFM). Also the multi-level FDM will be implemented.

REFERENCES [1] V. Rokhlin, “Rapid solution of integral equations of classical potential theory,” J. Comput. Phys., vol. 60, pp. 187–207, Sep. 1985. [2] R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: A pedestrian prescription,” IEEE Antennas Propag. Mag., vol. 35, pp. 7–12, Jun. 1993. [3] J. M. Song and W. C. Chew, “Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microwave Opt. Technol. Lett., vol. 10, no. 1, pp. 14–19, Sep. 1995. [4] J. M. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag., vol. 45, no. 10, pp. 1488–1493, Oct. 1997. [5] J. M. Taboada, M. G. Araújo, J. M. Bértolo, and L. Landesa, “MLFMA-FFT parallel algorithm for the solution of large-scale problems in electromagnetics,” Progr. Electromagn. Res., vol. 105, pp. 15–30, 2010. [6] C. C. Lu and W. C. Chew, “Fast far-field approximation for calculating the RCS of large objects,” Microwave Opt. Tech. Lett., vol. 8, no. 5, pp. 238–241, 1995.

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[7] A. McCowen, “Efficient 3-D moment-method analysis for reflector antennas using a far-field approximation technique,” Proc. Inst. Elect. Eng., vol. 146, pt. H, pp. 7–12, Feb. 1999. [8] C. C. Lu, J. M. Song, W. C. Chew, and E. Michielssen, “The application of far-field approximation to accelerate the fast multipole method,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Baltimore, MD, Jul. 21–26, 1996, vol. 3, pp. 1738–1741. [9] W. C. Chew, T. J. Cui, and J. M. Song, “A FAFFA-MLFMA algorithm for electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 50, no. 11, pp. 1641–1649, Nov. 2002. [10] T. J. Cui, W. C. Chew, G. Chen, and J. M. Song, “Efficient MLFMA, RPFMA, and FAFFA algorithms for EM scattering by very large structures,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 759–770, Mar. 2004. [11] S. N. Makarov, Antenna and EM Modeling with MATLAB. New York: Wiley, 2002. [12] J. Yeo, S. Köksoy, V. V. S. Prakash, and R. Mittra, “Efficient generation of method of moments matrices using the characteristic function method,” IEEE Trans. Antennas Propag., vol. 52, no. 12, pp. 3405–3410, Dec. 2004. [13] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, no. 3, pp. 409–418, May 1982. [14] D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag., vol. AP-32, no. 1, pp. 77–85, Jan. 1984. [15] J. D. Yuan, C. Q. Gu, and G. D. Han, “Efficient generation of method of moments matrices using equivalent dipole-moment method,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 716–719, 2009. [16] J. D. Yuan, C. Q. Gu, and Z. Li, “Electromagnetic scattering by arbitrarily shaped stratified anisotropic media using the equivalent dipole moment method,” Int. J. RF Microw. Comput.-Aided Engrg., vol. 20, no. 4, pp. 416–421, 2010. [17] Z. Y. Niu and J. P. Xu, “Near-field sparse inverse preconditioning of multilevel fast multipole algorithm for electric field integral equations,” presented at the IEEE Asia-Pacific Conf., 2005. [18] M. L. Hastriter, “A study of MLFMA for large-scale scattering problems,” Ph.D. dissertation, University of Illinois at Urbana-Champaign, Illinois, 2003.

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An Efficient Hybrid GO-PWS Algorithm to Analyze Conformal Serrated-Edge Reflectors for Millimeter-Wave Compact Range Alfonso Muñoz-Acevedo and Manuel Sierra-Castañer

Abstract—A method to analyze parabolic reflectors with arbitrary piecewise rim is presented in this communication. This kind of reflectors, when operating as collimators in compact range facilities, needs to be large in terms of wavelength. Their analysis is very inefficient, when it is carried out with fullwave/MoM techniques, and it is not very appropriate for designing with PO techniques. Also, fast GO formulations do not offer enough accuracy to reach performance results. The proposed algorithm is based on a GO-PWS hybrid scheme, using analytical as well as non-analytical formulations. On one side, an analytical treatment of the polygonal rim reflectors is carried out. On the other side, non-analytical calculi are based on effiorder 2-dimensional FFT. A combination of cient operations, such as these two techniques in the algorithm ensures real ad-hoc design capabilities, reached through analysis speedup. The purpose of the algorithm is to obtain an optimal conformal serrated-edge reflector design through the analysis of the field quality within the quiet zone that it is able to generate in its forward half space. Index Terms—Compact range, hybrid algorithm, millimeter wave, plane wave spectrum, serrated edge reflector.

I. INTRODUCTION Plane wave spectrum (PWS) theory offers a solution to Maxwell’s equations representing the electromagnetic field distributions as a sum of independent plane waves. The simplicity, in terms of mathematical treatment, for each one of these elementary contributions, makes PWS formulations quite attractive when dealing with half-space radiation and propagation schemes in both near and far field conditions. Reflectors have classically been analyzed with different order physics techniques. The two most common techniques are GO and PO. As commented in [1], PO techniques are high frequency solving techniques, which offer a realistic solution to most problems. GO is a much simpler approach, while, as stated in [2], does not offer enough accuracy to carry out design—stage tasks. Hybrid approaches to the problem combine different order electromagnetic physics in order to reach more accurate results and, if possible, computational speedup. As studied in [3], PO techniques are in some cases insufficient to analyze serrated edge reflectors, since they assume a high-frequency hypothesis for which surface currents are not affected by edge effects at the serrated reflector rim. As pointed out in [3], [4], a common solution consists on using MoM techniques to evaluate the reflector’s impressed currents and integrate them through PO techniques to calculate the radiated field. This approach is able to offer very accurate results of both near and far fields of the reflector, although it is highly inefficient Manuscript received April 29, 2010; revised October 04, 2010; accepted July 23, 2011. Date of publication October 20, 2011; date of current version February 03, 2012. This work was supported by a Spanish Government FPI scholarship for Ph.D. students and both CROCANTE (TEC2008-06736-C03-01/TEC) and TERASENSE (CSD2008-00068) projects. The authors are with the Radiation Group, Technical University of Madrid (UPM), Ciudad Universitaria, 28040 Madrid, Spain (e-mail: [email protected]. upm.es). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173100

for reflectors which are electrically very large, as those of CATRs are. Another hybrid approach to reach more realistic surface currents consists on assuming edge currents which also contribute to the radiated field (PTD techniques [2], [5]). Edge modeling is critical in the analysis and design of serrated-edge reflectors inside compact ranges. As commented in [6], simple approaches can offer sufficient results, while it is important to bound the maximum admissible error and also to be able to analyze different geometrical serrations alternatives [7]. GTD techniques [8] are an improvement of GO techniques, and are able to offer more accurate results by using information of the diffracted rays from the reflector’s edges, but its utility is limited to scattering angles distant from the reflector’s broadside, above a minimum reliable angle. GTD itself is not able to offer a complete solution to an arbitrary scattering problem. Common convergence criteria establish dense meshes (1x; y < o =30) over the surface currents [2]. Different techniques have been developed setting an adaptive mesh over the surface, or conformal meshing schemes [2], [9]. The evaluation of the Fresnel integral is itself a O(N4 ) complexity problem, given that for each one of the O(N2 ) points of interest, a O(N2 ) complexity integral must be performed, so PO techniques are a computational bottleneck when solving electrically large reflectors. The motivation of performing integration through a plane wave spectrum (PWS) approach is the substantial reduction of the computational cost. PWS formulations evaluate the radiated fields of arbitrary scatterers through complex variable integral formulations and Fourier analysis [10]–[12]. In particular, parabolic scatterers are modeled with their aperture field distribution [13], in a half-space problem, which suits simple planar geometry formulations. The treatment of the planar field distribution is carried out through Fourier analysis, and practical implementations use of FFT. Thus, if the sampling of the field is carried out at general Nyquist rate (1x; y = o =2), the discretization of the aperture field is dense enough [11], [13]. These lower density meshes are an intrinsic computational advantage of plane wave spectrum formulations. On the side of feed modeling, different order physics should be used depending on the particular setup to be analyzed. When low gain feeders and high F=D facilities are used, as in the case of CATRs, the reflector’s surface can be assumed to be in the far-field region of the feeder. Consequently, the impressed currents are calculated through GO techniques and feeder radiation pattern weighting. However, if this hypothesis is not accomplished, a complete near-field model of the feeder should be used to calculate the impressed currents distribution. Thus, PO techniques might be needed between the feeder and the reflector [15]. In the proposed algorithm, the impressed currents are projected to obtain the aperture distribution. As discussed in [15], this aperture field has an intrinsic phase error, due to the projection of the currents in the parabolic surface to the projected surface. The main consequence of computing the radiated field from this projected surface instead of the scatterer surface is the reduction of the angle validity in the radiated field. For the CATR case of study, scatterers with high F=D figures do not introduce high phase errors, while CATRs operate in a narrow margin of radiation angles. This communication is divided in five sections. After this introduction, Section II shows the mathematical formulation of the algorithm, explaining separately the PWS and the GO contributions. Section III offers field results of particular setups. Quantitative information is offered and field acquisitions are related to GRASP8 acquisitions. Section IV concludes some ideas for the use of this algorithm in the design of CATR facilities, and establishes the future lines.

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Fig. 2. Maximum usable angle seen by the reflector geometry. Fig. 1. Analysis problem: focused offset reflector CATR.

II. MATHEMATICAL FORMULATION OF THE ALGORITHM A. The Plane Wave Spectrum Representation As mentioned in the introduction, formulations of the Plane Wave Spectrum theory are shown for the planar acquisition surface canonical case. Clemmow studied deeply these formulations in [10]. There, it is shown that (1) and (2) constitute a solution to Maxwell’s equations in a free space source-free region, where r is the acquisition point vector, respect to the sources distribution coordinate system Os . This coordinate system is the one respect to which the plane wave spectrum of the fields is calculated

1 1 ~ (kx ; ky )e0j k1r dkx dky E 2 01 01 1 1 H(r) = 2 Z0 k0 1 1 ~ (kx ; ky )e0j k1r dkx dky : k E 01 01 1

E(r) =

1

1

(1)

1

2

(2)

Equations (1) and (2) become Fourier Transforms between the spatial (x; y ) domain and the spectral (kx ; ky ) one. This approach requires the definition of the longitudinal propagation constant kz , as in [12]. The transformed function of the field at the acquisition plane is the z-propagated plane wave spectrum distribution. The sources distribution is located at zs = 0 (Fig. 1), so the spectral distribution is obtained performing the inverse Fourier Transform shown in (3). A practical approach to this equation consists on setting a rectangular xy mesh of M times M points over the zs = 0 plane, with uniform distance between samples 1x = 1y = o =2. The continuous and vectorial E(xs ; ys ; zs = 0) field is sampled and separated in components, obtaining 3 scalar 2-dimensional discrete set of complex fields (for each one of the rectangular components x, y and z ). The Fourier transform of (3) becomes a DFT, which can be implemented with an efficient M 2 order 2-dimensional FFT

k ; ky ) =

~( x E

1 1 E(x ; y ; z = 0)ejk 1x ejk 1y dx dy: 01 01 s s s 1

1

(3)

B. Sampling Criteria As pointed out in [13], Nyquist sampling in the spatial domain x = 1y = o =2) is enough to avoid the aliasing effects, in the most general case. Moreover, the extension of the spatial domain xy must contain the region where the field is of interest. In particular

s  xy , where s is the minimum spatial support which includes the projected currents distribution, as defined in (4) and (5). If xy is (1

Fig. 3. Spectral domains involved in the propagation problem.

larger than s , there is a spatial “domain widening”, which might be understood as an oversampling of the transformed variables (kx ky ) whenever the space between spatial samples is kept

S =

0

xy =

0

Smin;x ; Smin;x 2

2

2

2

2

Sacq;x ; Sacq;x

2

0

0

Smin;y ; Smin;y 2

2

Sacq;y ; Sacq;y : 2 2

(4) (5)

For this case, and without loss of generality, an oversampling vector  can be defined as in (6)

Smin;y ;x  = [kx ; ky ] = 2 Smin Sacq;x ; 2 Sacq;y Mmin;x ; Mmin;y : = 2 Macq;x 2 Macq;y 1

1

1

1

(6)

A  component close to 0 implies that the corresponding spectral variable is strongly oversampled, and the field is being calculated out of the reflector’s projected surface for the transformed spatial direction. The modulus of the 2 vector states for the fraction of total field which is being calculated out of the reflector’s projected surface. It is recommendable to choose components kx , ky  0:5 that also make the number of samples per direction a power of two, in order to perform the calculations via FFT. For a fixed xy , the number of PWS modes that this domain is able to integrate is decreasing when increasing the zs distance between s and xy . This fact was already pointed out in [14], where some numerical examples for particular reflectors were given. It can be reached, moreover, an analytical expression for the required number of PWS modes to determine univocally the quiet zone field in xy . The explanation of this lies on the relationship between the spectral variables (kx ; ky ) and their corresponding geometrical angles (x ; y ),

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Fig. 4. Blocks scheme of the algorithm.

depending on the separation zs1 between s and xy . It is easily concluded that (kx ky ) do not reach their boundary values 6ko whenever zs > 0, and these bounds define the usable fraction of the whole complex spectrum (Fig. 2). Equations (7) and (8) relate (kx ky )max with the geometrical and electromagnetic parameters. The spectral domain of these contributing modes is described with (9) and the amount of usable samples in a spatial direction is stated by (10)

[kx ; ky ]max = k0 1 [sin(max;x ); sin(max;y )] (x=y ) + Smin;(x=y ) tan max;(x=y) = Sacq; 2 1 (zs 0 zoffset ) ;(x=y ) 1 0 = 4M1 (min 1 + 2 1 k1 zs 0 zoffset ) (x=y ) ~ QZ = [0kx;max ; kx;max ][0ky;max ; ky;max ] Musable(x=y) = int Mwid(x=y) 1 sin max;(x=y) :

(7)

(8) (9) (10)

C. GO Contributions to the Algorithm The setup to be analyzed, depicted in Fig. 1, consists of an offset parabolic shaped-edge reflector fed by a low gain horn, with an acquisition plane inside the Fresnel region of the reflector. There are three coordinate systems: Oo is centered in the vertex of the parabolic surface and z-directed according to the propagation of the scattered field by the reflector. Os is centered in the projected surface of the reflector, with the same z axis orientation as previous, but linearly shifted zoffset . Of is the feeder’s local system, z-directed with the maximum radiation pattern direction. The projected surface is located as close as possible to the reflector, but keeping all the surface currents at zs < 0 values. A block scheme of the whole hybrid algorithm is shown in the matrix of Fig. 4. The lower row corresponds to the calculation of the aperture fields (2nd column) through GO formulations (operation A) and current projection (op. C) over an analytical model of the CATR (1st column). A legend of the operations involved in the task flow is attached. The aperture field distribution (2nd column) is calculated through GO techniques, since the use of GO is a simple approach to the field representation. Two main hypothesis are assumed to use GO in the feed modeling: first the reflector is in the farfield of the feeder; the second one requires a large reflector, in terms of wavelength [16], to assume the high frequency hypothesis for the surface currents. At low frequencies, for which the GO approach would be insufficient, a higher order formulation for the aperture fields should be added. In our case of study, the serrations are assumed to be large (much larger than 5 wavelengths), so HF assumption can be taken.

D. Spectral Contributions to the Algorithm According to Fourier theory, the spatial window can be obtained by performing an IFT over the PWS of the projected surface. As presented in [12], [16]–[19], this calculation can be accurate and flexible for an arbitrary polygonal projected shape. In [14], it is approached through a convolution of the window’s spectral distribution with the Fourier coefficients of the feeder’s GO pattern projected over the acquisition plane, assuming that this pattern is spectrally very narrow banded. Our approach doesn’t work on this assumption but avoids the spectra convolution by performing the windowing in the spatial domain. Moreover, the use of the closed form expression proposed in [17] avoids the truncation of rim information through analytical analysis of the geometry. Truncation effects can be noticed in [14], and can also be minimized if a conformal mesh of the serrations is considered, as proposed in [2]. The proposed approach requires two M 2 -order O(M 2 log M 2 ) FFT calculations (Operations C,E), in addition with a O(M 2 )M times M multiplication (1st row, 1st column) to reach the spectral information of the reflector tapered by the feeder pattern. This spectral distribution is propagated towards the za = 0 acquisition plane multiplying each discrete spectral plane wave contribution by a propagation factor (Op. D). The electric field at the acquisition plane za = 0 is obtained performing a M 2 -order FFT (Op. E) over the propagated spectrum. Operation F evaluates if more acquisition planes are of interest and, in that case, loops back to Op. D. In practical terms, this means that the computation time used by the hybrid formulation is inelastic with respect to the number of acquisition planes and, thus, it is suitable for quiet zone volumetric characterization. It must be noted that the furthest the acquisition plane is set, the less time Ops. D, E take, since increasingly lower number of plane waves need to be processed. The tasks have been described for one field component, potentially the co-polar contribution. The process to obtain the other electric field components requires the analogous replication of columns 3rd and 4th for the other aperture field components. E. Accuracy of the Algorithm The hybrid algorithm is designed as a chain of different order Physics approximations based on certain hypothesis: the “feed in farfield” can be assumed for most CATR facilities and the “currents projection” approach should also be valid whenever the GO-PWS fields are of interest and the reflector is parabolic. However, the HF hypothesis is not valid if the frequency of operation is very low. Consequently, the accuracy analysis will focus on reaching an evaluator of the algorithm’s reliability for this hypothesis. A pragmatic approach to this evaluation deals with the achievement of a signal to error figure which may determine

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if the proposed algorithm is valid to analyze a CATR reflector, for a particular setup. It is realistic to obtain this signal to error figure through the energy ratio between two magnitudes (11). The numerator of this fraction is the best achievable approach to the quiet zone field (reference value). The denominator is the committed error when the quiet zone field is based on the HF approximation, as in (12).

snr

(x; y; z; )g = ffEEQZ 3 (x; y ; z; )g @

(11)

QZ 3 (x; y; z ; ) = E @ (x; y; z ; ) 0 E EQZ QZ QZ

(x; y; z; ):

(12)

It is convenient to express the numerator of (11) through Parseval’s theorem (13). This expression states that only the PWS modes inside

QZ contribute to the energy  of the reference distribution. It also states that the integration of the spectral distribution within QZ at an arbitrary zs > 0 plane leads to the same  figure as when the subintegral distribution is the spectrum existing at zs = 0.

( ;

) =

 fE x; y z;  g

xy

( ;

)

E x; y z;  1 dx 1 dy

= 4 112

~

~ ( ; = 0; )dkx 1 dky : (13)

E x; y zs

Fig. 6. Circular and serrated reflector projected rim respect to

Some numeric results can be offered and particular signal to error figures can be extracted. To this purpose, the formulation used as reference is the time-domain full wave technique implemented by CST software. The test structure is a 400 mm-wide PEC plate placed in xy plane, over which a y -polarized plane wave impinges. Periodic boundary conditions on its y -axis and freespace boundary conditions over its x-axis are set. This offers a realistic approach for a 1-D analysis along the x dimension. The purpose of this setup is to test the HF hypothesis for a fixed physical size scatterer between 5 GHz and 80 GHz. The induced homogeneous HF-currents Jind are compared with the full wave solution. These results are evaluated with (11) and (13) to reach the snr figures (Fig. 5(a)). The full-wave accuracy results are taken as the “reference” accuracy of the algorithm. It is relevant to point out the increase of accuracy obtained when moving up in frequency or shifting the acquisition plane away from the scatterer, as it is the algorithm’s scenario of interest: mmWave CATRs. The scatterer’s edge conformation is not considered with this canonical 1D problem. However, this lack of generality is overcome through the uniform illumination scheme, which is the HF hypothesis worst case scenario, given that it has the strongest impact on edge effects. An empirical expression of SNR as a function of frequency is extracted from the full wave results. The logarithmic figure of accuracy SN R(f [GHz ]) is best-fit to a second order polynomial, as in (14), and the ai coefficients (Fig. 5(b)) are calculated along the evaluated longitudinal shift domain. For sake of generality, it is also reached (15), which draws the SNR figure from the electrical size of the scatterer 3, in wavelengths. This is obtained bearing in mind that the physical size of the scatterer remains constant and equal to 400 mm among all the simulations

(

SN RdB fGHz SN RdB

Fig. 5. (a) Logarithmic SNR vs. frequency and acquisition plane distance using as reference the CST calculation and (b) the polynomial best fit SNR figure, as a function of the acquisition plane distance.

)=

(3) =

2

i=0 2

i=0

(

ai 1 fGHz ai 1

)i

3 1 3 i: 4

(14) (15)

O

coord. sys.

Fig. 7. Normalized amplitude, horizontal and vertical cuts.

III. RESULTS—VALIDATION Our purpose is to obtain plane acquisitions of electric field inside the Fresnel region of a serrated-edge reflector. Thus, the results to be studied will be E/H-plane cuts of the diffracted electric field, in a compact-range, where the feeder is pointing to the center of the reflector. The obtained results are compared to electric field results from GRASP8, that is the reflector analysis tool more widely used to carry out PO analysis over reflector antennas [22]. A. Circular Offset Reflector A first test structure is a circular offset scatterer, with the acquisition setup proposed in Fig. 1. The corresponding parameters of simulation are indicated in Table I. The circular projected surface is modeled by constructing a 60 vertices piecewise polygonal rim. Isotropic feeder is chosen in order to weight as uniformly as possible all the surface contributions on the scatterer and, thus, avoid weighting down certain regions, which could mask algorithm eventual inaccuracies. Normalized amplitude and phase for both E and H planes are depicted in Figs. 7 and 8. As it can be deduced from these figures, good agreement is achieved, in general terms, between PO and GO-PWS results.

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Fig. 8. Normalized phase, horizontal and vertical cuts.

TABLE I TEST REFLECTOR A: CIRCULAR REFLECTOR SIMULATION

Fig. 10. Normalized phase, horizontal and vertical cuts; serrated reflector.

ment. The analysis performed in [16] using MoM, observes similar discrepancies between PO and MoM. In any case, this hybrid method can give accurate results to bound the ripple and the tapering in a single reflector CATR, if the hypotheses introduced in Section II-C are accomplished. C. Performance of the Algorithm

TABLE II TEST REFLECTOR B: SERRATED CIRCULAR REFLECTOR SIMULATION

After simulation results have been shown, speedup notes must be added to offer a clear idea of the algorithm’s global performance. Simulation times are considered, as overall indicators. The algorithm has been implemented in FORTRAN code, and uses high performance FFT external modules. Simulations were carried out in a 2.8 GHz double core Pentium 4 processor, with 4 GB of RAM. GRASP8 took 51.200 s and 31.500 s to reach the results concerning reflectors A and B respectively. On the other side, the implementation of the algorithm took 47 s and 11 s to reach its results. IV. CONCLUSIONS: APPLICATION TO CATR DESIGN

Fig. 9. Normalized amplitude, horizontal and vertical cuts; serrated reflector.

B. Serrated Circular Offset Reflector A circular offset reflector with linear serrated edge is now studied. The setup is the same as previous, with the definition of Table II. This second reflector has a higher F/D, which minimizes the amplitude slope in the vertical cut; 36 test serrations of 0.1 m length are placed around a circular rim with 0.2 m of radius. The feeder follows a Gaussian pattern, with 01 dB at  = 10 and non -dependent. Amplitude and phase cuts are shown in Figs. 9 and 10. A good agreement is observed for both magnitudes in the vertical cut, whereas horizontal acquisition has not the same level of agree-

The development of this algorithm has, as main goal, to speed up the design of conformal serrated edge reflectors operating as wave collimators in compact range facilities. By performing serrations in the edge of parabolic reflectors, amplitude as well as phase ripples of the diffracted electric field decrease. Thus, the field distribution gets closer to the local plane one, which is our ideal situation. Typical peak to peak values are 1 dB for amplitude and 10 degrees for phase. The design process uses as unknowns the geometrical variables which define the CATR facility, while the cost functions are the amplitude and phase ripple of the electric field inside the quiet zone. High time-consuming unitary simulations have as main consequence prohibitive or, at least, incomplete design processes. The use of efficient algorithms, such as the one proposed in this communication, is necessary if complete design processes are desired. In this work, an algorithm to analyze arbitrary piecewise rim reflectors has been presented. The propagation problem was divided into regions, for each one of which, a different analysis solution was used. To carry out joint optimization of the different order electromagnetic physics, a deep study of the field sampling was presented and analytical formulation of the sampling problem was proposed for the general case. These formulations were implemented in order to reach both more accurate results and a computational speedup. Some clear ideas were given about the extension of the algorithm to higher order physics. In order to test the algorithm, a closed form expression of the algorithm’s accuracy was reached and experimentally developed with a full wave method. Moreover, field acquisitions were compared with theoretical results obtained through GRASP8, which uses PO-order techniques, for two different canonical reflectors.

REFERENCES [1] H. F. Schluper, “Compact antenna test range analysis using physical optics,” in Proc. 1987 Antenna Measurement Techniques Association Symp., Seattle, WA, pp. 309–312.

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[2] Y.-C. Chang and J. I. M. Jim, “A PTD analysis of serrated edge compact range reflectors,” in Proc. 1994 Antenna Measurement Techniques Association Symp., Long Beach, CA, pp. 175–179. [3] F. Jensen, L. Giauffret, and J. Marti-Canales, “Modeling of compact range quiet-zone fields by PO and GTD,” in Proc. 1999 Antenna Measurement Techniques Association Symp., Monterey Bay, CA, pp. 242–247. [4] A. Miura and Y. Rahmat-Samii, “Spaceborne mesh reflector antennas with complex weaves: Extended PO/Periodic-MoM analysis,” IEEE Trans. Antennas Propag., vol. 55, no. 4, Apr. 2007. [5] A. Michaeli, “Elimination of infinities in equivalent edge currents, part I: Fringe current components,” IEEE Trans. Antennas Propag., vol. 34, no. 7, Jul. 1986. [6] F. Jensen and K. Pontoppidan, “Modeling of the antenna-to-range coupling for a compact range,” in Proc. 2001 Antenna Measurement Techniques Association Symp., Denver, CO, pp. 387–391. [7] S. C. van Someren Greve, L. G. T. van de Coevering, and V. J. Vokurka, “On design aspects of compact antenna test ranges for operation below 1 GHz,” in Proc. 2001 Antenna Measurement Techniques Association Symp., Monterey Bay, CA, pp. 248–253. [8] M. Ando, “PO and PTD analyses of offset reflector antenna patterns,” presented at the Antennas and Propagation Society Int. Symp., 1988. [9] S. Kulkarni, R. Lemdiasov, R. Ludwig, and S. Makarov, “Comparison of two sets of low-order basis functions for tetrahedral VIE modelling,” IEEE Trans. Antennas Propag., vol. 52, no. 10, Oct. 2004. [10] P. A. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields, Reissued ed. Piscataway, NJ: IEEE Press, 1996. [11] R. J. Pogorzelski, “Improved efficient field computation via fast Fourier transforms,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 27–30, 2005. [12] S. Liao and R. J. Vernon, “A fast algorithm for computation of electromagnetic wave propagation in half-space,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 2068–2075, Jul. 2009. [13] J. J. H. Wang, “An examination of the theory and practices of planar near-field measurement,” IEEE Trans. Antennas Propag., vol. 36, no. 6, pp. 746–753, Jun. 1988. [14] J. P. McKay and Y. Rahmat-Samii, “Compact range reflector analysis using the plane wave spectrum approach with an adjustable sampling rate,” IEEE Trans. Antennas Propag., vol. 39, no. 6, pp. 746–753, Jun. 1991. [15] Hu, M. Arrebola, R. Cahill, J. A. Encinar, V. Fusco, H. S. Gamble, Alvarez, Yu, and F. Las-Heras, “94 GHz dual-reflector antenna with reflectarray subreflector,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3043–3050, Oct. 2009. [16] F. Jensen, “Polarization dependent scattering from the serrations of compact ranges,” in Proc. AMTA 2007, St. Louis, MO, pp. 150–155. [17] S.-W. Lee and R. Mittra, “Fourier transform of a polygonal shape function and its application in electromagnetics,” IEEE Trans. Antennas Propag., vol. 31, no. 1, pp. 99–103, Jan. 1983. [18] T.-H. Lee and W. D. Burnside, “Performance trade-off between serrated edge and blended rolled edge compact range reflectors,” IEEE Trans. Antennas Propag., vol. 44, no. 1, pp. 87–96, Jan. 1996. [19] P. A. Beeckman, “Prediction of the Fresnel region field of a compact antenna test range with serrated edges,” IEE Proc. Microw. Antennas Propag., vol. 133, no. 2, pp. 108–114, Apr. 1986. [20] P. A. Beeckman, “Control of far-field radiation patterns of microwave reflector antennas by using serrated edges,” IEE Proc. Microw. Antennas Propag., vol. 134, no. 3, pp. 270–274, Jun. 1987. [21] G. Parini and M. Philippakis, “Use of quiet zone prediction in the design of compact antenna test ranges,” IEEE Trans. Antennas Propag., vol. 143, no. 3, pp. 193–199, Jun. 1996. [22] R. L. Lewis and A. C. Newell, “An efficient and accurate method for calculating and representing power density in the near-zone of microwave antennas,” IEEE Trans. Antennas Propag., vol. AP-36, no. 6, Jun. 1988. [23] E. B. Joy and R. E. Wilson, “Shaped edge serrations for improved compact range performance,” in Proc. Antenna Measurement Techniques Association Meeting, Ottawa, Sep. 1986, pp. 23–25. [24] A. Muñoz-Acevedo, M. Sierra-Castañer, and J. L. Besada, “Efficient and accurate hybrid GO-spectral algorithm to design conformal serrated-edge reflectors operating as collimators in millimeter wave compact ranges,” presented at the 2010 Proc. Antenna Measurement Techniques Association Symp., Atlanta.

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Time-Domain Microwave Imaging of Inhomogeneous Debye Dispersive Scatterers Theseus G. Papadopoulos and Ioannis T. Rekanos

Abstract—A time-domain inverse scattering method for the reconstruction of inhomogeneous dispersive media described by the Debye model is presented. The method aims to the simultaneous reconstruction of the spatial distributions of the optical and static permittivity as well as of the relaxation time. The reconstruction of the scatterer is based on the minimization of a cost functional, which describes the difference between measured and estimated values of the electric field. The fulfillment of the Maxwell’s curl equations is set as constraint by means of Lagrange multipliers in an augmented functional. The Fréchet derivatives with respect to the scatterer properties are derived analytically and can be utilized by any gradientbased optimization technique. The proposed reconstruction technique is based on the Polak-Ribière nonlinear conjugate-gradient algorithm, while the finite-difference time-domain (FDTD) method is employed for the solution of the direct and the adjoint electromagnetic problem. Numerical results for the reconstruction of one-dimensional layered scatterers illustrate the performance of the proposed method. Index Terms—Debye model, dispersive media, finite-difference time-domain (FDTD), inverse scattering, Lagrange multipliers.

I. INTRODUCTION Microwave imaging is an electromagnetic inverse scattering problem of significant interest because of its numerous applications in medical imaging, geophysical prospecting, nondestructive testing, etc. In general, the problem can be described as follows. Given a set of known electromagnetic wave excitations that illuminate a scatterer, the total or the scattered field is measured at locations around the scatterer domain. The objective is to reconstruct the spatial distribution of the electromagnetic properties of the scatterer by inverting the measurement data. This problem is nonlinear because the scattered field is a nonlinear function of the scatterer properties. Furthermore, it is an ill-posed problem because the operator that maps the scatterer properties to the scattered field is compact [1]. Depending on the time variation of the excitations used, two approaches and associated methodologies appear. In the first approach, namely the frequency-domain one, the excitation is considered monochromatic and the inverse scattering problem is formulated in the frequency domain by neglecting the time dependence of the fields [2]–[7]. In single frequency-domain microwave imaging, the case of dispersive scatterer properties is treated similarly to the case of nondispersive ones, because dispersion phenomena are not generated by monochromatic excitations. Finally, the reconstruction resolution of the scatterer properties is governed and actually limited by the wavelength of the excitation [5], [8]. It should be mentioned that frequency-domain microwave imaging could still be applicable to the case of dispersive scatterers, if multiple distinct excitation frequencies are utilized. To improve the reconstruction resolution and at the same time to cope with dispersive scatterers, a second microwave imaging approach, namely the time-domain one, has been proposed where the excitation field is wideband and the microwave imaging problem is Manuscript received April 19, 2011; revised June 30, 2011; accepted August 03, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. The authors are with the Physics Division, School of Engineering, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2173150

0018-926X/$26.00 © 2011 IEEE

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formulated in the time domain [9]–[14]. In time-domain inversion techniques, the reconstruction resolution depends on the bandwidth of the excitation. In particular, the resolution is limited by the shortest effective wavelength of the excitation field. Time-domain microwave imaging can enhance the reconstruction resolution of nondispersive scatterers and it is also capable to reconstruct the spatial distribution of the characteristic parameters of dispersive ones. Thus, it is possible to reconstruct the time-domain representation of the complex relative permittivity of a scatterer [9], or the spatial distributions of the characteristic parameters of dispersion models [13]. Furthermore, a time-domain inverse scattering technique to estimate the average dielectric and conductivity properties of Debye media has been proposed [14]. In this communication, a time-domain microwave imaging method for the reconstruction of the characteristic parameters of Debye scatterers is presented. The Fréchet derivatives of an augmented cost functional with respect to the scatterer properties are derived analytically utilizing the calculus of variations [15]. These derivatives are utilized by the Polak-Ribière nonlinear optimization algorithm [16], while the finite-difference time-domain (FDTD) method [17] is employed for the electromagnetic analysis. In previous attempts only the static and optical permittivities were reconstructed simultaneously, whereas the relaxation time was considered known [14], [18]. The novelties of the present work are, first, the derivation of the Fréchet derivative with respect to the relaxation time of the Debye medium and second, the simultaneous reconstruction of the static and optical relative permittivities as well as of the relaxation time. In Section II, the mathematical formulation of the problem is presented. The inversion algorithm is described in Section III. The proposed method is applied to the reconstruction of layered planar Debye scatterers and numerical results are given in Section IV. Finally, Section V provides the conclusions.

B. Augmented Cost Functional Our objective is to estimate the three parameters characterizing a Debye medium, i.e., "~ ; 1", and ~ (tilde is used to denote the original scatterer properties), by inverting the electric field measurements ~ ~ ik . This inverse scattering problem is treated as an optimization one. E In particular, if p = [" ; 1";  ] is an estimate of the scatterer properties, the scatterer reconstruction is achieved by minimizing the cost functional that describes the discrepancy between the measured and ~ ik , at the meaestimated electric field. The estimated electric fields, E surement positions, are derived by the solution of the direct scattering problem. Moreover, the fulfillment of the Maxwell’s curl equations as well as of the Debye polarization relation can be introduced in the cost functional by means of Lagrange multipliers. Thus, the augmented cost functional is given by

1

1

1 ~ H; ~ J; ~ ~ F (p; E; e; ~h; q~) = 2

I

T

+

0

i=1

V

I

K

i=1 k=1 0

T

kE~ ik 0 E~~ ik k2 dt

1 r 2 E~ i + @t H~ i )

[~ hi (

1 r 2 H~ i 0 "0 "1 @tE~ i 0 J~i 0 J~si ) ~ i )] dvdt +~ qi 1 (J~i +  @t J~i 0 "0 1"@t E +~ ei (

(4)

where J~si is the current density that generates the ith incidence and ~i is the polarization current density that arises inside the J~i = @t P scatterer, both corresponding to the ith incidence, and V is the doqi are the Lagrange main of computation. The vector fields ~ei ; ~hi , and ~ multipliers. C. Maxwell’s Equations for the Estimated Fields and the Lagrange Multipliers

II. MATHEMATICAL FORMULATION OF THE PROBLEM

The minimization of the augmented cost functional (4) requires that its first variation is zero, i.e.,

A. Debye Dispersion Model

~ H; ~ J; ~ ~ F (p; E; e; ~h; q~) = 0:

The relative complex permittivity of an inhomogeneous scatterer exhibiting Debye dispersion is given by [19]

1

"r (!; ~ r) = " (~ r) +

1"(~ r) 1 + j! (~ r)

(1)

1 is the optical relative permittivity,  is the relaxation time, 0 "1 ("s is the static relative permittivity), and ! is the

where "

1" = "s

angular frequency. The time-domain polarization relation for a Debye medium is given by

1

~ (t; ~ ~ (t; ~ ~ (t; ~ D r ) = "0 " (~ r)E r) + P r)

(2)

~ is the electric flux density, E ~ is the electric field intensity, and where D ~ is the polarization field satisfying the differential equation P ~ (t; ~ ~ (t; ~ ~ (t; ~ r )E r): P r ) +  @t P r) = "0 1"(~

(3)

We assume that the Debye scatterer is nonmagnetic ( = 0 ) and lies within the scatterer domain S . The domain S is excited by I incident waves, while for each incidence the electric field is measured at K positions around the scatterer for the time interval [0; T ]. Hence, a set of I 2 K electric field measurements is obtained, which are denoted as ~ ~ ik where i = 1; . . . ; I and k = 1; . . . ; K . We note that the time T of E measurement is selected in a way that the measured field at the farthest receiver has significantly faded out.

(5)

If we express F in terms of variations of the arguments of the cost functional, then we derive the relations that have to be fulfilled by the estimated fields and the Lagrange multipliers. This approach has been presented in [10] for the case of nondispersive scatterers. The first obvious result of the stationarity condition (5) is that, for each incidence ~ i , and J~i are derived by the solution of the ~ i; H i, the estimated fields E direct scattering problem, which is formulated by the coupled differential equations

r 2 E~ i + @t H~ i = 0 r 2 H~ i 0 "0 "1 @tE~ i 0 J~i 0 J~si = 0 ~ i = 0: J~i +  @t J~i 0 "0 1"@t E

(6) (7) (8)

The initial conditions for the direct scattering problem are

j

~ i t=0 = 0; E

j

~ i t=0 = 0; H

j

J~i t=0 = 0

(9)

while appropriate boundary conditions according to the particular geometry of the problem are set. Furthermore, the stationarity condition, F = 0, results in the equations that have to be satisfied by the Lagrange multipliers as well as

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their initial and boundary conditions. Actually, the Lagrange multipliers corresponding to the ith incidence are governed by the following equations:

r 2 ~e 0 @ ~h = 0 r 2 ~h + "0 "1 @ ~e 0 ~j + (E~ 0 E~~ ) = 0 =1 ~ ~ e = 0 j 0  @ j + "0 1"@ ~ ~ ~ h j = = 0; j j = = 0 ~ e j = = 0; i

t

i

(10)

K

t i

i

i

ik

ik

(11)

k

i

t i

t i

i t

T

i t

T

i t

T

(12) (13)

0 1

where ~ji = "0 " @t q~i . In contrast to the estimated fields, the initial conditions for the Lagrange multipliers are defined at t = T . From (10)–(12), it is clear that the Lagrange multipliers, the so-called adjoint fields, expose a similar to the electromagnetic waves attitude. The source of the adjoint fields is the difference between the measured and the estimated values of the electric field. Moreover, from the signs of the partial time-derivatives in (10)–(12), which are opposite compared to the corresponding signs in (6)–(8), we conclude that the Lagrange multipliers are waves propagating backwards in time. However, by time-reversing, i.e., if we change the time variable (t = t T ), the time-transformed Lagrange multipliers satisfy Maxwell’s equations similar to (6)–(8).

0

D. Fréchet Derivatives The Fréchet derivatives of the cost functional with respect to the scatterer properties are obtained from the terms of F that include the first-order variations of the scatterer properties. In particular, the Fréchet derivatives are given by

= "F1 = 0"0

G"

G1"

I

T

i=1

t

0

F

...;G

; G1" ;

"

. . . ; G1

; G ;

"

...;G



T

:

(19)

p

p( +1) = p( ) + ( ) v( ) l

v( ) = 0G( ) +

(15)

i

and

T

i

;

l

i

G

G"

G of the cost functional with re-

The steps of each iteration of the Polak-Ribière inversion algorithm are as follows. Given an estimate of the scatterer (l) where l denotes the iteration, the direct scattering problem is solved for each incidence and the estimated values of the electromagnetic fields are obtained at the measurement points as well as inside the scatterer domain S . The next step is the solution of the equations satisfied by the time-reversed Lagrange multipliers. Having calculated both the electromagnetic fields and the adjoint ones (using the FDTD method), we are in a position to compute the gradient of the cost functional (19) using (14)–(16) and (18). Then, the properties vector (17) is updated according to the scheme [16]

T

i

I

G=

p

(14)

i

= 1" = 0 11" (E~ 1 ~j ) dt =1 0 = F (J~ 1 ~j ) dt: = "011"  =1 0 I

Hence, we form the gradient vector spect to , i.e.,

l

l

l

(20)

where

(~e 1 @ E~ ) dt i

Fig. 1. Geometry of the layered planar Debye scatterer.

(16)

i

i

The derivatives (14)–(16) can be utilized by any gradient-based optimization algorithm to reconstruct " ; 1", and  .

1

(l)

l

= arg min

G( ) 1 G( ) 0 G( 01) ( 01) v kG( 01) k2 l

l

l

l

l

p( ) + v( ) l

F

l

:

(21)

(22)

It should be mentioned that for the first iteration of the algorithm (l = 1) we set v(1) = 0G(1) . The aforementioned scheme is repeated iter-

atively until the discrepancy between the estimated and the measured fields is lower than a predefined threshold or until a total number of iterations is performed.

III. INVERSION ALGORITHM In the proposed inversion approach, the reconstruction of the scatterer properties is achieved by use of the Polak-Ribière conjugate gradient algorithm. The scatterer domain S is partitioned into N subdomains, Sn : n = 1; 2; . . . ; N . Within each individual subdomain Sn , the scatterer properties are assumed constant and are represented by the variables "n ; 1"n , and  n . Thus, the discretized representation of the scatterer is described by the vector

f

g

1

p=

11 ; . . . ; "1 ; 1"1 ; . . . ; 1"

"

N

N

1

; ;

...;

N

T

(17)

which has 3N components. As a result, the derivative of the cost functional with respect to the parameter xn (x stands for " ; 1", and  ) is given by the spatial integral of the corresponding Fréchet derivative on the area of the subdomain Sn , i.e.,

1

Gx

=

Gx dv S

(18)

IV. NUMERICAL RESULTS In order to illustrate the validity of the proposed inverse scattering methodology we have applied it to the reconstruction of layered planar Debye scatterers (Fig. 1). The total width of the scatterer is d and is surrounded by air. The electric field of the plane wave excitation is polarized parallel to the layers, thus the direct scattering problem is one-dimensional. Two excitation sources (I = 2) are placed at distance equal to d=2 and two receivers (K = 2) at distance d=4 from both sides of the scatterer. The excitation current density is a modulated Gaussian pulse given by

( ) = e0 ( 04 ) sin(2f t)u(t) (23) = 52108 Hz and a = 5 2108 sec01 . As shown in Fig. 1 the Js t

a

t

=a

c

where fc computational domain is truncated by means of PML absorbers placed at distance d from each side of the scatterer. For each excitation, the two measurement time series at the receivers’ positions are simulated by solving the direct scattering problem using the FDTD method. We

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Fig. 2. Cost function vs. the number of iterations, for the case of a two-layered Debye scatterer.

mention that in order to avoid the “inverse crime” the space discretization during the simulation of measurements is two times denser compared to the one applied in the inversion procedure. Two reconstruction examples have been considered. In both examples, the total width of the scatterer is d = 0:15 m. For the simulation of the measurement the FDTD spatial grid size and the time step were 1z = 3:75 2 1003 m and 1t = 6:25 2 10012 sec, respectively. The corresponding discretization when the FDTD is applied in the inversion procedure is 1z = 7:521003 m and 1t = 1:25210011 sec. In the first example, the scatterer consists of two Debye layers with equal widths d=2 where their properties differ. Actually, the original parame1 ters of the first layer are "~1 = 2; 1"1 = 4, and ~1 = 1 ns, whereas the 2 2 = 4 ; 1" = 1, corresponding parameters of the second layer are "~1 and ~2 = 1 ns. It is assumed that the widths of the two layers are a priori known for the solution of the inverse problem. Consequently, the unknowns of the inverse problem are six; three for each layer. The initial estimate of the scatterer properties, which are the same for both 1;2 = 1:1; 1"1;2 = 1:1, and  1;2 = 2 ns. The presented layers, are "1 inversion algorithm has been applied for 1500 iterations, whose completion is the actual stopping criterion of the iterative procedure. The cost function versus the number of iterations is illustrated in Fig. 2. Table I presents the values of the original and the finally estimated scatterer properties. In addition, Table I exhibits the absolute and the relative reconstruction errors for each parameter of each layer. The relative reconstruction errors for each parameter versus the number of iterations are presented in Fig. 3. It is more than obvious that the properties of the two-layered Debye medium are accurately reconstructed. The second example involves the reconstruction of a four-layered Debye scatterer, where the layer widths are all equal to d=4. The original values of the properties are given in Table II. The initial estimates of the scatterer properties are "n 1 = 1:1; 1"n = 1:1, and  n = 2 ns (n = 1; 2; 3; 4). Table II presents the values of the original and the finally estimated scatterer properties for each layer as well as the absolute and the relative reconstruction errors after 1500 iterations. The cost function versus the number of iterations is presented in Fig. 4. The relative reconstruction errors for each parameter versus the number of iterations are presented in Fig. 5. From both Table II and Fig. 5 we conclude that the reconstruction of the properties of the four-layered Debye medium is very accurate. However, the reconstruction accuracy is lower compared to the two-layered case. This is a reasonable result because in the second example the number of unknowns is doubled (12 unknowns) and the scatterer structure is more complicated compared to the first example. Finally, the proposed method has been applied to reconstruct the four-layered

1

Fig. 3. Relative reconstruction error of (a) " , (b) ", and (c)  vs. the number of iterations, for the case of a two-layered Debye scatterer.

TABLE I RECONSTRUCTION RESULTS OF A TWO-LAYERED DEBYE SCATTERER— ORIGINAL PARAMETER VALUES (TILDED), ESTIMATED VALUES, ABSOLUTE AND RELATIVE RECONSTRUCTION ERRORS

Debye scatterer by inverting simulated noisy measurements. In particular, the measurements have been contaminated by additive white

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TABLE II RECONSTRUCTION RESULTS OF A FOUR-LAYERED DEBYE SCATTERER— ORIGINAL PARAMETER VALUES (TILDED), ESTIMATED VALUES, ABSOLUTE AND RELATIVE RECONSTRUCTION ERRORS

Fig. 4. Cost function vs. the number of iterations, for the case of a four-layered Debye scatterer.

TABLE III RECONSTRUCTION RESULTS OF A FOUR-LAYERED DEBYE SCATTERER USING NOISY MEASUREMENTS (SNR = 30 DB)—ORIGINAL PARAMETER VALUES (TILDED), ESTIMATED VALUES, ABSOLUTE AND RELATIVE RECONSTRUCTION ERRORS

Gaussian noise with signal-to-noise ratio (SNR) equal to 30 dB. The reconstruction results derived after 1500 iterations are presented in Table III. From Table III, we conclude that, even in the case of noisy measurements, the reconstruction of the original scatterer parameters is adequate; the relative reconstruction errors for all parameters are lower than 2.5%. However, by comparing the reconstruction using noiseless measurements (Table II) with the one using contaminated measurements (Table III), we observe that the presence of noise deteriorates significantly the reconstruction accuracy of the method. Thus, regularization techniques should be adopted. V. CONCLUSION In this communication, a time-domain microwave imaging technique for the reconstruction of Debye dispersive scatterers is proposed. The Fréchet derivatives of the cost functional with respect to the Debye scatterer properties (" ; ", and  ) are derived analytically. These derivatives can be utilized by any gradient-based inverse scattering technique along with any time-domain computational method employed for the solution of the electromagnetic problem. In the present work, the FDTD and the Polak-Ribière optimization algorithm have been combined and the reconstruction of the spatial distribution of all

1 1

1

Fig. 5. Relative reconstruction error of (a) " , (b) ", and (c)  vs. the number of iterations, for the case of a four-layered Debye scatterer.

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the properties of the Debye scatterer has been achieved. Moreover, a novelty of the present work is the fact that all three parameters characterizing Debye scatterers have been reconstructed simultaneously. Future work is focused on the investigation of 2D and 3D reconstruction problems as well as on regularization to cope with the ill-posedness of the problem.

REFERENCES [1] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Berlin: Springer-Verlag, 1992. [2] M. Pastorino, S. Caorsi, and A. Massa, “A global optimization technique for microwave nondestructive evaluation,” IEEE Trans. Instrum. Meas., vol. 51, no. 4, pp. 666–673, Aug. 2002. [3] G. Franceschini, M. Donelli, R. Azaro, and A. Massa, “Inversion of phaseless total field data using a two-step strategy based on the iterative multiscaling approach,” IEEE Trans. Geosci. Remote Sens., vol. 44, no. 12, pp. 3527–3539, Dec. 2006. [4] A. Abubakar and P. M. van den Berg, “Iterative forward and inverse algorithms based on domain integral equations for three-dimensional electric and magnetic objects,” J. Computat. Phys., vol. 195, no. 1, pp. 236–262, Mar. 2004. [5] A. G. Tijhuis, K. Belkebir, A. C. S. Litman, and S. P. de Hon, “Theoretical and computational aspects of 2-D inverse profiling,” IEEE Trans. Geosci. Remote Sens., vol. 39, no. 6, pp. 1316–1330, Jun. 2001. [6] I. T. Rekanos, T. V. Yioultsis, and C. S. Hilas, “An inverse scattering approach based on the differential E-formulation,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 7, pp. 1456–1461, Jul. 2004. [7] P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, “Evolutionary optimization as applied to inverse scattering problems,” Inv. Probl., vol. 25, no. 12, pp. 123 003:1–123 003:41, Dec. 2009.

[8] M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microw. Theory Tech., vol. 32, no. 8, pp. 860–874, Aug. 1984. [9] S. He, P. Fuks, and G. W. Larson, “An optimization approach to timedomain electromagnetic inverse problem for a stratified dispersive and dissipative slab,” IEEE Trans. Antennas Propag., vol. 44, no. 9, pp. 1277–1282, Sept. 1996. [10] I. T. Rekanos, “Time-domain inverse scattering using Lagrange multipliers: An iterative FDTD-based optimization technique,” J. Electromagn. Waves Applicat., vol. 17, no. 2, pp. 271–289, Feb. 2003. [11] I. T. Rekanos, “Inverse scattering in the time domain: An iterative method using an FDTD sensitivity analysis scheme,” IEEE Trans. Magn., vol. 38, no. 2, pp. 1117–1120, Mar. 2002. [12] M. Gustafsson and S. He, “An optimization approach to two-dimensional time domain electromagnetic inverse problems,” Radio Sci., vol. 35, no. 2, pp. 525–536, Mar.–Apr. 2000. [13] E. Abenius and B. Strand, “Solving inverse electromagnetic problems using FDTD and gradient-based minimization,” Int. J. Numer. Methods Eng., vol. 68, no. 6, pp. 650–673, Nov. 2006. [14] D. W. Winters, E. J. Bond, B. D. V. Veen, and S. C. Hagness, “Estimation of frequency-dependent average dielectric properties of breast tissue using a time-domain inverse scattering technique,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3517–3528, Nov. 2006. [15] H. Sagan, Introduction to the Calculus of Variations. New York, NY: McGraw-Hill, 1969. [16] E. Polak, Computational Methods in Optimization: A Unified Approach. New York: Academic Press, 1971. [17] K. S. Kunz and R. J. Luebbers, The Finite Differece Time Domain Method for Electromagnetics. Boca Raton, FL: CRC Press, 1993. [18] M. Gustafsson, “Wave Splitting in Direct and Inverse Scattering Problems,” Ph.D. Dissertation, Lund University, Lund, Sweden, May 2000. [19] A. Ishimaru, Electromagnetics Wave Propagation, Radiation, and Scattering. Englewood Cliffs, NJ: Prentice-Hall, 1991.

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