IEEE Transactions on Antennas and Propagation [volume 59 number 7]

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JULY 2011

VOLUME 59

NUMBER 7

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas Axial-Ratio-Bandwidth Enhancement of a Microstrip-Line-Fed Circularly Polarized Annular-Ring Slot Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.-Y. Sze and W.-H. Chen Design of Triple-Frequency Microstrip-Fed Monopole Antenna Using Defected Ground Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W.-C. Liu, C.-M. Wu, and Y. Dai High-Gain Reconfigurable Sectoral Antenna Using an Active Cylindrical FSS Structure . . . . . . . . . . . . . . . . . A. Edalati and T. A. Denidni Exact Dipole Radiation From an Oblate Semi-Spheroidal Cavity Filled With DNG Metamaterial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. N. Askarpour and P. L. E. Uslenghi Bandwidth and Impedance-Matching Enhancement of Fractal Monopole Antennas Using Compact Grounded Coplanar Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Naghshvarian Jahromi, A. Falahati, and R. M. Edwards Multi-Chip RFID Antenna Integrating Shape-Memory Alloys for Detection of Thermal Thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Caizzone, C. Occhiuzzi, and G. Marrocco Three-Shaped-Reflector Beam-Scanning Pillbox Antenna Suitable for mm Wavelengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. G. Hay, S. L. Smith, G. P. Timms, and J. W. Archer System Fidelity Factor: A New Method for Comparing UWB Antennas . . . . . . . . . . . . . . . G. Quintero, J.-F. Zürcher, and A. K. Skrivervik Arrays Wideband Low-Loss Linear and Circular Polarization Transmit-Arrays in V-Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Kaouach, L. Dussopt, J. Lantéri, T. Koleck, and R. Sauleau 60-GHz Electronically Reconfigurable Large Reflectarray Using Single-Bit Phase Shifters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Kamoda, T. Iwasaki, J. Tsumochi, T. Kuki, and O. Hashimoto Relation Between the Array Pattern Approach in Terms of Coupling Coefficients and Minimum Scattering Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Rubio and J. F. Izquierdo Harmonic Beamforming in Time-Modulated Linear Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Poli, P. Rocca, G. Oliveri, and A. Massa Numerical and Analytical Techniques Fast Directional Multilevel Algorithm for Analyzing Wave Scattering . . . . . . . . . . . . . . . . . . . . . . . H. Chen, K. W. Leung, and E. K. N. Yung Efficient Computation of the Off-Diagonal Elements of the Vector-Potential Multilayered Periodic Dyadic Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. L. Fructos, R. R. Boix, and F. Mesa A Scalable Parallel Wideband MLFMA for Efficient Electromagnetic Simulations on Large Scale Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Melapudi, B. Shanker, S. Seal, and S. Aluru

2450 2457 2464 2473 2480 2488 2495 2502

2513 2524 2532 2538 2546 2557 2565

(Contents Continued on p. 2449)

(Contents Continued from Front Cover) A Quasi-3D Thin-Stratified Medium Fast-Multipole Algorithm for Microstrip Structures . . . . . . . J. L. Xiong, Y. Chen, and W. C. Chew Analytic Fields With Higher-Order Compensations for 3-D FDTD TF/SF Formulation With Application to Beam Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Singh, E. L. Tan, and Z. N. Chen Scattering by a Two-Dimensional Periodic Array of Vertically Placed Microstrip Lines . . . . . . . . . . . . . . . . . . . . . . A. K. Rashid and Z. Shen On the Co-Polarized Phase Difference of Rough Layered Surfaces: Formulae Derived From the Small Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Afifi and R. Dusséaux Wireless Parametric Design of Compact Dual-Frequency Antennas for Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Genovesi, S. Saponara, and A. Monorchio Integrated Wide-Narrowband Antenna for Multi-Standard Radio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Ebrahimi, J. R. Kelly, and P. S. Hall Multiband Inverted-F Antenna With Independent Bands for Small and Slim Cellular Mobile Handsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. F. AbuTarboush, R. Nilavalan, T. Peter, and S. W. Cheung Rectenna Application of Miniaturized Implantable Antenna Design for Triple-Band Biotelemetry Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.-J. Huang, C.-M. Lee, C.-L. Chang, L.-K. Chen, T.-C. Yo, and C.-H. Luo Spherical Horn Array for Wideband Propagation Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. Franek and G. F. Pedersen Location Specific Coverage With Wireless Platform Integrated 60-GHz Antenna Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. L. Amadjikpè, D. Choudhury, G. E. Ponchak, and J. Papapolymerou 3D Polarized Channel Modeling and Performance Comparison of MIMO Antenna Configurations With Different Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.-T. Dao, V.-A. Nguyen, Y.-T. Im, S.-O. Park, and G. Yoon Subwavelength Radio Repeater System Utilizing Miniaturized Antennas and Metamaterial Channel Isolator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Sarabandi and Y. J. Song Improved Two-Antenna Direction Finding Inspired by Human Ears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Zhou, H. Zhang, and H. Xin Foliage Attenuation Over Mixed Terrains in Rural Areas for Broadband Wireless Access at 3.5 GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. L. Chee, S. A. Torrico, and T. Kürner

2578 2588 2599 2607

2619 2628 2636 2646 2654 2661 2672 2683 2691 2698

COMMUNICATIONS

Modification of Radiation Patterns of First Harmonic Mode of Slot Dipole for Dual-Frequency Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y.-C. Chen, S.-Y. Chen, and P. Hsu Low-Profile Composite Helical-Spiral Antenna for a Circularly-Polarized Tilted Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Nakano, N. Aso, N. Mizobe, and J. Yamauchi A High-Isolation Dual-Polarization Microstrip Patch Antenna With Quasi-Cross-Shaped Coupling Slot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Lu, Z. Kuai, X. Zhu, and N. Zhang Omnidirectional Circularly Polarized Antenna Utilizing Zeroth-Order Resonance of Epsilon Negative Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.-C. Park and J.-H. Lee A Compact Sequential-Phase Feed Using Uniform Transmission Lines for Circularly Polarized Sequential-Rotation Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S.-K. Lin and Y.-C. Lin A New Formula for the Pattern Bandwidth of Fabry-Pérot Cavity Antennas Covered by Thin Frequency Selective Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. A. Hosseini, F. Capolino, and F. De Flaviis Switchable Frequency Selective Slot Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Sanz-Izquierdo, E. A. Parker, and J. C. Batchelor Full-Wave Scattering From a Grooved Cylinder-Tipped Conducting Wedge . . . . . . . . . . . . . . . . . . . . . . A. C. Polycarpou and M. A. Christou Zeroth-Order Complete Discretizations of Integral-Equation Formulations Involving Conducting or Dielectric Objects at Very Low Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Ubeda, J. M. Tamayo, J. M. Rius, and A. Heldring Fast Convergence of Fast Multipole Acceleration Using Dual Basis Function in the Method of Moments for Composite Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. S. Tong and W. C. Chew Validation of Rain Spatial Classification for High Altitude Platform Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Zvanovec and P. Pechac

2707 2710 2713 2717 2721 2724 2728 2732 2735 2741 2746

COMMENTS

Comments on “3-D Numerical Mode-Matching (NMM) Method for Resistivity Well-Logging Tools” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Dun, Y. Kong, L. Zhang, and J. Yuan Comments on “Topology Optimization of Sub-Wavelength Antennas” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. L. Thal

2751 2752

CORRECTIONS

Corrections to “Traveling Waves on Three-Dimensional Periodic Arrays of Two Different Alternating Magnetodielectric Spheres” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Li and R. A. Shore Corrections to “Stability Analysis and Improvement of the Conformal ADI-FDTD Methods” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Dai

2753 2755

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Digital Object Identifier 10.1109/TAP.2011.2160833

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Axial-Ratio-Bandwidth Enhancement of a Microstrip-Line-Fed Circularly Polarized Annular-Ring Slot Antenna Jia-Yi Sze, Member, IEEE, and Wei-Hung Chen

Abstract—The design of a new microstrip-line-fed wideband circularly polarized (CP) annular-ring slot antenna (ARSA) is proposed. Compared with existing ring slot antennas, the ARSAs designed here possess much larger CP bandwidths. The main features of the proposed design include a wider ring slot, a pair of grounded hat-shaped patches, and a deformed bent feeding microstrip line. The ARSAs designed using FR4 substrates in the L and S bands have 3-dB axial-ratio bandwidths (ARBWs) of as large as 46% and 56%, respectively, whereas the one using an RT5880 substrate in the L band, 65%. In these 3-dB axial-ratio bands, impedance is also achieved. matching with

TABLE I CP BANDS AND SIZES OF SOME EXISTING RING AND WIDE SLOT ANTENNAS

VSWR 2

Index Terms—Annular-ring slot antenna, axial ratio, circular polarization.

I. INTRODUCTION

C

IRCULARLY polarized (CP) antennas, known for their capabilities of reducing polarization mismatch [1] and moderately suppressing multipath interferences [2], require larger bandwidths to support emerging wireless-communication applications, especially those involving satellite communications [3]. Because of the advantages of large impedance bandwidths and low profiles, printed slot antennas have received much attention in antenna design requiring enhanced 3-dB axial-ratio bandwidths (ARBWs). CP printed slot antennas can be divided into two categories: wide and ring slot types. For comparison, the antenna sizes and frequency bands of some CP ring slot antennas [4]–[6] and wide slot antennas [7]–[12] are summarized in Table I. In this table, is the center frequency of the 3-dB axial-ratio band (or called CP band for is the corresponding free-space wavelength. In short), and addition, RS in the table stands for ring slot and WS for wide slot. The antenna area (the slot area) is the area of the smallest rectangle that can enclose the low-profile antenna (the ring slot or wide slot). This table indicates that, in general, the 3-dB ARBWs (12–30.6%) of the CP wide slot antennas in [7]–[12]

Manuscript received November 09, 2009; revised October 22, 2010; accepted December 03, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported by the National Science Council of the Republic of China (on Taiwan) under Grant NSC 97-2221-E-606-024. J.-Y. Sze is with the Electrical and Electronic Engineering Department, Chung Cheng Institute of Technology, National Defense University, Taoyuan 335, Taiwan (e-mail: [email protected]). W.-H. Chen was with the Electrical and Electronic Engineering Department, Chung Cheng Institute of Technology, National Defense University, Taoyuan 335, Taiwan. He is now with Ayecomm Technology Co., Ltd., Hsinchu 300, Taiwan, R.O.C. Digital Object Identifier 10.1109/TAP.2011.2152314

tend to be larger than those (4.4–10%) of the CP ring slot antennas in [4]–[6]. However, the normalized antenna areas of the former are also larger than those of the latter, except for the antennas in [8], [9] and [12]. In particular, although the 3-dB ARBWs of the wide slot antennas in [10] and [11], respectively, are as large as 45% and 31%, their normalized antenna areas are comparatively large and cannot be reduced if the 3-dB ARBWs are to be maintained. This is because reducing the ground-plane size (and hence the antenna area) would raise the axial ratios in the lower-frequency range of their original CP bands, thus significantly shrinking the CP bands and shifting up the associated CP-band center frequencies. The goal of this study is to propose a method for designing a new CP annular-ring slot antenna (ARSA) that not only preserves the compact size of a ring slot antenna but also possesses a greatly enhanced 3-dB ARBW that is comparable to those of wide slot antennas. Note that the CP radiations of the wide slot antennas in [7]–[9] are mainly due to perturbation structures that are embedded in the wide slots and that are 45 and 135 away from the feeding lines. The rectangular patch in [7], the cross patches in [8], and the pair of grounded L-shaped strips in [9] can be regarded as perturbation structures. On the other hand, the CP radiations of the antennas in [10]–[12] are mainly due to bent feeding lines protruded into the wide slots. The above discussions suggest that in designing a broadband CP ARSA we should simultaneously embed a perturbation structure in the annular-ring slot and bend the feeding line. The perturbation structure is a pair of grounded hat-shaped patches for producing a wide CP band. The bent feeding line is deformed for improving impedance matching within the wide CP band. For the grounded hat-shaped patches of appropriate size to be embedded in the ring slot, the width of the ring slot

0018-926X/$26.00 © 2011 IEEE

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is the width of the ring In the figure, slot. The section is symmetrically widened to have a width and the section is extended toward the right side to of have a width of . If , a bent (or inverted-L) feeding microstrip line is formed, which can broaden the CP band of the CP ARSA fed by a straight feeding line and loaded with rectangular patches. With , the grounded feeding structure is referred to as a deformed bent feeding line, which can improve the impedance matching of the designed antenna in the broadened CP band. Finally, with the rectangular patches replaced by hat-shaped patches, the CP band can be further widened.

III. CONCEPTS AND PROCEDURE OF ANTENNA DESIGN

A. The Size of the Annular-Ring Slot

Fig. 1. (a) Configuration of the proposed CP annular-ring slot antenna fed by a deformed bent microstrip line. (b) A simple metal-box-backed CP antenna for producing unidirectional radiations.

is set equal to or larger than those in [4]–[6]. Hence, the large width of the ring slot used here is also a main feature of the proposed design. In addition, it is demonstrated that the new CP ARSA can be further developed to possess a unidirectional CP radiation pattern with a slightly downgraded 3-dB ARBW by placing a simple open metal box underneath.

II. ANTENNA CONFIGURATION Fig. 1(a) shows the geometry of the proposed CP ARSA, in which the deformed bent feeding line and the perturbation structure consisting of a pair of grounded hat-shaped patches are redrawn separately for more clarity. The antenna is to be fabricated on a square microwave substrate with side length , thickness , dielectric constant , and loss tangent . The radiating annular-ring slot of outer radius and inner rais fed from the direction by a -wide 50- midius crostrip line printed on the side of the substrate opposite to the slot. Each hat-shaped patch re-formed from a rectangular patch can be divided into two parts. One part having a size of is originally a rectangular patch and in the last step of the design procedure is modified to be a half ellipse having the major and , respectively. The other part is a and minor axes of rectangular patch. The rectangular patches are tangential to the outer circle of the ring slot at the intersection points of the 45 -oriented diametric line and the outer circle. The two hat-shaped patches can perturb the magnetic current distribution in the ring slot so as to produce two equal-amplitude resonant modes that have a phase difference (PD) of around 90 . The feeding microstrip line protrudes into the annular-ring slot , measured from the -most with a length of point of the slot to the upper edge of the protruded feeding line.

and of The antenna design begins with determining a microstrip-line-fed linearly polarized (LP) ARSA, which will be subsequently developed into a CP ARSA. At the center frequency of the fundamental mode of a conventional microstripline-fed LP ARSA where the width of the ring slot is only 22.2% the average radius of the slot [4], one guided wavelength is estimated to be the average perimeter of the ring slot. However, for a similar LP ARSA in [6] with the ring-slot width being 28.6% the average radius and with a ground-plane size of smaller than that of the antenna in [4], one guided wavelength is modified to be the outer perimeter of the ring slot. In fact, the ground-plane size and slot width complicatedly affect the resonant frequency of the fundamental mode of an LP ARSA. Nevertheless, to more accurately estimate the fundamental-mode resonant frequency of an LP ARSA, extensive simulations using Ansoft HFSS and experimental verifications for the substrates of small dielectric constant and low substrate thickness were performed again in this study. Results show that the fundamental-mode resonant magnetic current of the LP ARSA for one guided wavelength is mainly distributed between the average and outer perimeter of the ring slot at the fundamental resonant-band center frequency. An empirical formula can be found for certain ranges of structural parameters to be described below. As mentioned in the previous section, the lower bound of (where is the average radius) for the proposed antenna will be set equal to the largest value of for the antennas in [4]–[6], i.e., 0.286. Simulations and is larger than 0.643, the 3-dB experiments show that if ARBW of the designed CP antenna tends to decrease. Hence, in this study will be set between 0.286 and the value of 0.643. On the other hand, the ground plane should be as small as possible to preserve the compactness of the designed antenna yet should not be small enough to deteriorate the CP perforwas found mance of the antenna. A reasonable value of to be between 2.3 and 2.9. We have found that for the chosen and ranges of and for and (for the CP antennas to be designed in the following subsections, this ), the expression of the corresponds to

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with a center frequency of around 2.4 GHz, which matches with the pre-selected frequency. B. Grounded Rectangular Patches for CP Operation (the Reference Antenna)

Fig. 2. Simulated (a) return losses, (b) axial ratios, (c) amplitude ratios, and (d) phase differences for an LP ARSA, the reference antenna, type 1 (d = 6 mm) antenna, and type 1 (d = 4 mm) antenna.

fundamental-mode resonant frequency of an LP ARSA can be slightly modified from those in [4] and [6] to

(1) is the resonant frequency of the In this empirical formula, fundamental mode, the speed of light in free space, and the effective dielectric constant. Note that most CP slot antennas in [4]–[12], [13], [14] are designed in the frequency range of 2–3 GHz. For comparison, we would first design a CP ARSA in a similar frequency range to elucidate how the antenna is developed and then verify the design procedure by using different substrates and different substrate thicknesses to design L-band CP ARSAs. For cost saving, to choose commonly used FR4 substrates with design the proposed CP antenna developed from a microstripratio of 2.3–2.9. Now, seline-fed LP ARSA having a lect 2.4 GHz as the fundamental-mode resonant frequency of according to (1) is 14 mm, which the LP ARSA. Thus, is the same as that in [6]. Next, set the width of the ring slot ; and for the initial study set to be , the middle value of the 2.3–2.9 range. The LP ARSA is fed by a 50- straight microstrip line. For the above structural parameters, an optimized impedance match can be de. The retermined through simulation as turn loss simulated for this LP ARSA is shown in Fig. 2(a). The impedance band ranges from 2160 to 2640 MHz

In order to convert the above LP ARSA into a CP antenna, rectangular patches as the perturbachoose a pair of tion structure to be implanted in the ring slot. These patches are tangential to the outer circle of the ring slot at the intersection points of the outer circle and a diametric line 45 away from the -axis. For convenience, refer to this CP ARSA as the reference antenna, which will be further developed into a wideband CP antenna later. For simplification in the reference-antenna design, emphasis is placed on achieving optimized CP axial ratios and ) of the grounded rectby adjusting the dimensions ( angular patches. Hence, the feeding structure of the reference antenna is kept the same as that of the LP ARSA presented in the previous subsection, and improving the impedance matching is not attempted in the CP band. The return loss and axial ratio simulated also using Ansoft HFSS for the reference antenna are shown in Fig. 2 and are summarized in Table II. Note that in the captions of this and the remaining tables, the unit for the structural parameters is millimeter, the CP-band center frequency, and the lower- and upper-end edge frequencies of and the frequency band of interest, respectively. The reference andirection left-hand CP (LHCP) radiatenna produces in the tion with a CP band of 2515–3005 MHz. The 3-dB ARBW (relative to the center frequency 2760 MHz) is about 17.8%, which is already larger than those of the antennas in [4]–[6]. From the of and the PD between amplitude ratio and , in the direction the two orthogonal far fields, shown in Figs. 2(c) and 2(d), respectively, observe that around 2760 MHz the reference antenna has a slowly varying amplitude ratio of around 0 dB but a very steep PD variation against frequency. We expect that if the PD can also be made slowly , a wider CP band can be achieved. varying around C. ARBW Enhancement Using Bent Feeding Line (Type 1 Antenna) To enhance the 3-dB ARBW of the reference antenna, we first employ a bent feeding microstrip line as suggested in [10]–[12]. The bent feeding line is formed by extending the section of the reference antenna toward the right to have a width of . Refer to the resulting ARSA having a pair of grounded rectangular patches and a bent feeding microstrip line as a type 1 antenna. With the right edge of the adjusted section very close to the right edge of the ground plane, the smallest axial ratio is raised from 0.7 dB for the reference antenna at 2.7 GHz to 4.3 dB around 3 GHz because the amplitude ratio increases from 0 dB to 9 dB. Relevant curves for this antenna in Fig. 2 are distin.” However, the PD of the type guished by “Type 1 1 antenna is much more slowly varying around than that of the reference antenna in 2300–3500 MHz, which is even wider than the CP band of the reference antenna. This suggests that if the amplitude ratio can be lowered to the 0 ,a dB level while the PD remains slowly varying around CP band around this frequency range can be obtained. Indeed, the desired amplitude ratio and PD (see Figs. 2(c) and 2(d)) can

SZE AND CHEN: AXIAL-RATIO-BANDWIDTH ENHANCEMENT OF A MICROSTRIP-LINE-FED CP ARSA

TABLE II PERFORMANCES OF THE PROPOSED ANTENNA IN THE DEVELOPMENT PROCESS; " w : R ,w : ,` : R ,`

= 10( 0 71 )

=15

= 1( 0 07 )

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= 4:4, tan  = 0:019, h = 0:8, G = 45, R = 17:5, R = 10:5, = 6( 0:43R ), AND ` = 2( 0:14R )

( = 4 mm)

Fig. 3. Measured input resistances and reactances for the type 1 d antenna and the type 2 antenna.

be attained with reduced from 6 mm to 4 mm . Now, the type 1 antenna is simulated to have a 3-dB ARBW of 41% (with respect to the center frequency 3075 MHz), 2.3 times that of the reference antenna. D. Impedance Matching via Deforming the Bent Feeding Line (Type 2 Antenna)

Fig. 4. Simulated and measured (a) return losses and (b) axial ratios, and simulated (c) amplitude ratios and (d) phase differences for the type 2 and type 3 antennas.

Although the axial ratio of the type 1 antenna has been lowered to form a wider CP band than that of the type 1 antenna, the input impedance within the CP band is still mismatched due to the relatively small input resistance and steeply varying input reactance as shown in Fig. 3. In order impedance match in the CP band of to achieve a antenna, the antenna’s input resistance the type 1 and reactance have to be increased and smoothened, respectively. This can be accomplished by symmetrically widening an section of the bent feeding line to form a deformed bent and feeding line (or a tuning pad for short). With the , dimensions of this tuning pad properly chosen, the resulting CP ARSA (referred to as the type 2 antenna) now has a wide impedance band of 2140–4740 MHz (a center-frequency-referenced bandwidth of 75.6%) and an almost intact CP band of 2460–3740 MHz, as shown in Fig. 4 and Table II. To more clearly show the effects of this tuning pad on the input

impedance, the input impedance measured for the type 2 antenna is also shown in Fig. 3 for comparison. Although the CP band of the type 2 antenna is very close to antenna, its lower-end edge frethat of the type 1 quency is higher than the desired 2400 MHz. This is because in the frequency range lower than 2460 MHz the amplitude ratio (PD) of the orthogonal far fields tends to be greater (smaller) . To overcome this problem, examine the than 0 dB magnetic current distributions in the ring slot around 2400 MHz (not shown here for brevity). It is found that the magnetic currents are much more strongly distributed around the right-angle apexes of the grounded rectangular patches than in the rest of the ring slot except for the feeding area. This hints to us that if the magnetic currents can be made more smoothly distributed in the ring slot, the CP characteristics of the antenna in the lower-frequency range may be improved.

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TABLE III STRUCTURAL PARAMETERS, MEASURED 3-dB ARBWS, AND MEASURED VSWR 2 IMPEDANCE BANDS OF ANTENNAS 1–4; " h = 0:8, G = 45, w = 1:5, ` = 1, ` = 2, w = 23, AND w = 10



= 4:4, tan  = 0:019,

E. Further ARBW Enhancement Using Grounded Hat-Shaped Patches (Type 3 Antenna) To prevent the magnetic current distribution from peaking around the right-angle apexes of the grounded rectangular patches, one can reshape the acute apexes. An easier way is to replace part of each rectangular patch by a half ellipse, which in conjunction with the rest of the grounded patch forms a of the hat-shaped patch. In the half ellipse, the width the rectangular patches is adopted as the major axis, and semi-minor axis. Substituting the rectangular patches in the type 2 antenna with the hat-shaped patches thus results in a type 3 antenna. The return losses and axial ratios of several type , 3 antennas having different semi-minor axes ( etc.) have been studied (results not shown here for conciseness). leads to the Among these semi-minor axes, the value of largest 3-dB ARBWs. The simulated return loss and axial ratio are shown for the type 3 antenna with a semi-minor axis of in Fig. 4, and relevant measured results are listed in Table II. The measured CP band of 2175–3800 MHz, corresponding to a 3-dB ARBW of 54.4% relative to 2988 MHz, agrees reasonably well with the simulated result. From the simulated amplitude ratios and PDs shown in Fig. 4 for the type 2 and type 3 antennas, observe that the type 3 antenna can more significantly enhance the 3-dB ARBW than can the type 2 antenna. This is because, after the perturbation structures are modified from the rectangular shape for the type 2 antenna to the hat shape for the type 3 antenna, the PDs in and the ampli2100–3740 MHz are increased to around tude ratios of around 0 dB are extended to a lower frequency of around 2100 MHz. Summarizing from the PDs shown in Figs. 2 and 4, we conclude that both the reference and type 3 antennas direction. The CP band of produce LHCP radiation in the impedance band, the former is not enclosed by its whereas that of the latter is. It is noteworthy and satisfying that the CP band (2175–3800 MHz) of the type 3 antenna is wider than that (2515–3005 MHz) of the reference antenna. F. Effect of Ws on CP Band Besides the already presented type 3 antenna, we proceeded with designing three other type 3 antennas that have the same average radius but different slot widths. The four antennas with , 7 mm , 5 mm a slot width of 9 mm , and 4 mm , called Ants. 1, 2, 3, and 4, respectively, have optimized 3-dB ARBWs measured to be 56.8%, 54.4%, 41%, and 39.6% (see Table III). Because the four antennas with the same ground-plane side length of 45 mm have a very close CP-band center frequency of around 3 GHz, they

Fig. 5. Measured (a) return losses and (b) axial ratios for L-band antennas with different substrates and substrate thicknesses.

have roughly the same normalized antenna area of only about . Of the four antennas, Ant. 1 (4) with the largest (smallest) has the largest (smallest) 3-dB ARBW. All these CP bands are completely enclosed by their corresponding impedance bands. With the outer radii larger than 19.5 mm or and for the smaller than 16 mm (i.e., same value of , i.e., 45 mm), not only is the axial-ratio level rapidly increased but the input impedance also becomes mismatched. Although the 3-dB ARBW of Ant. 4 is only 39.6%, it is already 1.3 times those of the antennas in [9] and [12] and 4–9 times those of the antennas in [4]–[6]. Note that Ant. 4’s normalis nearly equal to those of the former and ized slot area 1.67–2.5 times those of the latter. G. Further Design Verification CP antennas are frequently installed on a global navigation satellite system (GNSS), for which the L band is used. To further verify the presented design procedure, we designed several L-band type 3 antennas. These antennas are divided into two groups: one fabricated on an RT5880 substrate and the other on , 0.8, and 1.6 mm. Measured results FR4 substrates with of these antennas are shown in Fig. 5 and Table IV. The measured 3-dB ARBWs are all maintained larger than 40% with respect to their CP-band center frequencies. Note that the CP band of each of these four L-band antennas is completely enclosed by impedance band, signifying that the presented its design procedure can be applied to antennas using substrates with a broad range of substrate thickness (i.e., ) and with a dielectric constant of as low as 2.2.

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TABLE IV PERFORMANCES OF THE PROPOSED ANTENNAS WITH DIFFERENT SUBSTRATES AND SUBSTRATE THICKNESSES

TABLE V PERFORMANCES OF THE PROPOSED ANTENNA WITH DIFFERENT BACK-METAL-BOX SIZES; " d : ,s : ,w : ,w : ,` : R

= 13 7 = 1 7

=15

=15

= 4:4, tan  = 0:019, h = 0:8, G = 90, R = 35, R = 21, = 5( 0 18 ), AND H = 50( 1:79R )

side length of the metal box is larger than 0.78 times one wavelength at the CP-band center frequency of the bidirectional antenna (i.e., should be larger than 160 mm) and if the structural parameters are fine tuned using the design procedure presented in the previous subsections. Moreover, the CP bands of the unidirectional antennas are slightly lowered as compared with that of the bidirectional counterpart. To make arduous work simpler, a more general back metal box for unidirectional CP antennas should be developed such that once a bidirectional CP slot antenna (for example, the ones in [6], [9], [12], and this paper) is successfully designed, the structural parameters of the bidirectional counterpart needs not undergo re-adjustment for preserving most of the bidirectional CP band. This is future work that remains to be done. Fig. 6. Measured (a) return losses and (b) axial ratios for L-band unidirectional CP antennas.

H. Unidirectional CP Radiation Design All the antennas presented so far radiate into both the and half spaces. For GNSS applications, unidirectionally radiated CP antennas (unidirectional CP antennas for short) are usually required. In this subsection, we use the L-band type 3 antenna with a 0.8-mm-thick FR4 substrate given in Table IV as an example and demonstrate that a unidirectional CP radiation pattern can be obtained by placing a simple top-wall-less open metal box underneath, as shown in Fig. 1(b), where the bottom surface of the bidirectional antenna coincides with the aperture surface of the metal box. The wall height (H) of the metal box is set approximately equal to one quarter wavelength at the CP-band center frequency of the chosen antenna. Measured results of unidirectional CP antennas with several side lengths of the metal box are depicted in Fig. 6 and are summarized in Table V along with the antennas’ structural parameters. From these results, we found that more than 68% of the CP band of the chosen bidirectional antenna can be preserved if the

IV. ANTENNA RADIATION CHARACTERISTICS The far-field radiation patterns of the fabricated unidirectional CP ARSA using the 0.8-mm-thick FR4 substrate in both and planes at 1150 and 1450 MHz were measured the by using the NSI-800F-10 far-field antenna measurement system. As can be seen in Fig. 7, the L-band CP ARSA with produces mainly LHCP radiations in the half space. The RHCP radiations in the directions are eliminated to result in small back radiations with a front-to-back ratio of larger than 20 dB. The measured results are found to agree well with the simulated ones. The radiation patterns of the proposed CP ARSA without a back metal box are bidirectional half space are similar to those in Fig. 7; and in the for brevity, these bidirectional patterns are not shown here. -directed antenna gains of three Presented in Fig. 8 are the CP antennas. The first one is a unidirectional CP antenna with , and the a 0.8-mm-thick FR4 substrate and with remaining two are bidirectional (one with a 0.8-mm-thick FR4 substrate and the other a 0.6-mm-thick RT5880 substrate). The bidirectional CP antenna using the RT5880 (FR4) substrate has a measured antenna gain of 3.3–5.1 dBic (2.7–4.1 dBic) in its CP band, whereas the unidirectional CP antenna has a

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ACKNOWLEDGMENT The authors thank the reviewers for careful review and valuable comments, and thank the associate editor for helpful suggestions. REFERENCES

Fig. 7. Far-field radiation patterns for the unidirectional L-band CP antenna at (a) 1150 MHz and (b) 1450 MHz. with R

= 190 mm

+z direction for three = 190 mm).

Fig. 8. Measured and simulated antenna gains in the L-band antennas (side length of the metal box is R

higher antenna gain of up to about 9 dBic. The measurement reasonably agrees with the simulation. Satisfying radiation characteristics of the designed L-band unidirectional CP antenna has been achieved.

V. CONCLUSION A design procedure has been developed for designing a CP ARSA having a 3-dB ARBW greatly enhanced to a level of comparable to that of a CP wide slot antenna. The normalized antenna areas (slot areas) of the designed antennas are as small , yet the CP bands have a 3-dB ARBW of as as large as 65% and can be completely enclosed by the correimpedance bands. With a simple open sponding metal box placed beneath the L-band CP ARSAs, the resulting unidirectional radiation patterns have rendered the designed antennas more suitable for GNSS applications.

[1] R. C. Johnson and H. Jasik, Antenna Engineering Handbook. New York: McGraw-Hill, 1984. [2] L. Boccia, G. Amendola, and S. Di Massa, “A dual frequency microstrip patch antenna for high-precision GPS application,” IEEE Antennas Wirel. Propag. Lett., vol. 3, pp. 157–160, 2004. [3] J. M. Samper, R. B. Peérez, and J. M. Lagunilla, GPS & Galileo: Dual RF Front-End Receiver and Design, Fabrication, and Test. New York: McGraw-Hill, 2009. [4] K. L. Wong, C. C. Huang, and W. S. Chen, “Printed ring slot antenna for circular polarization,” IEEE Trans. Antennas Propag.., vol. 50, no. 1, pp. 75–77, Jan. 2002. [5] J. S. Row, “The design of a squarer-ring slot antenna for circular polarization,” IEEE Trans. Antennas Propag.., vol. 53, no. 6, pp. 1967–1972, Jun. 2005. [6] J. Y. Sze, C.-I. G. Hsu, M. H. Ho, Y. H. Ou, and M. T. Wu, “Design of circularly polarized annular-ring slot antennas fed by a double-bent microstripline,” IEEE Trans. Antennas Propag.., vol. 55, no. 11, pp. 3134–3139, Nov. 2007. [7] K. L. Wong, J. Y. Wu, and C. K. Wu, “A circularly polarized patchloaded square-slot antenna,” Microwave Opt. Technol. Lett., vol. 23, pp. 363–365, Dec. 1999. [8] C. C. Chou, K. H. Lin, and H. L. Su, “Broadband circularly polarised cross-patch-loaded square slot antenna,” Electron Lett., vol. 43, pp. 485–486, Apr. 2007. [9] J. Y. Sze and C. C. Chang, “Circularly polarized square slot antenna with a pair of inverted-L grounded strips,” IEEE Antennas Wirel. Propag. Lett., vol. 7, pp. 149–151, 2008. [10] J. S. Row and S. W. Wu, “Circularly-polarized wide slot antenna loaded with a parasitic patch,” IEEE Trans. Antennas Propag., vol. 56, pp. 2826–2832, Sep. 2008. [11] T. Y. Han, Y. Y. Chu, L. Y. Tseng, and J. S. Row, “Unidirectional circularly-polarized slot antennas with broadband operation,” IEEE Trans. Antennas Propag., vol. 56, pp. 1777–1780, Jun. 2008. [12] J. Y. Sze, J. C. Wang, and C. C. Chang, “Axial-ratio-bandwidth enhancement of asymmetric-CPW-fed circularly-polarised square slot antenna,” Electron Lett., vol. 44, pp. 1048–1049, Aug. 2008. Jia-Yi Sze (M’99) was born in Kaohsiung, Taiwan, R.O.C., in 1963. He received the B.S. degree in automatic control engineering from Feng Chia University, Taichung, Taiwan, the M.S. degree in electronic engineering from Chung Cheng Institute of Technology, Taoyuan, Taiwan, and the Ph.D. degree in electrical engineering from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 1984, 1986, and 2001, respectively. Since 1986, he has been a Lecturer with the Department of Electrical Engineering, Chung Cheng Institute of Technology, National Defense University, Taoyuan, Taiwan, R.O.C., where, in 2001, he became an Associate Professor. His current research interests are in antenna design and electromagnetic wave propagation.

Wei-Hung Chen was born in Taipei, Taiwan, R.O.C., in 1984. He received the B.S. degree in communications, navigation, and control engineering from National Taiwan Ocean University, in 2007, and the M.S. degree in electrical and electronic engineering from Chung Cheng Institute of Technology, National Defense University, Taoyuan, Taiwan, R.O.C., in 2009. He is currently with Ayecom Technology Co., Ltd., (DNI’s subsidiary in Delta Group) Hsinchu, Taiwan, R.O.C., where he is a hardware RD Engineer.

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Design of Triple-Frequency Microstrip-Fed Monopole Antenna Using Defected Ground Structure Wen-Chung Liu, Senior Member, IEEE, Chao-Ming Wu, and Yang Dai

Abstract—A novel triple-frequency microstrip-fed planar monopole antenna for multiband operation is proposed and investigated. Defected ground structure (DGS) is used in this antenna, which has a rectangular patch with dual inverted L-shaped strips and is fed by a cross-shaped stripline, for achieving additional resonances and bandwidth enhancements. The designed antenna 2 , and operates over the has a small overall size of 20 30 frequency ranges, 2.14–2.52 GHz, 2.82–3.74 GHz, and 5.15–6.02 GHz suitable for WLAN 2.4/5.2/5.8 GHz and WiMAX 3.5/5.5 GHz applications. There is good agreement between the measured and simulated results. Experimental results show that the antenna gives monopole-like radiation patterns and good antenna gains over the operating bands. In addition, effects of both the length of the protrudent strips and the dimensions of the DGS for this design on the electromagnetic performance are examined and discussed in detail.

mm

Index Terms—Defected ground structure, monopole antenna, multi-band, triple-frequency, WiMAX, WLAN.

I. INTRODUCTION ECENTLY, the demand for the design of an antenna with triple- or multiband operation has increased since such an antenna is vital for integrating more than one communication standards in a single compact system to effectively promote the portability of a modern personal communication system. For this demand, the developed antenna must not only be with a triple/multiband operation but also have a simple structure, compact size, and easy integration with the circuit. Among the known triple/multiband antenna prototypes, the planar monopole antenna with various structures has become a familiar candidate because of its attractive characteristics including low profile and weight, low cost, and versatile structure for exciting wide impedance bandwidth, dual- or multiresonance mode, and desirable radiation characteristics. However, the difficulty in designing antenna challenges engineers when the size of the antenna reduces and the number of operating frequency bands increases. So far, for size reduction, bandwidth enhancement, and resonance-mode increment, numerous monopole antennas have been proposed by employing

R

Manuscript received November 22, 2009; revised February 26, 2010; accepted November 29, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported by the National Science Council of the Republic of China under Grant NSC 99-2221-E-150-029. W.-C. Liu and C.-M. Wu are with the Department of Aeronautical Engineering, National Formosa University, Taiwan, R.O.C. (e-mail: wencliu@nfu. edu.tw). Y. Dai is with the Department of Electrical Engineering, National University of Tainan, Taiwan, R.O.C. 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.2011.2152315

various promising feed structures such as the probe [1]–[5], the microstrip [6]–[9], and the coplanar waveguide (CPW) [10]–[14]. In these presented monopole antennas, a large solid ground plane having the shape of a square, rectangle, circle, or ellipse is usually adopted. Different from this, a notable ground structure named defected ground structure (DFG) has recently been investigated and found to be a simple and effective method to reduce the antenna size as well as excite additional resonance modes [15]–[18]. In this paper, a small and low-profile microstrip-fed monopole antenna for triple-frequency operation is proposed. The radiating element was modified by loading it with protrudent strips and feeding it with a cross-shaped stripline. In addition, unlike the conventional microstrip-fed antenna prototype using a solid ground plane, in this design the ground was cut out by shaped slots and thus forms a DGS. The above design skills are introduced to approach excitation of triple resonant modes accompanied with good impedance bandwidths over the operating bands. By properly selecting the dimensions of the proposed antenna, good triple-broad impedance bandwidths and radiation characteristics suitable for two multiband wireless communication systems such as the wireless local-area network (WLAN) 2.4/5.2/5.8 GHz and the worldwide interoperability for microwave access (WiMAX) 3.5/5.5 GHz operations can be achieved. In addition, with the DGS, the proposed antenna can avoid a large surface-wave loss and thus decrease its impact on the coupling effect when it is used as an array element. The antenna configuration and simulated data as well as the constructed prototype and measured data will be carefully examined and discussed in Sections II and III. II. ANTENNA DESIGN AND SIMULATION The schematic configuration of the proposed microstrip-fed planar monopole antenna with defected ground structure (DGS) for triple-frequency operation is shown in Fig. 1. For the design studied here, the radiator and ground plane are etched on the opposite sides of a printed-circuit board (PCB) with a dielectric constant of 4.4 and substrate thickness of 1.6 mm. A cross-shaped stripline, which comprises both the vertical and , and horizontal strips with dimensions of respectively, and a distance of between the horizontal strip and the feed point, is used for feeding the radiator. The basis of the radiator is a rectangular patch, which has the dimensions and width , and is protruded with two inverted of length L-shaped strips from the patch’s upper two sides. Each of the two strips comprises both the vertical and horizontal strips with and , respectively. As for the dimensions of ground plane, unlike the general use of a solid rectangular plane for a microstrip-fed antenna, it is defected by two equal-shaped

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TABLE I OPTIMAL GEOMETRICAL PARAMETERS OF THE PROPOSED TRIPLE-FREQUENCY MONOPOLE ANTENNA

Fig. 1. Schematic configuration of the proposed triple-frequency monopole antenna with defected ground structure.

Fig. 2. Simulated return loss against frequency for the proposed antenna with/ without DGS, protrudent strips and cross-shaped feedline.

slots, which were appropriately embedded from the ground’s left and right sides. These slots were introduced to increase the excitation of resonant modes and improve the impedance matching condition for the proposed antenna. The overall size , and each of the embedded slots has of the ground is as well as a horizontal section a vertical section of . Meanwhile, the slot is etched with a distance of of from the bottom of the ground and its vertical section has a distance of from the ground side edge. The electromagnetic solver, Ansoft HFSS, was used to numerically investigate and optimize the proposed antenna configuration. Fig. 2, trace (i), shows the simulated return loss of the proposed antenna with the optimized parameters as listed in Table I. Obviously, the simulation results show three resonant bands at frequencies of 2.26, 3.6, and 5.3 GHz with bandwidths, defined for 10-dB return loss, of about 380 MHz (2.13–2.51 GHz), 570 MHz (3.26–3.83 GHz), and 880 MHz (5.03–5.91 GHz), respectively, corresponding to an impedance bandwidth of 17%, 16% and 17% with respect to the appropriate resonant frequencies over the three operating bands. Apparently, the above obtained bandwidths simultaneously cover the WLAN

Fig. 3. Simulated results of the surface current distributions for the proposed triple-frequency monopole antenna studied in Fig. 2 at (a) 2.26, (b) 3.6, and (c) 5.3 GHz.

standards in the 2.4/5.2/5.8 GHz bands and the WiMAX standards in the 3.5/5.5 GHz bands. To further examine the appropriate impedance matching condition caused from addition of the DGS, the protrudent strips, or the cross-shaped feedline, the return loss for the proposed optimal design (case (i)) but without using the DGS, without having the protrudent strips, or without the horizontal strip for the feedline, which are denoted as curves (ii), (iii), and (iv), respectively, are also analyzed and presented

LIU et al.: DESIGN OF TRIPLE-FREQUENCY MICROSTRIP-FED MONOPOLE ANTENNA USING DEFECTED GROUND STRUCTURE

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Fig. 4. Simulated return loss against frequency for the proposed triple-fre; other parameters are the same quency monopole antenna with various as listed in Table I.

in Fig. 2. Note that in these cases all the unmentioned dimensions are the same as listed in Table I. For the case without using solid ground plane for the the DGS (i.e., using the 20 30 proposed antenna), a worse matching condition appears over the frequency band, while a resonant mode seems to form at about 3.6 GHz. It is the fundamental mode excited by the radiator of the rectangular patch loaded with the protrudent strips since the one-quarter wavelength when operating at 3.6 GHz (i.e., about 21 mm) is almost the same as the electric length along the side of the radiator. As for the case of the radiating patch without having the two protrudent inverted L-shaped strips (curve (iii)), the matching condition is still poor across the full band. However, two modes resonating at 2.3 and 5.04 GHz, respectively, become forming and without seeing the second resonant mode. This indicates that inclusion of the two inverted L-shaped strips in the proposed design will not only significantly improve the impedance matching conditions for the lowest and highest bands but also can excite an additional resonance at the second band. Finally, if we remove the horizontal strip from the cross-shaped feedline (curve (iv)), though the triple-resonance situation is almost not affected, the matching condition of the highest resonance becomes worse. For further examining the above excitation mechanism of the proposed antenna, the excited surface current distributions, obtained from the HFSS simulation, on both the radiator and the ground for the optimally designed antenna, as presented in Fig. 2, was studied. Fig. 3 shows the results for the three resonant frequencies at 2.26, 3.6, and 5.3 GHz. Obviously, for the lowest band excitation (2.26 GHz band), a large surface current density is observed along the central longitude portion of the DGS, whereas for the second-(3.6 GHz band) and the highest-band (5.3 GHz band) excitations, the current distribution becomes more concentrated along the protrudent strips and cross-shaped feedline, respectively. According to the observed phenomena in current distribution on the proposed antenna when operating at the three bands, we investigated the influence of the related geometrical dimensions on the impedance

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Fig. 5. Simulated return loss against frequency for the proposed triple-fre; other parameters are the same as quency monopole antenna with various listed in Table I.

L

Fig. 6. Simulated return loss against frequency for the proposed triple-fre; other parameters are the same as quency monopole antenna with various listed in Table I.

matching condition of the three resonant modes. First, the effect of the DGS to the matching condition at the highest operating band is studied. Fig. 4 presents the simulated results of , 8, and 8.5 the proposed antenna with slot width mm for the defected ground. We found that the lowest resonant mode is shifted toward the higher frequency band when increases, whereas the two higher resonant modes are slightly affected. As for the second resonant mode, according to the current distribution shown in Fig. 3(b), it has been shown that large current density is concentrated on the radiator. Therefore, this of the inverted L-shaped was verified by adjusting the length strip to be the values of 2, 3, 5, and 7 mm, and Fig. 5 gives is the obtained the simulated results. Note that optimal case. Obviously, for this design, varying the length of the patch’s protrudent strip, as excepted, does not significantly change the triple-resonant mode but does shift the frequency of the second resonance mode. The second resonant mode moves

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Fig. 7. Simulated return loss against frequency for the proposed triple-fre; other parameters are the same quency monopole antenna with various as listed in Table I.

toward lower frequency band with increasing because the effective electric length along the radiator is lengthened when increases, whereas both the lowest and the highest dominant modes are only slightly affected. Considering the third (highest) resonant mode from the current distribution on the antenna, as shown in Fig. 3(c), a large current density was observed along both the cross-shaped feedline, especially on the horizontal strip, and the appropriate ground plane on the opposite side of the substrate. Thus, the tuning effect of strip length of the cross-shaped feedline on the impedance matching condition is examined and shown in Fig. 6. The varying dimensions of are 2, 4, 6, 8, and 10 mm. Here, the case of represents that the proposed antenna is fed by a straight microstrip instead of a cross-shaped stripline. Obviously, varying the strip would seriously affect the highest operation band length impedance matching, whereas less change is seen for the first and second operation bands. Another parameter that may affect . Fig. 7 presents the the highest resonance is the slot width simulated results of return loss against the frequency of the pro, 1.5, 2 and 2.5 mm. Note that the posed antenna with means the slot of does not exist. case of The third (highest) resonant mode is clearly found to move toward the lower frequency band if the ground is cut with this , the lowest resonant freslot. With an increase in the width quency of the third mode decreases, whereas those of the other two modes are almost unchanged. The above analyzed results are very much in accordance with that observed from the current distributions on the antenna and also very vital for designing such an antenna to obtain the desired triple-frequency bands. III. EXPERIMENTAL RESULTS The prototype of the proposed antenna with optimal dimensions, as listed in Table I and depicted as curve (i) in Fig. 2, was constructed and experimentally investigated. The return loss against frequency for this triple-frequency antenna was measured by using the Agilent E5071C vector network analyzer

Fig. 8. Measured and simulated return loss against frequency for the proposed triple-frequency monopole antenna.

TABLE II SIMULATED AND MEASURED IMPEDANCE BANDWIDTHS OF THE PROPOSED TRIPLE-FREQUENCY MONOPOLE ANTENNA

and is presented in Fig. 8. For comparison, the simulated results are also plotted in this figure. Obviously, three resonant modes at frequencies of 2.31, 3.42, and 5.44 GHz were also obtained. The measured impedance bandwidths are about 380 MHz (2.14–2.52 GHz), 920 MHz (2.82–3.74 GHz), and 870 MHz (5.15–6.02 GHz), corresponding to an impedance bandwidth of 16%, 27%, and 16% with respect to the appropriate resonant frequencies over the three operating bands. For comparison, Table II lists the related simulated and measured data. Reasonable agreement between the simulation and the measurement is at the lowest achieved beyond a frequency deviation of resonant frequencies of the triple operating bands. However, when we took into account the effects of the SMA connector as well as the connecting cable, by which the antenna was fed, in HFSS simulation, such a frequency shift could not be improved. Contrarily, the frequency deviations at the two higher bands get larger. The difference may therefore be attributed to the mismatching between the connector and the antenna feeder. The radiation characteristics such as radiation pattern, efficiency, directivity, and peak gain across the three operating bands for the proposed antenna have also been measured in a far-field range anechoic chamber with dimensions of and a three-dimensional rotator. Fig. 9 presents the measured radiation patterns including and horizontal polarizations in the the vertical azimuthal direction ( - plane) and the elevation direction ( and - planes) when operating at 2.45, 3.5, 5.25 and 5.75 GHz

LIU et al.: DESIGN OF TRIPLE-FREQUENCY MICROSTRIP-FED MONOPOLE ANTENNA USING DEFECTED GROUND STRUCTURE

Fig. 9. Measured radiation patterns for the proposed triple-frequency monopole antenna. (E : x x x and (d) 5.75 GHz.

for WLAN and WiMAX applications. Because of the symmetry in structure, rather symmetrical radiation patterns are seen in the - and - planes as depicted in the plots. In addition, very monopole-like radiation patterns with nearly omnidirectional radiation in the azimuthal plane and nearly conical patterns in the elevation planes are observed. In addition, it is also found and components of the patterns in both that the

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; E : 000) at (a) 2.45 GHz, (b) 3.5 GHz, (c) 5.25 GHz,

and - planes for operating at 2.45 and 3.5 GHz are seemed to be more comparable than those operating at 5.25 and 5.75 GHz. This electromagnetic phenomenon can be ascribed to the more horizontal components of the surface current on the antenna when operating at 2.45 and 3.5 GHz than at 5.25 and 5.75 GHz, as shown in Fig. 3. Also note that measurements at other operating frequencies across the bandwidth of each band

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respectively, across the triple bands. Agreement between the measurement and simulation seems well beyond a difference over the highest frequency band. of IV. CONCLUSION A novel microstrip-fed antenna design based on the patch monopole for a triple-frequency operation has been presented with simulated and measured results. With the skills of defecting the ground plane, protruding the patch with dual protrudent strips, and feeding the patch with a cross-shaped stripline, the proposed antenna can excite triple-resonances and has a suitable radiation performance to cater to the operation requirements of both the 2.4/5.2/5.8 GHz WLAN and the 3.5/5.5 GHz WiMAX communication systems. The antenna prototype has been constructed and measured, and shown to match well with the numerical prediction. Also, the effects of existences of the DGS, the prudent strips, and the cross-shaped stripline, and varying the dimensions of these structures on the antenna resonant frequencies and impedance bandwidths have been presented. REFERENCES

Fig. 10. Measured and simulated (a) directivity and radiation efficiency, and (b) peak gain across operating bands for the proposed triple-frequency monopole antenna.

show radiation patterns similar to those plotted here. Finally, Fig. 10 shows the measured and simulated radiation efficiency, directivity and peak gain of the proposed design for frequencies across the triple operating bands. The average efficiencies for the three bands are about 50, 58, and 71%, respectively, in measurement, whereas 60, 62, and 74%, respectively, in simulation. For directivity, the measured average values are 5.5, 4.8, and 4.5 dB, whereas the simulated results are 4.7, 4.1, and 3.8 dB, respectively, over the three frequency bands. It indicates that as the operating band gets higher, the efficiency increases while the directivity decreases. The reason could be that when operating at the lower frequency, the current on the antenna is more in-phase and concentrated in the same direction to thus make a higher directivity. The peak antenna gains were measured by applying the gain comparison method, in which a precalibrated standard gain antenna is used to determine the absolute gain of the antenna under test, and shown in Fig. 10(b). The ranges of measured gains are about 2.2–2.6, 2.1–2.6, and 2.5–3.4 dBi with an average value of 2.46, 2.45, and 3.0 dBi,

[1] Y. JoongHan, “Fabrication and measurement of modified spiral-patch antenna for use as a triple-band (2.4 GHz/5 GHz) antenna,” Microw. Opt. Technol. Lett., vol. 48, no. 7, pp. 1275–1279, 2006. [2] J. Costantine, K. Y. Kabalan, A. El-Hajj, and M. Rammal, “New multiband microstrip antenna design for wireless communications,” IEEE Trans. Antennas Propag. Mag., vol. 49, no. 6, pp. 181–186, 2007. [3] S. C. Kim, S. H. Lee, and Y. S. Kim, “Multi-band monopole antenna using meander structure for handheld terminals,” Electron. Lett., vol. 44, no. 5, pp. 331–332, 2008. [4] H. Wang and M. Zheng, “Triple-band wireless local area network monopole antenna,” IET Microw. Antennas Propag., vol. 2, no. 4, pp. 367–372, 2008. [5] C. H. See, R. A. Abd-Alhameed, P. S. Excell, N. J. McEwan, and J. G. Gardiner, “Internal triple-band folded planar antenna design for third generation mobile handsets,” IET Microw. Antennas Propag., vol. 2, no. 7, pp. 718–724, 2008. [6] H. C. Go and Y. W. Jang, “Multi-band modified fork-shaped microstrip monopole antenna with ground plane including dual-triangle portion,” Electron. Lett., vol. 40, no. 10, pp. 575–577, 2004. [7] K. Seol, J. Jung, and J. Choi, “Multi-band monopole antenna with inverted U-shaped parasitic plane,” Electron. Lett., vol. 42, no. 15, pp. 844–845, 2006. [8] C. M. Wu, C. N. Chiu, and C. K. Hsu, “A new nonuniform meandered and fork-type grounded antenna for triple-band WLAN applications,” IEEE Antennas Wirel. Propag. Lett., vol. 5, no. 1, pp. 346–348, 2006. [9] W. C. Liu, M. Ghavami, and W. C. Chung, “Triple-frequency meandered monopole antenna with shorted parasitic strips for wireless application,” IET Microw. Antennas Propag., vol. 3, no. 7, pp. 1110–1117, 2009. [10] W. C. Liu, “Design of a multiband CPW-fed monopole antenna using a particle swarm optimization approach,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3273–3279, 2005. [11] W. S. Chen and Y. H. Yu, “Compact design of T-type monopole antenna with asymmetrical ground plane for WLAN/WiMAX applications,” Microw. Opt. Technol. Lett., vol. 50, no. 2, pp. 515–519, 2008. [12] Y. Jee and Y. M. Seo, “Triple-band CPW-fed compact monopole antennas for GSM/PCS/DCS/WCDMA applications,” Electron. Lett., vol. 45, no. 9, pp. 446–448, 2009. [13] S. Chaimool and K. L. Chung, “CPW-fed mirrored-L monopole antenna with distinct triple bands for WiFi and WiMAX applications,” Electron. Lett., vol. 45, no. 18, pp. 928–929, 2009. [14] S. Xiaodi, “Small CPW-fed triple band microstrip monopole antenna for WLAN applications,” Microw. Opt. Technol. Lett., vol. 51, no. 3, pp. 747–749, 2009. [15] J. P. Thakur and J. S. Park, “An advance design approach for circular polarization of the microstrip antenna with unbalance DGS feedlines,” IEEE Antennas Wirel. Propag. Lett., vol. 5, no. 1, pp. 101–103, 2006.

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[16] M. A. Antoniades and G. V. Eleftheriades, “A compact multiband monopole antenna with a defected ground plane,” IEEE Antennas Wirel. Propag. Lett., vol. 7, pp. 652–655, 2008. [17] K. H. Chiang and K. W. Tam, “Microstrip monopole mntenna with enhanced bandwidth using defected ground structure,” IEEE Antennas Wirel. Propag. Lett., vol. 7, pp. 532–535, 2008. [18] D. Nashaat, H. A. Elsadek, E. Abdallah, H. Elhenawy, and M. F. Iskander, “Multiband and miniaturized inset feed microstrip patch antenna using multiple spiral-shaped defect ground structure (DGS),” in Proc. IEEE AP-S Int. Symp., Jun. 2009, pp. 1–4. Wen-Chung Liu (S’89–M’02–SM’06) was born in Changhua, Taiwan, R.O.C., in 1964. He received the B.S. degree in electronic engineering from Tamkang University, Tamsul, Taiwan, in 1986, the M.Sc. in nuclear engineering from National Tsinghua University, Hsinchu, Taiwan, in 1988, and the Ph.D. in electrical engineering and electronics from Liverpool University, Liverpool, U.K., in 1999. From 1990 to 2000, he was a Lecturer and then an Associate Professor in the Department of Electronic Engineering, Chien-Kuo Institute of Technology, Changhua, Taiwan. In 2000, he joined the Department of Aeronautical Engineering, National Formosa University, Yunlin, Taiwan, R.O.C., where he is currently a Professor. From August 2007 to January 2008, he was a Visiting Professor in the Department of Electronic Engineering, The King’s College London. He received the Outstanding Research Award from National Formosa University in 2010. His major research areas of interest are in NVIS antenna, printed antenna, and application of optimization technique in antenna design. Prof. Liu is listed in Who’s Who in Asia, Who’s Who in Engineering and Science, and Who’s Who in the World.

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Chao-Ming Wu received the B.S. and Ph.D. degrees in electrical engineering from National Cheng-Kung University, Tainan, Taiwan, R.O.C., in 1989 and 1995, respectively. He is currently an Associate Professor with the Department of Aeronautical Engineering, National Formosa University, Yunlin, Taiwan, R.O.C. His research interests include digital communication, antenna design, and data hiding.

Yang Dai was born in Taichung, Taiwan, R.O.C., in 1987. He received the B.S. degree in aeronautical engineering from National Formosa University, Yunlin, Taiwan, R.O.C., in 2010. He is currently working toward the M.S. degree at the Department of Electrical Engineering, National University of Tainan, Taiwan, R.O.C. His main research interests are in printed antenna design and RF circuit design.

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High-Gain Reconfigurable Sectoral Antenna Using an Active Cylindrical FSS Structure Arezou Edalati, Student Member, IEEE, and Tayeb A. Denidni, Senior Member, IEEE

Abstract—A novel design of a high-gain reconfigurable sectoral antenna using an active cylindrical frequency selective surface (FSS) structure is presented. The FSS structure consists of metallic discontinuous strips with PIN diodes in their discontinuities, and it is placed cylindrically around an omnidirectional electromagnetically coupled coaxial dipole (ECCD) array. The cylindrical FSS structure is divided into two semi-cylinders. By controlling the state of diodes in each semi-cylinder, a directive radiation pattern is obtained that can be swept in the entire azimuth plane. The effect of the diode-state configuration and the radius of the cylindrical structure are carefully studied to obtain an optimum sectoral radiation pattern. In addition, a solution for increasing the matching bandwidth of the antenna is also proposed. An experimental prototype was fabricated, and the measured results show a beamwidth of 20 in elevation and 70 in the azimuth plane at 2.1 GHz with a gain of 13 dBi. With these features, the proposed antenna is suitable for base-station applications in wireless communication systems. Index Terms—Cylindrical antennas, dipole arrays, frequency selective surfaces, reconfigurable antennas, sectoral antennas.

I. INTRODUCTION

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ECTORAL antennas are a type of directional antenna with a narrow beamwidth in one plane (H- or E-plane) and a wide beamwidth in the other plane. They are typically used at the base station of cellular communication systems for point-tomultipoint communications to cover sectoral areas [1], [2]. Recently, sectoral antennas have been designed using an array of metallic rods on the top of a ground plane to create a Fabry-Perot resonator antenna with a high gain using a multi-feed technique [3]. In the same perspective, sectoral antennas with dual-band operation [4] and dual polarization have also been designed [5]. However, the latest mentioned sectoral antennas have the limitation of pattern-reconfigurability, which is widely required in base-station antennas. Pattern-reconfigurability extends the functionality of the antennas, adds more flexibility, and enhances the transmission quality in hostile environments. Moreover, it allows saving energy by directing the desired Manuscript received March 12, 2010; revised November 04, 2010; accepted January 31, 2011. Date of publication May 10, 2011; date of current version July 07, 2011. A. Edalati is with Energy, Materials and Telecommunications Center of Institut National de la Recherche Scientifique (INRS-EMT), Montreal, QC H5A 1K6, Canada (e-mail: [email protected]). T. A. Denidni is with Energy, Materials and Telecommunications Center of Institut National de la Recherche Scientifique (INRS-EMT), Montreal, QC H5A 1K6, Canada and also with the Université du Québec, 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.2011.2152327

signal toward the appropriate user and adapting the antenna radiation pattern to a specific environment [6]–[10]. In this paper, following the author’s recent contribution [11], a novel design method of a high-gain sectoral antenna using frequency selective surfaces (FSSs) with a reconfigurable pattern is proposed. FSSs are periodic structures composed of arrays of mostly identical elements with frequency dependent reflection and transmission coefficients [12]–[15]. The proposed antenna is composed of an array of discontinuous strips and PIN diodes inserted into their discontinuities and it is placed cylindrically around an omnidirectional electromagnetically coupled coaxial dipole (ECCD) array. To obtain a reconfigurable radiation pattern, the cylindrical structure is divided into two semi-cylinder sectors. In each step, the diodes in one sector are on, whereas in the other sector they are off. The semi-cylinder with the diodes on operates as an array of continuous strips with high reflection at low frequencies, whereas the other semi-cylinder with the diodes off operates as an FSS with an array of printed dipole below resonance frequency and allows the transmission of the incident electromagnetic (EM) waves. Therefore, the FSS structure converts the omnidirectional radiation pattern of the source into a directive pattern. By switching the diode states, the direction of the high and low reflective sectors changes; therefore, the radiation pattern of the antenna can be swept in the entire 360 azimuth plane. Indeed, the PIN-diodes enable us to change the transmission and reflection coefficients of the periodic structure; therefore, the radiation pattern of the antenna can be reconfigured [16], [17]. The advantage of this design compared to the pattern-reconfigurable antennas proposed in [18], [19] is using only one layer of periodic structure instead of multi-layers, which leads to less active elements, lower cost, and less complexity in terms of fabrication. The rest of this paper is organized as follows. In Section II, the design of the FSS unit cell with reconfigurable transmission and reflection coefficients is described. The schematic of the proposed sectoral antenna and the radiation mechanism are presented in Section III. In Section IV, the design of the ECCD array as a radiating element is presented. In Sections V and VI, parametric studies are presented in order to illustrate the effect of the diode configuration and the radius of the cylindrical FSS on the radiation pattern. Moreover, a novel technique is presented in Section VII to achieve the required antenna matching. Experimental results of the fabricated prototype are provided and discussed in Section VIII. Finally, concluding remarks are given in Section IX.

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flection coefficient in the on-state and a high transmission coefficient in the off-state; therefore, the diode should have a low value for the on-state resistance and a low off-state capacitance. It should be noted that the dimension of the gap is almost equal to the diode dimension. The dimensions of the strip are defined using comprehensive parametric studies on the unit cell behavior and the radiation patterns of the antenna. Fig. 1. (a) Schematic of the FSS unit cell (all the dimensions are in millimeters), (b) on-state equivalent circuit, (c) off-state equivalent circuit.

III. RECONFIGURABLE SECTORAL ANTENNA: SCHEMATIC AND RADIATION MECHANISM A. Antenna Schematic To obtain a sectoral radiation pattern, the radiation beamwidth in one plane must be much wider than the other plane. The proposed antenna is designed such that the H-plane beamwidth is wide and the E-plane radiation beamwidth is narrow. Fig. 3 shows the proposed reconfigurable sectoral antenna, consisting of a cylindrical FSS structure and an ECCD array. The FSS structure is composed of unit cells, described in the previous section, that are placed cylindrically with an and radius of 50 mm on a angular periodicity of dielectric substrate with a permittivity of 3, a thickness of 0.254 mm and a loss tangent of 0.0013 (the same as the fabricated prototype RO3003). The transversal periodicity is defined by:

Fig. 2. Magnitude of the transmission and reflection coefficients of the FSS unit cell in on- and off-states.

II. FSS UNIT CELL To design a reconfigurable pattern antenna using an FSS structure, the unit cell must have reconfigurable transmission and reflection coefficients. To do so, the proposed unit cell, presented in Fig. 1(a), is composed of a metallic discontinuous strip with a PIN diode in its discontinuity. The EM plane waves propagate in the normal direction ( direction) with the E-field parallel to the strip ( direction). The PIN diode is modeled with its equivalent RC circuit. In , the on-state, the diode is equal to the resistor of and in the off-state, the diode is modeled with a parallel RC and , as shown in circuit with Fig. 1(b) and (c). (the values of lumped elements are taken from the data sheet of the diode used in the fabricated prototype). The magnitude of the transmission (T) and reflection (R) coefficients of the FSS unit cell is presented in Fig. 2. The coefficients are calculated by using CST Microwave Studio [20]. As can be seen, in low frequencies, the structure shows a high reflection coefficient when the diode is on and a high transmission coefficient when the diode is off. It is important to note that the magnitude of the reflection coefficient in the on-state is inversely proportional to the resistance of the on-state diode. This means with the higher values of the on-state resistance, lower values for the magnitude of the reflection coefficient are achieved. Also, the magnitude of the transmission coefficient in the off-state is inversely proportional to the capacitance of the off-state, meaning that for higher values of the capacitance, lower values of the transmission coefficient are achieved. In our design, it is required to obtain an FSS structure with a high re-

(1) The ECCD array shown in Fig. 3(b) is a high-gain omnidirectional source placed at the center of the cylindrical structure. It is important to note that the cylindrical FSS structure has almost the same transmission and reflection coefficients as the flat unit cell described in the previous section because the chosen transversal period is the same as the periodicity in the direction. B. Radiation Mechanism The proposed antenna operates as follows: in each step, the diodes in certain adjacent columns of strips are on and the rest of the diodes in the other columns are off. The semi-cylinder sector with on-state diodes has a high reflection coefficient at low frequencies and the sector with off-state diodes has a high transmission coefficient (see Fig. 2), which means the on-state sector reflects the EM waves forward like a reflector, and the offstate sector allows to pass the incident EM waves; Therefore, the cylindrical FSS converts the omnidirectional radiation pattern of the source into a directive pattern. By switching the diode states, the direction of the high and low reflective sectors can be oriented; therefore, the antenna is able to sweep the entire azimuth plane. IV. DESIGN OF THE RADIATING SOURCE To obtain a high-gain reconfigurable sectoral antenna, an ECCD array was designed and placed at the center of the cylindrical structure. The ECCD shown in Fig. 3(b) illustrates the ECCD array. It consists of 4 dipoles fed electromagnetically on the outer conductor of by slot rings with a width of

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Fig. 3. (a) Reconfigurable sectoral FSS antenna, (b) ECCD array antenna (dimensions in mm).

TABLE I FINAL DIMENSIONS OF THE ECCD ARRAY (IN MILLIMETER)

the coaxial cable. Each dipole contains two pipes with a length and (in the final design, both pipes have the of and , respectively same length), and diameters of (1 for the pipe with the smaller radius, and 2 for the pipe with the larger radius). The thickness of pipes is 0.355 mm. An appropriate distance between the cylindrical pipes is chosen to obtain in-phase excitation of the array antenna elements. The coaxial cable has a dielectric constant of 2 with the inner conductor radius of 0.456 mm. The outer conductor has a radius of 1.79 mm and a thickness of 0.6 mm (same as the coaxial cable used in the fabricated prototype). The initial design of the ECCD array used the same procedure as described in [21], subsequently for optimization, a proper ECCD array was designed. In addition, coaxial transformer with in [21], the authors proposed a as a matching circuit to match the antenna to the feeding line. However, this antenna has a complex design and provides a narrow band matching. To overcome these issues, in our design, a novel solution was introduced for a wider matching using two layers of pipes instead of one, which will be described in Section VII. Table I lists the final dimensions of the proposed ECCD array which is used in the simulations and fabrication of the antenna. The E- and H-plane radiation patterns of the proposed ECCD array are shown in Fig. 4. The E-plane beamwidth is 20 , whereas the H-plane radiation pattern is omnidirectional. The non-symmetric side-lobe is due to the non-symmetric design of the ECCD array at the top and bottom of the coaxial cable. V. EFFECT OF THE DIODE CONFIGURATION ON THE RADIATION PATTERN OF THE ANTENNA To examine the effect of the diode configuration on the radiation pattern and directivity of the sectoral FSS antenna, para-

Fig. 4. Radiation pattern of the ECCD array antenna at 2.1 GHz.

metric studies were carried out. The radius of the cylindrical structure is . Fig. 5 illustrates the effect of the diode-state configuration on the antenna radiation patterns. As can be seen, the diodestate configuration affects the radiation patterns of the antenna. The side-lobe level increases when 7 columns of diodes are off, because the size of the reflector in this case is small and the energy leaks from its sides. The back-lobe becomes worse when 3 columns of diodes are off due to the smaller opening and more reflection to the back. As illustrated in Fig. 6, the highest directivity is achieved with 5 columns of diodes in the off-state, which is mostly due to the maximum effective aperture illumination compared to other cases. Based on these results, the diode-state configuration for the proposed antenna is chosen to be 5 adjacent columns of diodes off and 7 columns on, which gives the best radiation patterns and maximum directivity. From now on, all simulations are carried out under this assumption.

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Fig. 5. Effect of the diode-state configuration on the antenna radiation pattern, (a) E-plane, (b) H-plane at 2.1 GHz.

Fig. 7. Effect of the radius of the cylindrical FSS on the radiation pattern ( = 30 ) at 2.1 GHz, (a) E-plane, (b) H-plane.

Fig. 6. Effect of the diode-state configuration on the directivity of the antenna.

Fig. 8. Effect of the radius of the cylindrical FSS on the antenna directivity ( = 30 ).

VI. EFFECT OF THE RADIUS OF THE CYLINDRICAL FSS STRUCTURE The initial value of the radius of the cylindrical structure has to ( is the free space been chosen to be between wavelength at the operating frequency), which is the optimum radius of the solid metallic semi-cylindrical reflector antenna.

After that, the radius is optimized for the proposed cylindrical FSS structure. To study the effect of the radius on the radiation patterns and directivity of the antenna, two types of parametric studies are presented here. In the first case, the angular period( , is the number of icity is fixed to the columns of strips) and the radius is varied. In this case, the

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Fig. 10. Effect of the radius of the cylindrical FSS on the directivity when the transversal period of the FSS structure is fixed.

Fig. 9. Effect of the radius of the cylindrical FSS on the radiation pattern at 2.1 GHz when the transversal period of the FSS structure is fixed.

Fig. 11. Input impedance of the antenna; for 1 and 2 pipes.

B. Fixed Transversal Periodicity transversal periodicity of the FSS structure is changed, whereas in the second one the radius is varied when the transversal pe. In both cases, the radiation patterns riod is fixed to are compared. A. Fixed Angular Periodicity Fig. 7(a) and (b) show the effect of the radius of the cylindrical FSS structure on the E- and H-plane radiation patterns when . As illustrated, the radius of the cylindrical structure affects the side-lobe and back-lobe levels in both the E- and H-planes. In the E-plane, the lowest side-lobe level is achieved . In this case, the side-lobe is 10 dB and 3 dB when below the case of and , respectively. , is 5 dB below the case of The back-lobe, when and 2 dB higher than the case of . The 3-dB beamwidth in the E-plane is not affected by the radius of the FSS structure, whereas the H-plane beamwidth is changed. results in a narrower In the H-plane, the case of beamwidth and higher back-lobe, and the case of has a wider beamwidth and lower back-lobe level. The effect of the radius of the cylindrical FSS structure on the antenna directivity when is shown in Fig. 8. The case of gives the highest antenna directivity.

Here, the transversal period is fixed when the radius changes. The same transversal period means the transmission and reflection coefficients of the FSS structure are the same. To keep the transversal period fixed when the radius of the must be modified. We cylindrical structure changes, the demonstrate this in the following three cases: , , • ; • , , ; , , . • It was attempted to keep the radius of the structure the same (number of columns of as the previous case, but since the strips) must be an integer, the value of R cannot be exactly the same as the previous case. The E- and H-plane radiation patterns of the antenna in the three cases are shown in Fig. 9. As it can be with seen, the lowest side-lobe is achieved when , but the with case has the lowest back-lobe level. The directivity of the antenna versus frequency for different radii when the transversal period is fixed is shown in Fig. 10. The highest directivity is achieved when with . with From the previous parametric studies, the is chosen as the final design parameter value. It is

EDALATI AND DENIDNI: HIGH-GAIN RECONFIGURABLE SECTORAL ANTENNA USING AN ACTIVE CYLINDRICAL FSS STRUCTURE

Fig. 12.

Fig. 13.

L

S

of the antenna with one and two pipe layers.

S

of the Antenna, effect of the length of the second pipe when

= 56 mm.

also important to note that the radius of the FSS structure affects the antenna matching, which is not shown here. By decreasing the radius of the cylindrical structure, the antenna matching becomes difficult. It should be noted that the E-plane beamwidth is defined by the radiating source, whereas the H-plane beamwidth as well as the side-lobe and back-lobe levels can be modified by the cylindrical FSS configuration. VII. INPUT IMPEDANCE AND MATCHING OF THE CYLINDRICAL FSS ANTENNA The antenna matching is mostly related to the ECCD array elements. To show the effect of each design parameter on antenna matching, some parametric studies are presented in this section. Fig. 11 shows the effect of using one and two layers of pipes on the input impedance. The ECCD array with one pipe , and the ECCD array with two pipes has has and . As can be seen, by using two pipes, smoother real and imaginary parts of the input impedance are achieved. Indeed, the slot couples the energy to the first pipe and the second pipe is parasitically coupled to the first pipe. This results in wider impedance matching as presented less in Fig. 12. The matching bandwidth is about 6% for the

Fig. 14. Antenna S , effect of the length of the first pipe when the L

56 mm.

Fig. 15.

D

S

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=

of the antenna, effect of the diameter of the second pipe when

= 6 mm.

than 10 dB. It is important to note that the diameters of the pipes in both cases are optimized for the best matching performance. The effects of the pipe length when a second pipe is fixed on the antenna matching are shown in Figs. 13 and 14. According to the presented results, it can be noted that increasing the length of the pipes leads to shift down the antenna resonance frequency. The length of the second pipe has more effect on the resonance frequency of the ECCD array compared to the first one. The best result is achieved when both pipes are the same length and equal . to 0.39 Fig. 15 shows the effect of the diameter of the second pipe , and Fig. 16 when the first one is fixed to presents the effect of the diameter of the first pipe when the on diameter of the second pipe is fixed to the antenna matching. Based on these results, by modifying the is modified. The best result radii of the pipes, the level of the and . is achieved when VIII. FABRICATION AND MEASUREMENT RESULTS To validate the proposed idea, an antenna prototype was fabricated using RO3003 flexible substrate with a permittivity of 3, a thickness of 0.254 mm, and a loss tangent of 0.0013. High frequency PIN diodes GMP-4201 from Microsemi with

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S

Fig. 18. Simulated and Measured S

of the antenna.

of the antenna, effect of the diameter of the second pipe when .

= 16 mm

Fig. 17. (a) Photograph of the proposed reconfigurable FSS antenna, (b) photograph of the ECCD array.

and are used in this prototype [22]. A photograph of the fabricated antenna is shown in Fig. 17(a). Two narrow DC feeding lines are used at both sides of the strips for ) are used at supplying the diodes. High-value resistors (5 the top of each line between the strips and the DC feeding line to produce the same amount of current for all the diodes to obtain the same performance and to protect the diodes. The ECCD array was also fabricated with a UT-141B semi-rigid coaxial cable from Micro-Coax [23], as shown in Fig. 17(b). The pipes are formable Brass (Alloy 260) tubes from McMASTER-CARR [24] with the same dimensions as listed in Table I. To fix the space between the pipes and the central coaxial cable, foam in a

Fig. 19. Simulated and measured radiation pattern at 2.1 GHz, (a) E-plane, (b) H-plane.

cylindrical-shape and liquid glue are employed. In addition, the cylindrical shape foam is used to keep the ECCD array in the middle of the cylindrical FSS structure. The dimensions of the fabricated antenna are the same as simulated ones. For the measurements, we used five columns of diodes in the off-state and seven columns of diodes in the on-state. The power

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Fig. 21. Measurement and simulation results for antenna gain.

Fig. 20. Antenna reconfigurability, (a) H-plane radiation pattern, (b) diodestate configuration.

supply voltage for the off-state diodes is 0 V, when seven adjacent columns of diodes in the on-state supply as parallel lines with 37 V DC. Fig. 18 presents the simulated and measured of the antenna. The measured matching bandwidth is 6% for less than , and it is in a good agreement with the the simulated one. The simulated and measured antenna radiation patterns in the E- and H-planes at 2.1 GHz are shown in Fig. 19. The measured E- and H-plane radiation beamwidths are 20 and 70 , respec, which is in tively, and the measured side-lobe level is good agreement with the simulated ones. The back-lobe level of the antenna is due to the non-perfect on-state reflector and its relatively high transmission coefficient (see Fig. 2). To show the radiation pattern reconfigurability of the antenna in the azimuth plane, the simulated H-plane radiation pattern for , and is also preswitching the beam toward 0 , sented in Fig. 20(a). The presented radiation patterns correspond to the diode-state configurations shown in Fig. 20(b). The radiation pattern for case (I) is compared with the measurement, which agrees well with the simulated one. As the fabricated prototype in the azimuth plane has cylindrical symmetry, it can be concluded that the other cases will give the same performance as the simulation. It is important to note that the beam spacing in the proposed and it is possible to obtain antenna is equal to smaller beam spacing by optimizing the FSS design for a . smaller The gain of the antenna was measured using the gain comparison method. Fig. 21 shows the measured and simulated gain.

Fig. 22. Measured radiation pattern of the antenna at different frequencies, (a) E-plane, (b) H-plane.

The measured gain of the antenna is about 13 dBi and agrees well with the simulated value. To evaluate the antenna radiation pattern bandwidth, the radiation patterns of the antenna at 3 different frequencies (2 GHz, 2.1 GHz, 2.15 GHz) are presented in Fig. 22. As demonstrated, they are in a good agreement with each other, which shows the stability of the antenna radiation across the operating band.

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IX. CONCLUSION In this paper, a novel design of a reconfigurable sectoral antenna based on an active cylindrical FSS structure has been proposed. The effects of the diode configuration as well as the radius of the cylindrical FSS structure on the radiation pattern of the antenna have been studied to improve the performance of the proposed sectoral antenna. An ECCD array has been designed and employed as a high-gain omnidirectional source in the center of the cylindrical FSS structure. It has been shown that the E-plane radiation beamwidth is defined by the radiating source, whereas the H-plane radiation beamwidth, side-lobe and back-lobe levels of the antenna are modified by the optimization of the FSS structure. A novel solution for increasing the bandwidth of the ECCD array using two layers of metallic pipes has also been proposed. The proposed antenna has been fabricated and measured. The measured results show good agreement with the simulated ones. With these features, the proposed antenna can be used for base-station applications.

REFERENCES [1] P. Barachat, “Sectoral pattern synthesis with primary feeds, LA turbie,” IEEE Trans. Antennas Propag., vol. 42, no. 4, pp. 484–491, 1994. [2] W. Di, Y. Yingzeng, and G. S. Minjun, “Wideband dipole antenna for 3G base stations,” in Proc. IEEE Int. Symp. Microw. Antenna, Propag. and EMC Technol. for Wireless Commun., Aug. 2005, vol. 1, pp. 454–457. [3] D. Serhal, M. Hajj, R. Chantalat, J. Drouet, and B. Jecko, “Multifed sectoral EBG antenna for WiMAX applications,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 620–623, 2009. [4] M. Hajj, E. Rodes, and T. Monediere, “Dual-band EBG sectoral antenna using a single-layer FSS for UMTS application,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 161–164, 2009. [5] M. Hajj, D. Serhal, R. Chantalat, and B. Jecko, “New development around M-FSS antennas for sectoral coverage of telecommunication networks with dual polarization,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 670–673, 2009. [6] J. T. Bernahard, Reconfigurable Antennas. London, U.K.: Morgan & Claypool, 2007. [7] 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,” IET Microwaves, Antennas Propag., vol. 3, no. 1, pp. 47–54, 2009. [8] J. D. Boerman and J. T. Bernhard, “Performance study of pattern reconfigurable antennas in MIMO communication systems,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 231–236, 2008. [9] Y. Cai and Z. Du, “A novel pattern reconfigurable antenna array for diversity systems,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1227–1230, 2009. [10] A. Edalati and T. A. Denidni, “Reconfigurable beamwidth antenna based on active partially reflective surfaces,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1087–1090, 2009. [11] A. Edalati and T. A. Denidni, “Experimental investigation of a new reconfigurable sectoral antenna,” in Proc. IEEE AP-S Int. Symp., 2010, pp. 1–4. [12] B. A. Munk, Frequency Selective Surfaces Theory and Design. New York: Wiley, 2000. [13] R. J. Mittra, C. H. Chan, and T. A. Cwik, “Techniques for analyzing frequency selective surfaces—A review,” Proc. IEEE, vol. 76, pp. 1593–1614, 1988. [14] J. P. Montgomery, “Scattering by an infinite periodic array of thin conductors on a dielectric sheet,” IEEE Trans. Antennas Propag., vol. 23, pp. 70–75, Jan. 1975. [15] H. A. Kalhor, “Electromagnetic scattering by a dielectric slab loaded with a periodic array of strips over a ground plane,” IEEE Trans. Antennas Propag., vol. 36, pp. 147–151, Jan. 1988.

[16] J. M. Lourtioz, A. De Lustrac, F. Gadot, S. Rowson, A. Chelnokov, T. Brillat, A. Ammouche, J. Danglot, O. Vanbesien, and D. Lippens, “Toward controllable photonic crystals for centimeter and millimeter wave devices,” J. Lightwave Tech., vol. 17, pp. 2025–2031, 1999. [17] A. Edalati, H. Boutayeb, and T. A. Denidni, “Band structure analysis of reconfigurable metallic crystals: Effect of active elements,” J. Electromagn. Waves Applicat., vol. 21, no. 15, pp. 2421–2430, 2007. [18] G. Poilasne, P. Pouliquen, K. Mahdjoubi, L. Desclos, and C. Terret, “Active metallic photonic bandgap material MPBG: Experimental results on beam shaper,” IEEE Trans. Antennas Propag., vol. 48, no. 1, pp. 117–119, Jan. 2000. [19] H. Boutayeb, T. A. Denidni, K. Mahdjoubi, A. C. Tarot, A. R. Sebak, and L. Talbi, “Analysis and design of a cylindrical EBG based directive antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 211–219, 2006. [20] CST Microwave Studio 2006 B, CST Darmstadt Germany. [21] H. Miyashita, H. Ohmine, K. Nishizawa, S. Makino, and S. Urasaki, “Electromagnetically coupled coaxial dipole array antenna,” IEEE Trans. Antennas Propag., vol. 47, no. 11, 1999. [22] [Online]. Available: www.microsemi.com [23] [Online]. Available: www.micro-coax.com [24] [Online]. Available: www.mcmaster.com Arezou Edalati (S’10) received the B.S. degree from K.N.T.U. University of Technology, Tehran, Iran, in 1999, the M.Sc. degree from Azad University, Tehran, in 2002, and the Ph.D. degree from the Institute National de la Recherche Scientifique (INRS), Montreal, QC, Canada, in 2010, all in telecommunication engineering. She is currently a Postdoctoral Research Fellow at University of Michigan, Ann Arbor. Her research areas of interest are frequency selective surfaces (FSSs), electromagnetic bandgap (EBG) structures, metamaterials and their applications on design of antennas, UWB antennas and reconfigurable antennas. Dr. Edalati was awarded a FQRNT Ph.D. scholarship and a FQRNT Postdoctoral Fellowship which are Quebec government grants based on competitions.

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. 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, 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. Dr. Denidni is a member of the Order of Engineers of the Province of Quebec, Canada. He is also a member of URSI (Commission C). He was an Associate Editor for the IEEE Antennas and wireless Propagation Letters from 2006 to 2007. From 2008 to 2010, he served as an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

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Exact Dipole Radiation From an Oblate Semi-Spheroidal Cavity Filled With DNG Metamaterial Amir Nader Askarpour, Student Member, IEEE, and Piergiorgio L. E. Uslenghi, Life Fellow, IEEE

Abstract—An oblate semi-spheroidal cavity is flush-mounted under a metal plane and is coupled to the half-space above the plane via its circular interfocal aperture. The cavity is filled with a lossless metamaterial whose refractive index is negative and the opposite of the refractive index of the medium filling the half-space above the cavity, while the intrinsic impedances of the two media may be different. An exact solution is obtained in terms of infinite series of oblate spheroidal wave functions, when the primary field is an axially-oriented electric or magnetic dipole located on the symmetry axis of the structure. Numerical results are presented and discussed. Index Terms—Complex media, electromagnetic radiation, exact solution, metamaterials, spheroidal functions.

I. INTRODUCTION

inside the cavity, respectively. The solutions are expressed as infinite series of oblate spheroidal wave functions in the notation of Flammer [21], as used in [22]. The modal expansion coefficients are found analytically by imposing the boundary conditions. In particular, the radiated far field is obtained. In the derivations, use is made of properties of oblate spheroidal functions which, for brevity, are not repeated herein; the interested reader is referred to the Appendix in [18]. The analysis is conducted in phasor domain with the time-dependence factor omitted throughout. Extensive numerical results are presented and discussed in Section V. Some analytical derivations for the particular case of equal intrinsic impedances inside and outside the cavity have been presented at conferences, for electric dipole [23] and magnetic dipole [24] sources.

T

HERE exist only two open cavities for which an electromagnetic boundary-value problem is presently amenable to exact analytical solution. The first problem is two-dimensional and involves a semi-elliptical channel flush-mounted under a metallic plane and coupled to the half-space above the plane via the channel’s interfocal slit. This problem was solved by Uslenghi [1]–[3] and generated a flurry of related works [4]–[17]. The second problem involves an oblate semi-spheroidal cavity flush-mounted under a metallic plane and coupled to the half-space above it via its circular interfocal aperture; it was solved exactly when the media inside and outside the cavity are isorefractive to each other and the primary field is an axial dipole located on the symmetry axis of the cavity [18]. In this paper, we consider the geometry analyzed in [18]–[20] when the material inside the cavity is a lossless double-negative (DNG) metamaterial whose negative refractive index is the opposite of the refractive index in the double-positive (DPS) half-space above the cavity. The intrinsic impedances of the media inside and outside the cavity are both positive, but may have different values. The geometry of the problem is discussed in Section II, and exact analytical solutions are obtained in Sections III and IV for dipole sources outside and Manuscript received May 10, 2010; revised November 08, 2010; accepted December 02, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. A. N. Askarpour is with the Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL 60607 USA on leave from the University of Tehran, Tehran, Iran (e-mail: [email protected]). P. L. E. Uslenghi is with the Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL 60607 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2152334

II. GEOMETRY OF THE PROBLEM A cross-section through the symmetry axis of the oblate spheroidal cavity with metal walls flush-mounted under an infinite metal plane is shown in Fig. 1. The cross-section BGE of the cavity is a semi-ellipse with major axis BE and interfocal distance . The segment CD is the diameter of the circular hole that couples the cavity to the half-space above it. The segments BC and DE are the cross-section of a flat metallic annular ring of outer diameter BE, inner diameter CD, and infinitesimal thickness. We introduce the oblate spheroidal coordinates related to the rectangular coordinates by (1)

. The surfaces , are oblate spheroids with major axis and as symmetry axis. The circular aperture of minor axis plane, is centered at the origin, and diameter lies in the . The curved wall of the cavity corresponds corresponds to . The surfaces are to half the spheroid hyperboloids of revolution with z as symmetry axis, and for is the axis, and plane with the circle excluded. The surfaces is the are half-planes originating in the axis. has electric The lossless medium in the half-space permittivity and magnetic permeability , both real positive. The lossless medium inside the cavity is a DNG metamaterial where

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For fields inside the cavity, we add the subscript 1 to the field components, replace with and with in (5)–(7). If the dipole is located at ( ) with moment corresponding to an electric Hertz vector where is the distance of the observation point ( ) from the dipole, the primary magnetic field is [22]

(8)

Fig. 1. Geometry of the problem.

with electric permittivity and magnetic permeability real negative and such that

, both

(2) Under these conditions, causality dictates that the wavenumbers outside the cavity and inside the cavity be the opposite of each other

are radial oblate spheroidal functions of the first where and third kind, respectively, and are angular oblate spheroidal functions [21]. The quantity is the smaller (larger) between and . For details on the notation of the oblate spheroidal functions see, e.g., [18], [21] and [22]. The total field outside the cavity is the sum of three terms: the incident field, the field reflected by the infinite metal plate in the absence of the cavity, leading to [18]

(3) whereas the intrinsic impedances outside the cavity and inside the cavity are both positive, but in general different from each other

(9) and the diffracted field

(4) is extensively used. In the following, the parameter The primary source is an electric or magnetic dipole located either outside the cavity at ( ) on the positive axis (Section III) or inside the cavity at ( ) on the negative axis (Section IV), and axially oriented. The electric and magnetic dipole cases are not the dual of each other, because the metal boundaries are taken as perfect electric conductors in both cases.

(10) due to the cavity, that satisfies the radiation condition. The total field inside the cavity is

(11)

III. DIPOLE SOURCES OUTSIDE THE CAVITY A. Electric Dipole For an electric dipole source located on the axis outside the cavity and axially oriented, the field components outside the cavity are of the type

where it should be noted that . and are found The unknown modal coefficients by imposing the boundary conditions on the metallic surface and across the hole coupling the cavity to the half-space above it, yielding

(5) where (6)

(12)

(7)

(13)

ASKARPOUR AND USLENGHI: EXACT DIPOLE RADIATION FROM AN OBLATE SEMI-SPHEROIDAL CAVITY FILLED WITH DNG METAMATERIAL

where , and the prime means derivative with respect to . The surface current densities on the PEC surfaces are the total tangential magnetic field rotated by 90 , and are obtained from (9)–(11) following the procedure used in [18]. The diffracted far field in the half-space is given by

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(21)

(22) (14) where, by imposing the boundary conditions where and as . When the cavity recedes to infinity , one obtains the limit case of a circular hole in an infinite PEC plane separating the DPS ) half-space from the DNG half-space . From the previous formulas (12)–(13) and by assuming that has a vanishing small positive imaginary part, one finds that

(23) (24)

(15) (16) (17)

The tangential magnetic fields at the PEC surface are easily and , as done in obtained from the general expansion for [18]. The far field for is

and consequently, (25)

(18) while

is

obtained

by

replacing with

with

For a circular hole in an infinite metal plate that

and

we find

(26)

in (18).

(27)

B. Magnetic Dipole

(28)

The field components are obtained from those of the electric and follow from , dipole case using duality; hence and . A magnetic dipole axially oriented and located at ( ) on the positive axis, with moment corresponding to a magnetic Hertz vector , generates the primary field

and consequently

(29) while

is

obtained with

by

replacing

in (29). (19) With notation akin to the one in the preceding section, one finds

IV. DIPOLE SOURCES INSIDE THE CAVITY A. Electric Dipole

(20)

) For an electric dipole located on the z axis at ( inside the cavity and axially oriented, it is expedient to express the total field inside the cavity as the sum of three terms: the

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incident field, the field reflected at the metal plane when there is no coupling hole and the cavity recedes to infinity , and the perturbation introduced in the field inside the cavity by the presence of the cavity and the coupling hole. The first two terms combine to yield

where

(39) (30) while the third term may be written as (40)

(31) The field that enters the half-space is

through the hole

(41) and the modal coefficients are found by imposing the boundary conditions, yielding

(42)

(32) Imposition of the boundary conditions yields the modal expansion coefficients:

(43) (44)

(33) where

(45)

(34) (35)

The far field in the half-space

above the cavity is

where (46) (36) In particular, the far field in the half-space above the cavity is

(37) B. Magnetic Dipole A procedure similar to the one adopted for an electric dipole source allows us to write the total electric field produced by an axially oriented magnetic dipole located at ( ) inside the cavity as

(38)

V. NUMERICAL RESULTS We only consider the computation of the quantities and . All other field components can be evaluated from these two field components. In our numerical evaluation of the fields, we used closed form expressions for the incident and reflected field terms in the form of radiation from a dipole and its image. In this way, we lowered the computational burden of the problem and increased the accuracy of numerical results. Therefore, only the diffracted fields are computed using the series representation of the fields in terms of spheroidal functions. Radial and angular oblate spheroidal functions are computed using the methods discussed in [25]. In order to achieve a convergence error of less than 0.1 percent in our calculations, we considered up to fifteen terms in summations needed for computing the diffracted fields.

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Fig. 2. Geometry for the computation of field components. The figure shows . However, the locations of the dipole source along the coordinate line only one source is considered for each case.

Fig. 4. Total magnetic field due to an electric dipole located inside the ) and evaluated along the coordinate lines cavity at ( (solid line), (dashed line) and (dotted line). The plots and (b) . correspond to (a)

Fig. 3. Total magnetic field due to an electric dipole located outside the ) and evaluated along the coordinate lines cavity at ( (solid line), (dashed line) and (dotted line). The plots and (b) . correspond to (a)

The fields are computed along the coordinate lines , as shown in Fig. 2, to illustrate the penetration of the field into the cavity due to a dipole outside the cavity or, conversely, the diffraction of the field outside the cavity due to a dipole inside it.

In order to demonstrate fields on the coordinate lines continuously, we use the dimensionless variable instead of . In this way negative values of correspond to the interior of the cavity and positive values of correspond to the exterior of the cavity. Fig. 3 shows the magnitude of the magnetic field due to an electric dipole outside the cavity. In these examples, the cavity wall is located at and the dipole source at ( ). The dipole is excited at a wavelength equal to . The results shown are for in the Fig. 3(a) and are for in Fig. 3(b). Fig. 4 shows the magnetic field in the same configuration as before, the only difference being that the position of the source is now at ( ) inside the cavity. Again, Fig. 4(a) is for and Fig. 4(b) for . In examining the plots of Figs. 3 and 4, we observe two salient features. First, the field magnitude increases in the vicinity of the dipole source, as expected. Second, has a sharp maximum at the DPS-DNG boundary, especially as the metal edge of the coupling hole is approached; this behavior would not be expected for a magnetic field component parallel to the edge if the edge were immersed in a DPS material. However, in our case

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do not decrease sufficiently rapidly as the index goes to infinity where . In the special case of the series do not converge for fields near the boundary. This behavior at the boundary separating DPS from DNG media was previously reported in [26]. VI. CONCLUSION An exact analytical solution has been derived for the radiation of axially located and oriented dipoles in the presence of a flush-mounted oblate spheroidal cavity filled with DNG metamaterial. The angular oblate spheroidal functions are independent of the sign of the propagation constant, thus allowing for the explicit, analytical determination of the modal expansion coefficients in the infinite series of oblate spheroidal eigenfunctions that represent the components of the electromagnetic field. Numerical results have been shown and discussed. This novel canonical solution is important not only because it enriches the catalog of exact solutions for electromagnetic field problems, but also because it can be utilized to test the accuracy of computer codes developed to handle complex structures involving cavities, different penetrable materials, and sharp edges. It must be noted that the analysis presented in this work is valid only at those frequencies where the behavior of the DNG material is as postulated. Those frequencies are determined by the dispersive properties of the material (e.g., Drude model for permittivity and Lorentz model for permeability). Additionally, it is difficult at present to construct DNG materials with negligible losses; however, continuous progress is being made in this area, and the assumptions made in this work do not violate any physical law. REFERENCES Fig. 5. Contour plots of the electric field due to a magnetic dipole located ) and evaluated for and (a) outside the cavity at ( and (b) .

the edge is at the interface between DPS and DNG materials; a satisfactory understanding of the field behavior would require the analytical solution of diffraction by a PEC half-plane located at the interface separating DPS and DNG half-spaces, which is not presently available. Finally, contour plots of the electric field due to a magnetic dipole outside the cavity are shown in Fig. 5. The amplitude of incident and reflected field components are much larger than the amplitude of diffracted field due to the cavity. Therefore, only the diffracted fields are shown in Fig. 5. Also, because of the symmetry of the problem, only half of the cavity is shown. In these examples the cavity is the same as before, and the wavelength is equal to . A comparison between Fig. 5(a) and (b) shows the strong influence that the value of has on the amplitude of the diffracted field. It should be noted that as approaches unity, near field values in proximity of the boundary between the DPS half-space and the DNG metamaterial become large. Therefore, for values of near unity we should consider many more terms in the eigenfunction expansions to achieve convergence. This behavior is due to the fact that the modal coefficients and

[1] P. L. E. Uslenghi, “Exact scattering from a slotted semielliptical channel,” in Proc. IEEE-APS Int. Symp. on Antennas and Propagation, Chicago, IL, Jul. 1992, pp. 1849–1852. [2] P. L. E. Uslenghi, “Exact radiation and scattering from a slotted semielliptical channel filled with isorefractive material,” presented at the URSI National Radio Science Meeting, Boulder, CO, Jan. 1999. [3] P. L. E. Uslenghi, “Exact penetration, radiation and scattering for a slotted semielliptical channel filled with isorefractive material,” IEEE Trans. Antennas Propag., vol. 52, pp. 1473–1480, Jun. 2004. [4] D. Erricolo, M. Lockard, C. M. Butler, and P. L. E. Uslenghi, “Analytical formulas and integral equation methods: A study of penetration, radiation, and scattering for a slotted semielliptical channel filled with isorefractive material,” presented at the IEEE Antennas and Propagation Society Int. Symp. and URSI Meeting, Columbus, Ohio, Jun. 2003. [5] D. Erricolo and P. L. E. Uslenghi, “Electromagnetic behavior of a partially covered trench in a corner,” presented at the National Radio Science Meeting, Boulder, CO, Jan. 2004. [6] T. Larsen, D. Erricolo, and P. L. E. Uslenghi, “Low-frequency behavior of a slotted semielliptical channel,” presented at the National Radio Science Meeting, Boulder, Colorado, Jan. 2004. [7] D. Erricolo and P. L. E. Uslenghi, “Radiation from a line source inside a trench in a corner,” in Proc. URSI International Symp. on Electromagnetic Theory, Pisa, Italy, May 2004. [8] D. Erricolo and P. L. E. Uslenghi, “Radiation from an antenna in a partially covered cavity near a 2D or 3D corner,” presented at the IEEE-APS Int. Symp. and URSI Meeting, Monterey, CA, Jun. 2004. [9] D. Erricolo and P. L. E. Uslenghi, “Exact analysis of a 2D cavitybacked slot in a ground plane covered by an isorefractive lens,” presented at the IEEE-APS Int. Symp. and URSI Meeting, Washington, DC, Jul. 2005. [10] D. Erricolo, M. D. Lockard, C. M. Butler, and P. L. E. Uslenghi, “Numerical analysis of penetration, radiation and scattering for a 2D slotted semielliptical channel filled with isorefractive material,” Progr. Electromagn. Res. (PIER), vol. 53, pp. 69–89, 2005.

ASKARPOUR AND USLENGHI: EXACT DIPOLE RADIATION FROM AN OBLATE SEMI-SPHEROIDAL CAVITY FILLED WITH DNG METAMATERIAL

[11] D. Erricolo, M. D. Lockard, C. M. Butler, and P. L. E. Uslenghi, “Currents on conducting surfaces of a semielliptical-channel-backed slotted screen in an isorefractive environment,” IEEE Trans. Antennas Propag., vol. 53, pp. 2350–2356, Jun. 2005. [12] D. Erricolo and P. L. E. Uslenghi, “Penetration, radiation and scattering for a cavity-backed gap in a corner,” IEEE Trans. Antennas Propag., vol. 53, pp. 2738–2748, Aug. 2005. [13] M. Valentino and D. Erricolo, “Exact two-dimensional scattering from a slot in a ground plane backed by a semielliptical cavity and covered with an isorefractive diaphragm,” Radio Sci., vol. 42, Nov.–Dec. 2007, RS6S12, doi:10.1029/2006RS003547. [14] D. Monopoli, D. Erricolo, P. L. E. Uslenghi, and R. E. Zich, “Scattering by a slotted semielliptical channel containing DNG metamaterial,” presented at the Proc. XXIX General Assembly of the Int. Union of Radio Science (URSI), Chicago, IL, Aug. 2008. [15] O. Akgol, D. Erricolo, and P. L. E. Uslenghi, “Electromagnetic scattering by a semielliptical trench filled with DNG metamaterial,” presented at the Proc. Int. Conf. on Electromagnetics in Advanced Appls. (ICEAA’09), Torino, Italy, Sep. 2009. [16] O. Akgol, D. Erricolo, and P. L. E. Uslenghi, “Radiation of a line source by a slotted semielliptical trench filled with DNG metamaterial,” presented at the 3rd IEEE Int. Symp. on Microwave, Antenna, Propagation, and EMC Technologies for Wireless Communications (MAPE 2009), Beijing, China, Oct. 2009. [17] O. Akgol, D. Erricolo, and P. L. E. Uslenghi, “Electromagnetic radiation and scattering for a gap in a corner backed by a cavity filled with DNG metamaterial,” presented at the 9th Engineering Mathematics and Appls. Conf. (EMAC 2009), Adelaide, Australia, Dec. 2009. [18] C. Berardi, D. Erricolo, and P. L. E. Uslenghi, “Exact dipole radiation for an oblate spheroidal cavity filled with isorefractive material and aperture-coupled to a half space,” IEEE Trans. Antennas Propag., vol. 52, pp. 2205–2213, Sep. 2004. [19] D. Erricolo, P. L. E. Uslenghi, and M. Valentino, “Exact analysis of an oblate spheroidal cavity with a circular aperture in a ground plane covered by an isorefractive lens,” presented at the Int. Conf. on Electromagnetics in Advanced Appls.(ICEAA’05), Torino, Italy, Sep. 2005. [20] M. Valentino and D. Erricolo, “Exact radiation of a dipole in the presence of a circular aperture in a ground plane backed by a spheroidal cavity and covered with an isorefractive diaphragm,” Radio Sci., vol. 42, Nov.–Dec. 2007, RS6S13, doi:10.1029/2006RS003548. [21] C. Flammer, Spheroidal Wave Functions. Stanford, CA: Stanford Univ. Press, 1957. [22] J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes. Amsterdam: North-Holland Publishing, 1969. [23] A. N. Askarpour and P. L. E. Uslenghi, “Radiation from an electric dipole axially mounted above a spheroidal cavity filled with DNG metamaterial,” presented at the URSI Electromagnetic Theory Symp. (EMTS), Berlin, Germany, Aug. 2010, invited paper. [24] A. N. Askarpour and P. L. E. Uslenghi, “Radiation from a magnetic dipole axially located above a spheroidal cavity filled with DNG metamaterial,” presented at the CNC/USNC/URSI Radio Science Meeting, Toronto, ON, Canada, Jul. 2010. [25] S. Zhang and J. M. Jin, Computation of Special Functions. New York: Wiley, 1996. [26] A. Alu and N. Engheta, “Radiation from a travelling-wave current sheet at the interface between a conventional material and a metamaterial with negative permittivity and permeability,” Microw. Opt. Technol. Lett., vol. 35, pp. 460–463, Dec. 2002. Amir Nader Askarpour (S’08) received the B.S. degree in electrical engineering from Sharif Institute of Technology, Tehran, Iran, in 2004 and the M.S. degree in electrical engineering from University of Tehran, in 2006, where he is currently pursuing the Ph.D. degree. Since November 2009, he is a Visiting Research Scholar at the University of Illinois at Chicago. His research interests include numerical and analytical methods in electromagnetics, electromagnetic inverse scattering problems and planar antenna design.

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Piergiorgio L. E. Uslenghi (SM’70–F’90–LF’07) was born in Turin, Italy in 1937. He received the Laurea degree of Doctor in electrical engineering from the Polytechnic of Turin, and the M.S. and Ph.D. degrees in physics from the University of Michigan, Ann Arbor, in 1960, 1964, and 1967, respectively. He has been an Assistant Professor at the Polytechnic of Turin (1961), an Associate Research Engineer at Conductron Corporation, Ann Arbor, Michigan (1962–1963), and a Research Physicist at the Radiation Laboratory of the University of Michigan (1963–1970). In 1970, he joined the University of Illinois at Chicago, where he held a number of positions, including Founder and first Director of the Communications Laboratory (1976–1978), Founder and Director of the Electromagnetics Laboratory (1991–present), Professor of Electrical and Computer Engineering (1974–present), Associate Dean of the College of Engineering (1982–1987; 1994–present), and Adjunct Professor of Physics (2004–present). His research interests encompass antennas, microwaves, scattering theory and applications, electronic materials, modern optics, and applied mathematics. He has published five books and more than 250 journal and conference papers. Dr. Uslenghi is a member of the Phi Beta Kappa and Sigma Xi honorary societies, the Antennas and Propagation, Microwave Theory and Techniques, and Electromagnetic Compatibility Societies of IEEE, and is also a member of USNC-URSI Commissions D, E and K. In 1990 he was elected Fellow of the IEEE, in 2002 he became an EMP Fellow and in 2003 was inducted into the Accademia delle Scienze di Torino. In 2000 he was a recipient of the IEEE Third Millennium Medal and in 2007 he was awarded the IEEE Antennas and Propagation Society Distinguished Achievement Award. He was named University of Illinois Scholar in 2006 and Distinguished Professor in 2009. He served as Secretary-Treasurer, Vice-Chair and Chair of the Joint AP/MTT Chicago Chapter of IEEE twice, in 1975–1978 and in 1989–1992. He was the General Chair of the 1992 IEEE-APS International Symposium and URSI/NEM Meeting and the Organizer and Chair of the 1976 National Conference on Electromagnetic Scattering, both held in Chicago. He served as an elected member of the IEEE/APS Administrative Committee (1994–1996). He has been a member of the Joint Committee on Future Symposia of IEEE/APS and USNC-URSI (1994–2000), of the IEEE Heinrich Hertz Medal Committee (1992–1997), of the IEEE Technical Activities Board Publications Committee (2002), and of the IEEE Technical Activities Board Management Committee (2003). He served as Vice President (2000) and President (2001) of the IEEE Antennas and Propagation Society. He is a past editor of Electromagnetics and of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, the Founding Editor and first Editor-in-Chief of the IEEE Antennas and Wireless Propagation Letters, and has served for many years on the editorial boards of the Journal of Electromagnetic Waves and Applications, the European Journal of Telecommunications, and Alta Frequenza. He was Chairperson of the Technical Activities Committee (1994–1996), ViceChair (1997–1999) and Chair (2000–2002) of Commission B of the U.S. National Committee of the International Union of Radio Science (USNC-URSI). He was elected Secretary (2003–2005) and Chair (2006–2008) of USNC-URSI. He was the General Chair of the XXIX General Assembly of URSI, held in Chicago in August 2008. He has organized several special sessions at IEEE/APS International Symposia and URSI Meetings on such topics as nonlinear electromagnetism, scattering by wedges, advanced materials for electromagnetic applications, recent developments in scattering, microelectromechanical systems and novel mathematical techniques in electromagnetics. He is the Chair of the Scientific Committee of the International Conference on Electromagnetics in Advanced Applications.

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Bandwidth and Impedance-Matching Enhancement of Fractal Monopole Antennas Using Compact Grounded Coplanar Waveguide Mahdi Naghshvarian Jahromi, Abolfazl Falahati, and Rob. M. Edwards

Abstract—Grounded coplanar waveguides (GCPW) are used to enhance antenna characteristics, namely, reference to the bandwidth and the input impedance dynamic range. A printed fractal monopole such as the Sierpinski–Carpet (SC) antenna can be considered as a good candidate for comparing a coaxial-fed system with a GCPW-fed antenna. Conventional ), while SC can be applied at 6.25–8.4 GHz (ref to modified SC matches throughout the 4.65–10.5 GHz (ref to ). Clearly, this new feeding technique changes the behavior of the fractal elements from multiband to wideband. The GCPW-fed antennas have a low crosspolar field and a well-behaved pattern over the enhanced ones. This system of antennas is a possible competitor for the FCC ultrawideband radio short, fat pipe systems. In addition, time-domain analyses are performed to realize the proposed antenna time-domain application over ultrawideband.

VSWR VSWR

2

VSWR

2

2

Index Terms—Fractal antenna, grounded coplanar waveguide (GCPW), printed monopole; ultrawideband radio, Sierpinski– Carpet (SC).

I. INTRODUCTION HE explosive growth of the Internet and broadband systems over the last decade has led to an increasing demand for further high-speed and wideband technologies. Broadband mobile wireless access helps this growing need by offering anytime, anywhere Internet and high-density resolution multimedia access to homes and offices. There are many wireless applications that satisfy different needs for continuous, convenient, and flexible access to up-to-date information. One of them is the ultrawideband (UWB) technology. UWB brings with it the need for miniature antennas with bandwidths much larger than those required for the current mobile communication systems such as GSM and CDMA. This has been proposed as a technology suitable for wireless body area networks. In addition, UWB will

T

Manuscript received February 02, 2010; revised October 08, 2010; accepted November 27, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. M. Naghshvarian Jahromi was with the Electrical Engineering Department, Iran of University Science and Technology, Narmak, Tehran, Iran. He is now with the Antenna Design Group, Faramoj Pajouh Engineering Co., Tehran, Iran (e-mail: [email protected]). A. Falahati is with the Department of Electrical Engineering, Iran University of Science and Technology (IUST), Tehran (e-mail: [email protected]). R. M. Edwards is with the Department of Electrical Engineering, Loughborough University, Loughborough, 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.2011.2152321

offer a high data rate, short-range, low-power channel within a tightly specified range of frequencies [1]. Wideband antennas have been used in other applications, for example, the authors of [2], [3] discuss its application in groundpenetrating radar. In [4], several implementations of UWB systems are discussed including radiofrequency identity tags, impulse radar, and position finders. It is reasonable to assume these demands require a small antenna because of their applications. Earlier, the wideband antennas were classified by their shapes, namely, spirals, cones, etc. The term “slow wave” has been applied to the nonspecific family of wideband antennas. Many of these antennas were large and often required balanced feeds to work well. Antennas are particularly a challenging aspect of wideband technology as they are now required to be multioctave in response besides being small and for convenience unbalanced due to feeding constraints. To satisfy such requirements, various wideband antennas have been researched for UWB [5]–[10]. Further useful properties can be observed when fractal antennas are employed, such as their specification by the angle and the degree to which they are not closely coupled in resonance to the electrical length. Fractal geometry has been useful in designing small antennas [11], [12], multiband elements [13]–[17], and highly directive antennas [18]. They therefore tend to exhibit good bandwidth characteristics. In addition, they are efficient in terms of footprints of relatively long current paths that can be generated along in the fractal design a different number of iterations. Examples of some small lightweight fractal designs can be seen in [19]–[21]. The fractal shapes are used to design multiband antennas, but some of them have potential wideband properties too. The Penta-Gasket-Koch (PGK) is a fractal shape that was first introduced in [22] and was implemented for a third iteration version fractal design in [23], [24]. It is used to compare a conventional standing antenna and a planar monopole antenna. These papers ) show a wideband input impedance match (ref to to achieve 2–20 GHz; the same characteristics are also proved for the SC monopole antenna with the third iteration fractal design [25]. However, it is noticed that there are some occasional mismatches in operating bands. Also, specific dimensions and a number of iterations must be used for the fractal shape to achieve a wideband characteristic. In [26], at first, GCPW is introduced and utilized for modifying the PGK and SC fractal shape. In this paper, a GCPW is employed to improve the performance of the printed fractally designed antenna. The fractal antenna is first modeled in an industry-standard electromagnetic package and then measured. The proposed SC antennas

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Fig. 2. The modified antenna before mounting on ground plane: (a) Modified Sierpinski-Carpet radiation element, (b) ground plane of the proposed antennas.

Fig. 1. Sierpinski-Carpet antenna configurations: (a) Radiation element and substrate of conventional antenna, (b) all elements of the modified antenna.

are designed in two forms. One, within a conventional-fed manner using a probe through a ground plane created from the conductor center of a coaxial cable. Conventional SC can ). However, be applied at 6.25–8.4 GHz (ref to the other form, the modified SC, matches throughout the ). Clearly, this new feeding 4.65–10.5 GHz (ref to technique changes the multiband behavior of fractal elements to wideband behavior. One advantage of the modified antenna is that it is possible to control the start operating frequency control with CPW length variations. In this paper, the start operating frequency is decreased from 6.25 GHz to 4.65 GHz during measurements. Hence, the combination of the two given advantages can result in the application of such antenna types in UWB systems. II. PROPOSED ANTENNA DESIGN Two types of fractal monopole antennas are designed: those employing the conventional feed and those with GCPW. Fig. 1(a) and (b) show these antennas bearing second-iteration SC geometry as a radiation element. All the mentioned antennas are constructed on a Rogers , a relative RO4003 substrate with a thickness of , and a loss tangent about 0.0027 dielectric constant with the same square ground plane width of and substrate dimensions of 20.4 18 . The two proposed antennas have the same radiation element , , and . widths of Furthermore, the modified SC antenna has special dimensions , that produce the following CPW feeding: , , and (gap). It should be emphasized that the CPW ground is perpendicularly connected to main ground plane of this antenna. The modified antenna manufactured before being placing over the ground plane is shown in Fig. 2(a). Fig. 2(b) depicts the main ground plane of both proposed antennas. The conventional and the modified fractal antennas can be observed in Fig. 3. The left-hand side of the picture is the conventional standing monopole antenna, while the right-hand side is the modified standing monopole antenna.

Fig. 3. The manufactured proposed antennas: conventional Sierpinski-Carpet monopole (left), modified Sierpinski-Carpet monopole (right).

III. REALIZATION OF THE EFFECT OF NEW FEEDING TECHNIQUE ON STANDING FRACTAL MONOPOLE ANTENNA BANDWIDTH Fig. 4(a)–(c) shows the planar, conventional, and modified SC monopole antennas, respectively. If the modified antenna is observed, one can notice a hybrid of the conventional monopole antenna and the planar monopole antenna. The equivalent circuit of input impedance of this antenna is proportional to parallel input impedance of planar antenna and conventional antenna, but the weight effect of each impedance is variable with frequency, which is shown in Fig. 5. The CPW line is designed with an approximately 50- impedance characteristic. The electrical length of this line is found to vary from 7.1 to 24.1 for the 3.1–10.6 GHz frequency. These parameters can be calculated from the relationships given in [27]. Because of the low electrical length of the CPW line at a lower frequency band, its effect on the input impedance of the modified monopole antenna at low frequencies can be regarded as negligible. However, at higher frequencies, the effect of CPW line is found dominant and the modified antenna behaves as a planar monopole antenna. These are clear in Figs. 6 and 7, which show the comparison of the input impedance of planar with the conventional and the modified SC antennas. These are achieved by employing the CST Microwave Studio [28] commercial software. As a result, the authors selected the coefficient of the impedance and (Fig. 5) to satisfy this physical concept. At the beginning, must be close to 0 but close to 1, as at the start of the frequency band, the modified antenna behaves as a conventional monopole antenna. Toward the end of the frequency band, the must be close to 1 effect of the CPW line is dominant and close to 0. but A very simple method is proposed to calculate the impedance for the modified monopole antenna from that of the planar monopole antenna and the conventional monopole antenna. and variations with A linear approximation is used for

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Fig. 4. The proposed antennas: (a) planar Sierpinski-Carpet monopole antenna, (b) conventional Sierpinski-Carpet monopole antenna, (c) modified SierpinskiCarpet monopole antenna.

Fig. 7. Imaginary parts of input impedance of the proposed antennas.

Fig. 5. Modified antenna is a hybrid of a planar monopole antenna and a conventional monopole antenna.

Fig. 8. The linear approximation for monopole antenna estimation.

n

and

n

coefficients for the modified

Fig. 6. Real parts of input impedance of the proposed antennas.

the frequency band, which is indicated in Fig. 8. Matlab codes used to calculate the input antenna characteristic results are printed in Appendix A. Figs. 9 and 10 show the real and the imaginary parts of input impedance of the modified antenna, respectively. These figures also compare the results obtained for both Matlab and EM simulations. Now, considering these two curves, very good agreements are achieved. Furthermore, Figs. 11 and 12 are produced for the magnitude and the phase of parameter, respectively, to specify such approximations the that should be enough to design the modified monopole antenna based on conventional and planar monopole antennas. IV. EXPERIMENTAL RESULTS Fig. 13(a) and (b) demonstrate the parameter magnitude (dB) results obtained for the conventional SC and the modified

Fig. 9. Comparison of the real parts of the modified monopole antenna and the estimated impedance in an equivalent circuit.

SC by simulations and measurements. These plots reveal good resemblance between the simulated and the measured results. (Fig. 13), the conventional SC can be Referring to applied for the 6.25–8.4 GHz and the modified SC is matched throughout the 4.65–10.5 GHz band. Clearly, this new feeding

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Fig. 10. Comparison of the imaginary parts of the modified monopole antenna and the estimated impedance in an equivalent circuit.

Fig. 11. Comparison of S11 magnitudes of the modified monopole antenna and the estimated impedance in an equivalent circuit.

Fig. 12. Comparison of S11 phase of the modified monopole antenna and the estimated impedance in an equivalent circuit.

Fig. 13. S parameter magnitudes for the proposed antennas, (a) conventional antenna, (b) modified antenna, (c) comparisons of the measured results of the two proposed antennas.

technique changes the multiband behavior of the fractal element to the wideband (Fig. 13(c)) behavior. The reason for this event is depicted in the previous section. To get further insight into the effect of the GCPW feeding technique on these types of antennas, the measured input im-

pedances are also shown in Fig. 14. Fig. 14(a) presents the real parts and Fig. 14(b) the imaginary parts of the input impedance from 2–11 GHz. These figures clearly demonstrate the important reasons for frequency mismatches that can be modified by this technique, especially in the first resonance frequency and

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Fig. 15. Radiation patterns of the modified antenna, copolarization (continuous line), cross-polarization (dash line). (a) Phi = 0 (x-z plane), (b) Phi = 90 (y -z plane), (c) Theta = 90 (x-y plane).

Fig. 14. Input impedance of conventional and modified SC with a closer look at the real and the imaginary parts (a) Input resistance, (b) reactance.

about midband by increasing and/or decreasing the real and the imaginary parts of the input impedance, respectively. Radiation patterns of the modified antenna at frequencies of 4.65, 7.5, and 10.5 GHz are shown in Fig. 15. Because of symmetry and flatness, good radiation patterns are obtained for the three bands that display a dipole radiation mode at 4.65 GHz and a monopole radiation mode at 7.5 GHz and 10.5 GHz. It can be observed that, for most frequency ranges, the crosspolarization is at least 10 dB below that of copolarization (except the cut in Fig. 15(c). However, it should also be noted that the effect of the finite size of the ground plane must be taken into account when analyzing the patterns on these figures [29]. For instance, those at upper bands follow a ripple-like characteristic, which is due to diffraction at the edges of the plane. The ripple variations are faster when the frequency is increased. The reason is that the squared plane is not self-scalable and the edges are spaced at a longer distance in terms of the corresponding wavelength. In addition, the expected null in the -axis direction is hidden by the contribution of the antisymmetrical mode of the ground plane overall radiated power [29]. The peak gain (dBi) of both proposed antennas is depicted in Fig. 16. The peak gain is very small over most of the required

Fig. 16. Peak gain of both proposed antennas in dBi.

frequency band, which in comparison is a very wide band indeed, i.e., 4.65–10.5 GHz. This is because the gain variation is less than 3.5 dB for the conventional SC and the modified SC throughout the pass band (4.65–10.5 GHz). In addition, in most of the band frequency, the gain of the modified SC is greater than that of the conventional SC at about 4.0–7.0 GHz and 8.0–9.75 GHz frequency bands. These results show that a good antenna matching input impedance increases the antenna gain. V. TIME-DOMAIN RESPONSE As shown in the previous section, the proposed antenna (the modified SC antenna) has a wide bandwidth. However, this does

NAGHSHVARIAN JAHROMI et al.: BANDWIDTH AND IMPEDANCE-MATCHING ENHANCEMENT OF FRACTAL MONOPOLE ANTENNAS

Fig. 17. Time and frequency input responses: (a) Antenna input signal, (b) Fourier transform of the input signal amplitude.

not necessarily ensure that the antenna behaves well in time domain too; that is, the antenna does not widen a narrow time-domain pulse. Some multiresonant wideband antennas, such as log-periodic antennas and the proposed antenna, due to multiple reflections and large discontinuities within their structures, widen the narrow pulse in the time domain [30]. Fig. 15 shows the dependence of the radiation patterns on the frequency. The antenna does not have a flat transfer function. Therefore, to ensure the usefulness of the proposed antenna for time-domain applications, the time-domain responses of antennas must be examined, too. Furthermore, the proposed antennas are assumed to be excited by the wideband signals corresponding to 3.1–10.6 GHz band. This band is selected to examine the time-domain response of the proposed antennas, which is the first step for further optimization of the modified antennas to fulfill FCC standards in UWB applications. This wideband signal is the 5thderivative of the Gaussian pulse and is given by

(1) Here, has to be 51 psec to ensure that the shape of the spectrum complies with the spectral mask. The received signals, along with their Fourier transform, are shown in Fig. 17. The

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Fig. 18. Time and frequency terminal responses: (a) Signal at the terminals of the receiving antenna for the case of the proposed pair of Sierpinski-Carpet antennas operating in free space, (b) Fourier transform of the signal amplitude.

simulation is done by means of a finite integration technique (FIT), employing the commercially available CST microwave studio. Therefore, the communication link is made between the two identical proposed antennas in free space. The distance between the transmitting and the receiving antennas is assumed to be 10 cm. The Tx/Rx antennas for all the proposed antennas are assumed to have the same orientation. The signal at the receiving antenna at 50- output terminals, along with its Fourier transform, for the case of the two identical proposed antennas operating in free space are shown in Fig. 18. To determine the correlation coefficient between the signal at and the input signal the terminals of the receiving antenna , the following equation can be used:

(2)

where is a delay and is changed to make the numerator in (2) a maximum value [31], [32]. The values of the correlation coefficient obtained for each band are summarized in Table I. By this new feeding technique, it is seen that the correlation factor for the SC antennas does not change much. Moreover,

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TABLE I CALCULATED CORRELATION COEFFICIENT VALUES FOR THE PROPOSED ANTENNAS

; % Obtaining S11, amplitude and phase, and comparing with EM simulation ; % In EM simulation

the correlation between the time-domain transmitting antenna’s input signal and the receiving antenna’s output signal shows approximately 95% antipodal resemblance.

; % In EM simulation

VI. CONCLUSION

;%

In this paper, the SC fractal shape with second iteration and miniature sizes are employed. The results are compared with the conventional standing monopole antenna and further with the modified standing monopole antenna bearing new feeding techniques, i.e., the combination of the planar CPW-fed and the conventional monopole-fed antennas. The effect of this technique parameter magnican provide a very good improvement in tude considering SC geometry throughout the 4.65–10.5 GHz bandpass. The qualitative analysis was performed to propose a high-speed method for designing the modified SC based on planar monopole and conventional monopole input impedance rather than employing EM simulation method. In fact, this will make the impedance-matching optimization straightforward. In addition, the antennas were assumed to be excited by the 5thderivative of the Gaussian pulse wideband signal in 3.1–10.6 GHz, which means that these antennas can be used as a wideband antenna in time-domain applications such as impulse radio communication systems. In the course of achieving it, the correlation between the time-domain transmitting antenna’s input signal and the receiving antenna’s output signal results in approximately 95% antipodal resemblance. Furthermore, if optimization analyses are performed, a domestic UWB can be also designed employing such antennas. Moreover, with such antenna designs, a wide range of wireless communication systems considering frequency-selective channel characteristics with multimedia transmission in mind can be provided with good spectral shapes. APPENDIX The following are the Matlab code instructions for the parameter computational analyses of the modified SC input impedance that is obtained by the planar SC and the conventional SC input impedance: ; %n2 for conventional SC ; %n1 for planar SC % Building the input impedance of antennas that obtain by CST Microwave Studio ; % Planar SC ; % Modified SC ; % Conventional SC % Obtaining approximated input impedance for modified antenna by planar and %conventional SC

Approximation ; % Approximation REFERENCES [1] FCC 802 Standards Notes, “FCC first report and order on ultra-wideband technology” 2002. [2] J. Young and L. Peter, “A brief history of GPR fundamentals and applications,” in Proc. 6th Int. Conf. Ground Penetrating Radar, 1996, pp. 5–14. [3] D. J. Daniels, “Surface penetrating radar, IEE radar sonar navigation avionics series 6,” IEEE Press, pp. 72–93, 1996. [4] R. J. Fontana, “Recent system applications of short pulse ultra-wideband (UWB) technology,” IEEE Trans. MTT, vol. 52, no. 9, pp. 2087–2104, 2004. [5] M. Naghshvarian Jahromi, “Compact bandnotch UWB antenna with transmission-line-fed,” in Progr. Electromagn. Res. B (PIER B), 2008, vol. 3, pp. 283–293. [6] K.-H. Kim and S.-O. Park, “Analysis of the small band-rejected antenna with the parasitic strip for UWB,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1688–1692, Jun. 2006. [7] Y. J. Cho, K. H. Kim, D. H. Choi, S. S. Lee, and S. O. Park, “A miniature UWB planar monopole antenna with 5 GHz band-rejection filter and the time-domain characteristics,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1453–1460, May 2006. [8] K.-H. Kim, Y.-J. Cho, S.-H. Hwang, and S.-O. Park, “Bandnotched UWB planar monopole antenna with two parasitic patches,” Electron. Lett., vol. 41, no. 14, pp. 783–785, Jul. 2005. [9] Y.-C. Lin and K.-J. Hung, “Compact ultrawideband rectangular aperture antenna and band-notched designs,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3075–3081, Nov. 2006. [10] C. T. P. Song, P. S. Hall, H. Ghafouri-Shiraz, and D. Wake, “Fractal stacked monopole with very wide bandwidth,” Electron. Let, vol. 35, no. 12, pp. 945–946, 1999. [11] J. Anguera, E. Martínez, C. Puente, and E. Rozan, “The fractal Hilbert monopole: A two-dimensional wire,” Microw. Opt. Technol. Lett., vol. 36, no. 2, pp. 102–104, Jan. 2003. [12] J. P. Gianvittorio and Y. Rahmat-Samii, “Fractal antennas: A novel antenna miniaturization technique, and applications,” IEEE Antennas Propag. Mag., vol. 44, no. 1, pp. 20–36, Feb. 2002. [13] C. Puente, J. Anguera, C. Borja, and J. Soler, “Fractal-shaped antennas and their application to GSM 900/1800,” J. Inst. British Telecommun. Eng., vol. 2, pt. 3, Jul. 2001. [14] D. H. Werner, R. L. Haupt, and P. L. Werner, “Fractal antenna engineering: The theory and design of fractal antenna arrays,” IEEE Antennas Propag. Mag., vol. 41, no. 5, pp. 37–58, Oct. 1999. [15] J. Anguera, C. Puente, C. Borja, and J. Soler, K. Chang, Ed., “FractalShaped antennas: A Review,” in Wiley Encyclopedia of RF and Microwave Engineering, 2005, vol. 2, pp. 1620–1635. [16] J. Anguera, C. Puente, C. Borja, and J. Soler, “Dual frequency broadband stacked microstrip antenna using a reactive loading and a fractalshaped radiating edge,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 309–312, 2007. [17] Y. Kim and D. L. Jaggard, “The fractal random array,” Proc. IEEE, vol. 74, no. 9, pp. 1278–1280, Sep. 1986. [18] J. Anguera, C. Puente, C. Borja, R. Montero, and J. Soler, “Small and high directivity bowtie patch antenna based on the Sierpinski fractal,” Microw. Opt. Technol. Lett., vol. 31, no. 3, pp. 239–241, Nov. 2001.

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[19] A. Falahati, M. Naghshvarian Jahromi, and R. M. Edwards, “Dual band-notch CPW-ground-fed UWB antenna by fractal binary tree slot,” in Proc. 5th Int. Conf. on Wireless and Mobile Communications (ICWMC 2009), Cannes/La Bocca, France, Aug. 2009, pp. 385–390. [20] J. Anguera, J. P. Daniel, C. Borja, J. Mumbrú, C. Puente, T. Leduc, N. Laeveren, and P. Van Roy, “Metallized foams for fractal-shaped microstrip antennas,” IEEE Antennas Propag. Mag., vol. 50, no. 6, pp. 20–38, Dec. 2008. [21] M. Naghshvarian-Jahromi, “Novel miniature semi-circular-semifractal monopole daul band antenna,” J. Electromagn. Wave Applicat. (JEMWA), vol. 22, no. 2, pp. 227–237, 2008. [22] M. L. Lapidus and E. P. J. Pearse, “Complex dimensions and the Steiner formula,” presented at the AMS Nat. Meeting, San Antonio, TX, 2006. [23] M. Naghshvarian-Jahromi and N. Komjani, “Novel fractal monopole wideband antenna,” J. Electromagn. Wave Applicat. (JEMWA), vol. 22, no. 2, pp. 195–205, 2008. [24] M. Naghshvarian-Jahromi, “Novel wideband planar fractal monopole antenna,” IEEE Trans. Antennas Propag., vol. 56, no. 12, Dec. 2008. [25] M. Naghshvarian-Jahromi and N. Komjani, “Analysis of the behavior of Sierpinski-Carpet monopole antenna,” ACES J., vol. 24, no. 1, 2009. [26] M. Naghshvarian Jahromi and A. Falahati, “Classic miniature fractal monopole antenna for UWB applications,” presented at the ICTTA’08, Syria, Apr. 2008. [27] R. N. Simouns, Coplanar Waveguide Circuits, Components, and Systems. New York: Wiley-Interscience, 2001. [28] Computer Simulation Technology, 2008. [29] H. Jasik, Antenna Engineering Handbook. New York: McGraw-Hill, 1961, pp. 2.10–2.13. [30] N. Behdad and K. Sarabadani, “A compact antenna for ultrawideband applications,” IEEE Trans. Antennas Propag., vol. 53, no. 7, 2005. [31] N. Telzhensky and Y. Leviatan, “Novel method of UWB antenna optimization for specified input signal forms by means of genetic algorithm,” IEEE Trans. Antennas Propag., vol. 54, no. 8, 2006. [32] R. L. Peterson, R. E. Ziemer, and D. E. Borth, Introduction to Spread-Spectrum Communications. Englewood Cliffs, NJ: Prentice-Hall, 1995. [33] Ansoft Corporation, High frequency structure simulation (Ansoft HFSSTM v10.0), 2005. Mahdi Naghshvarian Jahromi was born in Jahrom, Iran, in 1985. He received the B.Sc. degree in electrical engineering (communications) from Iran University of Science and Technology (IUST), Tehran, in 2008. Since 2007, he has been Head of the Antenna Design Group, Faramoj-Pajoh-Engineering Company. His scientific fields of interest include antennas, arrays, filters and all passive microwave device especially with UWB behavior. He is the author of 14 journal and four conference papers.

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Abolfazl Falahati was born in Tehran, Iran. He received the B.Sc. (Hons) degree in electronics engineering from Warwick University, U.K., in 1982, and the M.Sc. degree in digital communication systems and the Ph.D. degree in digital communication channel modeling from Loughborough University, U.K., in 1984 and 1988, respectively. In 1993, he was a Postdoctoral Researcher in HF channel signaling, Rotherford Appleton Laboratory, Oxford, U.K. Since 1994, he has been an Associate Professor and faculty member with Department of Electrical Engineering, Iran University of Science and Technology (IUST), Tehran. His research interests are ultrawideband communication systems and antenna designs, mimo channel modeling and relay networks, cognitive radio and wireless sensor networks, mimo relay network modeling and simulation, information theory and channel coding techniques, cryptography, secure communication system managements and applications, universal mobile for telecommunication system (UMTS) in adaptation with GSM mobile system and WiMAX systems.

Rob M. Edwards received the B.Eng. degree (Hons.) in electronic engineering (communications) and the Ph.D. degree from the University of Sheffield, Sheffield, U.K. Sheffield University belongs to the Russell Group. The Russell Group is an association of 20 major research-intensive universities of the United Kingdom. He is the former Director of Sheffield’s Centre for Mobile Communications Research (C4MCR), and moved to Loughborough University with his research group in September 2004. Dr. Edwards has reviewed for several of the technology programs organized by the U.K. Government and also reviews numerous publications relating to mobile communications, signal processing and the possibility of radio frequency radiations on humans.

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Multi-Chip RFID Antenna Integrating Shape-Memory Alloys for Detection of Thermal Thresholds Stefano Caizzone, Cecilia Occhiuzzi, and Gaetano Marrocco

Abstract—Low-cost wireless measurement of objects’ temperature is one of the greatest expectation of radiofrequency identification technology for the so many applications in cold supplychain control and safety assessment in general. In this context, the paper proposes a dual-chip UHF tag embedding shape memory alloys (SMA) able to transform the variation of the tagged item’s temperature into a permanent change of antenna radiation features. This event-driven antenna is hence able to selectively activate the embedded microchips according to the temperature above or below a given threshold. A general design methodology for the resulting two-ports tag antenna is here introduced and then applied to prototypes able to work at low (around 0 ) and high (80 ) temperatures.

C

C

Index Terms—Radiofrequency Identification (RFID), shape memory alloy, temperature sensor.

I. INTRODUCTION ASSIVE UHF tags for radiofrequency identification (RFID) are often considered as digital devices which, when interrogated and energized by a reader, may send back their own ID or other resident information through a backscattering modulation of the incident continuous wave [1]. While the most assessed use is the item labeling, there is a growing interest in new applications devoted to sense the variation of the tagged objects as well as of the nearby environment [2]–[6]. One of the most interesting parameters to be sensed is the temperature variation for a large variety of scopes, in logistic, safety and medical contexts. In particular, cold supply chain management for goods that have to be kept at constant temperature has a severe need for electronic tags able to continuously detect possible thermal anomalies [7], [8]. Nowadays, the most common devices for temperature monitoring are active data loggers [9] which, by means of active sensors and local data storage, are able to store all the time-history of the temperature. The relative high costs and the limited lifetime before recharge make these systems only useful for an environmental-level monitoring, leaving unsolved the problems related to the pervasive item-level control. An alternative and cheaper approach is represented by semiactive RFID temperature tags [10], having a longer lifetime and

P

Manuscript received June 21, 2010; revised October 26, 2010; accepted November 19, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported by the Italian Ministry of University under project PRIN-2008- RFID MULTITAG. S. Caizzone is with the Universita di Roma Tor Vergata, DISP, 00133 Roma, Italy and also with the Antenna Group, Institute of Communications and Navigation, German Space Agency (DLR), 82230 Wessling, Germany. C. Occhiuzzi and G. Marrocco are with the Universita di Roma Tor Vergata, DISP, 00133 Roma, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2152341

able to transmit the instantaneous value of the local temperature following the reader’s interrogation. Their main drawback is that the thermal event can be detected only in the presence of a close-distance reader and hence they are not suited to assess the thermal safety all along the supply chain. A possibility to reduce cost, size and maintenance of RFID sensor systems is to focus the attention to a completely passive technology where the challenge is to design a proper passive sensor component able to perform the sensing activity without batteries and to store the occurrence of the temperature hot spot independently on the reader interrogation. Following this idea, a UHF passive sensing tag which integrates a temperature printed nano sensor has been recently proposed in [11]. The sensor works as a chemical thermal fuse able to record the thermal event by irreversibly changing its resistivity through the melting of the polymeric part of the conducting ink. However, it currently looks suitable only for high temperature applications, approximately over 60 , and not for typical cold supply chain conditions. In [6] a similar idea was instead applied to low temperature, wherein the switch effect was obtained by the melting of an iced region placed in the close proximity of the antenna and hence the device is restricted to applications around 0 without many tuning possibility. This paper proposes a completely different mechanism to achieve one-shot RFID thermal seals, fully suitable to high and and low temperatures, in a range approximately between 100 . A shape-memory-alloy (SMA) compound, introduced into an RFID tag as sensitive material, modifies its geometry depending on the local thermal boundary conditions and consequently it is able to modulate the RFID communication. The sensing tag is here conceived as an antenna provided with two RFID microchips so that the first one (chip-1) is able to transmit its digital identifier (ID1) regardless the temperature status, while the second chip (chip-2) transmits or not its code (ID2) depending on the local temperature. Hence ID1 has the meaning of item’s identifier while ID2 gives its status information, e.g. a thermal alarm (above or below a threshold temperature), according to the ID-modulation paradigm [12]. For this purpose, a portion of the antenna (Fig. 1) is fabricated in SMA whose shape variation with respect to the temperature will be such to match or mismatch the chip-2, leaving mostly unmodified the impedance condition seen by chip-1. The occurrence of a hot spot is hence registered into the device by a shape change, independently on the presence of the reader. The design of such a new family of devices deserves some care for what concerns the management of multi-chip RFID tags, and a proper methodology is here proposed able to take into account the two-state response of the tag.

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Fig. 2. Geometry of the Nitinol temperature switch in the martensite and austenite states. Fig. 1. Functional scheme of a two-chip temperature sensing tag embedding regular conductors of shape S and Shape Memory compounds whose shape S is strongly affected by the local temperature.

The paper is organized as follows: Section II introduces a possible configuration of thermal switch, suited to antenna integration and discusses, by means of some experiments, the repeatability of its thermal response. Section III formulates the design problem of the two-port RFID tags including the event-driven sensor’s status. A tag prototype is then designed and electromagnetically characterized in Section IV and its response is finally experimented in Section V at hot (around 80 ) and cold (around 0 ) temperature conditions. II. THE NITINOL THERMAL SWITCH Shape memory alloys [13] are characterized by two crystallographic phases, a martensitic one and an austenitic one, in which the structural and mechanical properties of the alloy greatly change. By heating, martensite transforms into austenite, the only stable phase at high temperature. SMAs are easily deformable in the martensitic phase; however they recover the prefixed austenitic shape every time they are heated above the transition temperature (Austenite Start), with an increasing recovery as the temperature rises up to (Austenite Finish). According to their property to recover the “memorized” shape, the SMAs can be employed both as sensors and/or as actuators. The shape-memory effect has been found in many alloys such as CuAlNi, CuZnAl, AuCd, but NiTi, also known as Nitinol [14], is the most commonly used thanks to its good characteristics of corrosion resistance, ductility, high recoverable deformation, relatively high electrical conductivity ( , [15]) and biocompatibility. Nitinol is nowadays widely used in medicine and robotics mainly to build actuators. In the Electromagnetic context Nitinol wires are instead used to fabricate flexible whip antennas for mobile terminals. It is worth mentioning that the transition temperature is tunable by changing the alloy percentage composition: therefore transitions can be obtained in a very wide range of temperatures, i.e. between and more than 100 . This unique feature makes the Nitinol an enabling compound to assess the integrity of pharma, blood sacs, frozen foods and to monitor fever rush or industrial processes in general. A. Switch Geometry There are several options to achieve a temperature-sensitive tag taking advantage of the shape recovering feature of the Nitinol, but one of the easiest is to conceive a thermal switch which is normally closed when the temperature and

Fig. 3. Example of fabricated Nitinol switch and graphical representation of the switch-off with respect to the temperature rising.

. The switch can be placed in series or in open for parallel to the chip-2 (see again Fig. 1) to strongly affect its activity. According to the last case, the tag’ states will be: ID1 under thermal threshold, and ID1+ID2 over the thermal threshold. A possible implementation of a thermal SMA-based switch, suitable to RFID antenna integration, includes a Nitinol wire segment processed to have straight shape in the austenitic (stable) phase. The wire is bent (Fig. 2) in the martensitic phase, , and fixed across the gap of a copper trace e.g., for (wherein the RFID microchip will be placed) by a crimped connector1 soldered at one of the gap’s sides, and more softly, by a silver glue, at the other side. When the temperature exceeds the threshold , the wire begins to recover its original straight shape, breaking the conducting glue and hence opening the switch. Since the austenitic shape is stable, a successive fall of the temperature below the threshold will not modify the wire’s shape anymore. The Nitinol switch hence acts as a one-way element, configuring itself as an unidirectional thermal fuse. B. Prototype and Characterization Fig. 3 shows a fabricated example of a Nitinol switch over a FR4 substrate (thickness 1.58 mm). Two Nitinol wires having a nominal transition temperature and different diameters {0.3 mm, 0.5 mm}, respectively, have been considered in separate devices. The same figure gives a graphical rendering of the switch-off capability of the device, when the temperature is linearly raised from 60 to 85 in a thermostatic chamber. The switch-off event has been collected by monitoring the DC resistance at the switch’s terminals. A sharp variation of input DC resistance from low (switch on) to an extremely high value (switch off) indicates the transition from martensite to austenite, 1The Nitinol can not be easily soldered and hence a possible fixing is achieved by crimping the Nitinol wire within a conducting small cylinder and soldering this one on the antenna trace.

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The open circuit voltages depend on the geometry of the tag but also on the mutual orientation between reader and tag. Therefore, as in the case of a regular array, the active impedances of the tag are strongly affected by the interrogation modality. The general formulation of this kind of multi-chips device, also denoted as “RFID grids,” is out of the scope of this paper and can be found in [19]. Under broadside incidence condition (e.g., the plane of the tag is perpendicular to the reader’s antenna beam), it is possible to assume and the above system equation gives

TABLE I SWITCH-OFF TEMPERATURE DETECTED BY THE CONSIDERED NITINOL SWITCHES

(4) Fig. 4. Two-port model of the two-chip RFID tag in receiving mode.

and hence the required open circuit. The experiments have been repeated five times for each kind of switch and the detected transition temperatures in Table I demonstrate that the device mechanism has good repeatability, with a temperature uncertainty of temperature is dependent on the 1–2 . The switch-off diameter of the wire so that a thicker Nitinol wire will recover the stable state at a lower temperature in comparison with a thinner wire and hence this one is more sensible. From an electrical point of view, the Nitinol wire can be , and in simply modeled by a lumped impedance particular as an ideal open circuit when in the austenite phase , and as a real inductor with parameters (1)

Accordingly, the active impedances for broadside interrogation are (5) (6) where . The maximum power transfer from the antenna to the microchip loads imposes [20] the following conjugate matching conditions at the ports: (7) (8) By recalling the ID-modulation [12] scheme to be implemented in the sensing RFID tag, e.g. that only the chip-1 is matched in the ON state , while both the microchips are active in the OFF state , the matching conditions to enforce on the two-ports tag are

(2) (9) in the martensite phase, where and are the length and the radius of the Nitinol wire segment, respectively [16]. (10) III. TWO-CHIP TAG ANTENNA DESIGN The passive RFID communication between reader and tags requires the input impedance of the antenna to be matched to the conjugate of microchip impedance for maximum power transfer (and hence for maximum reading range) [1]. Since the two RFID microchips are connected over a same radiating structure, the electromagnetic coupling can not be neglected and hence the input impedances to be controlled are the active impedances of the system [17]. The design problem can be conveniently formulated in terms of a two-port antenna system in receiving mode. The corresponding network model is given in Fig. 4: is the load impedance at port-1, wherein the chip-1 is the terminating is connected, while impedance of port-2, which depends on the ON/OFF status of the Nitinol switch, and hence on the sensed temperature. are the open circuit voltages collected at the ports and produced by the interrogating field emitted by the remote reader. is the impedance matrix of the unloaded two-port antenna. The network equations [18] are

(3)

The geometrical parameters of the two port antenna have to be therefore designed such to achieve the best trade-off among the above conditions. The following fitness function

(11) has thus to be minimized by means of a conventional optimizaare constant weights), having assumed tion tool ( as discussed in the previous section. IV. PROTOTYPE A first prototype of the UHF RFID thermal seal consists of a layout as in Fig. 5 originated from the single-port design in [21] where a planar dipole has been partly folded to reduce the overall area to a typical credit card size. The port impedances are controlled by the aspect ratio of two symmetric T-match circuits. The Nitinol switch has been fabricated as previously described by using a Nitinol wire of radius and length . Accordingly, the closed-state impedance from (1) and (2) is . The RFID microchips

CAIZZONE et al.: MULTI-CHIP RFID ANTENNA INTEGRATING SHAPE-MEMORY ALLOYS FOR DETECTION OF THERMAL THRESHOLDS

Fig. 5. Layout of the two-chip tag whose double T-match aspect ratio (a; b) has to be optimized to achieve the required ID modulation.

Fig. 6. Map of the fitness function F (a; b) in equation (11) calculated by FDTD simulations.

are NXP TSS OP8 with nominal input impedance at 869 MHz and power sensitivity . For the sake of simplicity, the T-match circuits are assumed to be of equal sizes whose values are selected by minimization of the fitness function in (11) with respect to , . The antenna’s weights matrix, for each shape of the T-match, is evaluated by an FDTD : model [22] of the tag layout. Fig. 6 shows the map of a solution suited to fabrication is and . The prototype has been cut on adhesive copper sheet as shown in Fig. 7 with a detail of the Nitinol wire insertion. The electromagnetic tag performances in ON and OFF states have been experimentally characterized with respect to the realized gain which directly imposes the tag’s read distance. For this purpose a customized UHF long-range reader based on the ThingMagic M5-e ASIC has been used. The sensing tag is interrogated by means of a 6 dBi circular polarized patch antenna, connected to the reader. The tag is placed in front of the reader’s antenna, at 1.5 m from ground and the floor reflections are minimized by using absorbing panels. Under the assumption collected by the th of free-space channel, the power port of the tag may be obtained using the Friis formula as (12) is the gain of the reader antenna, is the realized where gain of the tag which includes the impedance mismatching between the th port and the microchip as well as the polarization

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Fig. 7. Adhesive copper-sheet prototype of the tag embedding two microchips and the Nitinol switch, deduced from Fig. 5 with a = 12 mm and b = 25 mm. The Nitinol wire length and diameter are l = 1 cm and r = 0:5 mm, respectively.

Fig. 8. Measured and simulated realized gain for the sensor-tag when the temperature is under the threshold (T < A ) and only chip-1 is responding.

mismatch with respect to the reader’s field. is the power accepted by the antenna of the reader unit. The realized gain is measured by means of the turn-on power method [23], e.g by provided to the reader’s anrecording the minimum power tenna that enables the microchip on the tag to be activated and to send back its own identifier. In that condition and hence, for a fixed reader-tag distance , the realized gain is found to be (13) , at Fig. 8 and Fig. 9 give the measured 870 MHz, over the horizontal plane in both the Nitinol states. When the switch is closed and hence only the chip-1 is active, the realized gain over the H-plane is rather omnidirectional, with average value close to 1 dB. When the switch is open both the microchips are responding and the active radiation diagrams become less omnidirectional with specular gains of the two chips with respect to . About 3 dB ripple and lower values are evident with respect to the single port case due to the presence of the inter-port coupling. Under the assumption of free space conditions and interrogation power 3.2 W EIRP, the measured realized gain achieves a maximum read disand tance (by inverting (13)) given by , below and above the threshold temperature, respectively.

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Fig. 11. Digital response from sensing tag with A ture raise and time.

' 80

C versus tempera-

Fig. 9. Measured and simulated realized gain for the sensor-tag when the temperature is over the threshold (T > A ) and both chips are responding.

Fig. 12. Digital response from sensing tag with A increase and time.

Fig. 10. Measurement setup comprising the remotely controlled reader, the multimeter connected to the thermocouple and the RFID tag under test (TUT), placed into the thermostatic chamber.

'0

C versus temperature

TABLE II SWITCH-OFF TEMPERATURES OF THE SENSING TAGS

V. THERMAL CHARACTERIZATION OF THE TAG The proposed RFID sensor has been tested in real conditions in order to verify the effective communication and sensing performances. Two different scenarios have been reproduced: a first one characterized by a high-temperature environment and a second one with a very low-temperature condition as for the logistics of frozen items. The Nitinol switch has been accordingly fabricated with wires of nominal transition temperature and , respectively. The same antenna layout, shown in Fig. 7 has been used for both the experiments. The tag has been attached on a cardboard box and heated inside a temperature controlled chamber, while being monitored by a thermal probe and continuously interrogated by the reader antennas also placed inside the chamber itself (Fig. 10). The tag’s digital responses have been registered while the local temup to obperature was progressively raised from a serve the martensite to austenite transition. The temperature was recorded through a thermocouple connected to a multimeter and correlated to the digital responses of the tag. The reader collects only the ID1 below the temperature. When both ID1 and ID2 are received for the first time it means that the Nitinol switch has can be acbroken the glue and the switch-off temperature cordingly detected through the thermocouple reading.

In case of the low-temperature Nitinol, particular care was devoted to the integration of the Nitinol wire into the tag and to have the conducting glue getting solid. The fabrication of the switch was hence performed inside the thermostatic chamber itself. In both cases the measurements have been repeated five times to perform averaging. Examples of collected data in both the experiments are shown in Fig. 11 and Fig. 12 and the estimated switch-off temperatures can be found in Table II. The standard deviation is the same (3 ) in both the configurations even if it is percentually more significant in the low-temperature case. This is probably due to the more difficult integration of the nitinol wire at temperature below zero degrees. The conducting glue adhesion reliability proved to be poor due to the presence of water/ice condensation between the nitinol and copper trace which makes it difficult to replicate the same device conditions in successive tests. VI. CONCLUSIONS The Nitinol compound demonstrates to be suitable for integration into UHF tags for the control of both cold and hot goods

CAIZZONE et al.: MULTI-CHIP RFID ANTENNA INTEGRATING SHAPE-MEMORY ALLOYS FOR DETECTION OF THERMAL THRESHOLDS

and its response to temperature variation seems to be stable and repeatable, especially at high temperatures. The experiments however highlight the importance of having a stable fastening method for the Nitinol wire to the antenna and further research is required to improve the stability of the device, particularly below 0 . Moreover the self-sensing tag should be provided with a lock mechanism to keep the Nitinol wire in the curved shape before use even at temperatures exceeding the austenite transition. This is also important in case the tags are placed over the items before they freeze. Finally, the described design methodology for two-chip tags is more general than the considered temperature application, and it may be extended to the integration of other families of passive sensors, provided that their impedance model is known and that the fitness function is properly defined according to the required data encoding. ACKNOWLEDGMENT The authors wish to thank F. Nanni and C. Del Gaudio for inspiration and support on high-temperature measurements, SAES-GETTER for providing Nitinol wires and hosting the low-temperature experiments and finally but not least, G. Manzi of NXP for technical support with microchips and readers. REFERENCES [1] D. M. Dobkin, The RF in RFID: Passive UHF RFID in Practice. Amsterdam: Elsevier, 2007. [2] Y. Li, Z. Rongwei, D. Staiculescu, C. P. Wong, and M. M. Tentzeris, “A novel conformal RFID-enabled module utilizing inkjet-printed antennas and carbon nanotubes for gas-detection applications,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 653–656, 2009. [3] J. Sidn, X. Zeng, T. Unander, and H.-E. Nilsson, “Remote moisture sensing utilizing ordinary RFID tags,” in Proc. IEEE Sensors, Atlanta, GA, Oct. 2007, pp. 308–311. [4] J. Virtanen, L. Ukkonen, and L. Sydnheimo, “Ink jet printed RFID humidity sensor,” presented at the RFIDay 2010, Tampere, Finland. [5] A. Vaz, A. Ubarretxena, I. Zalbide, D. Pardo, H. Solar, A. Garca-Alonso, and R. Berenguer, “Full passive UHF tag with a temperature sensor suitable for human body temperature monitoring,” IEEE Trans. Circuits Syst.-II_ Expr. Briefs, vol. 57, no. 2, pp. 95–99, Feb. 2010. [6] R. Bhattacharyya, C. Floerkemeier, and S. Sarma, “RFID tag antenna based temperature sensing,” in Proc. IEEE Int. Conf. on RFID, 2010, pp. 8–15. [7] M. C. O’Connor, “Coldchain project reveals temperature inconsistencies,” RFID J. Dec. 2006 [Online]. Available: http://www.rfidjournal. com/article/view/2860/1 [8] A. Dada and F. Thiesse, “Sensor applications in the supply chain: The example of quality-based issuing of perishables,” Lecture Notes Comput. Sci., vol. 4952, pp. 140–154, 2008. [9] R. Kuchta and R. Vrba, “Wireless and wired temperature data system,” in Proc. Int. Conf. on Systems ICONS, 2007, p. 49. [10] S. Kim, J. H. Cho, H.-S. Kim, H. Kim, H. B. Kang, and S. K. Hong, “An EPC Gen 2 compatible passive/semi-active UHF RFID transponder with embedded FeRAM and temperature sensor,” in Proc. IEEE Asian Solid-State Circuits Conf., Jeju, Korea, 2007, pp. 4–6. [11] J. Gao, J. Sidn, and H. E. Nilsson, “Printed temperature sensors for passive RFID tags,” presented at the Progress In Electromagnetics Research Symp., Xi’an, China, 2010. [12] M. Philipose, J. R. Smith, B. Jiang, A. Mamishev, S. Roy, and K. Sundara-Rajan, “Battery-free wireless identification and sensing,” IEEE Pervasive Comput., vol. 4, no. 1, pp. 37–45, 2005. [13] H. Funakubo, Shape Memory Alloys. New York: Gordon and Breach Publishers, 1987.

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[14] E. Hornbogen, “Review thermo-mechanical fatigue of shape memory alloys,” J. Mater. Sci., vol. 39, no. 2, pp. 385–399, 2004. [15] AA.VV, Niti Smart Sheet-Properties [Online]. Available: http:// web.archive.org/web/20030418012213/http://www.sma-inc.com/NiTiProperties.html [16] E. B. Rosa, “The self and mutual inductances of linear conductors,” Bulletin Bureau of Stand., vol. 4, no. 2, p. 301, 1908. [17] G. Marrocco, L. Mattioni, and C. Calabrese, “Multi-port sensor RFIDs for wireless passive sensing—Basic theory and early simulations,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2691–2702, Aug. 2008. [18] J. R. Mautz and R. Harrington, “Modal analysis of loaded N-port scatterers,” IEEE Trans. Antennas Propag., vol. 21, no. 2, pp. 188–199, March 1973. [19] G. Marrocco, “Electromagnetic theory of RFID grids,” IEEE Trans. Antennas Propag., submitted for publication. [20] C. A. Desoer, “The maximum power transfer theorem for n-ports,” IEEE Trans. Circuit Theory, vol. 20, no. 3, pp. 328–330, May 1973. [21] B. Yang and Q. Feng, “A folded dipole antenna for RFID tag,” in Proc. Int. Conf. on Microwave and Millimeter Wave Technology, 2008. [22] G. Marrocco and F. Bardati, “BEST: A finite-difference solver for time electromagnetics,” Simulation Practice Theory, no. 7, pp. 279–293, 1999. [23] G. Marrocco and F. Amato, “Self-sensing passive RFID: From theory to tag design and experimentation,” in Proc. 39th Eur. Microwave Conf., Rome, 2009, pp. 1–4.

Stefano Caizzone received the M.Sc. degree in telecommunications engineering from the University of Rome “Tor Vergata,” in 2009, where he is working part-time toward the Ph.D. degree. His main research interests concern small antennas for RFIDs and navigation, antenna arrays and grids with enhanced sensing capabilities. He is now with the Antenna Group, Institute of Communications and Navigation, German Space Agency (DLR), where he is responsible for the development of innovative miniaturized antennas.

Cecilia Occhiuzzi received the M.Sc. degree in medical engineering from the University of Rome “Tor Vergata,” where she is currently working toward the Ph.D. degree. In 2008, she was at the School of Engineering, University of Warwick, U.K., as a Postgraduate Student, working on design and implementation of wireless SAW sensors. In 2010, she was a Visiting Researcher at the Georgia Institute of Technology, Atlanta. Her research was mainly focused on the design of passive RFID sensors for structural health monitoring and gas detection by means of CNT-based tags.

Gaetano Marrocco was born in Teramo, Italy, on August 29, 1969, He received the Laurea degree in electronic engineering (laurea cum laude and academic honour) and the Ph.D. degree in applied electromagnetics from the University of L’Aquila, Italy, in 1994 and 1998, respectively. Since 1997, he has been a Researcher at the University of Rome “TorVergata,” Rome, Italy, where he currently teaches antenna design and medical radiosystems, manages the Antenna Lab and is Advisor in the Geo-Information Ph.D. program. In October 2010, he achieved the degree of Associate Professor of Electromagnetics. In summer 1994, he was at the University of Illinois at Urbana-Champaign as a

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Postgraduate Student. In autumn 1999, he was a Visiting Researcher at the Imperial College in London, U.K. In 2008, he joined the Ph.D. program of the University of Grenoble (FR). His research is mainly directed to the modeling and design of broad band and ultrawideband (UWB) antennas and arrays as well as of sensor-oriented miniaturized antennas for biomedicine, aeronautics and radiofrequency identification (RFID). He has been involved in several Space, Avionic, Naval and Vehicular programs of the European Space Agency, NATO, Italian Space Agency, and the Italian Navy about the analysis and the design of non conventional antennas and systems. He holds eight patents on broadband naval antennas and structural arrays, and on sensor RFID systems.

Prof. Marrocco currently, he serves as an Associate Editor of the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS and is a Reviewer for the IEEE TRANSACTION ON ANTENNAS AND PROPAGATION, IEEE PROCEEDINGS, IEEE MICROWAVE THEORY AND TECHNIQUES, and is a member of Technical Program Committee of several International Conferences. In 2008, he was the General Chairman of the first Italian multidisciplinary scientific workshop on RFID: RFIDays-2008: Emerging Technology for Radiofrequency Identification. He was the Co-Chair of the RFIDays-2010 International Workshop in Finland and Chairman of the Local Committee of the V European Conference on Antennas and Propagation.

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Three-Shaped-Reflector Beam-Scanning Pillbox Antenna Suitable for mm Wavelengths Stuart G. Hay, Senior Member, IEEE, Stephanie L. Smith, Member, IEEE, Greg P. Timms, Senior Member, IEEE, and John W. Archer, Fellow, IEEE

Abstract—We describe the design and measured performance of an improved beam-scanning fan-beam antenna suitable for mm wavelengths. The fan beam is scanned in the direction orthogonal to the fan and can be polarized either in or orthogonal to this direction. The pillbox design is improved by the introduction of a third cylindrical reflector and optimum shaping of all reflector profiles using an iterative diffraction analysis that includes aberration and blockage effects. Good agreement between the calculated and measured results is obtained, confirming a beamwidth of 0.3 and a scan range of 13.5 (45 beamwidths) for both polarizations at 186 GHz. Index Terms—Beam-scanning antennas, mm-wave antennas, pillbox antennas, reflector antennas, shaped reflectors.

I. INTRODUCTION REVIOUSLY we described the design and testing of a prototype beam-scanning fan-beam antenna for a mm-wave imaging system [1], [2]. The antenna was of pillbox type, comprising two cylindrical reflectors and a horn feed between two parallel plates. The fan-shaped beam was scanned in the direction orthogonal to the fan by rotation of the small subreflector. The imaging system requires two such antennas, with orthogonally intersecting beams, to scan in two dimensions. Measurements of the prototype antenna indicated the feasibility of this approach to scanning, using either of two orthogonally polarized modes in the multi-mode parallel-plate structure. However, although optimization of the reflector profiles was undertaken, the antenna suffered significant gain loss and beam broadening with scan, limiting the scan range to less than that required for the application. In this paper, we describe the design of an improved antenna for the same application [3]. The new antenna also is of pillbox type, but features three optimized shaped reflectors that produce significantly better beam-scanning characteristics than the tworeflector prototype. These improvements include a larger scan range and higher gain, obtained while maintaining small size of the rotating subreflector as required for rapid scanning. We extend pioneering work by Wertz et al. [4], [5] on the imaging-reflector approach to beam scanning by subreflector

P

Manuscript received June 04, 2010; revised October 22, 2010; accepted November 29, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. S. G. Hay, S. L. Smith and J. W. Archer are with the CSIRO ICT Centre, 1710, Epping, New South Wales, Australia (e-mail: [email protected]). G. P. Timms is with the Tasmanian ICT Centre, 7001, Hobart, Tasmania, 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.2011.2152335

motion. In [4], they proposed and analyzed a three-reflector system comprising a paraboloidal main reflector, a fixed ellipsoidal subreflector and a small rotating tertiary reflector. The tertiary reflector was approximately the image of the main reflector in the ellipsoidal subreflector, and its surface was shaped to eliminate aberration for one particular direction near the center of the field of view. In [5], this approach was applied to cylindrical reflectors and an array feed was used to reduce the aberrations that occur as the tertiary reflector is rotated from its designed aberration-free position, resulting in a beam-scan range of 10 or approximately 40 beamwidths. Here, we also use cylindrical reflectors but constrain the feed to be a single horn and shape all three reflector profiles to optimize the beam radiation patterns throughout the field of view. The optimization process uses a physical optics analysis that includes both aberration and blockage effects, which is important for maximizing the scan performance of the antenna whilst satisfying constraints on its size. The objective of the optimization takes the form of maximizing the minimum beam gain within the scan range whilst maintaining all beam sidelobes less than 15 dB of the corresponding beam maximum. By optimally shaping the three reflectors, we obtain a beam-scan range of 13.5 or 45 beamwidths (slightly larger than that in [5]) with the advantage of simplicity of the single-horn feed. Although developed specifically for the mm-wave imaging application, the antenna may also be of interest for other systems such as mm-wave radar or communications. The optimization of the design and the measurement of performance makes the paper a useful reference on this type of scanning antenna. II. ANALYSIS Fig. 1 illustrates the mm-wave imaging system which uses two antennas to produce two orthogonal scanning fan beams. The antennas are of pillbox type, each consisting of three cylindrical reflectors and a horn feed between two parallel plates, as illustrated in Fig. 2. Except for the feeds the two antennas are and the other in the parthe same; one operates in the allel-plate mode, thereby providing the orthogonal fan beams with the same polarization. Fig. 13 in the Appendix shows one of the fabricated antennas. Fig. 3 illustrates the beam scanning operation of the antenna. is rotated around To scan the beam, the small subreflector the indicated axis. This subreflector has two similar reflecting surfaces, producing two scans of the field of view in each 360 rotation of the reflector. In the imaging-reflector interpretation [4], [5], the aperture of the antenna is approximately the image after reflection in the two fixed of the reflecting surface of and . Thus the main effect of rotating is a reflectors

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Fig. 3. Illustration of path of the feed-axis ray from the feed aperture to the aperture of the pillbox for various rotation angles of the rotating reflector.

between the parallel plates, and radiation-pattern analysis of the pillbox aperture [1]. The extended analysis includes radiation pattern analysis for a variety of multi-reflector scattering paths, accounting for the blockage effects of the subreflectors and scattering from the side-walls of the pillbox structure. For example, the path Fig. 1. Illustration of the apertures (a) and beams (b) of the two pillbox antennas. The arrows in the apertures indicate the direction of the electric field of the H and E parallel-plate modes. The dimensions of the pillbox apertures are 5 mm by 480 mm. The arrows on the beams indicate the scan directions.

Fig. 2. Illustration of reflector profiles and rays of the central beam.

linear change in the phase of the aperture field, scanning the beam whilst maintaining efficient illumination of the antenna and can also be regarded aperture. The two reflectors as a dual-reflector feed system in the side-fed offset Cassegrain (SFOC) configuration that is known for its wide field of view in multibeam applications [6]. In the latter interpretation, and correspond to the main reflector and subreflector respecin the field of the tively of the SFOC configuration; rotating feed scans the focal point of the field reflected by , similar to the effect produced by varying the direction of incidence on the fixed main reflector of the SOFC system. For optimum design of the antenna, we have extended an analysis procedure employing mode-matching analysis of the horn feed, physical optics analysis of the reflectors, using the appropriate Green’s functions for the various propagating modes

is included to account for the blockage of on the field reflected by . The radiation pattern of the antenna is obtained by summing the individual radiation patterns of all the considered paths. Mutual coupling between the feed and reflectors has not been included in the present analysis. To efficiently evaluate the reflector scattering and aperture radiation, we have extended the two-reflector Gaussian series analysis [1], [8] to apply to an arbitrary number of surfaces. The approach is to expand each incident field and surface current into a Gaussian series so that the scattered or radiated field can then be efficiently evaluated as a sum of corresponding edge diffracted Gaussian beams. The number of terms in the series is reduced by individually phase matching the Gaussian terms to significant contributions to the incident field, determined by grouping and summing the incident beams within finite intervals of propagation direction. Details of the representation are given in the Appendix. III. DESIGN CONSIDERATIONS The analysis is performed for a number of subreflector rotation angles, giving the radiation patterns of a number of beams throughout the scan range. The multibeam radiation pattern analysis is performed at each iteration of a numerical optimization procedure to find best shapes for the profiles of the three reflectors and also the best rotation angles of the rotating subreflector. The reflector profiles are parameterized in terms of the coefficients of cubic spline basis functions and these coefficients are used as optimization variables. The objective in the optimization was to maximize the minimum gain in the required 13.5 scan range whilst maintaining sidelobes less than 15 dB relative to each beam peak. This objective was handled by way of a penalty-function approach, with iterative

HAY et al.: THREE-SHAPED-REFLECTOR BEAM-SCANNING PILLBOX ANTENNA SUITABLE FOR mm WAVELENGTHS

improvement of error-weighting factors in order to achieve the sidelobe constraints. Two different optimization techniques have been used: simulated annealing and gradient search, in an effort to obtain a globally optimum solution. As a starting point for the reflector optimization, the profiles and were chosen to be elliptic, concave hyperbolic of , and parabolic respectively, with the parameters selected to give zero aberration in the center of the field of view, and with the central point of the antenna aperture being the image of the cenand . tral point on the rotating reflector after reflection in The optimization of the reflectors was important, increasing the minimum gain within the scan range by 5 dB and significantly decreasing the sidelobe levels. The horn feeds are of rectangular cross section with a WR5 waveguide input. The variation of the cross section with distance along the horn axis was selected so that the aperture field of the horn contains mainly the fundamental waveguide mode of the required polarization. The dimensions of the horn aperture were selected by iteration so that the pillbox beams had the required 15 dB relative to the beam peak. This analysis sidelobes feed should be indicated that the aperture dimensions of the feed 5 mm (E plane) 8.5 mm (H plane) and those of the should be 5 mm (H plane) 6 mm (E plane). Feeds of these dimensions were manufactured. mode Some errors were detected in the manufacture of the feed, and so the feed from the prototype antenna [1] was used instead for this mode. Fig. 4 shows the profiles of the feeds used in the initial testing of the antennas reported here. The H-plane mode feed is less than ideal, causing some diameter of the sidelobes of the pillbox beams to be larger than 15 dB relative to the beam peak. Fig. 4.

IV. TEST RESULTS The radiation patterns of the pillbox antennas have been measured using a compact range [7]. To facilitate comparison with the two-reflector prototype [1], the new antennas were also designed for maximum gain at 38 m from the antenna, considerably less than the 260 m minimum far-field distance. Therefore the compact range was defocused, moving its feed toward its reflector. Further details of the compact range measurement technique can be found in [11] including extension to measurement of phase. Figs. 5 and 6 show the calculated and measured radiation patterns of the central beam of the antenna in the plane orthogonal to the pillbox plates. Very good agreement between the calculated and measured results is seen. The cross-polarization of and modes was measured in this plane and the both the scan plane parallel to the plates, and was found to be at or less than 35 dB relative to the co-polar peak. These results are important because they show that although the 5 mm plate separation of the pillbox makes it capable of propagating a number of different modes, only the desired modes significantly exist in each polarization case. The 5 mm plate separation is important for obtaining the required beamwidth of the fan beam in the direction of the fan and reducing ohmic loss in the parallel plates of the pillbox [1]. Figs. 7 and 8 show the measured scan plane radiation patand mode beams respectively throughout terns of the

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Profiles of horn feeds for H (a) and E (b) modes.

Fig. 5. Comparison of the E -mode calculated and measured radiation patterns in the plane orthogonal to the pillbox plates at 186 GHz.

the field of view. Positive angles refer to anticlockwise rotation from the z axis of Fig. 4. As shown in Fig. 9, rotation of the subover a 26 range causes the pillbox beam to scan reflector and 7.35 , giving a 13.5 field of view. Within between this scan range the measured beam gain is constant to within 0.5 dB for both polarization modes. The full-width half-power

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Fig. 6. Comparison of the H -mode calculated and measured radiation patterns in the plane orthogonal to the pillbox plates at 186 GHz.

Fig. 7. H -mode measured (solid) and calculated (dashed) radiation patterns in the plane of the pillbox plates at 186 GHz.

beamwidth in the scan plane is 0.3 , so the field of view is 45 beamwidths in size. At larger positive and negative scan angles, the beam gain is reduced due to blockage effects of the feed and respectively. mode, the sidelobes are as desired being 15 dB less For the mode, than the beam peak throughout the scan range. For the some sidelobes are larger than 15 dB. This is due to the available feed at the time of the measurements being non-ideal, having a larger beamwidth and subreflector edge illumination than desired. Figs. 10 and 11 compare the calculated and measured scanplane radiation patterns for three beams in the field of view for and modes respectively. At levels less than 15 dB the relative to the beam peak there are some differences between the calculated and measured patterns, particularly in the case of mode. This may be due to errors in the manufacture of the the feeds or multiple interactions between the reflectors or the mode reflectors and the feed. The calculated sidelobes of the mode feed suggesting that feed are larger than those of the sidelobe-scattering effects may be involved in the differences.

Fig. 8. E -mode measured (solid) and calculated (dashed) radiation patterns in the plane of the pillbox plates at 186 GHz.

Fig. 9. Comparison of calculated (solid line) and measured (circles) beam scan angle as a function of the subreflector rotation angle.

Repeating the analysis with additional scattering paths produces noticeable effects. However, the totality of significant paths is not easily determined. Additionally, paths involving reaction of reflectors on the feed are not included in the present analysis. The gains of the pillbox antennas also have been measured, using the antenna substitution method. Table I compares the measured and calculated gains of the central beam for both polarization modes. Included in the calculated gains are ohmic and modes respectively, losses of 2.4 and 1.1 dB for the calculated using the perturbation method for the horns and the complex propagation constants of the parallel-plate modes, and . The calculated gains assuming a conductivity of are consistent with the measurements. The performance of the new three-reflector pillbox antenna may be compared to that of the two-reflector prototype antenna for the mm-wave imaging system [1]. The latter has a scan range of 10 (25 beamwidths) with gain varying between approximately 33 and 30 dBi in this range. The new antenna gives a significant improvement, with a similarly-sized aperture yielding gain greater than 35 dBi throughout a 13.5 (45 beamwidth) field of view.

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Fig. 10. Comparison of the calculated (solid) and measured (dashed) radiation patterns of E -mode beams in the plane of the parallel plates at 5:3 (a), 0.7 (b), and 6.3 (c).

Fig. 11. Comparison of calculated (solid) and measured (dashed) radiation patterns of H -mode beams in the plane of the parallel plates at 5:2 (a), 0.7 (b), and 6.2 (c).

V. CONCLUSION

TABLE I COMPARISON OF CALCULATED AND MEASURED GAIN (dBi) OF THE CENTRAL PILLBOX BEAM AT 186 GHz

0

We have described the design and test of a new three-reflector beam scanning fan beam antenna for a mm-wave imaging system. The antenna has been compared to a two-reflector prototype for the same system and found to have significantly increased gain and beam-scan range.

0

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Good agreement was obtained between the predicted and measured radiation patterns in the required range within 15 dB of the beam peak. If lower sidelobes were required then more accurate analysis or manufacture may be necessary. An analysis method accounting for all mutual interactions between the reflectors and feed may be required. Feed fabrication may be easier if the high-gain horn were replaced by a smaller horn and an additional reflector. The antennas have been refocused at a range length of 10 m by insertion of new main reflectors. Analysis showed negligible change in the radiation patterns from those presented here. The refocused antennas have been used in a study of active imaging at 186 GHz. Some initial results can be found in [9].

Fig. 12. Illustration of parameters of the Gaussian series representation of the current on the contour S .

APPENDIX The parallel-plate mode field scattered by a cylindrical reflector within the pillbox antenna is given by

(1) where is the induced current on the reflector contour , and where is the vector from the point on to the observation point, is the wave impedance of free space and is the wavenumber of the parallel-plate mode under consideration [1]. We use the physical-optics approximation where is the unit vector normal to the surface and pointing is the incident magnetic toward the illuminated side and field. Following [8], the reflector current is represented as a Gaussian series. We extend this approach to any number of reflectors treating each successively, evaluating the incident field and expanding the corresponding current in non-edge-diffracted Gaussian elements. As illustrated in Fig. 12, the current is expanded as

Fig. 13. Image of the pillbox antenna.

[8] by applying the method of successive projections to obtain equality of the left- and right-hand sides of (2) at a grid of points over the reflector surface. Radiation from the pillbox aperture is treated similarly using an equivalent magnetic current and the free-space Green’s function. In the current application, param, , eter values that have been found useful are with the points up to 1 cm apart. These parameters define the summation limits in (2). The summation on the limit is then determined by the number of points reflector contour and the limit is determined by the number of propagation-direction sets with a relative field contribution greater than . The field produced by the reflector current is determined as is assumed sufficiently follows. The beamwidth parameter small for the exponent in (1) to be approximated by a secondorder expansion around , giving, e.g.,

(2) on . Thus is an index where is a parameter of points of the maximum-magnitude points of the Gaussians whereas and are indices of quadratic and linear phase terms. At each point , the incident beams from the previous reflector or feed are located and summed within contiguous sets of propagation directions of angular width where is given in (2) with . This choice of is consistent with the representation of the scattered field as the product of a phase-center factor and a sampling expansion [10]. If the field contribution from any such propagation-direction set is greater than a specified fraction of the total field then this set is uniquely identified with one value of the summation index in (2). The phase parameters and are then determined by least-squares fitting the phase model in (2) to the phase of the field from the identified set over a grid of points around . The coefficients are then determined as previously

(3) where

The integral (3) has a complex stationary phase point and two end points and . Since

with

HAY et al.: THREE-SHAPED-REFLECTOR BEAM-SCANNING PILLBOX ANTENNA SUITABLE FOR mm WAVELENGTHS

only the end point closest to can be significant. Assuming the rest of the integrand varies linearly between this point and the stationary phase point, the integrals can be completed in closed form yielding

where

if or

, and . In these equations, respectively and

if

and or 0 if

ACKNOWLEDGMENT The authors wish to thank J. Tello, K. Smart and F. Ceccato for assistance with the antenna measurements; R. Forsyth and J. Flores for design of the antenna mounts and R. Forsyth for assisting with the alignment of the reflectors. REFERENCES [1] S. G. Hay, J. W. Archer, G. P. Timms, and S. L. Smith, “A beam-scanning dual-polarized fan-beam antenna suitable for millimeter wavelengths,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2516–2524, 2005. [2] G. P. Timms, S. L. Smith, J. W. Archer, and S. G. Hay, “Effect of subreflector and wall thickness on the performance of a scanning pillbox antenna,” in Proc. 9th Australian Symp. on Antennas, Sydney, Feb. 16–17, 2005, p. 52. [3] S. G. Hay, “Multibeam reflector approach to beam-scanning feed for a pillbox antenna,” in Proc. 9th Australian Symp. on Antennas, Sydney, February 16–17, 2005, p. 16.

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[4] P. C. Wertz, W. L. Stutzman, and K. Takamizawa, “A high-gain trireflector antenna configuration for beam scanning,” IEEE Trans. Antennas Propag., vol. 42, no. 9, pp. 1205–1214, 1994. [5] P. C. Wertz, M. C. Bailey, K. Takamizawa, and W. L. Stutzman, “An array fed tri-reflector system for wide angle beam scanning,” in Proc. IEEE Int. Symp. Antennas Propag., 1992, pp. 8–11. [6] R. Jørgensen, P. Balling, and W. J. English, “Dual offset reflector multibeam antenna for international communications satellite applications,” IEEE Trans. Antennas Propag., vol. 33, no. 12, pp. 1304–1312, 1985. [7] S. J. Barker, C. Granet, A. R. Forsyth, K. J. Greene, S. G. Hay, F. G. Ceccato, K. W. Smart, and P. Doherty, “The development of an inexpensive high-precision mm-wave compact antenna test range,” presented at the 27th Annu. Symp. of the Antenna Measurement Techniques Association (AMTA 2005), Newport, RI, 2005. [8] S. G. Hay, “A double-edge diffraction Gaussian series method for efficient physical optics analysis of dual-shaped-reflector antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2597–2610, 2005. [9] G. Hislop, S. Hay, and A. Hellicar, “Efficient sampling of electromagnetic fields via the adaptive cross approximation,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3721–3725, 2007. [10] O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag., vol. 37, no. 7, pp. 918–926, 1989. [11] S. L. Smith, J. W. Archer, K. W. Smart, S. J. Barker, S. G. Hay, and C. Granet, “A millimeter-wave antenna amplitude and phase measurement system,” IEEE Trans. Antennas Propag., submitted for publication. Stuart G. Hay (SM’10) received the B.E. and Ph.D. degrees in electrical engineering from the University of Queensland, Brisbane, Australia, in 1985 and 1994, respectively. From 1986 to 1989, and since 1994, he has been with the CSIRO ICT Centre, Epping, NSW, Australia, where he is currently a Principal Research Scientist. He has published over 100 research papers and technical reports in various aspects of electromagnetics and antennas including shaped reflectors, wide field-of-view beam-scanning and multibeam antennas and connected arrays. Dr. Hay currently serves as Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and was Co-Guest Editor of the June Special Issue on Antennas for Next Generation Radio Telescopes.

Stephanie L. Smith, photograph and biography not available at the time of publication.

Greg P. Timms, photograph and biography not available at the time of publication.

John W. Archer, photograph and biography not available at the time of publication.

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System Fidelity Factor: A New Method for Comparing UWB Antennas Gabriela Quintero, J.-F. Zürcher, and Anja K. Skrivervik

Abstract—The main purpose of the System Fidelity Factor (SFF) is to incorporate frequency and time domain characteristics of an antenna system into a comparison method for ultrawideband (UWB) antennas. The SFF is an interesting tool because both simulations and measurements can be done in a simple and straight-forward manner. Simulations of a single antenna are combined into a two-antennas system analysis by means of a simple post-processing, where the transfer function of the transmitting and receiving antennas are calculated. Measurements of the SFF are done using a two port Vector Network Analyzer (VNA). The polar representation of the SFF allows an equitable comparison between antennas. The procedure to derive the SFF is described in detail in the paper. Two examples are given where the UWB performance of three antenna systems are compared. In the first example antenna systems of two identical monopoles are studied. In the second example the transmitting antenna is a Vivaldi and the receiving antenna a monopole. Index Terms—Fidelity factor, planar circular monopole, ultrawideband (UWB) antenna.

I. INTRODUCTION

I

N the last years ultrawideband (UWB) antennas have been of strong interest among antenna designers. A large amount of UWB antenna designs are produced every year, each publication using a different parameter to show the efficiency of its design. UWB antennas are always defined in first place by their return dB in the entire frequency band. loss, which has to be below The radiation pattern is commonly plotted at specific frequency points of the UWB band. The gain pattern is often represented in a frequency vs. angle ( or ) plot [1], where the weak radiation points can be identified. Some designers show the impulse response in time domain, which gives a general idea of the distortion of the pulse [2]. The group delay (or phase derivative) of an antenna system is represented in a time-frequency plot, describing the time needed by the signal to travel from one antenna to the other. The parameters above describe an antenna either in frequency or in time domain at a given point in the space, but they do Manuscript received January 26, 2010; revised November 11, 2010; accepted November 15, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported in part by the National Competence Center in Research on Mobile Information and Communication Systems (NCCR-MICS), a center supported by the Swiss National Science Foundation under Grant 5005-67322. The authors are with the Laboratory of Electromagnetics and Acoustics (LEMA), Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland (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.2011.2152322

not present all these important characteristics together. The only parameter that analyzes time, space and frequency together is the Fidelity Factor [3]. There are several limitations to calculate this parameter as the radiated E-field in time is needed. Not all simulation tools can directly obtain the time domain E-field, and special measurement equipment is required, which in most cases has limited dynamics. A time domain software was used in [4] to calculate the Fidelity Factor. The authors showed that the measured correlation factors of a two-identical antenna system is approximated with the square of the Fidelity Factor. The method used has several restrictions, including the distance between the antennas that is short (400 mm) due to software capabilities and that the same antenna is used at transmission and reception. Measurements were done only at two points in the azimuth plane and the results were presented in a table form, making the comparison difficult. The System Fidelity Factor is not subject to these restrictions. in the The SFF makes use of the transmission coefficient frequency domain to calculate the correlation between the input and output pulses of an antenna system. The distance between the antennas and the characteristics of the channel can be freely modified. The analysis is done for a full azimuthal plane and does not take more time, effort or computer speed than analyzing a single antenna. The results are shown in a polar plot, facilitating with this the visual comparison of different antenna systems. In order to demonstrate the usefulness of the SFF the planar circular monopole [4]–[7] will be analyzed, comparing the SFF with some of the parameters mentioned earlier in this section. Three different ground plane configurations are studied and compared using the proposed method. Two different antenna systems are studied, the first using the same monopole as Tx and Rx antenna, and the second using a Vivaldi antenna as the transmitting antenna. Some results of the first case were presented by the authors in [8]. In Section II, the proposed method will be described, together with the measurement and simulation techniques used. Section III introduces the studied antennas, depicting some of the frequency characteristics that are mentioned before and that are commonly used to describe UWB antennas. In Section IV, results of the studied monopoles are compared using the SFF and the group delay. II. OBTENTION OF THE SFF The System Fidelity Factor quantifies the dispersion produced by any two antenna system in a plane of radiation. The system can be composed of any type of antennas (identical or different) and can be located at any distance apart from each

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other. In our case, simulations are done with the Finite Element Method software Ansoft HFSS [9] and a HP 8720D Network analyzer is used for the measurements. A description of the method is presented in the following subsections. First the simulation and measurement methods to calculate the system transfer function, followed by the post-processing used to calculate the time domain signals, and finally the SFF is calculated from the transmitting and received signals.

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Equation (2) is then simplified, assuming no polarization losses (antennas located in the same plane) and that , as in this case the Tx and Rx antennas are identical. The system transfer function, as mentioned before, is the ratio between the received and transmitted voltages; therefore is the square root of the equation above. Knowing that and that the antenna impedances are identical , the amplitude of the transfer function can be defined as

A. Transfer Function The system transfer function or transmission paramis essential to obtain the impulse response, and plays eter the main role in the simulations and measurements results. It is composed of the transfer functions of the transmitting antenna , the receiving antenna and the channel . It is normally represented in the frequency domain as (1) can be directly measured, The transmission coefficient but the simulated value has to be calculated using the parameters is of special obtained by the software. This definition of interest (in simulations) as the three transfer functions are independent from each other. Therefore each function can be calculated separately and multiple combinations can be achieved by the exchange of the different elements. The following subsection describes the simulations and post processing needed to obtain the transfer function. The measurement setup is described in the subsequent subsection. of an antenna system can be ob1) Simulation: The tained directly from simulations for a limited distance between antennas and at a large computing cost [10]. One simulation is required for each desired angle in the plane, increasing with this the amount of time needed to obtain the results. A simple simulation technique to obtain the transfer function is presented here. From simulations of a single antenna, the transmission coefficient between two identical antennas located at any distance and at any angle of rotation (in the azimuth plane) is calculated. From the two identical antennas transfer function , the transfer functions of each antenna and are obtained. The transfer function of an antenna system composed of two different antennas can be directly obtained by combining and of the desired elements. the The transfer function is defined as the ratio between the voltage received at the Rx-antenna terminals and the voltage at the input of the Tx-antenna [11]. This ratio can be obtained from the Friis’ Transmission equation

(3) , the phase of the channel In order to obtain the phase of and the phase distortion inside the antennas should be calculated separately. The phase change in the channel is obtained from the , which gives the radial dependance of universal function a radiated field. The phase distortion produced at each antenna and ) has an important effect on the pulse shape and ( the added time delay is considerable. The phase of the radiated E-field contains this information and it can be calculated by the simulator. The phase of the transfer function will be given by the channel and the antennas radiated E-field as follows:

(4) is the radiated electric In HFSS for instance, the value field multiplied by the radial distance. Depending on the orientation of the antenna, the theta or phi component is used in ang_rad, which is the phase in radians. The simulation parameters (Gain, Return Loss and radiated E-field phase) should be obtained over a band larger than the desired band in order to have enough accuracy in the time domain signals. The antenna gain and E-field phase should as well be calculated at every angle of the desired plane. From (3) and (4) the channel transfer function is defined as

(5) The transfer function of two identical antennas are related through the reciprocity theorem [12]

(6) Substituting (5) and (6) in (1) and (4) (7) (8)

(2) is the polarization efficiency of the system, deWhere notes the mismatch at each antenna terminal, is the total anwavetenna gain and the last terms define the path loss ( distance between antennas). length, Obtaining and from simulations of a single antenna, the of a two identical antennas system can be calculated.

and solving for

and

using (3) (9) (10)

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This is a straightforward procedure and any input signal can be used. The pulse to be transmitted is defined in Matlab and a fast Fourier transform FFT is done to obtain its frequency response. This information is multiplied with the transfer function of the antenna system to obtain the received signal in frequency domain. Applying the inverse Fourier transform IFFT the Rx-signal in time is obtained. The following equations denote the mentioned steps (11) (12)

Fig. 1. Measurement setup inside an anechoic chamber.

The measured received signal is obtained by substituting the with the measured . The received signal will simulated present the distortion and dispersion produced by both antennas and the channel. The best way to quantify this effect is to make a correlation between both input and received pulses. The following subsection describes the method proposed, which uses this approximation. C. System Fidelity Factor

From the equations above, the transfer function of the two antennas (Tx and Rx) can be calculated for any point in the space. The system transfer function can then be obtained with and the combination of the two antennas, substituting of the desired antennas and the channel in (1). In this paper, the free-space channel is used in order to characterize the antenna properties only. However, the three transfer functions are independent from each other; therefore the free-space channel can be substituted by any other if required. If the transmitting antenna and the receiving antenna is rotated every is oriented at point, we can characterize the antenna system in the azimuthal plane of the Rx antenna. The same information is obtained from the measurement setup explained in the following subsection. measurements are straightforward 2) Measurement: with a Vector Network Analyzer (VNA) inside an anechoic chamber. The measurement setup used in this work is shown in Fig. 1. Each antenna is mounted at one end of a foam frame that is 1.09 m long. One of the antennas is fixed (Tx) while the in the azimuthal other antenna (Rx) is rotated with a step measurement is done every . Each antenna plane. A new is connected to a port of the HP 8720D Network analyzer by a 3 m long high quality cable. The VNA is located outside the chamber and was previously calibrated at the antennas ports using an electronic kit to cover all its working band (50 MHz–20.05 GHz). B. Time Signal Ultra-wideband Impulse Radio (UWB-IR) sends very short ns) covering a very broad frequency specpulses in time ( trum. As the pulses are narrow, they are greatly affected by diswill never persion. The pulse at the receiving antenna port . However, the receiver be the same as the transmitted pulse should recognize the incoming pulse. For this reason a time domain analysis of the transmitted pulse is done, in order to predict the distortion produced by the system. The antenna analysis is done in frequency domain and this information is post-processed to obtain the time domain signal.

The fidelity factor, as defined in [13], quantifies the degree to which the radiated E-field waveform of a transmitting antenna resembles the driving voltage. This is calculated with the cross-correlation of the radiated E-field and the input signal, as stated in [3]. Nevertheless, it is not always easy to calculate in real applications due to the difficulty to measure or simulate the radiated E-field. The System Fidelity Factor SFF on the other parameter to compare the received hand, uses the standard and transmitted signal between two antennas, quantifying the degree to which the antenna system affects the input pulse. It takes into consideration the distortion induced by the two antennas, whereas the Fidelity Factor takes into consideration the transmit antenna effect only. The received and the transmitted pulses, calculated in the subsection above, are normalized as shown in (13) and (14). The normalization is done in order to compare only the shape of the is expected to be pulses, and not their magnitude, as the much lower than (13)

(14) The cross-correlation between both signals is done at every point in time and the maximum value of this correlation is obtained when both pulses overlap (15) The SFF in (15) has to be solved for every angle and plotted in a polar plot. Because of the normalization of the signals, the results of the cross-correlation are between 0 and 1. An SFF value of 1 indicates that the received pulse is identical to the input pulse, hence no dispersion occurred during transmission. A value of 0 means the received pulse is completely different

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Fig. 2. Planar circular monopoles [8]. (a) Monopole A, (b) Monopole B, (c) Monopole C.

Fig. 3. Measured (dashed lines) and simulated (solid lines) return loss.

than the one at the Tx-antenna port. The required SFF value needed to accurately detect a desired pulse depends on the application, but we consider that a distortion higher than 50% will make the pulse almost unrecognizable. III. STUDIED ANTENNAS The usefulness of the SFF in comparing UWB antennas is demonstrated with the three monopoles depicted in Fig. 2. They mm) printed on a consist of a circular monopole (radius mm) substrate. They are fed by FR4-epoxy ( a 50 microstrip line of 1.4 mm width. The size and position of the ground plane varies for the three antennas, as is indicated on the figure. From the structure of a circular monopole it can be assumed that the best radiation will be in the direction normal to the an. Nevertheless, this is not true tenna surface for most planar circular monopoles, as found in [4], [6], [14]. The second antenna is the Vivaldi antenna [1], used as a transmitting antenna for the three monopoles described before as

Fig. 4. Vivaldi Antenna [1].

receivers. This is done in order to prove the efficiency of the method to represent antenna systems composed of two different antennas. The Vivaldi antenna used is shown in Fig. 4. Its return dB over all the UWB. loss is below The gain pattern graph, showing the gain versus frequency and angle in one scan plane, gives a global overview of the radiation performances of an antenna over a frequency band. The azimuth Gain patterns of each antenna are given in Fig. 5. A large difference between the patterns can be observed. dB) Monopole A (Fig. 5(a)) has a much lower gain ( , from frequencies above 8.2 GHz and up to 10 around GHz. The other monopoles (5(b) and 5(c)) present as well an important decrease in gain but at different frequencies and angles. As expected, the gain pattern of Monopole B is not symmetrical, as it is for the other two. The lowest gain points of the C antenna are located at frequencies above 10 GHz. The Vivaldi antenna

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Fig. 6. Upper view of Monopole–Monopole system setup.

Abrupt changes in the gain of an UWB antenna promote distortion of the transmitted pulse. This distortion will not be the same in all systems. Gain pattern is a good method to identify where is the optimum radiation of an antenna, but gives no direct information on pulse distortion. Phase is an important factor that must not be forgotten when analyzing a time signal, but it cannot be obtained with a conventional radiation pattern measurement. One way to obtain the , or the transfer funcphase in a transmission is to measure tion of the system, which includes both magnitude and phase. This value is the basis to calculate the SFF. The SFF of the three monopoles, together with its intermediate steps (transfer function and time domain signal), are shown and discussed in the following section. IV. UWB ANTENNA SYSTEM CHARACTERIZATION In this section the simulated values obtained from the expressions in Section II are compared to measurements. The group delay of the systems will be calculated from the transfer function’s phase. In the first subsection the antenna systems composed of two identical monopoles will be analyzed. The second subsection presents the mixed antenna systems, where the Vivaldi antenna is the transmitting antenna and the monopoles the receiving elements. A. Monopole–Monopole Systems

Fig. 5. Antenna Gain Pattern. (a) Monopole A, (b) Monopole B, (c) Monopole C, (d) Vivaldi.

has an almost constant gain over the band within the 50 main beam.

1) Transfer Function: The transfer functions of the antenna systems, consisting of two identical monopoles (A, B or C) are presented in this subsection. The transmitting antenna is fixed on one end of the foam support and is facing the Rx antenna . The Rx antenna is rotated every 10 . The setup is depicted in Fig. 6. A frequency independent gain is an important characteristic of an ideal UWB antenna. The transfer function of two antennas system should be constant over frequency in order to transmit over the desired a pulse without distortion. By plotting band, we can localize a magnitude drop. If the dip corresponds to a nonlinearity in phase, the transmitted pulse will be distorted. How much and in which measure will the pulse be distorted? alone. Distortion depends This cannot be answered with on where in the band the dip occurs, its magnitude and phase nonlinearity. This is a further motivation to analyze the system in time domain as well, which contains both phase and magnitude, and pulse distortion can be quantified. The transfer function is obtained at one point in space. If it is analyzed only at some angles, the “bad radiation” point might

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not be seen. Therefore it is important to analyze the transfer function in the entire plane. This can be done by plotting for different angles in the same graph, but a conclusion on radiation performance might not be directly obtained and it might be difficult to visualize results at every radiation angle. To ilwas obtained at three points in the azimuth lustrate this, and 180 ) and plotted together plane ( with the measured in Fig. 7. This was done for the three antennas. The dips in magnitude and phase nonlinearities correspond, as expected, to the low gain points in Fig. 5. 2) Group Delay: Group delay is commonly used to characterize two port devices (filters, amplifiers, mixers, etc.), and has been used on UWB antenna systems as well. It measures the total phase distortion of the antenna system, and the dispersion of the transmitted signal can be inferred. The average of the group delay is equal to the time needed for a signal (at a given frequency) to travel from one antenna terminal to the other. It is defined as the derivative of the phase response versus frequency [15] (16) The group delay of all studied systems was calculated (from the phase of the obtained transfer function) at the azimuthal an, 90 and 180 . This was done in order to emphasize gles the benefits of the SFF, compared to this conventional characterization tool. Group delay of the Monopole–Monopole systems are shown in Fig. 8(a)–(c). A fair agreement is seen between simulations and measurements. The measured plots seem to be more noisy while the simulated ones present less abrupt changes. Nevertheless, the strong group delay variations appear at the same frequencies. It is difficult to conclude at which angle the pulse distortion will be higher only by looking the graphs. Monopole A system has relatively strong group delay variations in the 8–10.5 GHz range, while variations in Monopole B system continue until 12 GHz and in Monopole C they appear above 9.5 GHz. These frequencies correspond to the encountered phase non-linearities in Fig. 7, as group delay is another way of representing the phase of the antenna system. The maximum deviation can be calculated, but this will be only at one frequency and on one direction. The frequency at which this maximum occurs and the range where the strongest variations occur, directly affect the distortion of the pulse. Nevertheless there is not a direct method to calculate the distortion produced by the system only by analyzing its group delay. 3) Time Signal: The next step to calculate the SFF is to obtain the received signal in the frequency domain to later transform it into the time domain. This is done, as mentioned in Section II.B, ) and the input pulse by multiplying the transfer function (or spectrum. An IFFT is applied to the frequency domain signal and the Rx pulse obtained. Again, this result does not infer directly how distorted the transmission will be at any given point. The signal can be obtained for several angles in the plane. This was done for each antenna system, at the same angles used in the previous subsec, 90 and 180 ). As seen in Fig. 9, the tion (

=0

Fig. 7. Transfer function at ( , 90 and 180 A, (b) Monopole B, (c) Monopole C.

;  = 90

): (a) Monopole

measured and simulated received pulses are almost identical. This confirms the quality of the proposed method. The Rx signal should be compared to a reference signal, which in our case is the input signal at the Tx antenna port. This is done in order to know how much they differ from each other and in which measure the signal is distorted. In the plots,

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Fig. 8. Group delay: Simulated (dashed lines) and measured (solid lines). (a) Monopole A–Monopole A, (b) Monopole B–Monopole B, (c) Monopole C–Monopole C, (d) Vivaldi–Monopole A, (e) Vivaldi–Monopole B, (f) Vivaldi–Monopole C.

the Rx signals are clearly different and more dispersed in time than the input signal, however an objective conclusion cannot be drawn. Correlating both signals is one way of quantifying the difference. The SFF uses this method and the results are shown below. 4) System Fidelity Factor: The SFF of the three antennas is plotted in Fig. 10. Monopole A has the lowest SFF, which is below 0.75 at every angle. Its radiation in all the UWB band is . clearly not omnidirectional as the SFF is higher at The simulated values are lower than the measured ones, but both plots are similar in shape. The difference is amplified on the figure by the scale used, having 0.5 at the center of the diagram. This scale is used because we considered that an antenna system having SFF values below 0.5 will distort the pulse in such a way that the received pulse will no longer be recognized. The main characteristic of antenna B (Fig. 10(b)) is its asymmetry. From the plot it is clear that the side of the antenna closer presents a better SFF. The difto the ground edge ference between the two plots can be attributed to simulation inaccuracies, as phase nonlinearities are not detected by the simulator, as shown in Fig. 7(b). Nevertheless, in both lines a dip and . The lowest SFF values is seen between are found there as well. The front and backward radiation are asymmetrical as well, which is only seen in this antenna. The A and C antennas have a symmetric SFF seen from both and planes. Antenna C presents the highest SFF in the front and back directions. A difference of 10% is found between simulations . The reason of this and measurements at the sides behavior is not very clear, as the simulated transfer function (plotted in Fig. 7(c)) does not differ much from at

measurements. On the other hand, the simulated time domain signal at this point (Fig. 9(c)) is wider than the measured one. The time resolution to obtain the IFFT of the pulse is limited. Limited resolution added to phase inaccuracy of the simulator lead, in this case, to miss an important point of the time domain signal (a ripple is missed and the pulse seems to be wider). The same occurs in other cases, but the position of the point is not very crucial (is not inside the main pulse) and therefore the SFF is not affected considerably.

B. Vivaldi–Monopole Systems In this subsection the SFF of three antenna systems, each composed of a transmitting Vivaldi antenna and a receiving monopole, will be calculated and discussed. The antenna system setup is depicted in Fig. 11. 1) Transfer Function: The Vivaldi antenna has an almost constant gain at bore-sight over all the frequency band. At frequencies above 6 GHz, its gain is slightly higher than at lower frequencies. On the other hand, the three monopoles have lower gains at high frequencies. Therefore, using the Vivaldi antenna as a transmitter will improve the performance of the system, if the same monopole is used as a receiver. This is first seen when calculating the system transfer functions (Fig. 12). The magniis higher than that of the Monopole–Monopole systude tems and is almost constant over all the band. The dips found in looks almost Fig. 7 are only slightly appreciated, and constant for some angles. Phase nonlinearities are still present, but their position in frequency and intensity are different.

QUINTERO et al.: SYSTEM FIDELITY FACTOR: A NEW METHOD FOR COMPARING UWB ANTENNAS

Fig. 9. Tx signal and Rx signal (simulated and measured): (a) Monopole A, (b) Monopole B, (c) Monopole C.

2) Group Delay: The group delay plots of Vivaldi–Monopole systems are shown in Fig. 8(d)–(f). The simulated values are clearly more linear than the Monopole–Monopole systems, as are the phases of the transfer functions in Fig. 12.

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Fig. 10. System Fidelity Factor: Simulated (dashed lines) and measured (solid lines). (a) Monopole A, (b) Monopole B, (c) Monopole C.

Strong variations are found in measurements, which do not correspond to the measured phases, as they seem to be linear. This is an indicative that a better analysis of an antenna system phase is achieved using the group delay. The frequency ranges where the strong variations occur, as well as the maximum

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Fig. 11. Upper view of Vivaldi–Monopole system setup.

deviations are smaller than in the Monopole–Monopole systems. Comparing Monopole A and B systems, it can be directly concluded that the Vivaldi–Monopole systems distort less, in every azimuthal angle, the transmitted UWB pulse. However, Monopole C has the same maximum deviation in both graphs, but at different frequencies. Showing that the information given by the group delay is not enough to conclude which system produces more distortion. 3) Time Signal: Comparing the receiving signals of the Vivaldi–Monopole systems with those of the Monopole–Monopole systems only by looking at the results in Fig. 13 and 9 is not evident. The received pulses in Fig. 13 are slightly delayed and narrower than those in Fig. 9, but again the simulated signals overlap perfectly with the measured ones. 4) System Fidelity Factor: The high directivity and constant gain of the Vivaldi antenna are indices of a non dispersive UWB transmission. The Vivaldi antenna working as the Tx antenna helps to normalize the SFF. As the Tx antenna is the same for all the systems, the performance of the monopoles working as Rx antenna can be compared. This is not the case for the Monopole–Monopole system case, where the transmission characteristics of each monopole are as well considered. The SFF plot of Monopole A (Fig. 14(a)) is clearly improved in all directions. An increase of 20% on the sides and 10% on the front and back directions is achieved. This shows that the than antenna has a better UWB performance on the at . Monopole B has again a non-symmetric pattern seen from both and planes. The increase on the SFF is about 10% in every direction, reaching values of 0.9 at several points. The improvement on the SFF plot is high but still less than the one achieved by Monopole A. The improvement on Monopole C SFF is not as important as for the other two monopoles but it is the only having a real omnidirectional behavior. The SFF is higher than 0.8 in all directions, but does not reach the 0.9 value that is achieved by Monopoles A and B. The agreement between measurements and simulations is very good. Monopoles A and B have a large ground plane and a not omnidirectional SFF. Monopole C, in the other hand, has a small ground plane and omnidirectional SFF which is lower than the

= 0

180

= 90

Fig. 12. Transfer function at ( , 90 and ;  ). (a) Vivaldi–Monopole A, (b) Vivaldi–Monopole B, (c) Vivaldi–Monopole C.

other monopoles. It could be concluded that Monopole C has the , whereas the other best transmission characteristics at monopoles have their best transmitting performance at other angles. If the antennas are used in a mobile network system, the Monopole–Monopole case should be studied. Using Monopole

QUINTERO et al.: SYSTEM FIDELITY FACTOR: A NEW METHOD FOR COMPARING UWB ANTENNAS

Fig. 13. Tx signal and Rx signal (simulated and measured). (a) Vivaldi– Monopole A, (b) Vivaldi–Monopole B, (c) Vivaldi–Monopole C.

C will assure a good transmission of the UWB pulse in every direction, while Monopole A might have bad performance at some

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Fig. 14. Vivaldi–Monopole SFF: Simulated (dashed lines) and measured (solid lines). (a) Vivaldi–Monopole A, (b) Vivaldi–Monopole B, (c) Vivaldi–Monopole C.

directions. Monopole B could be interesting to use when asymmetries of the radio or the environment are present. If the radios are to connect to a base station, the abnormalities of radiation

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of the mobile antenna could be canceled if a good directive antenna is used at the base station, like in the Vivaldi–Monopole case. Optimal pairing could be found by analyzing the SFF of the two antennas. V. CONCLUSION The SFF allows a fair and systematic comparison of UWB antennas, taking in consideration frequency and time domain characteristics of the studied antennas in a given plane of radiation. This is achieved by using conventional frequency domain simulation tools and measurement equipment. The described simulation method converts the frequency information of a single antenna into the transfer function of a two-antennas system. and are independent from each other, allowing a free combination of the three . Our method has then one functions to obtain the desired degree of freedom more than the Fidelity Factor method in [4], where only the two antennas are independent. With our method, the channel can be modified as well and substituted with other non free-space channels. The system transfer function is multiplied by the desired input pulse spectrum to obtain the received pulse in time domain. The SFF is calculated from the cross-correlation between the reference pulse (input at the Tx antenna terminals) and the received pulse. Measurement proceedings to calculate the SFF were as well described. Two cases were studied to prove the efficiency of the SFF: 1) antenna systems composed of two identical monopoles, 2) antenna systems with a Vivaldi antenna at transmission and a monopole at reception. The simulated and measured values of the system transfer function, group delay, time domain signal and SFF were shown. A good agreement between simulations and measurements was achieved in both cases. It was shown that a better SFF can be achieved when a directive antenna is working as a transmitter, as with the Vivaldi–Monopole systems. REFERENCES [1] W. Sorgel, C. Waldschmidt, and W. Wiesbeck, “Transient responses of a Vivaldi antenna and a logarithmic periodic dipole array for ultra wideband communication,” presented at the IEEE Antennas and Propagation Society Int. Symp., 2003. [2] S. Licul and W. Davis, Unified frequency and time-domain antenna modeling and characterization Sep. 2005, pp. 2882–2888. [3] D. Lamensdorf and L. Susman, “Baseband-pulse-antenna techniques,” IEEE Antennas Propag. Mag., vol. 36, no. 1, pp. 20–30, Feb. 1994. [4] Q. Wu, R. Jin, J. Geng, and M. Ding, “Pulse preserving capabilities of printed circular disk monopole antennas with different grounds for the specified input signal forms,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2866–2873, Oct. 2007. [5] H. Schantz, The Art and Science of Ultrawideband Antennas. Norwood, MA: Artech House, 2005. [6] J. Liang, L. Guo, C. Chiau, and X. Chen, “Time domain characteristics of UWB disc monopole antennas,” in Proc. Eur. Conf. on Wireless Technology, Oct. 3–4, 2005, pp. 289–292. [7] Y.-L. Zhao, Y.-C. Jiao, G. Zhao, L. Zhang, Y. Song, and Z.-B. Wong, “Compact planar monopole UWB antenna with band-notched characteristic,” Microw. Opt. Technol. Lett., vol. 50, no. 10, pp. 2656–2658, Oct. 2008. [8] G. Quintero, J. F. Zürcher, and A. Skrivervik, “Omnidirectional pulse dispersion of planar circular monopoles,” in Proc. IEEE Int. Conf. on Ultra-Wideband ICUWB, Sep. 9–11, 2009, pp. 395–399.

[9] Ansoft High Frequency Structure Simulator (HFSS) v12.0 (2009) [Online]. Available: http://www.ansoft.com/products/hf/hfss/ [10] Z.-A. Zheng and Q.-X. Chu, “A simplified modeling of ultrawideband antenna time-domain analysis,” in Proc. IEEE Int. Conf. on Ultra-Wideband ICUWB, Sep. 9–11, 2009, pp. 748–752. [11] D. M. Pozar, “Waveform optimizations for ultrawideband radio systems,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2335–2345, Sept. 2003. [12] M. Kanda, “Transients in a resistively loaded linear antenna compared with those in a conical antenna and a tem horn,” IEEE Trans. Antennas Propag., vol. 28, no. 1, pp. 132–136, Jan. 1980. [13] D.-H. Kwon, “Effect of antenna gain and group delay variations on pulse-preserving capabilities of ultrawideband antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 8, pp. 2208–2215, Aug. 2006. [14] G. Q. D. de Leon and A. K. Skrivervik, “Analysis of UWB antennas for carrier-based UWB impulse radio,” in Proc. 2nd Eur. Conf. on Antennas and Propagation EuCAP, Nov. 11–16, 2007, pp. 1–5. [15] X. Zhu, Y. Li, S. Yong, and Z. Zhuang, “A novel definition and measurement method of group delay and its application,” IEEE Trans. Instrum. Meas., vol. 58, no. 1, pp. 229–233, Jan. 2009. Gabriela Quintero received the B.Sc. degree in electrical engineering from the Instituto Teconológico y de Estudios Superiores de Monterrey (ITESM-CCM), Mexico City, Mexico, in 2003, the M.Sc. degree in electrical engineering from Chalmers University of Technology, Göteborg, Sweden, in 2005, and the Ph.D. degree from the Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland, in 2010. She is currently working in Switzerland as an Application Engineer. Her research interests are applied electromagnetics, particularly the design, optimization, miniaturization and measurement of microstrip antennas.

J.-F. Zürcher was born in Vevey, Switzerland, in 1951. He graduated with the degree of Electrical Engineer from Ecole Polytechnique Fédérale de Lausanne (Lausanne Institute of Technology), Switzerland, in 1974. He is presently employed as a permanent Scientific Associate with the Laboratoire d’Electromagnétisme et d’Acoustique EPFL, where he is the Manager of the Microwave Laboratory. His main interest lies in the domain of microstrip circuits and antennas. In 1988, he invented the SSFIP concept (“Strip Slot Foam Inverted Patch antenna”), which became a commercial product. He is presently developing instrumentation and techniques for the measurement of near fields of planar structures and microwave materials measurement and imaging. He is the author or coauthor of about 140 publications, chapters in books and papers presented at international conferences. He is a coauthor of the book Broadband Patch Antennas (Artech House, 1995). He holds nine patents.

Anja K. Skrivervik received the Electrical Engineering degree and Ph.D. degree from Ecole Polytechnique Fédérale de Lausanne, Switzerland, in 1986 and 1992, in respectively. After a stay at the University of Rennes as an Invited Research Fellow and two years in the industry, she returned part time to EPFL as an Assistant Professor in 1996, where she is now a Professeur Titulaire. Her teaching activities include courses on microwaves and antennas. Her research activities include electrically small antennas, implantable and on body antennas, multifrequency and ultrawideband antennas, numerical techniques for electromagnetics and microwave and millimeter wave MEMS. She is the author or coauthor of more than 100 scientific publications. She is very active in European collaboration and European projects. Dr. Skrivervik received the Latsis Award. She is currently the Chairperson of the Swiss URSI, the Swiss representative for COST action 297 and a member of the board of the Center for High Speed Wireless Communications of the Swedish Foundation for Strategic Research.

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Wideband Low-Loss Linear and Circular Polarization Transmit-Arrays in V-Band Hamza Kaouach, Laurent Dussopt, Senior Member, IEEE, Jérome Lantéri, Thierry Koleck, and Ronan Sauleau, Senior Member, IEEE

Abstract—Several linearly-polarized and circularly-polarized transmit-arrays are designed and demonstrated in the 60-GHz band. These arrays have a fairly simple structure with three metal layers and are fabricated with a standard printed-circuit board technology. The simulation method is based on an electromagnetic model of the focal source and the unit-cells, associated to an analytical modeling of the full structure. A theoretical analysis is presented for the optimization of the power budget with respect to ratio. Several prototypes are designed and characterized the in -band. The experimental results are in very good agreement with the simulations and demonstrate very satisfactory characteristics. Power efficiencies of 50–61% are reached with a 1-dB gain bandwidth up to 7%, and low cross-polarization level. Index Terms—Discrete lens, unit-cell, millimeter-wave antennas, transmit-array.

Fig. 1. Operation principle of a transmit-array in transmit mode. The transmitarray consists of two Rx and Tx antenna arrays connected by phase shifters. The latter are used to modify the signal phase delay and thus to control the main beam direction of the transmit-array.

I. INTRODUCTION ECHNOLOGY advances in low-cost millimeter-wave integrated circuits have triggered a lot of application perspectives for wireless communication systems in - and -bands [1]–[3]. In particular, wide license-free frequency bands are available worldwide in the 57–66 GHz range, and several standardization groups are working on regulatory rules for wireless high-definition video transmission [4], [5], wireless personal and local area networks [6]–[8], and radio-over-fiber networking solutions [9]. The main common requirements for these applications are the following: low-cost, high efficiency antenna solutions [10], and ease of integration in user terminals or base stations (cell phone, laptop, set-top box, etc.). In some cases like “gigabit offices” applications, antennas with medium or high directivity, or multiple-beam [11] and shaped beam [12], [13] capabilities are needed for long range transmission or better radio coverage. Another important application is point-to-point multi-gigabit wireless communications for

T

Manuscript received May 02, 2010; revised November 11, 2010; accepted December 28, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. H. Kaouach was with CEA, LETI, Minatec, F38054 Grenoble, France. L. Dussopt is with CEA, LETI, Minatec, F38054 Grenoble, France (e-mail: [email protected]). J. Lantéri was with CEA, LETI, Minatec, F38054 Grenoble, France. He is now with the University of Nice-Sophia Antipolis, France (e-mail: [email protected]). T. Koleck is with the CNES (French Space Agency), Toulouse, France. R. Sauleau is with the Institute of Electronics and Telecommunications of Rennes (IETR), UMR CNRS 6164, University of Rennes 1, 35042 Rennes, France (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.2011.2152331

metropolitan area networks in -band (70–80 GHz) where highly directive beams are necessary [14]–[16]. To date, numerous high-directivity antenna structures have been demonstrated at millimeter-wave frequencies: (i) Some commercial solutions are based on horn antennas and dielectric lenses. The latter are very efficient, but expensive and bulky [13], [17]; (ii) Printed antenna arrays are generally easy to integrate in base stations, but suffer from a very limited efficiency due to excessive transmission-line losses in the beamforming network [18]; (iii) More recent solutions based on leaky-wave structures (e.g., [19]), electromagnetic band-gap materials (like FSS-based resonator antennas [20], [21]), or Fabry-Perot resonators [22] are intrinsically narrow-band structures and can not meet the above-mentioned bandwidth requirements; (iv) Reflect-arrays combine the benefits of quasi-optical structures (high efficiency) and printed antennas (light weight and cost-effective fabrication) [23], [24]. However, in some cases, the position of the focal source in front of the reflecting surface is not suitable for close integration of integrated circuits, as usually desired in commercial systems. In addition, the possible shadowing effect due to the feed cluster (for multiple beam applications) and its supporting struts may severely degrade the radiation performance for medium-directivity antennas. An alternative solution consists in using transmit-arrays or discrete lenses [25]–[27]. Their generic configuration is schematized in Fig. 1. While keeping the advantages of reflect-arrays in terms of efficiency, light-weight and cost, this antenna technology is more favorable for close integration of the focal source with the transceiver on the main printed-circuit board and for mounting on host platforms (in this case the radiating panel shares the platform surface, and the transceiver is integrated inside the platform).

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Transmit-arrays have been known for a long time and are currently used in military long-range electronic beam-steering radars at the low microwave frequency bands. Active arrays have been investigated as well for power-combining applications in order to mitigate the power limitations of solid-state integrated circuits [28], [29]. Recently, several research groups have studied such structures in Ku- and Ka-bands for various wireless applications [27], [30]–[32]. Electronic beam-steering has been investigated as well, but with limited success due to the integration complexity of phase-shifters and switches inside the array unit-cell [33], [34]. To our best knowledge, this work is the first one investigating the radiation performance of transmit-array antennas operating in the 57–66 GHz band. This paper is organized as follows. The geometry and characteristics of the proposed unit-cell are presented in Section II. Then we describe the modeling technique for the analysis of electrically-large transmit-arrays (Section III.A) and apply this approach to design linearly- and circularly-polarized transmit-arrays (Sections III.C and III.D). This study also includes a theoretical discussion on the power budget and antenna ratio (Section III.B). performance as a function of the The fabricated prototypes are characterized experimentally in Sections IV (linear polarization) and V (circular polarization). Finally, conclusions are drawn in Section VI. II. DESCRIPTION OF THE UNIT CELLS: OPERATION PRINCIPLE AND DESIGN The proposed transmit-arrays are based on the unit-cell configuration represented in Fig. 2. Each cell has a size of 2.5 at the center frequency 60 mm 2.5 mm (i.e., GHz); it consists of two identical square patch antennas (1.55 mm 1.55 mm) connected by a metalized via hole ( mm) and separated by a 17 m-thick copper ground plane. The upper patch (free space side) is etched on a 254 m-thick Rogers RO3003 substrate , and the lower substrate (focal source side) is composed of a Rogers , thickness 100 4403 bonding film m) and a Rogers RT/Duroid 6002 ( , thickness 127 m). As the top and bottom substrates have nearly the same thickness and permittivity, identical patch antennas can be used on each side, and they exhibit similar bandwidth. Due to the rotation of the upper patch, the polarization of the transmitted wave is tilted at an angle . It is easily understood that choosing two values of separated by 180 will provide two unit-cells with the same linear polarization but with a 180 differential phase shift. Therefore, to generate a linearly-polarized beam, these two unit-cells can be used to synthesize a desired phase distribution across the array aperture with a 180 (1-bit) phase quantization. This approach is adopted in Sections III.C and IV ( 90 ) through the design and characterization of four linearly-polarized transmit-arrays. Moreover, it is also well-known that a linearly-polarized antenna array can be used to generate a circularly-polarized beam provided the antenna elements are distributed with a sequential physical rotation and phase excitation [35], [36]. The unit-cell proposed in Fig. 2 can be implemented for this purpose if the angle is properly defined as a function of the phase of the

Fig. 2. Geometry of the proposed unit-cell (with an arbitrary rotation angle  of the top patch antenna located on the free space side). (a) Cross-section view. : mm, W : mm, d : mm, (b) Top view. Dimensions: a ; : mm.

= 01

= 25

= 1 55

= 0 05

incoming wave and the desired phase distribution in the radiating aperture. In principle, the ideal (discrete) phase distribution can be obtained since any value of is achievable. However, this would require designing a large number of unit-cells. In practice, an array with a 90 (2-bit) phase quantization (i.e., , 90 , 180 , and with only four different unit-cells: 270 ) is enough for proof-of-concept with limited design complexity. Such an approach is followed in Sections III.D and V. , 90 , Throughout this paper, the four unit-cells with 180 , and 270 will be referred to as unit-cells #1, #2, #3, and #4, respectively. These four cells will be used for the circularly-polarized arrays, whereas only cells #2 and #4 are needed for the linearly-polarized arrays. The -parameters of the unit-cells are computed with the Ansoft-HFSS simulation tool using Floquet ports and periodic boundary conditions. They are represented in Fig. 3. Under normal incidence, the four cells exhibit very similar performance with a good impedance matching ( dB between 59.5 GHz and 61.9 GHz) and low insertion loss dB at 60.4 GHz). The 1-dB transmission bandwidth equals 3.9 GHz (6.5%). Although a wideband transmit-array unit-cell would ideally exhibit a linear phase response, it is interesting to note that this design leads to a rather compact and wideband unit-cell with a well-controlled constant phase-shift across the whole bandwidth in contrast to other examples based on resonant structures with large transmission-phase variations [32]. The radiation patterns computed at 60 GHz (Fig. 4) show a dB beamwidth of 88 and maximum gain of 4.85 dBi and a 131 in E- and H-planes, respectively. A sensitivity study has been done on this unit-cell through m on the electromagnetic simulations. A dispersion of patch length and via diameter, which are the most critical dimensions, results in a variation of the resonant frequency of GHz % and GHz % respectively. In both dB, cases, the impact on the insertion loss is lower than dB at resoand the reflection coefficient remains below nance. These tolerance values ( m) are well within standard industrial production capabilities, and this sensitivity study show that the impact is clearly smaller than the bandwidth performances demonstrated further in Sections IV and V.

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Fig. 5. (a) Geometry of the transmit-array. (b) Equivalent model of one unitcell.

Fig. 3. S -parameters of the four unit-cells illuminated by a plane wave under normal incidence (simulation results).

Fig. 4. Co-polarization components computed at 60 GHz for the T x patch of unit-cell #1 ( = 0 ). This computation is done with a small lumped port excitation between the ground plane and the feed via, and with periodic boundary conditions on the sides. The radiation patterns of the Rx patch are identical; those of the other three unit-cells are very similar, except for the polarization rotation. The cross-polarization components are not shown since their level is lower than 30 dB.

0

III. MODELING AND DESIGN OF LINEARLY- AND CIRCULARLY-POLARIZED TRANSMIT-ARRAYS A. Array Configuration and Modeling The focal source is a linearly-polarized pyramidal horn antenna (Flann Microwave 25240-10) with a gain equal to 10.2 dBi . A fixat 60 GHz and a full half-power beamwidth of 72

ture composed of a metal plate and four dielectric struts (Delrin, mm) is used to hold and align the array and the focal source. The length of the struts defines the focal distance (Fig. 5(a)). An absorbing material has been used on the feed metal plate to reduce multiple reflections between this plate and the transmit-array. These reflections did not affect the radiation patterns at frequencies close to the nominal frequency but significantly reduced the ripples in the frequency responses presented further. The simulation method of this antenna structure is a mix of analytical modeling and full-wave electromagnetic simulations. The focal source is simulated with a 3D finite-element software are exported and and its complex radiation patterns are normalsaved. In this paper the radiation patterns ized to the square root of the isotropic gain so that the following relation is met

where is the antenna efficiency. As shown in Section II, each unit-cell is also simulated sepaand its comrately in order to compute its -parameters plex radiation patterns on the focal source side and free space side . From this set of simulation results, an analytical model described hereafter allows the calculation of all the characteristics of the antenna structure, including the power budget and the radiation patterns. Each unit-cell is modeled by two antennas connected through a 2-port network (Fig. 5(b)) which contains the -parameters simulated under normal incidence with periodic conditions (Fig. 3). Further simulations have been performed with incident in both principal planes; they have evidenced no angles at significant variation of the -parameters. The actual incident which takes power captured by each cell is determined by into account the reduction of the effective area and the effect of adjacent cells (periodicity) as a function of the incidence direction.

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Assuming the focal source fed with the input power received by unit-cell is given by incident wave

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, the

(1) is the distance between the focal source and unit-cell where , and (resp. ) is the complex radiation pattern of the focal source (resp. unit-cell ) in the direction of the unit-cell (resp. focal source). Here the radiation pattern of the unit-cell is shown in Fig. 4 and is identical for all unit-cells. Given the incident phase distribution (phase of ), each unitcell type is selected to synthesize the best fit of the required side), i.e., a uniphase distribution in the radiating aperture ( form or linear phase distribution. The reflected and transmitted and are computed as follows: waves (2a) (2b) Finally, the radiation pattern of the transmit-array is computed by summing the contribution of and each cell, accounting for their transmitted wave radiation pattern on the free-space side (3) Note here that the radiation patterns of the unit-cells on the side are very similar to the one of unit-cell #1 (Fig. 4), except for polarization rotation. B. Power Efficiency and Spill-Over The power efficiency of the transmit-array antenna is defined as the ratio of the total radiated power on the free-space side to the focal source input power (4) Its maximum value is limited by the following loss factors: i) losses in the focal source (negligible for the horn antenna used in this work), ii) spill-over losses, i.e., power radiated by the focal source outside the angular sector covered by the transmit-array, iii) insertion loss of the transmit-array cells. Since all unit-cells have very similar insertion loss (Fig. 3), the power efficiency of the transmit-array antennas presented in this article is independent of the unit-cell distribution and depends only on the ratio which determines the spill-over losses. According to the definitions given above, the spill-over losses are calculated as follows

(5a) (5b)

Fig. 6. Efficiency and spill-over loss of the transmit-array as a function of the F=D ratio (D mm, f GHz).

= 50

= 60

where is the incident power on the aperture and is the . In (5b), the incident surface of each unit-cell power on each cell is the product of the power density radiated and the apparent surby the focal source in direction . face of the cell viewed from the focal source Fig. 6 presents the spill-over loss and power efficiency of the ratio, with mm transmit-array as a function of the , 20 20 cells). For small focal lengths, the max( imum power efficiency is limited to about 63%: this is mainly due to a residual 1.5 dB spill-over loss corresponding to the sideand back-radiation of the focal source. Therefore, the power efficiency of the transmit-arrays designed in this paper is primarily limited by the low-directivity of the focal source. It is imporratios tant to note that the simulation results obtained for close to zero should be considered as estimates since the numerical model described in Section III.A does not take into account near-field radiation patterns. Moreover, as expected for values, it is seen that the spill-over loss increases large monotonically with , and the power efficiency decreases accordingly. C. Transmit-Arrays in Linear Polarization Four linearly-polarized transmit-arrays have been designed in each case and fabricated, with a different value of , or 1). One of these prototypes ( is shown in Fig. 7(a), and the four cell distributions are represented in Figs. 7(b) to (e). (computed) The variations of the antenna directivities and gains (measured and computed) are plotted in Fig. 8 as func. The ripples observed on the predicted gain and tions of directivity curves are due to the phase quantization and limited number of cells. It could be shown that these curves are smoother for larger arrays and higher bit resolution. As the maxof a transmitimum theoretical directivity array with a uniform amplitude and phase distribution equals 31 dBi, we can conclude from Fig. 8 that the 1-bit resolution used in these four designs induces at least 4 dB quantization loss because the peak directivity reaches only 27 dBi for large values. The antenna gains have been measured using a

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Fig. 8. Gain and directivity of the linearly-polarized transmit-array as functions of the F=D ratio (f = 60 GHz).

TABLE I SIMULATED POWER BUDGETS AND MEASURED GAINS OF THE FOUR LINEARLY-POLARIZED PROTOTYPES AT 60 GHz

Fig. 7. Linearly-polarized transmit-arrays. (a) Photograph of one of the fabricated prototypes (F=D = 0:5). (b)–(e) Distribution of the unit-cells #2 (white pixel) and #4 (gray pixel) for the four prototypes: (b) F=D = 0:25, (c) F=D = 0:5, (d) F=D = 0:75, and (e) F=D = 1.

20 dBi standard gain horn. The agreement between the experimental and simulated gain values is very satisfactory. In particular, Fig. 8 reveals that the maximum gain (23.3 dBi) is obtained . The corresponding directivity, spill-over loss for and power efficiency equal 26 dBi, 2.24 dB, and 53.6%, respecratio in the range 0.4–0.9 leads to a gain value tively. An only 1 dB below the maximum, indicating thereby a rather low sensitivity of the array performances to the focal source position range is determined primarily in this range. The optimal by the radiation pattern of the focal source; for the sake of compactness and integration for future applications, it is desirable to minimize this parameter using a custom focal source with a wide synthesized beam optimizing the power distribution across the array and minimizing spill-over losses. The main features and power budget of these four prototypes are summarized in Table I. The quantization loss and taper loss in dB are defined as follows (6a) (6b)

where is the theoretical antenna directivity. and are the directivities computed assuming in the aperture are uniform in that the distributions phase and amplitude, respectively. Table I shows that the quantization loss due to the 180 phase quantization is in the range 3.5 to 4.4 dB, and that the taper down to 0.3 dB loss decreases from 3.3 dB . The resulting directivities of the four transmitarrays vary between 23.1 and 26.8 dBi. The spill-over losses (between 1.72 and 4.26 dB) have a significant impact on the antenna power budget as discussed in Section III.B (Fig. 6). The insertion loss is about 0.46 dB at 60 GHz (Fig. 3). The simulated gain values equal 20.9–23.3 dBi and reach their maximum for , as shown in Fig. 8. Table I also shows that the difference between the measured and computed gain values is less than 1 dB; note that the largest uncertainty is obtained for , which corresponds to the range where the antenna gain is quite sensitive to the actual position of the focal source (Fig. 8). Finally, the aperture efficiency, defined as the ratio between , is in the range 9.8–17%. Optithe measured gain and mizing this figure of merit would require a specific focal source with minimal back-radiation and a shaped-beam so as to provide a quasi-uniform illumination of the array.

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TABLE II SIMULATED POWER BUDGETS AND MEASURED GAINS OF THE TWO CIRCULARLY-POLARIZED PROTOTYPES AT 60 GHz

Fig. 9. Circularly-polarized transmit-arrays. (a) Gain and directivity as a funcGHz). (b), (c): cell distributions for F=D : tion of the F=D ratio (f and F=D : .

= 05

= 60

= 0 25

D. Transmit-Arrays in Circular Polarization Two circularly-polarized transmit-arrays have been designed and fabricated: ( mm) and ( mm). The building blocks are the four unit-cells #1–4. The antenna directivities, gains, and cell distributions are given in Fig. 9. Despite the smaller phase quantization (90 ), the maximum antenna directivity reaches 27 dBi for large values (Fig. 9(a)); this peak value is close to the one obtained with the linearly-polarized transmit-arrays (Fig. 8) because applying the sequential rotation principle with linearly-polarized sources results in directivity loss [35], [36]. In addition, the theoretical curves plotted in Fig. 9 are smoother than in Fig. 8 because of the 2-bit resolution in circular polarization. The maximum theoretical gain (23.2 dBi) is obtained for . The corresponding directivity, spill-over loss, and power efficiency are 26 dBi, 2.40 dB and 51.9%, respectively. The main characteristics and power budget of these two prototypes are summarized in Table II. It is seen that the quantization losses are slightly higher than in the linear-polarization case: this is due to the sequential rotation of linearly-polarized elements. The resulting directivities, gains, and efficiencies are comparable to the linear-polarization arrays since the spill-over and insertion losses are identical. The authors believe that quantization losses may be significantly reduced using circularly-polarized elements and appropriate rotation angle for each cell [38]. Considering the case, a maximum theoretical aperture efficiency of 43.7% would be expected in the ideal case of zero quantization loss.

Fig. 10. Computed and measured gains in the broadside direction for the four linearly-polarized prototypes.

TABLE III SIMULATED 1-dB BANDWIDTH OF THE FOUR LINEARLY-POLARIZED PROTOTYPES AT 60 GHz

IV. LINEARLY-POLARIZED TRANSMIT-ARRAYS: EXPERIMENTAL RESULTS A. Frequency Response in Gain Fig. 10 represents the gains of the four linearly-polarized prototypes measured and simulated from 50 to 70 GHz. Their simulated 1-dB bandwidths are given in Table III, and are seen to increase from 4.41% to 7.55% with the focal distance. The bandwidth values could not be extracted from the measurements due to the experimental uncertainties, but the general good agreement between simulated and experimental responses in Fig. 10 confirms this trend. As for reflectarrays [23], the bandwidth of transmit-arrays primarily depends on phase errors in the radiating aperture (as well as on the frequency response of the unit-cell). It is seen from Fig. 7(b)–(e) that long focal arrays

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= 0 25

Fig. 11. Simulated (a), (c), (e), (g) and measured (b), (d), (f), (h) radiation patterns of the linearly-polarized prototypes at 60 GHz in H-plane with F=D : (a), (b), F=D : (c), (d), F=D : (e), (f), and F=D (g), (h). The theoretical cross-polarization components are not given since their level is lower than dBi. The discrepancy between the simulated and measured cross-polarization levels is believed to be primarily due to unaccounted scattering effects on the array edges and dielectric struts supporting the array.

020

=05

= 0 75

=1

have larger zones and therefore will be less frequency-sensitive than short focal arrays.

The impedance matching of the focal source in the presence of the transmit-array has been measured and remains below

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TABLE IV SIMULATED GAIN AND DIRECTIVITY OF THE LINEARLY-POLARIZED TRANSMIT-ARRAY AS A FUNCTION OF THE STEERING ANGLE IN H-PLANE

(F=D = 0:5)

= 0:5. Measured radia-

Fig. 12. Linearly-polarized transmit-array with F=D tion patterns in H-plane at 58 GHz (a) and 62 GHz (b).

dB in the pass-band for each focal distance, so that the measured realized gain is close to the absolute gain. Only for , the reflection coefficient increases up to dB out of the pass-band and may result in a measured realized gain reduction by less than 0.75 dB. B. Radiation Patterns The computed and measured radiation patterns at 60 GHz are represented in Fig. 11 in H-plane. A good agreement between simulations and experiments is obtained. For instance, with (Fig. 11(c) and (d)), the measured and predicted values for the beamwidth and sidelobe levels are (sim./meas.) and dB (sim./meas.), respectively. Although not reported here, a very good matching between experimental and numerical data has been obtained in E-plane as well. The frequency dependence of the patterns is illustrated in Fig. 12 which demonstrates that the radiation patterns at 58 and 62 GHz are very close to those obtained at 60 GHz (Fig. 11). Finally, in order to assess the sensitivity of these prototypes to fabrication tolerances, three identical prototypes have been fabricated and characterized. The fabrication was subcontracted and done using a standard class-4 industrial process with a minimum feature size of 100 m for the via diameter and 300 m for the apertures etching in the ground plane. The measurement campaign has shown that the gain variation dB; this indicates the satisfactory robustwas less than ness of these designs although a thorough investigation would require several fabrication runs with different manufacturers.

= 05

Fig. 13. Transmit-array prototype with F=D : and tilted beam at 30 in H-plane; (a) cell distribution, (b) simulated and measured gain frequency response.

C. Prototypes With Tilted Beam Electronic beam steering is an important functionality in modern antenna systems. The main beam-steering method requires to tune the phase of each unit-cell using active devices (MEMS switches or capacitances, diodes, etc.) or tunable materials [33], [34]. This approach is very promising in its principle because the illumination of the array by the focal source remains constant and optimal. Nevertheless the implementation of reconfigurable unit-cells is very challenging (integration, biasing [39]) and often leads to significant insertion losses, especially at millimeter waves. In order to demonstrate the beam steering capabilities of the linearly-polarized transmit-arrays based on unit-cells #2 and #4, the gain and directivity of the antenna prototype with have been simulated as a function of steering angle (Table IV). This Table shows that scan angles up to 60 can be achieved in H-plane with only a 3-dB gain reduction,

KAOUACH et al.: WIDEBAND LOW-LOSS LINEAR AND CIRCULAR POLARIZATION TRANSMIT-ARRAYS IN V-BAND

Fig. 14. Simulated (a) and measured (b) radiation patterns of the transmit-array : and tilted beam at 30 in H-plane. with F=D

=05

in agreement with the simulated beamwidth of the unit-cells (Fig. 4). To validate these numerical results, a passive prototype has been designed for a main beam direction at 30 . The array cell distribution and the frequency response in gain are represented in Fig. 13. The simulated 1-dB bandwidth is 3.3 GHz (5.5%); this value is comparable to the one obtained with the prototype radiating at broadside with the same focal distance. The simulated and measured radiation patterns are in good agreement (Fig. 14). For the sake of completeness, it is important to mention the alternative beam steering technique based on the orientation of the focal source with respect to the array. By moving the feed along the focal arc, the distribution of the incoming wave [30], [32], [37] is modified and results in a beam steered at the corresponding angle. The drawbacks of this solution are to rely on a mechanical displacement or multiple switched feeds, and to result in increased spill-over losses if the feed is tilted by more than 30–40 off the normal direction. V. CIRCULARLY-POLARIZED TRANSMIT-ARRAYS: EXPERIMENTAL RESULTS An interesting advantage of transmit-arrays as compared to dielectric lenses for instance is the possibility to play with dif-

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Fig. 15. Gain and axial ratio (simulation and measurement) for both circularly: . : , (b) F=D polarized transmit-arrays. (a) F=D

= 0 25

=05

ferent polarizations on each side of the array and thus to obtain a very good polarization purity [29]. This was achieved in Sections III and IV where the linearly-polarized -fields radiated by the focal source and radiated beam were perpendicular to each other. A more demonstrative example is presented here using a transmit-array converting (and collimating) the incident linearly-polarized wave into a circularly-polarized narrow beam; the design of the corresponding prototypes have and 0.5). been described in Section III.D ( The measured and computed gains and axial ratios are represented in Fig. 15 from 50 to 70 GHz. The 1-dB gain bandwidths of both prototypes are 6.5 GHz (10.7%) and 11.1 GHz (17.6%), respectively; they are much higher than for the linear-polarization prototypes; this is attributed to the better phase resolution. The axial ratios exhibit small variations across the whole band . The experimental and is clearly lower than 2 dB for , response is in good agreement with simulations for whereas the case shows more discrepancies. This is attributed to a higher sensitivity to the relative position of the focal source and the array, and the far-field approximation of the illumination by the focal source. The simulated and measured radiation patterns are plotted in configuration. This Figure conFig. 16 for the firms the very good agreement between simulations and experiments: the computed and measured gain, beamwidth and side

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Fig. 16. Circularly-polarized transmit-array with F=D principal planes.

= 0:5. Simulated (a), (b) and measured (c), (d) co- and cross-polarization components at 60 GHz in both

lobes equal 22.8/23 dBi (sim./meas.), (sim./meas.), and dB (sim./meas.), respectively. The cross-polarization level in the main beam is lower than dB in simulation and close to dB experimentally. This good polarization quality is obtained across the whole bandwidth of the array thanks to the sequential rotation configuration. VI. CONCLUSION Several transmit-array antennas operating in the 60-GHz band have been studied numerically and experimentally in linear and circular polarization. A new unit-cell configuration with only three metallization layers and a 1-dB gain bandwidth of 6.5% has been proposed. Their fabrication process relies on standard PCB technologies, which makes the corresponding transmit-arrays very attractive for low-cost integration in all sorts of terminals, vehicles or buildings. The antenna design and optimization procedures have been presented with a detailed power budget analysis. The experimental results are in good agreement with the simulations in terms of gain, frequency response and radiation patterns. Gain levels in the range of 22–23 dBi have been demonstrated with radiating apertures of and a 10 dBi focal source. The power efficiency and measured 1-dB gain bandwidth reach 50% and 7%, respectively. Such characteristics make these antennas very promising for high-data rate communications in - and -bands. REFERENCES [1] A. M. Niknejad, “Siliconization of 60 GHz,” IEEE Microw. Mag., vol. 11, no. 1, pp. 78–85, Feb. 2010.

[2] S.-Q. Xiao, M.-T. Zhou, and Y. Zhang, Millimeter Wave Technology in Wireless PAN, LAN, and MAN. Boca Raton, FL: CRC Press, 2008. [3] P. Smulders, “Exploiting the 60 GHz band for local wireless multimedia access: Prospects and future directions,” IEEE Commun. Mag., Jan. 2002. [4] WirelessHD [Online]. Available: http://www.wirelesshd.org [5] Standard ECMA-387 (2008, Dec.). High Rate 60 GHz PHY, MAC and HDMI PAL [Online]. Available: http://www.ecma-international. org/publications/standards/Ecma-387.htm [6] Millimeter Wave Alternative PHY [Online]. Available: http://www. ieee802.org/15/pub/TG3c.html IEEE 802.11 Working Group, Wireless PAN Task Group 3c (2009) [7] [Online]. Available: http://www.ieee802.org/11/Reports/tgad_update. htmIEEE 802.11 Working Group. Very high throughput in 60 GHz [8] Wireless Gigabit Alliance [Online]. Available: http://wireless-gigabitalliance.org [9] T. Chen, H. Woesner, Y. Ye, and I. Chlamtac, “WiGEE: A hybrid optical/wireless Gigabit WLAN,” in Proc. IEEE Global Telecommunications Conf. (Globecom’07), Washington, DC, Nov. 26–30, 2007, pp. 321–326. [10] Ph. Coquet, R. Sauleau, K. Shinohara, and T. Matsui, “Multi-Layer microstrip antennas on quartz substrates. Technological considerations and performance at 60 GHz,” Microw. Opt. Techn. Lett., vol. 40, no. 1, pp. 41–47, Jan. 2004. [11] J. Ala-Laurinaho, A. Karttunen, J. Säily, A. Lamminen, R. Sauleau, and A. V. Räisänen, “Mm-wave lens antenna with an integrated LTCC feed array for beam steering,” presented at the Eur. Conf. Antennas Propag., Barcelona, Spain, Apr. 12–16, 2010, EuCAP, 2010. [12] G. Godi, R. Sauleau, L. Le Coq, and D. Thouroude, “Design and optimization of three dimensional integrated lens antennas with genetic algorithm,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 770–775, Mar. 2007. [13] R. Sauleau and B. Barès, “A complete procedure for the design and optimization of arbitrarily-shaped integrated lens antennas,” IEEE Trans. Antennas Propagat., vol. 54, no. 4, pp. 1122–1133, Apr. 2006. [14] FCC, Millimeter Wave 70–80–90 GHz Service [Online]. Available: http://wireless.fcc.gov/services/index.htm?job=service_home [15] T. Kosugi, A. Hirata, T. Nagatsuma, and Y. Kado, “MM-wave longrange wireless systems,” IEEE Microwave Mag., vol. 10, no. 2, pp. 68–76, Apr. 2009.

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[16] J. Wells, “Faster than fiber: The future of multi-Gb/s wireless,” IEEE Microw.Mag., vol. 10, no. 3, pp. 104–112, May 2009. [17] N. T. Nguyen, R. Sauleau, and C. J. Martínez Pérez, “Very broadband extended hemispherical lenses: Role of matching layers for bandwidth enlargement,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 1907–1913, Jul. 2009. [18] D. Liu, B. Gaucher, U. Pfeiffer, and J. Grzyb, Advanced MillimeterWave Technologies: Antennas, Packaging and Circuits. Hoboken, NJ: wiley, 2009. [19] J. L. Gómez-Tornero, F. D. Quesada-Pereira, and A. Álvarez-Melcón, “Analysis and design of periodic leaky-wave antennas for the millimeter waveband in hybrid waveguide-planar technology,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2834–2842, Sept. 2005. [20] 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. [21] 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. [22] R. Sauleau, , K. Chang, Ed., “Fabry Perot resonators,” in Encyclopedia of RF and Microwave Engineering. Hoboken, NJ: Wiley, May 2005, vol. 2, pp. 1381–1401, ISBN:0-471-27053-9. [23] J. Huang and J. A. Encinar, Reflectarray Antennas. Piscataway-Hoboken, NJ: Wiley-IEEE Press, Oct. 2007. [24] D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeterwave microstrip reflectarrays,” IEEE Trans. Antennas Propag., vol. 45, no. 2, pp. 287–296, Feb. 1997. [25] L. Schwartzman and L. Topper, “Analysis of phased array lenses,” IEEE Trans. Antennas Propag., vol. 16, no. 6, pp. 628–632, Nov. 1968. [26] D. T. McGrath, “Planar three-dimensional constrained lenses,” IEEE Trans. Antennas Propag., vol. 34, no. 6, pp. 46–50, Jun. 1986. [27] D. M. Pozar, “Flat lens antenna concept using aperture coupled microstrip patches,” Electron. Lett., vol. 32, no. 23, pp. 2109–2111, Nov. 1996. [28] Z. Popovic and A. Mortazawi, “Quasi-optical transmit/receive front ends,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 11, pp. 1964–1975, Nov. 1998. [29] R. A. York and Z. Popovic, Active and Quasi-Optical Arrays for SolidState Power Combining. New York: Wiley, 1997. [30] M. B. Perotoni, S. Rondineau, R. Lee, D. Consonni, and Z. Popovic, “X-band discrete lens array for a satellite communication ground station antenna,” in Proc. SBMO/IEEE MTT-S Int. Conf. on Microwave and Optoelectronics, Jul. 25–28, 2005, pp. 197–200. [31] P. Padilla de la Torre, M. Sierra-Castañer, and M. Sierra-Perez, “Design of a double array lens,” in Proc. Eur. Conf. Antennas Propag., Nice, France, Nov. 2006. [32] A. Abbaspour-Tamijani, K. Saranbandi, and G. M. Rebeiz, “A millimeter-wave bandpass filter-lens array,” IET Microw. Antennas Propag., vol. 1, no. 2, pp. 388–395, Apr. 2007. [33] A. Muñoz-Acevedo, P. Padilla de la Torre, and M. Sierra-Castañer, “Ku-band active transmitarray based on microwave phase shifters,” presented at the Eur. Conf. Antennas Propag., Berlin, Germany, Mar. 23–27, 2009. [34] C.-C. Cheng, B. Lakshminarayanan, and A. Abbaspour-Tamijani, “A programmable lens-array antenna with monolithically integrated MEMS switches,” IEEE Trans. Microwaves Theory Tech., vol. 57, no. 8, pp. 1874–1884, Aug. 2009. [35] P. S. Hall, “Application of sequential feeding to wide bandwidth, circularly polarised microstrip patch arrays,” in IEE Proc. Microw., Antennas, and Propag., Oct. 1989, vol. 136, no. 5, pp. 390–398. [36] P. S. Hall, J. S. Dahele, and J. R. James, “Design principles of sequentially fed, wide bandwidth, circularly polarised microstrip antennas,” in IEE Proc. Microw., Antennas, and Propag., Oct. 1989, vol. 136, no. 5, pp. 381–389. [37] H. Kaouach, L. Dussopt, R. Sauleau, and Th. Koleck, “Design and demonstration of 1-bit and 2-bit transmit-arrays at X-band frequencies,” presented at the Eur. Microwave Conf. (EuMC 2009), Roma, Italy, 28 Sep.–1 Oct. 2009. [38] R. H. Phillion and M. Okoniewski, “An array lens for circular polarization with emphasis on aperture efficiency,” presented at the Eur. Conf. Antennas Propag., Barcelona, Spain, Apr. 12–16, 2010. [39] J. Y. Lau and S. V. Hum, “Design and characterization of a 6 6 planar reconfigurable transmitarray,” presented at the Eur. Conf. Antennas Propag., Barcelona, Spain, Apr. 12–16, 2010.

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Hamza Kaouach received the M.S. degree in high frequency communications systems from the University of Paris-Est Marne-la-Vallée, France, in 2006 and the Ph.D. degree in signal processing and telecommunications from the University of Rennes 1, France, in 2009. From 2006 to 2009, he was a Research Assistant at CEA-LETI, Grenoble, France. His research interests include quasi-optic reconfigurable antennas at millimetre-wave frequencies.

Laurent Dussopt (S’00–A’01–M’03–SM’07) received the M.S. and Agrégation degrees in electrical engineering from the Ecole Normale Supérieure de Cachan, France, in 1994 and 1995, the Ph.D. degree in electrical engineering from the University of Nice-Sophia Antipolis, France, in 2000, and the “Habilitation à Diriger des Recherches” degree from the University Joseph Fourier, Grenoble, France, in 2008. From September 2000 to October 2002, he was a Research Fellow with the University of Michigan at Ann Arbor. Since 2003, he is a Research Engineer at CEA-LETI, Grenoble, France. His research interests include reconfigurable antennas, millimetre-wave integrated antennas and antenna arrays, RF-MEMS devices and systems. Dr. Dussopt received the Lavoisier Postdoctoral Fellowship from the French government in 2000 and was a co-recipient of the 2002 Best Student Paper Award (Second Prize) presented at the IEEE Radio Frequency Integrated Circuit (RFIC) Conference. Jérôme Lanteri received the M.S. and Ph.D. degrees in electrical engineering from the University of Nice-Sophia Antipolis, France, in 2004 and 2007, respectively. He was a Postdoctoral Researcher at the University of Nice-Sophia Antipolis, France, from 2007 to 2008, and at CEA-LETI, Grenoble, France, from 2008 to 2010. Since September 2010, he is an Associate Professor at the University of Nice-Sophia Antipolis, France. His research interests include integrated antennas for gigabit wireless communications, reflectarrays and transmitarrays at millimetre-wave frequencies. Thierry Koleck, photograph and biography unavailable at the time of publication. 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, 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, France, in 1999 and 2005, respectively. He was an Assistant Professor and Associate Professor at the University of Rennes 1, between September 2000 and November 2005, and between December 2005 and October 2009. He has been a full Professor in the same University since November 2009. His current research fields are numerical modelling (mainly FDTD), millimeter-wave printed and reconfigurable (MEMS) antennas, lens-based focusing devices, periodic and non-periodic structures (electromagnetic bandgap materials, metamaterials, reflectarrays, and transmitarrays) and biological effects of millimeter waves. He has received four patents and is the author or coauthor of more than 80 journal papers and more than 190 contributions to national and international conferences and workshops. Dr. 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.

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60-GHz Electronically Reconfigurable Large Reflectarray Using Single-Bit Phase Shifters Hirokazu Kamoda, Member, IEEE, Toru Iwasaki, Jun Tsumochi, Takao Kuki, Member, IEEE, and Osamu Hashimoto, Member, IEEE

Abstract—A large electronically reconfigurable reflectarray antenna that has 160 160 reflecting elements was designed, fabricated, and evaluated so that it could be applied to a millimeterwave imaging system operating in the 60-GHz band. To make it feasible to construct such a large reflectarray, the reflecting element structure had to be simple and easily controlled; therefore, a reflecting element consisting of a microstrip patch and a single-bit digital phase shifter using a p-i-n diode was employed. A large reflectarray antenna was fabricated using the reflecting elements. The measured radiation patterns and antenna gain were in good agreement with those that were calculated. Furthermore, the nearfield beam focusing capabilities, which was required to image nearfield objects, were also verified through an experiment. Finally, the response time for beamforming was measured, which was far less than the system requirements. Index Terms—Beamforming, millimeter wave, phase shifter, p-i-n diode, reflectarray.

I. INTRODUCTION N recent years, radars and imaging systems at millimeterwave frequencies have been extensively developed, due to the advances in and availability of millimeter-wave components. Most radars and imaging systems need antennas with large apertures, or narrow beamwidths, and fast beam-scanning capabilities, because narrower beamwidths yield higher angular resolutions, and fast beam-scanning capabilities allow fast updates of the images that are obtained, or high frame frequency. While conventional phased-array antennas have been a promising candidate for such applications, electronically reconfigurable reflectarray antennas have recently evolved into acceptable alternatives [1]. A typical reconfigurable reflectarray antenna consists of an array of reflecting elements with electronically controllable phase shifters and a feed antenna to illuminate the array. The

I

Manuscript received June 10, 2010; revised November 02, 2010; accepted December 28, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. H. Kamoda, J. Tsumochi, and T. Kuki are with Science & Technology Research Laboratories, Japan Broadcasting Corporation, Tokyo 157-8510, Japan (e-mail: [email protected]). T. Iwasaki is with the Engineering Administration Department, Japan Broadcasting Corporation, Tokyo 157-8510, Japan and also with the Department of Electrical and Communication Engineering, School of Engineering, Tohoku University, Sendai-shi 980-8579, Japan. O. Hashimoto is with the Department of Electrical Engineering and Electronics College of Science and Engineering, Aoyama Gakuin University, Sagamihara, Japan. Digital Object Identifier 10.1109/TAP.2011.2152338

illuminated elements reflect the incident field with phase shifts to obtain the required phase front over the array aperture. Because a reflectarray is spatially fed, no complex, high-loss, power dividers/combiners are required, and therefore very little insertion loss is encountered at the feed. This feature is extremely advantageous for millimeter-wave radars and imaging systems that need an array antenna with a large numbers of elements. Various types of reflecting elements for reconfigurable reflectarrays have been reported. Reflecting elements containing liquid crystal, whose dielectic constant can be changed with an applied bias voltage, have been investigated at 77 GHz [2]. A one-dimensional beam scan was demonstrated with an array of 16 16 elements. In [3], an element was formed by a rectangular patch with two halves connected by two varactor diodes. Beam-scanning capabilities have been demonstrated at 5.8 GHz with an array of 10 7 elements. These two cases used analogphase shifters, whereas reflecting elements using digital-phase shifters have also been reported. A 35-GHz reflectarray with three-bit ferrite phase shifters has been developed [4]. The reflectarray is hexagonal and has a 0.64-m diameter containing 3,600 elements. With the use of ten MEMS capacitors, a five-bit digital phase shifter has been formed in a reflecting element [5]. In [6], four p-i-n diodes were used for the reflecting element to achieve a two-bit phase shifter. However, neither [5] nor [6] reported the fabrication of reflectarrays but only the designs and characterizations of the reflecting elements. These digital-phase shifters have advantages in operational stability [5], [7] and compatibility with digital-control circuits. Their main drawback is that there is a trade-off between simplicity in the design of elements and antenna directivity when choosing the number of bits for phase quantization. A digitalphase shifter with a large number of bits can minimize the degradation in antenna directivity; however, it increases the complexity of the structure of reflecting elements [7], which hinders the design of large reflectarrays. A small number of bits also allows the reflecting elements to have a simple structure, whereas the degradation of antenna directivity increases because of the increase in phase-quantization errors, which, for a single bit, is 3.9 dB [8]. We are currently developing a millimeter-wave imaging system [9], where a narrow antenna beam is scanned two-dimensionally to obtain two-dimensional images. We have been focusing on a reflectarray antenna for a narrow beam that can be scanned rapidly to achieve this. A simplified element structure and control circuit are primary concerns in terms of the practical fabrication of such large reflectarrays. Hence, we

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KAMODA et al.: 60-GHz ELECTRONICALLY RECONFIGURABLE LARGE REFLECTARRAY USING SINGLE-BIT PHASE SHIFTERS

Fig. 1. Geometry of reflectarray antenna.

TABLE I DESIGN PARAMETERS FOR REFLECTARRAY ANTENNA

previously designed a reflecting element with a single-bit phase shifter using a p-i-n diode operating in the 60-GHz band, and obtained preliminary results [10]. We fabricated and evaluated a large reflectarray antenna using the reflecting elements we previously designed [10] with the aim of directly applying it to an actual millimeter-wave imaging system. The fabricated reflectarray antenna has as many as 160 160 elements, which to the best of the authors’ knowledge, is the largest electronically reconfigurable reflectarray operating in the 60-GHz band ever discussed in the literature. Furthermore, we demonstrated the control circuitry for numerous elements that need to be independently controlled and measured the response time for beam-forming, which, in practice, plays an important role in the operation of such large reflectarrays. The refectarray antenna design is described in Section II in terms of the requirements of the imaging system. The reflecting element design, its characterization, and the control circuitry are also presented. Section III describes the fabrication of the large reflectarray and discusses the experimental results for the radiation patterns, antenna gain, and the response time for beamforming. Section IV summarizes the paper. II. REFLECTARRAY ANTENNA DESIGN The geometry of the reflectarray antenna is outlined in Fig. 1. A planar square array of reflecting elements forms the reflectarray aperture, the center of which has been placed at the origin of the coordinates. The array aperture has dimensions of along the x- and y-axes. A feed antenna has been placed on the . z-axis at the point of The design parameters were determined by using the specifications of the imaging system, such as angular resolution, field of view, and frame frequency. Table I summarizes the required design parameters.

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To obtain a beamwidth of 0.6 , the required aperture size, , has to be about , where represents the wavelength at the operating frequency, taking into consideration the addition of an illumination taper with an edge level of 10 dB, which yields a relatively low sidelobe level and optimum aperture efficiency in terms of spillover and illumination efficiency [11], [12]. Note that the evaluation of these efficiencies will be discussed in Section III-C. Therefore, was set to 574 mm. A limiting factor in the bandwidth for reflectarray antennas is the operational bandwidth of the reflecting elements. As the required bandwidth of 450 MHz only corresponds to about a 0.75-% fractional bandwidth, it would be easy for a microstrip patch element to cover the bandwidth [11]. The other limiting factor in the bandwidth for reflectarray antennas is differential spatial-phase delay, which arises from the difference in path lengths between the feed antenna and the elements across the aperture [11]. A maximum path-length difference occurs between the feed-to-center-element path and the feed-to-cornerratio is set to unity, element path in this design. If the the maximum path-length difference in this case will be 25.83, at the lowest, center, and highest frequen25.93, and cies. When phase adjustment for beam-forming is performed at the center frequency, the phase error for the corner elements becomes 35 at the lowest and highest frequencies. However, this is the maximum phase error and less than the quantization interval which is 180 assuming single-bit quantization. Therefore, the degradation in gain due to differential spatial-phase delay can be ignored within the required bandwidth. Hence, the ratio was determined to be unity, i.e., . Then, a corrugated scalar horn antenna for the feed antenna that had a 10-dB beamwidth of 52 both on the E- and H-planes was chosen so that the edge level became 10 dB. , The element spacing, , was determined to be less than or 3.5 mm, from the beam-scanning range and the half-subtended angle of the aperture viewed from the feed antenna, so that no grating lobes appeared in the visible region. Then, the aperture size and the element spacing determined the number of reflecting elements, which was 160 160. The response time for beam-forming is a factor determining how fast the antenna can scan the beam, and is defined here as the time elapsed between the moment at which a control command for beam-forming is issued by the host system and the moment at which the beam has actually been formed. A. Reflecting Elements The number of reflecting elements was going to be as many as . As has previously been mentioned, it is very important to have a simplified structure for the reflecting elements and the corresponding control circuit to implement a large aperture. Therefore, a microstrip reflecting element with a single-bit phase shifter was chosen. To make the phase shifter electronically controllable, an off-the-shelf p-i-n diode was used, which generally can be expected to form the beam within the required response time. Therefore, we designed the reflecting elements as follows. The basic model for the elements is outlined in the schematics in Fig. 2. A rectangular microstrip patch is connected to a shortcircuited stub loaded with a p-i-n diode. The p-i-n diode acts as

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Fig. 2. Basic model of reflecting elements [10].

an RF switch, enabling the stub input impedance to be changed in response to the on/off state of the p-i-n diode. The dimensions of the stub and/or the patch are determined such that the reflection phase changes by 180 when the p-i-n diode switches. Thus, a single-bit phase shifter was implemented in the elements. A via was added to the middle of the patch and this was connected to the control circuit to control the p-i-n diode, as shown in Fig. 3. The ring-shaped gap on the ground plane (layer 2) and the circular disc formed on layer 3 worked to choke the RF signals. The dimensions of the entire element structure including the via and the choke structure were adjusted with full-wave analysis, where an infinite array of elements was assumed to take into account the effect of coupling with neighboring elements by setting periodic boundaries for the element unit cell. The unit cell was 3.5 3.5 mm, which was determined from the element spacing that was previously discussed. We chose a substrate with a low-loss tangent for substrate 1 and a rigid substrate for the other substrates to prevent the laminate from warping. The dielectric constant of substrate 1 (Arlon Diclad880) was 2.17 and the loss tangent was 0.0009. Substrates 2 and 3 had a dielectric constant of 4.25 and a loss tangent of 0.021. The thickness of the substrates were 0.254, 0.07, and 0.8 mm for substrates 1, 2, and 3, respectively. Layers 1 to 4 thick. The p-i-n diodes were made of copper that was 18used were a beam-lead type (M/A-COM MA4AGBLP912). The , equivalent circuit constants of the p-i-n diodes were , , which were measured at 60.25 GHz and used in full-wave analysis. The phase and magnitude of the measured and simulated reflection coefficients as functions of the frequency are plotted in Figs. 4(a) and (b). As can be seen, the phase change was 159 at a frequency of 60.25 GHz, which was smaller than that in the simulated results. This was due to the resonant frequency of the fabricated patch shifting toward a higher frequency. The reflection loss was 5.3 dB for the on state and 2.7 dB for the off state. within the 450-MHz The phase change varied by 10 to bandwidth, which is not significant considering single-bit quantization. Note that the deviation in phase change of 20 from the ideal of 180 led to degraded antenna directivity by approximately 0.14 dB on average. Note that an unwanted resonance for the off state was observed at 63.0 GHz in the experiment and also at 62.5 GHz in the

Fig. 3. Layer structure of reflecting element [10]. (a) Elevational view and (b) top view.

Fig. 4. Reflection coefficients of reflecting element [10]. (a) Phase and (b) magnitude.

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Fig. 6. Fabricated reflectarray antenna.

Fig. 5. Control circuitry for reflectarray.

simulation, although there were discrepancies between them. This resonance may be associated with the bias network structure, which was examined in the full-wave simulation, where the ring-shaped gap was removed or filled with the copper ground plane, i.e., the bias network structure was equivalently removed, and the results revealed there was no unwanted resonance in the phase curve. The p-i-n diode used is capable of handling more than 10 dBm of power according to the datasheet. Because the power received by a single reflecting element is obviously much lower than the feed power of 10 dBm, the p-i-n diodes were expected to operate normally. B. Control Circuitry As the reflectarray had 160 160 p-i-n diodes that needed to be independently controlled to form the beam, an equivalent number of control lines would have needed to be wired between the p-i-n diodes and the host system. However, it was too complex and virtually impossible to wire such an enormous number of control lines. Therefore, we employed the active matrix architecture shown in Fig. 5. The control data distributor was implemented in a field programmable gate array in the host system, which sent 160-parallel control data that corresponded to the rows of the p-i-n diodes. The shift register, driven by the control data distributor set the row-select lines high one by one, which enabled the latches to retain the control data. The latch outputs directly drove the p-i-n diodes. Thus, the p-i-n diodes were sequentially controlled row by row. A p-i-n diode control circuit consisting of the shift register and the latches was placed at the back side of the reflectarray, so that the number of wiring configurations from the back of the reflectarray was only 162. Because it was clocked at 16.7 MHz, to send all the control data to the latches; it took about 10 however, this architecture effectively reduced the complexity of the wiring. Each p-i-n diode draws about 10 mA of current in the on state and negligible current in the off state. As approximately half the p-i-n diodes were considered to be in the on-state in operation, the control circuitry was required to supply a total

current of about 130 A at about 1.8 volts, which meant the power dissipated by the p-i-n diodes was roughly 234 watts in total. III. EXPERIMENTAL RESULTS A. Fabrication Fabricating the large reflectarray on one piece of the substrate would have been very difficult considering the yield rate for the p-i-n diode attachment. Therefore, the reflectarray was constructed by creating 16 sub-reflectarrays, each of which had 40 elements on a laminate of 143.5 143.5 mm, and 40 concatenating them in a four-by-four configuration. There is a photograph of the reflectarray antenna we fabricated in Fig. 6. The p-i-n diode control circuit was placed and encased just behind the reflectarray. To transfer the heat generated by the p-i-n diodes into air, each sub-reflectarray had an aluminum heatsink on the back side of the laminate and some fans for ventilation in the casing were installed. The array aperture was covered with a radome, which was made of 3-mm thick polycarbonate, to protect the p-i-n diodes. As can be seen from Fig. 6, there are blank spaces where the elements should have been placed at the joints of the subreflectarrays. This is because the sub-reflectarray laminate needed margins at the sides for manufacturing reasons and these margins became blank spaces. However, these defects in the reflecting elements may safely be ignored, because the number of defects only represents about 3% of the entire number of elements. B. Radiation Patterns First, to test and verify how reconfigurable the reflectarray antenna was, the main beam was formed in various directions and the radiation patterns were measured for each direction of the beam. The control data for the p-i-n diodes were determined as follows. First, from the required beam direction, the required phase compensation, , for each reflecting element ( th element) was calculated by (1) where is the free space wave number, is the position th element, is that of the feed antenna, and vector of the

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Fig. 8. Radiation patterns for beam scanned every five degrees in (a) azimuth and (b) elevation.

Fig. 7. Calculated (left) and measured (right) radiation patterns for different beam angles of (a) (0,0 ), (b) (25, 0 ), (c) (0, 25 ), and (d) (25, 25 ).

is the unit vector of the required beam direction. Then, is converted into a value between 0 and by modulo operation and quantized as

.

(2)

Finally, the two quantized values of are assigned to the p-i-n diode states, ‘on’ and ‘off’. Therefore, the control data are determined and the beam is formed by using the control circuitry. The planar near-field measurements were performed at a distance of 755 mm from the reflectarray aperture, which was just

behind the feed antenna. Then, the near fields that were obtained were transformed into far-field radiation patterns [13]. Fig. 7 plots the measured and calculated two-dimensional radiation patterns for four beam directions at 60.25 GHz. The calculated radiation patterns were obtained using conventional array theory. The main beam angle could be successfully controlled. The sidelobe patterns differed slightly between the measurements and calculations. This was partly due to the reflected fields from the radome and from the ground plane of the reflecting elements, which were not controlled by the reflecting elements and were not taken into account in the calculations. The azimuth and elevation cut patterns are also plotted in Fig. 8, which were measured while the beam was scanned every five degrees in azimuth and elevation. The measured beamwidth ranges from 0.55 to 0.63 , depending on the scanning angle, whereas the calculated one ranges from 0.53 to 0.58 . The measured beamwidths are slightly wider than the calculated ones; however, they approximately meet the requirements. The average sidelobe level is 25 dB in the measurements, which is 5-dB higher on average than that from the calculations. However, the tendencies of the sidelobes are similar. The main beams in both the azimuth and elevation have lower scanned at power than that of the other beams. This is because a grating lobe is just entering the visible region. The radiation patterns previously described were measured focusing the beam at infinity, or the far field. The boundary of [13] is about the far field and the near field given by 132 m away from the reflectarray. Therefore, in practice, the

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Fig. 10. Radiation pattern in x-y plane at z = 3 m when beam is focused at z = 3 m.

Fig. 9. Radiation patterns in x-z plane at y = 0 when beam is focused at near field. (a) z = 1:5 m, (b) z = 3 m, and (c) z = 5 m.

reflectarray antenna for the imaging system must focus the beam at the near field to take images of objects closer than 132 m. Here, we actually tried to focus the beam in the near field, where the required phase compensation for each reflecting element (1) must be substituted by (3) is the position vector of the required focal point. where Near-field measurements were done in the same way as those for the far-field radiation patterns. Then, the measured fields were transformed to the fields at near-field distances of interest by using a Fourier-optics based algorithm [14], which converted the measured fields to an angular spectrum through a Fourier transform, and it then added phase shifts to the fields during propagation from the measured plane to the plane of interest, and it finally converted them back to spatial domain fields through an inverse-Fourier transform. Fig. 9 shows the near-field radiation patterns in the x-z plane obtained with the previously mentioned calculation from the measured fields, focusing at distances z of 1.5, 3, and 5 m. Aliasing due to the Fourier and inverse-Fourier transform conducted on the limited size of calculation planes can be seen at and greater; however, this is sufficient to observe the

features of the main beam. As can be seen, the beamwidth becomes smallest at those respective distances that correspond to about 0.6 of the angular beamwidth. Furthermore, the nearwhen the field radiation pattern in the x-y plane at beam was focused at is shown in Fig. 10. The sidelobe level is similar to that of the beam pattern focused at infinity, as shown in Fig. 7(a), which means constant image quality can be obtained for the imaging system, regardless of the distances that were focused at. Then, cross-polarization discriminations (XPDs) for the , (0 , ), beam angles of ) while focusing the beam at a dis(20 , 0 ), and (20 , tance of 3 m were also measured at their respective focused beam positions using a standard gain horn antenna. The XPDs at the first two positions were more than 35 dB and those at the other two positions were 13 dB; the XPDs for the beam scanned at the azimuth angle of 0 were excellent, while those at the azimuth angle of 20 were degraded. However, because the imaging system does not require polarimetric analysis, this is not of any concern. Thus, we could confirm the fabricated reflectarray was reconfigurable. C. Antenna Gain Here, we discuss the antenna gain of the reflectarray antenna. The formula for antenna gain is well known and given by

(4) where represents the aperture area. The is the aperture efficiency, which for a reflectarray antenna, is essentially decomposed into the five factors of

(5) where , , , , and correspond to efficiency based on phase errors, the return loss of the reflecting elements, spillover loss, illumination distribution across the aperture, and radome loss.

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Fig. 11. Experimental set up for measuring response time.

As we discussed earlier, is approximately 41% is determined by the ( 3.9 dB) for single-bit quantization. average return loss of the reflecting elements when the p-i-n diodes are on and off, assuming the number of “on” and “off” elements is nearly equal, which is then 40% ( 4.0 dB) from and the measured results at 60.25 GHz shown in Fig. 4(b). are determined by the feed antenna’s radiation pattern and the geometry of the reflectarray aperture, which are common and are numerically factors for reflector antennas [12]. calculated to be 88% for the former and 84% for the latter. is estimated to be approximately 79% ( 1 dB). Hence, the overall aperture efficiency, , is calculated to be 9.5%. Consequently, the antenna gain is estimated to be 42.0 dBi. The antenna gain was then measured at 60.25 GHz with a method of substitution using a standard gain-horn antenna. As a result, when a beam was formed in the broadside direction, 41 dBi was obtained, which approximately matched the estimate. Therefore, successful beam-forming was also demonstrated in terms of this evaluation of antenna gain.

D. Response Time for Beamforming The response time for beamforming was measured using the experimental set up outlined in Fig. 11. The beam direction was switched from one position to another. A square-law detector with a horn antenna was placed at one of the positions, and the received power variations over time were observed using an oscilloscope. As the elapsed time was measured from when a beam control command was issued by the host system, the response time included the distribution of control data by the control circuitry. Fig. 12 plots the received power variations as a function of time. Note that the absolute power is not important here; hence, the raw voltage from detector output was taken to be the measure of received power. The received power started to significantly vary at around 5 , reached 90% variation at 13 , and finally achieved 100% variation at 28 . The same measurements were repeated for the other pairs of beam directions and almost the same results were observed. Therefore, the response time for beam-forming was considered to be 28 , which is far less than the required response time of 200 .

Fig. 12. Measured received power variations over time when beam is switched between two directions.

IV. CONCLUSION We successfully demonstrated a large electronically reconfigurable reflectarray antenna that had 160 160 reflecting elements that were applied to a millimeter-wave imaging system operating in the 60-GHz band. A single-bit digital phase shifter was employed for the reflecting elements to make it feasible to construct such a large reflectarray. The reflecting elements were formed with a microstrip patch and a stub loaded with a p-i-n diode that switched the reflection phase. The measured radiation patterns for different beam directions and antenna gain were in good agreement with those that were calculated, which confirmed the capabilities of the antenna for reconfiguration. Furthermore, the capabilities for near-field beam focusing were also verified through an experiment in which images of near-field objects were taken. Finally, the response time for beam-forming was measured, which was far less than that in the system requirements. Large DC power dissipation due to the current required to bias p-i-n diodes was a drawback in using p-i-n diodes for such a large reflectarray. Some field-effect-transistor switches with a smaller footprint that could have fitted the limited space available for the reflecting element would have been preferable to p-i-n diodes in terms of power dissipation, but these were not commercially available at the time of design. ACKNOWLEDGMENT The authors would like to thank Dr. T. Derham who provided them with helpful comments and suggestions. Special thanks also go to Prof. K. Ito from Chiba University, Dr. K. Shogen, and their colleagues who greatly encouraged them and offered invaluable advice. REFERENCES [1] J. Huang and J. A. Enchinar, Reflectarray Antennas, M. E. El-Hawary, Ed. Hoboken, NJ: Wiley-IEEE Press, 2008. [2] R. Marin, A. Moessinger, F. Goelden, S. Mueller, and R. Jakoby, “77 GHz reconfigurable reflectarray with nematic liquid crystal,” presented at the Proc. Eur. Conf. Antennas Propag., Edinburgh, U.K., Oct. 2007. [3] S. V. Hum, M. Okoniewski, and R. J. Davies, “Modeling and design of electronically tunable reflectarrays,” IEEE Trans. Antenna Propag., vol. 55, no. 8, pp. 2200–2210, Aug. 2007.

KAMODA et al.: 60-GHz ELECTRONICALLY RECONFIGURABLE LARGE REFLECTARRAY USING SINGLE-BIT PHASE SHIFTERS

[4] A. A. Tolkachev, V. V. Denisenko, A. V. Shishlov, and A. G. Shubov, “High-gain antenna system for millimeter-wave radars with combined electrical and mechanical beam steering,” in Proc. IEEE Symp. on Phased Array System and Technology, Boston, MA, Oct. 1996, pp. 266–271. [5] 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. [6] M. Barba, E. Carrasco, J. E. Page, and J. A. Encinar, “Electronic controllable reflectarray elements in X band,” Frequenz, vol. 61, no. 9–10, pp. 203–206, 2007. [7] B. Wu, A. Sutinjo, M. E. Potter, and M. Okoniewski, “On the selection of the number of bits to control a dynamic digital MEMS reflectarray,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 183–186, 2008. [8] B. D. Steinberg, Principles of Aperture and Array System Design. New York: Wiley, 1976. [9] T. Derham, H. Kamoda, and T. Kuki, “Frequency-encoding technique for active MMW imaging,” in Proc. Asia-Pacific Microwave Conf., Yokohama, Japan, Dec. 2006, pp. 1833–1836. [10] H. Kamoda, T. Iwasaki, J. Tsumochi, and T. Kuki, “60-GHz electrically reconfigurable reflectarray using p-i-n diode,” in Proc. IEEE MTT-S Int. Microwave Symp. Dig., Boston, MA, Jun. 2009, pp. 1177–1180. [11] J. Huang, “Analysis of a Microstrip Reflectarray Antenna for Microspacecraft Applications,” JPL TDA Progress Report 42-120, Feb. 1995, pp. 153–173. [12] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design. New York: Wiley, 1981. [13] D. Slater, Near-Field Antenna Measurements. Norwood, MA: Artech House, 1991. [14] J. W. Goodman, Introduction to Fourier Optics Third Edition. Greenwood Village, CO: Roberts and Company, 2005. Hirokazu Kamoda (M’97) received the B.E. and M.E. degrees from the Tokyo Institute of Technology, Tokyo, Japan, in 1995 and 1997, respectively. In 1997, he joined the Japan Broadcasting Corporation (NHK) and worked at the Nagano Broadcasting Station. Since 2001, he has been with the Science & Technical Research Laboratories, NHK, where he has been engaged in research and development on millimeter-wave functional devices, wireless HDTV transmission system, and millimeter-wave imaging. He is now a Principal Research Engineer in the Science & Technology Research Laboratories, NHK. Mr. Kamoda is a member of IEICE and ITEJ. He received the Young Engineer Award from IEICE in 2002.

Toru Iwasaki received the B.E. and M.E. degrees in electrical communication engineering from Tohoku University, Miyagi, Japan, in 1990 and 1992, respectively. In 1992, he joined the Japan Broadcasting Corporation (NHK) and worked at the Nagoya Broadcasting Station. Since 1995, he has been with NHK’s Science & Technical Research Laboratories, where he has been engaged in research and development work on active slot-array antennas, millimeter-wave antennas for application to broadcasting systems, and metamaterials for antenna applications. In 2009, he was transferred to NHK’s Engineering Administration Department. Since then, he has been engaged in work on the introduction of digital terrestrial broadcasting relay stations. He is now a Senior Engineer in NHK’s Engineering Administration Department. Mr. Iwasaki is a member of ITEJ.

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Jun Tsumochi received the B.E. and M.E. degrees from Toyohashi University of Technology, Aichi, Japan, in 2002 and 2004, respectively. In 2004, he joined the Japan Broadcasting Corporation (NHK) and worked at the Osaka Broadcasting Station. Since 2008, he has been with the Science & Technology Research Laboratories of NHK, where he has been engaged in research and development on millimeter-wave imaging system.

Takao Kuki (M’04) was born in Tochigi, Japan, in 1961. He received the B.E. and M.E. degrees from the University of Electro-Communications, Tokyo, Japan, in 1983 and 1985, respectively, and the D.Eng. degree from the Tokyo Institute of Technology, in 2004. In 1985, he joined the Japan Broadcasting Corporation (NHK) and worked at the Asahikawa Broadcasting Station. Since 1988, he has been with the Science & Technical Research Laboratories, NHK, where he has been engaged in research and development work on electroluminescent, magnetostatic-wave, and microwave-functional devices. He has recently developed an interest in millimeter-wave application systems. Dr. Kuki is a member of IEICE, IEEJ, and ITEJ.

Osamu Hashimoto (M’86) received the B.E. and M.E. degrees in applied electronics engineering from the University of Electro-Communications, Tokyo, Japan, in 1976 and 1978, respectively, and the D.Eng. degree from the Tokyo Institute of Technology, in 1986. In 1978, he joined Toshiba Corporation. In 1981, he joined the Defense Technical Development Laboratory. In 1991, he moved to the Department of Electrical Engineering and Electronics, Aoyama Gakuin University, Japan, as an Associate Professor, where he is currently a Full Professor. From 1994 to 1995, he was with the University of Illinois as a Guest Researcher. He has been engaged in research on electromagnetic computation, microwave and millimeter-wave absorbers, planar filters, and measurement and analysis of radar cross-sections. He is the author/ coauthor of 28 books in Japanese including Introduction to Microwave Absorbers, Introduction to the Finite-Difference Time Domain Method, and Technologies and Applications of Wave Absorber II. He has had more than 500 papers published in reviewed journals and international conferences. Prof. Hashimoto is a member of the IEICE, IEEJ, JIEP, and AIJ. He received the 2006 Electronics Society Award from the Institute of Electronics, Information, and Communication Engineers and was awarded the 2006 Prize at the Asia Pacific Microwave Conference (APMC).

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Relation Between the Array Pattern Approach in Terms of Coupling Coefficients and Minimum Scattering Antennas Jesús Rubio and Juan F. Izquierdo

Abstract—The well-known approach for the calculation of the radiation pattern of a finite array from the scattering parameters between the array feeding ports (coupling coefficients) is connected to the theory of minimum-scattering antennas (MS antennas) by making use of spherical mode expansions. It is shown how this approach for array radiation patterns involves an MS Antenna approximation in the first order of reflections (mutual coupling) between the elements in the array. For this purpose the generalized scattering matrix (GSM) formalism for a finite array in terms of spherical modes is used. Finally, the MS antenna theory is applied to estimate the radiation pattern of an array of waveguide-fed apertures. Index Terms—Finite array, generalized scattering matrix, minimum-scattering antennas, mutual coupling, spherical mode expansion.

array feeding ports and the isolated element patterns, can be understood from the perspective of MS antennas. A relationship between the aforementioned array pattern approach and MS antennas will be found by means of the generalized-scattering-matrix (GSM) formulation for a finite array in terms of spherical modes [14], [15]. The GSM formulation of a finite array and the MS antenna theory are summarized in Sections II and III. The array pattern approach in terms of coupling coefficients for voltage- and current-driven elements will be connected with short- and open-circuited MS antennas respectively. This is shown in Section IV. Finally, the proposed theory will be used to calculate the radiation pattern of an array of waveguide-fed apertures in Section V. II. GSM FORMULATION OF A FINITE ARRAY

I. INTRODUCTION

I

T has been shown in previous works that the radiated field of a finite array can be expressed in terms of the scattering matrix of coupling coefficients and the isolated element patterns [1]–[5]. Thus, this relationship has been used by Steyskal and Herd for mutual coupling compensation in small arrays of waveguide-fed apertures [1]. It has also been used by Josefsson and Persson for conformal array synthesis [4], and by Pozar, to find a relation between the active input impedance and the active element pattern of a phased array [5]. On the other hand, the theory of minimum-scattering antennas (MS antennas) was introduced by Dicke more than 60 years ago [6] and considerably extended later [7]–[11]. The main property of MS antennas is that they do not scatter at all when their ports are terminated in a particular set of reactive loads. When these reactive loads are open circuits they are referred to as Canonical MS Antennas and it has been recently proved that they provide a good approximation for certain types of antenna such as blades, dipoles or probe-fed patch antennas [12], [13]. The purpose of this work is to show that the array pattern approach, given in terms of the scattering parameters between the

Manuscript received June 21, 2010; revised October 27, 2010; accepted December 10, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported by MICINN, Spain, under project TEC201020249-C02-02. The authors are with the Departamento de Tecnología de Computadores y de las Comunicaciones, Escuela Politécnica de Cáceres, Universidad de Extremadura, s/n 10071 Cáceres, Spain (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2152342

Consider a finite array of radiating elements with internally uncoupled excitation ports fed by matched generators, where the radiation of each element can be described in terms of spherical modes. The GSM of each isolated antenna of this array can be expressed as follows [16]

(1) where , , and are column vectors and their elements are respectively the complex amplitudes of incident and reflected modes on the feeding ports, and incoming and scattered spherical modes on the radiation port. , , , and are respectively the antenna reflection, reception, transmission and scattering matrices as defined in the classic theory [6] and denotes the identity matrix. Assuming no incoming field towards it is possible to calculate the far-field patthe antenna tern by means of the following expression: (2) [16], where is a row vector of far-field pattern functions and is the number of these functions required to fit the radiation pattern. In order to take into account the mutual coupling effects between the elements in the array, the radiation ports of every element must be interconnected. For this purpose, every radiation port must be connected with the other radiation ports while it must also be an external port to receive the direct field coming from outside the array. With this aim, the incident field on each radiation port will be expanded in terms of incoming spherical

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modes with complex amplitudes given by the column vectors . This incident field can be calculated as the contributions of the scattered field coming from the other radiating elements in the array and the direct field from outside of the array (3)

where and are column vectors with the complex amplitudes of the spherical mode expansion of, respectively, the direct field from outside the array and the field scattered by each element translated to the position of element . The interconnection between radiation ports is achieved by relating incoming modes in element coming from element , given by , with through the genscattered modes from element , given by , which is obtained by using rotation eral translation matrix and translation of spherical waves [16] (4) By introducing (4) in (3), substituting the result in (1) for every radiating element and following the mathematical procedure explained in [14], it is possible to obtain the overall GSM of the finite array, defined as (5) ,

according to Fig. 1, where

,

and

are given by

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

Fig. 1. Graphical definition of the overall GSM of a finite array.

is given by (6)

and the submatrices , , , and represent the finite array reflection, reception, transmission and scattering matrices when the mutual coupling is taken into account. These submatrices are given by

where is the identity matrix. In this case , are diagonal block-matrices

,

.. . .. .

..

.

..

.

..

.

..

.

..

.

.. . .. .

(9)

Specifically, is the scattering matrix between the array feeding ports that provides the reflection and coupling parameters directly. As proved in [14], the transmission matrix can otherwise be expressed as

(7)

(10)

(8)

which is more suitable for the aim of this work since the first order of reflections (mutual coupling) between the elements has been extracted from the inverse operator, which provides every one of the different orders of reflections between the elements. The radiation pattern of the array can be obtained analytically as

and

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follows by assuming no incoming field towards the array in (5)

be expressed as a function of their transmission, reception and reflection properties

(11) where

is the following row vector: (12)

with being the wave number vector and vector.

(16) It should be noted that this expression fits with the expression given in [7] for Canonical (open-circuited) MS antennas by . making The electric short dipole is a member of this type of antenna.

the element location IV. DERIVATION OF RELATIONS

III. MINIMUM SCATTERING ANTENNAS

A. Array of Voltage-Driven Elements

MS antennas have been dealt with widely in many previous papers [6]–[11]. As shown in [7], these antennas do not scatter at all when the local ports are terminated in a particular set of reactive loads. Although MS antennas were initially required to be matched, in [11] Rogers extended the application of the MS antenna theory to mismatched antennas. In this section two particular cases are shown: short-circuited MS antennas and open-circuited MS antennas. A. Short-Circuited MS Antennas The scattering response of a short-circuited antenna can be obtained from (1) by imposing the short-circuit condition in the terminal

First, consider an isolated voltage-driven element, i.e., an antenna which produces a far-field pattern proportional to the terminal voltage [2], [5]. For this element the far-field pattern can be expressed as

(17) with being a row vector of far-field element patterns, where each pattern is generated by the terminal voltage of each feeding is a column vector mode in the element excitation port, and of the corresponding terminal voltages, given by

(18) (13) where is the identity matrix. Since the short-circuited MS antennas do not scatter at all under the short-circuit condition , we get

Assuming no incoming field from outside the isolated element in (1), (18) can be expressed as

(19) (14) Therefore, the scattering response of short-circuited MS antennas can be expressed as a function of their transmission, reception and reflection properties. An example of this type of antenna is the magnetic short dipole. B. Open-Circuited MS Antennas The scattering response of an open-circuited antenna can be obtained identically from (1) by imposing the open-circuit conin the terminal dition

By substituting (19) into (17) and comparing with (2), be given in terms of far-field pattern functions as

can

(20) On the other hand, the relationship between the far-field pattern of a finite array and the scattering coefficients between feeding ports has been provided for voltage-driven elements in several works [1]–[5]. This relationship can be expressed in matrix form as

(21) (15) Since they do not scatter at all when they are open-circuited, the scattering response of open-circuited MS antennas can also

where

is the identity matrix and is defined as

(22)

RUBIO AND IZQUIERDO: RELATION BETWEEN THE ARRAY PATTERN APPROACH IN TERMS OF COUPLING COEFFICIENTS

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Equation (21) can be expressed in terms of far-field pattern functions, by making use of (20), (8) and (12), as follows

can

By introducing (30) into (28) and comparing with (2), be expressed in terms of far-field pattern functions as

(23)

(31)

Now, if the radiation of each isolated element can be degiven by scribed in terms of spherical modes, the value of (7) can be introduced in (23) to obtain

For current-driven elements the relationship between the farfield pattern of a finite array and the scattering coefficients between feeding ports in matrix form is given as

(24)

(32)

which can otherwise be written as

with defined as in (22). Equation (32) can be expressed in terms of far-field pattern functions as follows by using (31), (8) and (12)

(25) By comparing with (10) and (11) we find that the following approximation is assumed for the first order of reflections between the elements in the array:

(26) Since these matrices are diagonal-block matrices, it means that for each element

(27) which is the expression given for short-circuited MS antennas in (14). Therefore, a short-circuited MS antenna approximation is used implicitly for the first order of mutual coupling between voltage-driven elements when the radiation pattern of the array is given in terms of coupling coefficients and the isolated element patterns. As a consequence, the array pattern approach in terms of coupling coefficients will be exact if the voltage-driven elements of the array are short-circuited MS antennas.

(33) By introducing the value of given by (7) in (33), supposing that the radiation of each isolated element can be described in terms of spherical modes, and after a few operations, it becomes

(34) Thus, by comparing with (10) and (11), we find that the following approximation is assumed implicitly for the first order of reflections between the elements in the array

(35) For each element it means that

(36) B. Array of Current-Driven Elements For this kind of element the far-field pattern can be expressed as:

(28) with being a row vector of far-field element patterns, where each pattern is generated by the terminal current of each feeding mode in the element excitation port, and is a column vector of the corresponding terminal currents given as

(29) If we assume no incoming field from outside the isolated element in (1), (29) can be expressed as:

(30)

which is the expression given for open-circuited MS antennas in (16). In this case, an open-circuited MS antenna approximation is used implicitly for the first order of mutual coupling between current-driven elements to approach the radiation pattern of the array in terms of coupling coefficients. Therefore, the array pattern approach in terms of coupling coefficients will be exact if the current-driven elements of the array are open-circuited MS antennas. V. APPLICATION TO WAVEGUIDE-FED APERTURES The array pattern approach in terms of coupling coefficients has been previously used to estimate the radiation pattern of arrays of waveguide-fed apertures on a ground plane since their elements are voltage-driven [1], [3]. As shown in Section IV, this implies a short-circuited MS antenna approximation only for the first order of reflections. In this section, we will use the short-circuited MS antenna approach for all orders of reflections, so that the coupling coefficients are not required.

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Fig. 2. Active element directive gain pattern of the central element in a 7 array of circular apertures. E-plane configuration.

27

Fig. 3. Active element directive gain pattern of the central element in a 7 array of circular apertures. H-plane configuration.

27

Therefore, by using a spherical mode expansion on a hemispherical radiation port surrounding each element, the far-field pattern of the array will be calculated from (11), (7) and (26) as (37) As in [1] and [3], first only the propagating modes will be used to feed the apertures since the far-field patterns of isolated elements are usually known only for these modes. As explained in [16], can be obtained directly from . On the other hand, can be calculated from (20) if the reflection coefficient and the far-field for the isolated elements are known. For this purpose the coefficients of can be determined using the orthogonal properties of far-field functions [16]. Thus, in order to approach the array pattern by means of (37), it suffices to know the reflection coefficients and the far-field patterns of the isolated elements. As an advantage, this approach does not require the a priori knowledge of the scattering parameters between the feeding ports in contrast to the coupling coefficient-based approach. As an example, a 7 7 array of empty circular apertures on an infinite perfect metallic plane, with radius 1.905 cm and a center-to-center separation of 4 cm on a square lattice, is studied at 5 GHz. Both the coupling coefficient-based approach and the MS antenna approach are compared with the full-wave response obtained by applying two different methods: the GSM formulation of a Finite Array as proposed in [14], and a purely numerical 3-D Finite Element Method (3-D FEM) for the whole array. Figs. 2 and 3 show the active element directive gain pattern of the co-polarized components for the central element in E and H-planes, respectively. A good agreement between all the responses is found bearing in mind that only the propagating modes have been used to feed the apertures in both approaches. The response of the isolated element is also included in the figures to notice the mutual coupling effects in the array pattern. A very accurate solution can be achieved by increasing the number of the excitation modes in the computation of (37). A convergence test has been carried out to show that the scattering matrix of the aperture obtained from the minimum-scattering

Fig. 4. The RMS Error in the approximation of the scattering matrix of a circular aperture by using the MS antenna approach as a function of the number of excitation modes ordered by their cut-off frequency.

antenna approach by using (27) converges to the full-wave result, which is taken as a reference solution. For this purpose, the RMS Error function, defined as

(38) is shown in Fig. 4 as a function of the number of excitation modes, ordered by their cut-off frequency, which are used in the aperture to approximate its scattering matrix. In (38) is the scattering matrix obtained from the MS is the scattering matrix antenna approach and obtained numerically by using the 3-D finite element method given in [14]. We have observed that for a number of excitation the active element modes greater than 7 directive gain pattern obtained by means of the MS antenna approach practically coincides with the full-wave results.

RUBIO AND IZQUIERDO: RELATION BETWEEN THE ARRAY PATTERN APPROACH IN TERMS OF COUPLING COEFFICIENTS

VI. CONCLUSION In this work, the array pattern approach in terms of coupling coefficients between feeding ports has been explained from the viewpoint of the MS antenna theory. It has been proved that the array pattern approach in terms of coupling coefficients involves an MS antenna approach for the first order of reflections (mutual coupling) between the elements in the array. Therefore, in those cases where the array elements can be approximated as shortor open-circuited MS Antennas, the array pattern approach in terms of coupling coefficients provides good results. Moreover, if a MS antenna approach is used for all orders of reflections, the array pattern can be estimated from the reflection and transmission properties of isolated elements.

ACKNOWLEDGMENT The authors would like to thank Prof. J. Zapata for his helpful comments.

REFERENCES [1] H. Steyskal and J. Herd, “Mutual coupling compensation in small array antennas,” IEEE Trans. Antennas Propag., vol. 38, pp. 1971–1975, Dec. 1990. [2] D. M. Pozar, “The active element pattern,” IEEE Trans. Antennas Propag., vol. 42, pp. 1176–1178, Aug. 1994. [3] R. J. Mailloux, Phased Array Antenna Handbook. Norwood, MA, Artech House: , 1994. [4] L. Josefsson and P. Persson, “Conformal array synthesis including mutual coupling,” Electron. Lett., vol. 35, no. 8, pp. 625–627, Apr. 1999. [5] D. M. Pozar, “A relation between the active input impedance and the active element pattern of a phased array,” IEEE Trans. Antennas Propag., vol. 51, pp. 2486–2489, Sep. 2003. [6] C. G. Montgomery, R. H. Dicke, and E. M. Purcell, “Principles of microwave circuits,” in Radiation Laboratory Series. New York: McGraw-Hill, 1948. [7] W. K. Kahn and H. Kurss, “Minimum-scattering antennas,” IEEE Trans. Antennas Propag., vol. AP-13, pp. 671–675, Sep. 1965. [8] W. Wasylkiwskyj and W. K. Kahn, “Scattering properties and mutual coupling of antennas with prescribed radiation pattern,” IEEE Trans. Antennas Propag., vol. AP-18, pp. 741–752, Nov. 1970. [9] W. Wasylkiwskyj and W. K. Kahn, “Mutual coupling and element efficiency for infinite linear arrays,” Proc. IEEE, vol. 56, pp. 1901–1907, Nov. 1968.

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[10] W. Wasylkiwskyj and W. K. Kahn, “Theory of mutual coupling among minimum-scattering antennas,” IEEE Trans. Antennas Propag., vol. AP-18, pp. 204–216, Mar. 1970. [11] P. G. Rogers, “Application of the minimum scattering antenna theory to mismatched antennas,” IEEE Trans. Antennas Propag., vol. 34, pp. 1223–1228, Oct. 1986. [12] O. E. Allen and W. Wasylkiwskyj, “Comparison of mutual impedance of blade antennas with predictions based on minimum-scattering antenna theory,” IEEE Trans. Electromagn. Compat., vol. 42, no. 4, pp. 326–329, Nov. 2000. [13] H. A. Abdallah and W. Wasylkiwskyj, “A numerical technique for calculating mutual impedance and element patterns of antenna arrays based on the characteristics of an isolated element,” IEEE Trans. Electromagn. Compat., vol. 53, no. 10, pp. 3293–3299, Oct. 2005. [14] J. Rubio, M. A. González, and J. Zapata, “Generalized-scattering-matrix analysis of a class of finite arrays of coupled antennas by using 3-D FEM and spherical mode expansion,” IEEE Trans. Antennas Propag., vol. 53, pp. 1133–1144, Mar. 2005. [15] J. Rubio, J. Córcoles, and M. A. González, “Inclusion of the feeding network effects in the generalized-scattering-matrix formulation of a finite array,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 819–822, 2009. [16] Spherical Near-Field Antenna Measurements, J. E. Hansen, Ed. London, U.K.: Peter Peregrinus Ltd., 1988. Jesús Rubio was born in Talavera de la Reina, Toledo, Spain, in 1971. He received the Ingeniero de Telecomunicación degree in 1995 and the Ph.D. degree in 1998, both from the Universidad Politécnica de Madrid, Spain. Since 1994, he has collaborated with the Departamento de Electromagnetismo y Teoría de Circuitos of the Universidad Politécnica de Madrid. At present, he is working with the Departamento de Tecnología de los Computadores y de las Comunicaciones, Universidad de Extremadura, Cáceres, Spain, as a Professor. His current research interests are in the application of the finite element method and modal analysis to antennas and passive microwave circuits problems.

Juan F. Izquierdo was born in Quintana de la Serena, Badajoz, Spain. He received the Ingeniero de Telecomunicación degree from the Universidad de Sevilla, Spain, in 2002. He is currently working towards the Ph.D. degree at the Universidad de Extremadura, Cáceres, Spain. Since 2007, he has been with the Departamento de Tecnología de los Computadores y de las Comunicaciones, Universidad de Extremadura, as an Assistant Professor. His research interests include the application of numerical and analytical techniques to antennas and microwave passive circuits.

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Harmonic Beamforming in Time-Modulated Linear Arrays Lorenzo Poli, Graduate Student Member, IEEE, Paolo Rocca, Member, IEEE, Giacomo Oliveri, Member, IEEE, and Andrea Massa, Member, IEEE

Abstract—In this paper, the synthesis of simultaneous multibeams through time-modulated linear arrays is studied. Unlike classical phased arrays where the antenna aperture is usually shared to generate multiple beams, the periodic on-off sequences controlling the static excitations are properly defined by means of an optimization strategy based on the Particle Swarm algorithm to afford desired multiple patterns at harmonic frequencies to make practical application of these harmonic beams which are typically regarded as an undesirable effect in time-modulated arrays. The synthesis of simultaneous broadside sum and difference patterns, flat-top and narrow beam patterns, and steered multibeams is enabled as assessed by a set of selected results reported and discussed to show the potentialities of the proposed method. Comparisons with previously published results are reported, as well. Index Terms—Harmonic beams, particle swarm optimization, time-modulated linear arrays.

I. INTRODUCTION ULTIBEAM antennas are radiating systems devoted to generate multiple patterns from the same aperture. Originally, they were used for surveillance and tracking in radar systems [1], but multiple beam antennas are recently also playing a key role in communication systems installed on satellites and ground stations [2]. As for radars, widebeam patterns are required for search and detection functions, while narrow beams with high angular resolution capabilities are needed for tracking purposes. On the other hand, the advantages of using multiple beams for communication purposes lie in the possibility to manage more communications/services from different spatial directions and to operate over multiple frequency bands. Moreover, patterns with different shapes can be used in adaptive systems to maintain reliable wireless links in the presence of jammers or interference signals or to properly address different requests of service. Two different solutions have been principally considered in designing multibeam antennas. The former considers reflector antennas, either parabolic structures or reflectarrays, where the

M

Manuscript received January 31, 2010; revised October 29, 2010; accepted November 26, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work supported in part by the Italian National Project: Wireless multiplatform MIMO Active access networks for QoS-demanding multimedia delivery (WORLD), under Grant 2007R989S. The authors are with the Department of Information Engineering and Computer Science, University of Trento, Povo 38050 Trento, Italy (e-mail: [email protected]; [email protected]; giacomo.oliveri@disi. unitn.it; [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.2011.2152323

primary source is composed by multiple feeds whose output signals can be shared between adjacent patterns to reduce the costs and the complexity of the feeding structure [1]. The other exploits antenna arrays and, nowadays, is the preferred solution since it allows a direct control of the illumination on the aperture and an electronic beam steering. Moreover, array architectures generally have a lighter weight and they can be made conformal. Dealing with arrays, many strategies have been proposed to synthesize switchable [3] and reconfigurable arrays [4]–[6]. The excitations of switchable arrays are selected among a list of predetermined configurations, whereas the excitations are adaptively controlled (usually only the phase weights to simplify the hardware implementation) in reconfigurable arrays. Since the whole set of array elements is optimized to generate each pattern, high radiation performances can be yielded even though only one beam can be generated at each time instant. To avoid this drawback, other solutions have been proposed to simultaneously radiate multiple patterns. Sub-arrayed antennas as well as interleaved arrays have proved to work properly in both radar [7]–[10] and biomedical [11], [12] applications. Another approach for the simultaneous generation of multiple-patterns is based on time-modulation (TM). Although its theoretical formulation [13] and the first practical implementations to obtain average ultra-low sidelobe patterns [14] date back to the 1960s, there is nowadays a renewed and growing interest towards such a solution as confirmed by the number of contributions in the reference literature on antennas as well as the applications where timed arrays have been successfully applied. The main limitation of these systems is related to the presence of the sideband radiation (SR) due to the periodic on-off commutations of the RF switches of the beam forming network. However, optimization strategies based on evolutionary algorithms (e.g., differential evolution [15], simulated annealing [16], genetic algorithms [17], and particle swarm optimizer [18]) have recently provided efficient switching schemes to properly address such an issue. It is also worth noting that the theory of TM arrays has been recently revised and formalized in a rigorous mathematical fashion [19], [20], [21] in order to derive analytical relationships for exactly quantifying the power dispersions associated to the SR. For completeness, let us consider that further studies have dealt with the evaluation of the directivity and gain [22] and their optimization [23]. As for the applications, time-modulated arrays have been employed to generate ultra-low sidelobe patterns for the suppression of interferences and clutters [14] and shaped beams [24] with reduced dynamic range ratios. The synthesis of pulse doppler radars [25] and compromise patterns for monopulse radars [26] has been also analyzed, but experimental prototypes have been implemented [17], [27] in a

0018-926X/$26.00 © 2011 IEEE

POLI et al.: HARMONIC BEAMFORMING IN TIME-MODULATED LINEAR ARRAYS

few cases and further researches are expected in such a framework. In these works, the harmonic beams are power losses to be minimized, while other approaches consider a different point of view by exploiting the generation of simultaneous patterns at different sidebands. More specifically, patterns with the same shape generated at different harmonics and pointing at different angular directions have been used in radar systems to enable the electronic beam scanning by means of a progressive on-off switching of the array elements [28]. The same concept has been recently revisited in [29] and extended to the estimation of the directions of arrival [30]. Accurate radar localization and tracking capabilities have been assessed in [31] by synthesizing a two-element TM array providing a monopulse-like behavior where two beams (one with a peak and another with a null along the broadside direction) have been generated at the central and first harmonic, respectively. In this paper, an optimization-based technique for the synthesis of TM linear arrays able to generate multiple beams at different sidebands with different shapes and features is presented. Such a strategy can be used for multi-function radars devoted to surveillance and security purposes [28], [31] as well as for selectively receiving different signals at different sidebands. Moreover, the diversity of different beam patterns with various shapes can be of interest in mobile communications for designing innovative base stations as well as in multiple-input multiple-output (MIMO) systems. The harmonic beamforming is obtained by determining the periodic on-off time sequence that modulates the static excitations of the array by considering an iterative optimization evolving according to the particle swarm (PS) strategy [32]. For a numerical assessment of the proposed approach, simultaneous broadside sum and difference patterns, flat-top and narrow beams, as well as steered multibeams are synthesized. The paper is organized as follows. The problem is mathematically formulated in Section II where the synthesis procedure is described, as well. Selected results from a set of numerical experiments are presented (Section III) to point out the potentialities of the proposed multibeam strategy. Eventually, some conclusions are drawn (Section IV). II. MATHEMATICAL FORMULATION Let us consider a time-modulated linear antenna array (TMLA) where a set of radio-frequency switches is used to modulate the static excitations of amplitudes , , and phases , . Such a linear arrangement of isotropic sources displaced along the axis at the positions , , radiates the following field [19]

(1)

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is the central angular frequency, is the where free-space wavenumber, being the speed of light in vacuum, and denotes the angular direction. The RF switches are con, , trolled by means of digital signals, of period mathematically modeled as rectangular pulse funcwithin the normalized (with retions with values , being spect to ) duration the switch-on (raise) and switch-off (fall) time instants, respectively, and for the remaining part of the modulation period. Because of the time periodicity, each pulse function can , be expanded into a Fourier series, where and the expansion coefficients are equal to . Accordingly, (1) can be rewritten as an (ideally) infinite number of radiated patterns from shifted of

(2) whose

th

term

is

given by , being the equivalent excitation. In particular, the pattern at the central frequency is given by and the corresponding Fourier coefficients are real and equal to , . Otherwise, those at the harmonic frequencies are complex and given by [23] (3), shown at the bottom of the page. It follows that to synthesize desired patterns at the central frequency and at its harmonics the set of the pulse durations, , and the values of the switch-on instants, , can be properly optimized. Whether necessary, the sets of amplitudes, , and phases, , of the static excitations can be also properly tuned. However, it is worthwhile to point out that without suitable countermeasures the power associated to the SR decreases as the harmonic index increases [19] and only the first harmonic modes are expected to generate “useful” patterns. In order to overcome this drawback in view of multiple harmonic beams, the following PS-based optimization procedure is proposed: • Definition of the Static Excitations— and The values of the static excitations are generally a-priori defined according to the following motivations depending on the problem at hand. The use of unitary amplitude weights (i.e., , ) is generally preferred in order to simplify the complexity of the beam

.

(3)

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forming network [16], [17] and to avoid additional amplifiers/attenuators as shown in synthesizing ultra-low sidelobe patterns [14], [33] with a reduced dynamic range. Differently, non-uniform (complex) weights can be used to generate a desired quiescent (without time-modulating the element excitations) pattern at the central frequency . From an algorithmic point of view, a smaller number of control variables reduces the computational burden required by the optimization process. Otherwise, a better matching of the synthesized solutions with the user-defined requirements can be achieved when more degrees-of-freedom are available; • Definition of the TM Pulse Sequence— and The parameters modeling the rectangular digital signals, namely the pulse durations, , and the switch-on instants, , are determined by minimizing the cost function, , aimed at enforcing the desired patterns both at the central fre, and at the first sideband (upper and quency, lower) radiations, ,

(4)

where is the set of unknown quantities (i.e., ) and is the Heaviside function. Moreover, denotes the target/desired value along the direction of the sidelobe level , the first null beamwidth , the sideband level , and the ripple level of the th harmonic pattern. The same notation holds true for the . Furthermore, is a real and current value positive coefficient weighting each term of the cost function according to the problem requirements. In order to minimize (4), a standard PS optimization algorithm [32] is used by iteratively ( being the iteration index, ) applying the following updating equations

(5)

is the inertial weight coefficient and and where are the cognitive and social acceleration coefficients, respectively. Moreover, and are uniformly-distributed random variables. As far as the swarm is concerned, is the index of the th particle, is the personal best particle, while is the global best of the whole swarm at the th iteration.

0

N

d = jhj

Fig. 1. 6 1 configuration ( = 16, = 2)—optimized (a) pulse sequence and (b) relative power patterns at the central frequency ( = 0—sum = 1—difference pattern). pattern) and at the fundamental frequency (

h

III. NUMERICAL RESULTS The first experiment deals with the generation of a sum pattern at the central frequency and a difference pattern at the funuseful in monopulse radars for damental one search-and-track purposes. A linear array with equally, , spaced elements located at being the inter-element distance, is the reference architecture. Amplitude-only static excitations with unitary gains and , ) have been chosen (i.e., to yield a simple beamforming network. According to the guidelines in [31], here applied to the synthesis of arrays with more than two elements [34], it is enough to energize the two halves of in the time pulse sequence to obtain the array with a shift of a pattern with a null on boresight. Therefore, the pulse durations , , have been set to and only half elements of the TM array have been optimized by minimizing (4) with and , , 1, the reference values being fixed to , , and . Moreover, the initial configuration of the unknowns has been set to afford a at . Dolph-Chebyshev pattern with Fig. 1(a) shows the configuration of the switch-on times obof the PS procedure carried tained after

POLI et al.: HARMONIC BEAMFORMING IN TIME-MODULATED LINEAR ARRAYS

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TABLE I Sum-Difference Configurations ( ,

N = 16 d = =2)—PATTERN INDEXES

601 jhj = 1

N = 16 d = =2

Fig. 3. configuration ( , )—relative power patterns at the central frequency ( —sum pattern) and at the fundamental frequency —difference pattern) synthesized with the PS optimization and in [34]. (

601

N = 16 d = =2

Fig. 2. configuration ( , )—percentage of the individual power associated to the harmonic radiations.

out with a swarm of and the control paramand eters set to a standard configuration [23]: . The radiated patterns are given in Fig. 1(b) for and , while the corresponding pattern indexes are reported in Table I ( configuration). As expected, most of the power is radiated at the central frequency and at the funand damental one with a percentage equal to of the total power ( being the power associated to the pattern at the th harmonic radiation), respectively. Such an outcome is further pointed out in Fig. 2 where the power content of each pattern up to the 9th harmonic mode is reported. The amount of power rapidly tends to zero when the harmonic index grows. For completeness, the solution with the proposed approach and that in [34] are compared in Fig. 3 to also point out that an unavoidable trade-off exists. As a matter of fact, if can be oba more accurate control of the sidelobes for tained through the PS optimizer, on the other hand, the SLL at increases with respect to that in [34]. As for the minimization process, Fig. 4 gives an indication of the optimization process by showing the plot of the cost function throughout the iterations. The second experiment still considers the synthesis of sum and difference patterns, even though now the sum beam is

601

h=0

N = 16 d = =2

Fig. 4. configuration ( , )—behavior of the cost function throughout the optimization process and its the individual terms.

, while the difference one at the central synthesized at . Towards this end, the odd symmetry frequency of the excitations is enforced to generate a difference beam , by adding to the phase values of half array (i.e., ). The value has been set to 1 . Moreover, dB since only one main beam is required at the thresholds have been chosen and all terms of the cost function have been equally weighted. The optimized switch-on times and the corresponding and difference patterns are shown sum in Fig. 5(a) and (b), respectively. From the quantitative configuration), it turns pattern indexes in Table I ( out that both patterns are quite similar in terms of sidelobe ) and sideband levels (i.e., . Moreover, and values (Fig. 6). Concerning the sideband radiation, the values of the SBLs of the first 30 harmonic modes of the solution are reported in Fig. 7(a) and compared with configuration, as well. Whether for those of the the SBLs have similar behaviors, the advantages of the solution are non-negligible at the fundamental frequency (i.e., vs. —Table I).

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106

N = 16 d = =2 jhj = 1

Fig. 5. configuration ( , )—optimized (a) pulse sequence and (b) relative power patterns at the central frequency ( —difference pattern) and at the fundamental frequency ( —sum pattern).

h=0

106

N = 16, d = =2)—relative power patterns of jhj = 1, 2, 3.

Fig. 6. configuration ( the sideband radiation for

For completeness, the radiation patterns at , 3 are displayed in Fig. 7(b). The following experiments are concerned with two design solutions suitable for intercepting signals impinging on an array of elements spaced by from different angular directions without phase shifters in the array architecture. In the

N = 16, d = =2)—comparison of SBL when h 2 [1; 30] and (b) the

Fig. 7. Sum-difference configurations ( (a) the behavior of the sideband levels harmonic patterns radiated at , 3.

jhj = 2

first example, the approach exploits the guidelines in [28], recently also considered in [29], where the beam steering at the first harmonic mode is forced by controlling the pulse sequence. Likewise [29], the pattern is requested to be directed at broadand are the diside at , while and rections of the desired beams at the first upper lower sideband, respectively. Towards this purpose, the cost function has been defined by setting . Moreover, the reference level of the and has been fixed to 20 dB. secondary lobes at The PS-optimized solution is summarized in Fig. 8 where the on-off sequence [Fig. 8(a)] and the corresponding radiated patterns are shown [Fig. 8(b)]. The indexes in Table II (Harmonic Beam Steering) confirm the effectiveness of the proposed strategy in yielding a satisfactory solution that fits the user requirements. The possibility to selectively and simultaneously receive signals from three different directions is the main advantage of such a solution, even though an ad-hoc receiver able to separate each harmonic component [29] would be necessary. An alternative solution is presented in Fig. 9 where only one pattern, characterized by two main beams pointing at , is generated at the first sideband either at and not at or vice versa. Although it is not possible to distinguish whether the signals impinge from or , the

POLI et al.: HARMONIC BEAMFORMING IN TIME-MODULATED LINEAR ARRAYS

N = 16 d = =2 ( = 0 ) ( = 030 ) h = 01

Fig. 8. Harmonic beam steering ( , )—optimized (a) pulse sequence and (b) relative power patterns at the central frequency [ —broadside beam ] and at the fundamental frequen—steered beam ; —steered beam cies [ ].

h =0 h = 1 ( = 30 )

TABLE II Harmonic Beam Steering—Double Sum Configuration ( )—PATTERN INDEXES

d = =2

N = 16,

HW implementation is less complex in this case since the received power is collected only at and . The indexes describing the synthesized Double Sum Beam are summarized in Table II where the performances of the Harmonic Beam Steering solution are also reported for comparison. The last test case is a representative example aimed at showing some other potentialities of the proposed approach. and More specifically, the synthesis of a flat-top pattern at of a sum pattern at is addressed by considering an array

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N = 16 d = =2 ( = 0 )  = 630

Fig. 9. Double sum configuration ( , )—optimized (a) pulse sequence and (b) relative power patterns at the central frequency ( —broadside beam ] and at the fundamental frequency ]. —steered beam, [

h=0 jhj = 1

TABLE III Flat top and Sum Configuration ( , )—INITIAL SETTING [35] FOR THE ITERATIVE OPTIMIZATION

N = 12 d = =2

(k = 0)

of half-wavelength spaced elements with uniform , ). The amplitude weights (i.e., phase weights, , and switch-on times, , have been initialized to afford a flat-top beam as in [35] and their normalized values are reported in Table III. The synthesis targets have been fixed to and . The pulse sequence determined with the PS is shown in Fig. 10(a) together with the synthesized two patterns [Fig. 10(b)] whose descriptive indexes are reported in

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to avoid, have been profitably exploited to design an antenna system providing simultaneous multiple patterns. Towards this purpose, an optimization procedure has been applied to determine the time-domain descriptors of the digital time sequence that controls the RF switches devoted to implement the time-modulation of the array. Since the power associated to the harmonic modes rapidly reduces for higher harmonic modes, the desired patterns have been synthesized at the central and the first harmonics. Although further investigations are currently under development, the results of the numerical assessment provide a proof that time-modulated arrays are an effective and reliable architectural solution for those applications requiring a multiple and simultaneous beamforming. ACKNOWLEDGMENT The authors are grateful to the anonymous reviewers for their technical comments and suggestions. REFERENCES

N = 12 d = =2 h =1

Fig. 10. Flat top and sum configuration ( , )—optimized (a) pulse sequence and (b) relative power patterns at the central frequency ( —flat-top pattern) and at the fundamental frequency [j j —sum pattern)].

0

h=

TABLE IV Flat top and Sum Configuration ( ,

N = 12 d = =2)—PATTERN INDEXES

Table IV. As it can be observed, the beams satisfy the design constraints with the level of the secondary lobes equal , , and the peak to of the sum beam with value . Moreover, with a maximum the flat-top region presents a ripple level . oscillation of IV. CONCLUSIONS In this paper, the synthesis of multiple harmonic beams in time modulated linear arrays has been addressed. The sideband radiations arising from the periodic time-modulation of the static excitations of the array, usually regarded as power losses

[1] I. M. Skolnik, Radar Handbook, 3rd ed. New York: McGraw-Hill, 2008. [2] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. Hoboken, NJ: Wiley, 2005. [3] M. Barba, J. E. Page, J. A. Encinar, and J. R. Montejo-Garai, “A switchable multiple beam antenna for GSM-UMTS base stations in planar technology,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3087–3094, Nov. 2006. [4] M. Durr, A. Trastoy, and F. Ares, “Multiple-pattern linear antenna arrays with single prefixed amplitude distributions: Modified WoodwardLawson synthesis,” Electron. Lett., vol. 36, no. 16, pp. 1345–1346, Aug. 2000. [5] A. Trastoy, Y. Rahmat-Samii, F. Ares, and E. Moreno, “Two-pattern linear array antenna: Synthesis and analysis of tolerance,” IEE Proc. Microw. Antennas Propag., vol. 151, no. 2, pp. 127–130, Apr. 2004. [6] M. Comisso and R. Vescovo, “Multi-beam synthesis with null constraints by phase control for antenna arrays of arbitrary geometry,” Electron. Lett., vol. 43, no. 7, pp. 374–375, Mar. 2007. [7] R. L. Haupt, “Interleaved thinned linear arrays,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2858–2864, Sep. 2005. [8] I. E. Lager, C. Trampuz, M. Simeoni, and L. P. Ligthart, “Interleaved array antennas for FMCW radar applications,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2486–2490, Aug. 2009. [9] G. Oliveri and A. Massa, “Fully-interleaved linear arrays with predictable sidelobes based on almost difference sets,” IET Radar Sonar Navigat., vol. 4, no. 5, pp. 649–661, 2010. [10] G. Oliveri, P. Rocca, and A. Massa, “Interleaved linear arrays with difference sets,” Electron. Lett., vol. 46, no. 5, pp. 323–324, 2010. [11] A. Austeng and S. Holm, “Sparse 2-D arrays for 3-D phased array imaging—Design methods,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 49, no. 8, pp. 1073–1086, Aug. 2002. [12] G. Oliveri and A. Massa, “ADS-based array design for 2D and 3D ultrasound imaging,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 57, no. 7, pp. 1568–1582, Jul. 2010. [13] H. E. Shanks and R. W. Bickmore, “Four-dimensional electromagnetic radiators,” Canad. J. Phys., vol. 37, pp. 263–275, Mar. 1959. [14] W. H. Kummer, A. T. Villeneuve, T. S. Fong, and F. G. Terrio, “Ultra-low sidelobes from time-modulated arrays,” IEEE Trans. Antennas Propag., vol. 11, no. 6, pp. 633–639, Jun. 1963. [15] S. Yang, Y. B. Gan, and A. Qing, “Sideband suppression in time-modulated linear arrays by the differential evolution algorithm,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 173–175, 2002. [16] J. Fondevila, J. C. Brégains, F. Ares, and E. Moreno, “Optimizing uniformly excited linear arrays through time modulation,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 298–301, 2004. [17] S. Yang, Y. B. Gan, A. Qing, and P. K. Tan, “Design of a uniform amplitude time modulated linear array with optimized time sequences,” IEEE Trans. Antennas Propag., vol. 53, no. 7, pp. 2337–2339, Jul. 2005. [18] L. Poli, P. Rocca, L. Manica, and A. Massa, “Handling sideband radiations in time-modulated arrays through particle swarm optimization,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1408–1411, Apr. 2010.

POLI et al.: HARMONIC BEAMFORMING IN TIME-MODULATED LINEAR ARRAYS

[19] J. C. Brégains, J. Fondevila, G. Franceschetti, and F. Ares, “Signal radiation and power losses of time-modulated arrays,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1799–1804, Jun. 2008. [20] L. Poli, P. Rocca, L. Manica, and A. Massa, “Pattern synthesis in timemodulated linear arrays through pulse shifting,” IET Microw., Antennas Propag., vol. 4, no. 9, pp. 1157–1164, 2010. [21] L. Poli, P. Rocca, L. Manica, and A. Massa, “Time modulated planar arrays—Analysis and optimization of the sideband radiations,” IET Microw. Antennas Propag., vol. 4, no. 9, pp. 1157–1164, 2010. [22] S. Yang, Y. B. Gan, and P. K. Tan, “Evaluation of directivity and gain for time-modulated linear antenna arrays,” Microw. Opt. Technol. Lett., vol. 42, no. 2, pp. 167–171, Jul. 2004. [23] L. Manica, P. Rocca, L. Poli, and A. Massa, “Almost time-independent performance in time-modulated linear arrays,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 843–846, 2009. [24] S. Yang, Y. B. Gan, and P. K. Tan, “A new technique for power-pattern synthesis in time-modulated linear arrays,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 285–287, 2003. [25] G. Li, S. Yang, and Z. Nie, “A study on the application of time modulated antenna arrays to airborne pulsed doppler radar,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1579–1583, May 2009. [26] J. Fondevila, J. C. Brégains, F. Ares, and E. Moreno, “Application of time modulation in the synthesis of sum and difference patterns by using linear arrays,” Microw. Opt. Technol. Lett., vol. 48, no. 5, pp. 829–832, May 2006. [27] A. Tennant and B. Chambers, “Time-switched array analysis of phaseswitched screens,” IEEE Trans. Antennas Propag., vol. 57, no. 3, pp. 808–812, Mar. 2009. [28] H. Shanks, “A new technique for electronic scanning,” IRE Trans. Antennas Propag., vol. 9, no. 2, pp. 162–166, Mar. 1961. [29] G. Li, S. Yang, Y. Chen, and Z. Nie, “A novel electronic beam steering technique in time modulated antenna arrays,” Progr. Electromagn. Res., vol. 97, pp. 391–405, 2009. [30] G. Li, S. Yang, and Z. Nie, “Direction of arrival estimation in time modulated antenna arrays with unidirectional phase center motion,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1105–1111, Apr. 2009. [31] A. Tennant and B. Chambers, “Two-element time-modulated array with direction-finding properties,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 64–65, 2007. [32] J. Kennedy, R. C. Eberhart, and Y. Shi, Swarm Intelligence. San Francisco, CA: Morgan Kaufmann, 2001. [33] S. Yang, “Study of low sidelobe time modulated linear antenna arrays at millimeter-waves,” Int. J. Infrared Milli. Waves, vol. 26, no. 3, pp. 443–456, Mar. 2005. [34] A. Tennant and B. Chambers, “Control of the harmonic radiation patterns of time-modulated antenna arrays,” presented at the IEEE AP-S Int. Symp., San Diego, CA, Jul. 5–12, 2008. [35] Y. U. Kim and R. S. Elliot, “Shaped-pattern synthesis using pure real distributions,” IEEE Trans. Antennas Propag., vol. 36, no. 11, pp. 1645–1649, Nov. 1988. Lorenzo Poli (S’10) received the M.S. degree in telecommunication engineering from the University of Trento, Italy, in 2008. He is with the International Graduate School in Information and Communication Technologies, University of Trento and a member of the ELEDIA Research Group. His main interests are the synthesis of the antenna array and electromagnetic inverse scattering.

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Paolo Rocca (M’09) received the B.S., M.S., and Ph.D. degrees in telecommunications engineering from the University of Trento, Italy, in 2004, 2005, and 2009, respectively. He is with the International Graduate School in Information and Communication Technologies, University of Trento and a member of the ELEDIA Research Group. His main interests are in the framework of antenna synthesis and design, electromagnetic inverse scattering, and optimization techniques for electromagnetics.

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 Electromagnetic Diagnostic Laboratory, 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.

Andrea Massa (M’03) received the “Laurea” degree in electronic engineering and Ph.D. degree in electronics and computer science from the University of Genoa, Genoa, Italy, in 1992 and 1996, respectively. From 1997 to 1999, he was an Assistant Professor of electromagnetic fields at 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 and, 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, University of Trento and Deputy Dean of the Faculty of Engineering. 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 a member of the PIERS Technical Committee, 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).

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Fast Directional Multilevel Algorithm for Analyzing Wave Scattering Hua Chen, Kwok Wa Leung, Fellow, IEEE, and Edward K. N. Yung

Abstract—A new method named fast directional multilevel algorithm is proposed for analyzing wave scattering. Similar to a fast multipole method, an oct-tree structure is used in the new method. Unlike the conventional MLFMA, the kernel is evaluated by a new method, instead of approximating the Green’s function by multipole expansion. The fast directional multilevel algorithm is first used to solve the wave scattering problem via combined field integral equation (CFIE) using Rao-Wilton-Glisson (RWG) basis functions. The low rank representations are extended to apply in the low frequency regime which is proved efficient and stable. The numerical results demonstrate that the computational complexity of this new multilevel algorithm can achieve the complexity of ( log ). This algorithm is robust and performs well on other oscillatory kernels because of its not depending on the explicit form of kernels. Index Terms—Electromagnetic (EM) scattering, fast directional multilevel algorithm, low frequency, low rank representation.

I. INTRODUCTION HE method of moments (MoM) [1], [2] is preferred in many occasions because of its robustness. However, for a problem with numerous unknowns , it is impractical to solve resultant matrix equation because the memory required is of the and the corresponding computational requirements order of . For solving wave scattering from an are of the order of electrically large body of arbitrary shape, a number of successful techniques associated with iterative methods [3]–[16] are used in general. Unfortunately, we always experience problems in the computational process in solving electrically small bodies, the so-called low-frequency breakdown. In fact, one of the major objectives in writing this paper is to alleviate this breakdown. Surprisingly, the trick used to solve the problem due to an electrically small scatterer is similar to that of large bodies. After all, the aim is to complete the computational process satisfactorily. For large scatterers, many techniques have been developed for speeding up the MoM. For example, the current basis functions in the adaptive integral method [3] and the pre-corrected fast Fourier transform method [4], [5] are replaced by higher-

T

Manuscript received November 11, 2010; accepted November 16, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported in part by the Major State Basic Research Development Program of China (973 Program: 2009CB320201) and in part by the Natural Science Foundation Grants 60871013, 60701004. The authors are with the Department of Communication Engineering, Nanjing University of Science and Technology, Nanjing 210094, 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.2011.2152354

order equivalent sources such that the resultant impedance matrix would resemble a block Toeplitz structure; thus, the computation could be accelerated by FFT. In general, these neo-FFT methods are more efficient in solving volume integral equations on planar structures [6]. In fact, the physics of Green’s function could be exploited to develop many compression-based methods such as the efficient electrostatic and electromagnetic simulation (IES3) [7], the UV multilevel partitioning method [8], the multilevel QR decomposition method [9], and the multilevel matrix decomposition algorithm (MLMDA) [10], [11]. While the IES3 algorithm is designed to tackle static and electrically small scatterers; the MLMDA is efficient only for planar or piecewise planar objects [12]. Although the multilevel UV decomposition and the multilevel QR methods are applicable in more situations, their uses are restricted because they require sophisticated matrix manipulations. Nowadays, one of the most popular fast algorithms in electromagnetics is the multilevel fast multi-pole algorithms (MLFMA) [13]–[16] since the memory cost is reduced to the . Unfortunately, baring simple cases, impleorder of mentation of MLFMA is not easy because it may involve many complex mathematical manipulations, including partial wave expansion, exponential expansion, filtering, and interpolation of the spherical harmonics. The situation would become even more complex when MLFMA is used to solve a multi-layered medium. Similar to the compression-based methods, the interaction between two well-separated regions has a rank-deficient characteristic, as a result, the fast directional multilevel algorithm (FDMA) is developed [17], [18]. In essence, the far field region is partitioned into a group of directional regions; therefore the new method is called directional. Formulation of the methodology will be presented in the subsequent section. It is used to solve a simple problem via the combined field integral equation using RWG basis functions. The method is first used to solve the wave scattering from large scatterers. It is then switched to analyze electrically small scatterers in low frequency regime. The paper is organized as follows. Section II gives a brief introduction to the processing of near interaction of FDMA and the CFIE formulation for electromagnetic wave scattering. Section III describes the theory and implementation of the fast directional multilevel algorithm in more details. Numerical experiments with a few electromagnetic wave scattering problems are presented to demonstrate the efficiency of the fast directional multilevel algorithm in Section IV. Section V is concluded by remarks on the present development and comments on further developments.

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II. FORMULATION

III. FAST DIRECTIONAL MULTILEVEL ALGORITHM

The CFIE formulation of electromagnetic wave scattering problems using planar RWG basis functions for surface modeling is presented in [2]. The resulting linear systems from CFIE formulation after Galerkin’s testing are briefly outlined as follows:

In many literatures, it is pointed out that typical Green’s functions vary smoothly with distance [10]–[16]. Similar to the methods in [10]–[16], the FDMA also exploits the limited number of degrees of freedom (DoF) [19] that characterize a field observed over a domain that is well separated from a source domain. In our paper, the far field of FDMA is partitioned into multiple wedges in the high frequency regime, thus the interaction between the source and the observed boxes in each wedge will satisfy the directional parabolic separation condition which has a directional low rank property. As a result, the computation of the interactions between a source box and the observed boxes in a given wedge can be accelerated using the low rank representations.

(1) where, see equation at the bottom of the page. Here, is refers to the Green’s function in free space and is the column vector containing the unknown coefficients of the surface current expansion with RWG basis functions. Also, as usual, and denote the observation and source point and are the incident excitation plane locations. wave, while and denote the free space impedance and wave number, respectively. The combination parameter ranges from 0 to 1 and can be chosen to be any value within this range. Once the matrix (1) is solved by numerical matrix equation solvers, the expansion coefficients can be used to calculate the scattered field and RCS. In the following, is used to denote the coefficient matrix in (1), and for simplicity. Then, the CFIE matrix (1) can be symbolically rewritten as: (2) To solve the above matrix equation by an iterative method, the matrix-vector products are needed at each iteration. Physically, a matrix-vector product corresponds to one cycle of interactions between the basis functions. Using the fast directional multilevel can be written algorithm (FDMA), the matrix-vector product as:

Here, is the near part of and the far part of . reIn the FDMA, the calculation of matrix elements in mains the same as in the MoM procedure. However, those elements in are not explicitly computed and stored. Hence they are not numerically available in the FDMA. When the FDMA is implemented in multilevel, the total cost can be reduced further to .

A. The Construction of Directional Low Rank Representations In the low frequency regime, for a box with the width w as shown in Fig. 1(a), its far field is light dark area as defined in MLFMA. In the high frequency regime, the far field of box is partitioned into multiple wedges and each wedge corresponds to a low rank separated representation. We define as a wedge’s direction with the spanning angle , and its far field is in away. For simplicity, we take two-dimension the distance of case as in Fig. 1(b) for example. The low rank representations are based on random sampling, which is proved efficient and accurate [18], [19]. 1. Sample the box randomly and densely to obtain a set , we use about 4 to 5 points of samples are sampled per wavelength and the number of samples grows linearly with the area of the box. Especially, more points are sampled at surface than inside the box. Sample the region similarly with a set of samples . Compute matrix by scalar Green’s function (3) Here, the dimension of matrix is . For a fixed error , there exists a low rank factorization of with rank . We first make QR factorization of matrix , then we regard the rank of matrix as the . It is noticed that the here is just an approximate value. Using this knowledge, the following randomized method constructs such an approximate factorization (4)

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where is of size , is . In the next few steps, we construct such a factorization. It is worth mentioning here is computed according to kernels, therefore that the matrix free space Green’s function will be replaced by the multilayered Green’s function here if we deal with layered media problems. be a set of columns taken from at random, and 2. Let is of rank , usually, is selected about three times larger than . is decomposed by the singular value de3. The matrix composition into (5a) where Let the diagonal of than the threshold ,

with . contains only the singular values greater and we get hence the (5a) can be replaced by (5b)

where and the samples associated with these columns are . The is expected of size and is denoted about . be a set of m rows taken from at random, 4. Let is the submatrix formed by these rows. Similarly, let be the with the same rows. The samples associated submatrix of with these rows are denoted . where stands for the pseudo-inverse 5. Set . is much smaller than , if 6. Check whether this is the case, the factorization is accepted. Otherwise, double and restart the process from Step 2. 7. Finally, we have the approximation

of

Fig. 1. (a) A box Y with its far field X in low frequency regime. (b) A box Y with its far field X is first partitioned into four wedges when the box size is equal or larger than 1 in the size of wavelength.

Green’s function is translation invariant, the points of and for a set centered at an arbitrary point can be obtained from those of the set centered at the origin position by translation. However, the matrix remains the same. And because of the rotation invariant, for a fixed size of box in every level, we only and for a fixed direction need to generate the points of and rotate these quantities for any other direction.

(6) B. The Low Rank Representations for CFIE Formulation We define to be the matrix and , then

sandwiched between

(7) has only rows and columns. Use The matrix denote the entries of , we can rewrite (7) in the form

Using the low rank representation generated above, we have a group of and and the matrix for every level. The CFIE is formed by electric field integral equation (EFIE) and magnetic field integral equation (MFIE), which is defined by the relation (10)

(8)

We describe the algorithm for the rapid evaluation of EFIE formulation

The vector Green’s function is just the gradient of the scalar Green’s function,

(11) (9) , It can be seen from (8) and (9) that only the locations and the matrix need to store for every level. This costs in storage for a fixed error threshold . Since the only

If the surface bounding the object is closed, it can also be described using MFIE (12)

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The unknown current is expanded in RWG basis functions , where is the unknown expansion coefficient and is unit tangent at any given point on S. The impedance matrix of EFIE is

Fig. 2. Consider the source is in the box Y , and the region X is the interaction far field of box Y .

(13)

Expand the right side of the (13) using low rank representation

Suppose the box and its far field is in the cone-shaped area as shown in Fig. 2. Change into and into herein. Rewrite the expression (16) corresponding to Fig. 2 as follows:

(14)

(17)

It can be seen that a set of basis function can be placed as source in box in order to reproduce the impedance tested by the basis function in region .

The impedance matrix of MFIE is expanded as

(18) (15)

In above equation, is defined as . For convenience, we uniformly write the impedance matrix of CFIE into

is called the outgoing equivalent densities of . The points and are respectively called the outgoing equivalent points and the outgoing check points of . Especially, we defined outgoing check potentials of as (19)

(16)

It can be seen that if a box is in the far field of another box, then the two boxes are definitely in each other’s far field. Therefore, reverse the situation as shown in Fig. 3. The corresponding expression is

where

is equal to for the magnetic vector potential and for the scalar potential.

(20)

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Fig. 3. Consider the source is in the region Y , and the box X is the interaction far field of region Y .

Fig. 5. Consider the box A is in the wedge of B indexed by ` while the box B is in the wedge of A indexed by ` . Ac is the children box of A, and Bc is the children box of B.

of wavelength. Consider the box A is in the wedge of B indexed is the by while the box B is in the wedge of A indexed by . children box of A (see Fig. 5). In other words, box A and B are on each other’s far interaction lists. We take two levels FDMA for example herein, in which the lowest level is none direction and the top level is directional. The directional multilevel algorithm contains following steps: Travel up in the octree: Step 1. Compute the outgoing equivalent densities by (18). are none directional because the is less than 1 here width of the Fig. 4. The relationship of wedges between parent box and children box.

We consider source basis function in the region in order to reproduce the impedance which is tested by the basis function in box (21) is defined as incoming equivalent densities of . The points and are respectively called incoming equivalent points and incoming check points of . Especially, we defined incoming check potentials of as (22)

C. The Data Structure for FDMA and the Detail of FDMA Similar to the MLFMA, the whole object is first enclosed in a smallest cubical box, and then constructing a hierarchical subdivision of that box, in which it is divided into eight boxes of equal size (also cubes), each of which is likewise subdivided. The subdivision process continues recursively until the lowestlevel boxes. A box, which is divided into eight boxes, is termed “parent” and the eight boxes are termed its “children”. In the rest of this paper, B denotes a box in the octree and stands for its width in size of wavelength. The partition of the far field is similar to [17] and the wedges of box B can be contained in the wedges of B’s children as shown in Fig. 4. belongs to the finest leaf boxes with Suppose the box with width 1 in size width 0.5 and B is the parent box of

(23)

Step 2. The far field of B is partitioned into a group of wedges because the width is 1. Use outgoing equivalent as the source points of B and the points as the densities of B. The directional outgoing equivalent densities of B in direction is defined . We use “ , ” to distinguish “ , ” for the top level

(24)

Step 3. Compute directional incoming check potentials of in direction and the is used as the densities denotes the incoming check points of A in the direction and denotes the incoming equivalent points of B in the direction (25)

Travel down in the octree. Step 4. Compute directional incoming equivalent densities multiplying the matrix of A through

(26)

CHEN et al.: FAST DIRECTIONAL MULTILEVEL ALGORITHM FOR ANALYZING WAVE SCATTERING

Step 5. Use the parent box’s directional incoming equivalent as the children boxes’ densities densities to get the incoming check potentials (27) The first term on right side of (27) stands for the contribution of ’s far field, which is calculated by (25). The second term on right side of equation stands for the contribution of ’s parent box. Compute the incoming equivalent densities of by multiplying the incoming check potentials

(28) Step 6. Finally, the impedance matrix of CFIE can be expressed as in (29), shown at the bottom of the page. From implementation of above procedures, it can be seen that there are not multipole expansions of the Green’s function but some kernel evaluations. D. Complexity of FDMA It is the same as MLFMA that there are also three translations called far-to-far translation, far-to-local translation and local-to-local translation [17]. For the far-to-far translation, we compute the directional outgoing equivalent densities of B from its child box . For the far-to-local translation, the two boxes A and B at same level in each other’s interaction list are considered. The far-to-local translation is to evaluate the difrom B to A. The rectional incoming check potentials last step of local-to-local translation is to let the construction of parent boxes disaggregation to every child box. The process of local-to-local translation can be carried out through above (22) to (25). In FDMA, most of time is consumed for three translations. For a box B with width , the wedges of a box are . far-to-far translations for box B’s all the 1) There are wedges. The complexity is while far-to-far translaoperations for each wedge. tion takes 2) The local-to-local translation is the reverse case of the . far-to-far translation, hence the complexity is also 3) All the boxes in B’s interaction are included in a fan-shaped to , so that there are region with radius between boxes in this region. Each far-to-local translation operations and therefore the complexity is also takes .

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4) Suppose there is a circle with smallest radius R to enclose boxes of size . Therethe object. There are fore, the number of steps spent on each level is . Finally, summing over all levels, the complexity of this multidirectional algorithm is as follows:

E. The Low Frequency Stability of FDMA It is well known that when the box size of the multilevel fast multipole algorithm (MLFMA) is less than 0.2 ( is the wavelength in free space), the MLFMA will suffer from “low frequency breakdown” [4] or “subwavelength breakdown” [20]. The low frequency breakdown can be understood by looking at the translation operator of the MLFMA

where , and is the translation vector. Because the spherical Hankel function increases exponentially as a f unction of if , the terms with a low are eventually swamped in the high order terms. Obviously, the loss of the low order terms is catastrophic because they contribute the most to the addition theorem. As a result, MLFMA is inefficient when applied to sub-wavelength problems where the principal dimension of the domain is less than or order of a wavelength only. As an attempt for a possible remedy, low frequency multilevel fast multipole algorithm (LF-MLFMA) has been developed to modify the MLFMA [21]. However, when the operating frequency increases to certain – , the number of multipoles suddenly value around increases to keep the certain accuracy and accordingly both CPU time per iteration and memory consumption will increase. The box is regarded in low frequency regime when the box’s size is smaller than 1 . The far field of each box are not divided into a group of directions but considered similar in MLFMA as shown in Fig. 1(a). The low rank representations are still applied in the low frequency regime. Although the electrical size of the box is small in low frequency regime, the interaction about the well-separated boxes still has a “low rank” property as long as the density of unknowns is high. The matrix formed by the well-separated boxes can also be compressed even though the rank of the matrix in low frequency regime is larger than that in high frequency counterpart. The rank will close to the full rank when the box size is further reduced. In our investigation, the box size of the finest level can be reduced to around 0.025 without loss of accuracy, but it will incur divergence when the

(29)

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Fig. 6. Bistatic RCS for a cylinder (bottom radius is 0.5 m and length is 4 m) with 78294 unknowns at 1.2 GHz.

box size is further reduced due to the numerical error. If double precision is utilized, the box size of the finest level can be reduced to around 0.005 without loss of accuracy. It must be emphasized that the expansion of the Green’s function for FDMA at low frequency regime is unchanged. Simply put, only kernel evaluation is needed and it involves no treatment of singularity due to the spherical Hankel function as found in MLFMA, the major cause of breakdown in the sub-wavelength studies. IV. NUMERICAL RESULTS In this section, several numerical examples are presented to demonstrate the efficiency of the FDMA for solving linear systems arising from the discretization of CFIE formulation of electromagnetic wave scattering problems. In the implementation of the FDMA, the restarted version of GMRES algorithm [22] is used as the iterative method. The restarted version of GMRES(30) algorithm is used as iterative method. All experiments are performed on a Core-2 6300 with 1.86 GHz CPU and 1.96 GB RAM in single precision. The iteration process is terminated when the normalized backward error is reduced for all examples. All numerical results are calculated by using CFIE and the combination parameter is set 0.5. A. PEC Sphere and Cylinder The performance of FDMA for vertical polarization is first investigated on a cylinder for bistatic RCS calculation. The in, at 1.2 GHz. As cident angles of plane wave are shown in Fig. 6, the length of cylinder (bottom radius is 0.5 m) is 4 m. The cylinder is discretized with 78294 unknowns, which has an average discretized size 0.1 . The sets of angles for the bistatic RCS vary from 0 to 180 degree in elevation direction when azimuth angle is fixed at 0 degree. When applying the octree structure, the finest box size is 0.25 and four levels FDMA is used. Due to the box size of both levels is 0.25 and 0.5 respectively, none direction is used in both the first level and the second level. Since the box size is 1 at the third level, 6 directions are used for each nonempty box and each direction is further divided into four directions with the total 24 directions for the fourth level. As shown in Fig. 6, the bistatic RCS at 1.2 GHz for the cylinder with 78294 unknowns is given. Both the

Fig. 7. Bistatic RCS for a sphere (radius is 4 m) with 100296 unknowns at 300 MHz.

results of MLFMA and FEKO are provided for the comparison with the FDMA. It can be found that there is a good agreement with them. To be noted, MLFMA of five levels is used here for the cylinder. The second example is a sphere (radius is 4 m) for vertical polarization at 300 MHz. The sphere is discretized with 100296 unknowns, which have an average discretized size around 0.05 . The sets of angles of interest for the bistatic RCS vary from 0 to 180 degree in elevation direction when azimuth angle is fixed at 0 degree. The finest box size is 0.25 and four levels FDMA is used and the number of directions in every level is the same as the first example. As shown in Fig. 7, the bistatic RCS for a sphere at 300 MHz with 100296 unknowns is given. It can be seen that there is a good agreement between the Mie series and FDMA. B. The Analysis of Complexity for FDMA In order to check both the memory requirement and the CPU time, the log-log plots for the memory requirement of the proposed FDMA are provided in Fig. 8(a) for the bistatic RCS of the cylinder. The cylinder with its electrical size increased is excited by a normally incident plane wave of vertical polarization with frequency 300 MHz. The length of cylinders is from 1 to 8 and the ratio of bottom radius to the length of cylinder is 0.125. It can be observed that the curve of memory requirement for FDMA is almost parallel with that of MLFMA. In this paper, the memory requirement of MoM without acceleration is tested when the number of unknowns is small. However, the memory requirement is estimated when the number of unknowns is more than 10000. The memory requirement of FDMA is about 1.8 times larger than the MLFMA. This demonstrates that the for the memory requirement of FDMA is about cylinder. The CPU time of one matrix-vector operation for both FDMA and MLFMA is shown in Fig. 8(b) with their both finest level box size 0.25 . And the total CPU times of matrix-vector time with the same accuracy is also provided. It can be seen in Fig. 8(b) that its CPU time of per iteration is close to the MLFMA for this narrow and long cylinder. As shown in Fig. 9(a), the log-log plots for the memory requirement of the proposed FDMA are given for the bistatic RCS of the thin plate. The thin plate with its electrical size increased

CHEN et al.: FAST DIRECTIONAL MULTILEVEL ALGORITHM FOR ANALYZING WAVE SCATTERING

Fig. 8. (a) Plot of the memory consumption for cylinder example. (b) CPU times for one matrix-vector operation versus the number of unknowns for cylinder example.

is excited by a normally incident plane wave of vertical polarization at 300 MHz. The electrical length of the thin plate is from 1 to 8 and the ratio of the width to the length is 0.1, while the ratio of the thickness to the length is 0.025. It can be observed that the curve of memory requirement for FDMA is almost parallel with that of MLFMA. The memory requirement of FDMA is about 1.6 times larger than the MLFMA. This demonstrates that the memory requirement of FDMA is about for the thin plate. The CPU time of one matrix-vector operation for both FDMA and MLFMA is shown in Fig. 9(b) with their both finest level box size 0.25 . Here the total CPU times for matrix-vector time with the same accuracy is also provided. It can be seen in Fig. 9(b) that its CPU time of per iteration is close to the MLFMA for this narrow and thin plate. As shown in Fig. 10(a), the log-log plots for the memory requirement of the proposed FDMA are given for the bistatic RCS of the ogive at 300 MHz with a normally incident plane wave of vertical polarization when its electrical size increases. The electrical length of the ogive is from 1 to 8 ; and the radius of largest cross section is used as cross section radius. The ratio of cross section radius to the length is 0.5. It can be observed that the curve of memory requirement for FDMA is almost parallel with that of MLFMA. The memory requirement of FDMA is about 2 times larger than the MLFMA. This

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Fig. 9. (a) Plot of the memory consumption for thin block example. (b) CPU times for one matrix-vector operation versus the number of unknowns for thin block r example.

demonstrates that the memory requirement of FDMA is about for the ogive. The CPU time of one matrix-vector operation for both FDMA and MLFMA is shown in Fig. 10(b) with their both finest level box size 0.25 . Here the total CPU times for matrix-vector time with the same accuracy is also provided. It can be seen in Fig. 10(b) that its CPU time of per iteration is analogous to the MLFMA for this ogive. The average discretization element size is 0.04 for the examples in Figs. 8–10. The dyadic Green’s function is used in the MLFMA for the addition theorem expansion while the FDMA utilizes the expansion of scalar Green’s function. It can be inferred that the memory storage of FDMA will be nearly the same as the MLFMA if the MLFMA also utilizes the expansion of scalar Green’s function. It can be found from the Fig. 10(b) that the CPU time of FDMA for ogive is lager than those of both the cylinder in Fig. 8(b) and thin plate Fig. 9(b). It is because that the ratio of the cross section radius of the ogive to the length is larger than both the cylinder and thin plate. Then the number of the directions is larger than both the cylinder and thin plate, which results in the more time requirement. From the examples investigated, it can be concluded that the FDMA is more efficient for the analysis of the narrow and long objects as described in [17]. For the narrow and long objects, the number of directions for a box to address is much smaller than the isotropic objects. When the objects are elongated such as cylinders, cones,

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Fig. 10. (a) Plot of the memory consumption for ogive example. (b) CPU times for one matrix-vector operation versus the number of unknowns for ogive example.

Fig. 11. (a) Bistatic RCS for a sphere with radius 0.5 m with 3852 unknowns at 100 MHz with finest level box size is 0.05 . (b) Bistatic RCS for a sphere with radius 1 m with 9339 unknowns at 30 MHz with finest level box size is 0.025 .

strips, the nonempty boxes almost aline in few directions, thus there are few directions need to be addressed for FDMA. C. The Low Frequency Performance of FDMA As shown in Fig. 11(a), a sphere of radius 0.5 m with 3852 unknowns is tested at 100 MHz, and FDMA is investigated with the finest level box size 0.05 . It can be found from Fig. 11(a) that the result of FDMA is in a good agreement with that of MIE series when the group size is less than 0.1 . However, there is a large difference between the MLFMA and Mie series since the MLFMA suffers from low frequency breakdown when the finest box size is 0.05 . Two levels FDMA is investigated with the finest level box size at 0.025 for the frequency at 30 MHz. As shown in Fig. 11(b), there is also a good agreement between Mie series and FDMA. It is well known that the MLFMA suffers from subwavelength breakdown when the finest level box size is smaller than 0.2 . As shown in Fig. 12, the relative errors of both MLFMA and FDMA are given for the size of finest level group. The relative error is defined by the following formulation as:

where denotes the RCS computed by Mie series and is the RCS by MLFMA or FDMA. The direction of incident wave

Fig. 12. Relative error of MLFMA and FDMA for a sphere corresponding to the size of finest level group for monstatic scattering.

is

, and the observation point is located at , . It can be found that the relative error of MLFMA is much larger than the FDMA when the size of finest level group is less than 0.2 . It also can be seen that the relative error of the FDMA is relatively acceptable even when the finest level group size is 0.025 . When double precision is utilized, the box size of the finest level can be reduced to around 0.005 without loss of accuracy as shown in Fig. 13. In Fig. 13, a sphere of radius 1 m with 9339 unknowns is tested at 1 MHz and 3 MHz, corresponding to the finest level group size are 0.002

CHEN et al.: FAST DIRECTIONAL MULTILEVEL ALGORITHM FOR ANALYZING WAVE SCATTERING

Fig. 13. Double precision is utilized for a sphere of radius 1 m at 1 MHz and 3 MHz.

and 0.005 respectively. It can be seen that the results of FDMA are still agree with those of the Mie series. Therefore, it can be concluded that the FDMA is more robust than the MLFMA in low frequency regime. The FDMA is easier to be implemented than the MLFMA since only kernel evaluation is required in FDMA. V. CONCLUSION In this paper, the fast directional multilevel algorithm is first applied for electromagnetic scattering problems by solving CFIE equation using RWG basis functions. The derivation of CFIE form for FDMA is presented. The low rank representations are extended to apply in the low frequency regime which is proved also efficient. Though efficient, the proposed algorithm is more suitable for analyzing the electromagnetic scattering from the narrow and long objects. Numerical results demonstrate that the FDMA is stable for both low frequency and mid-frequency. In our study, we have tried and successfully reduced the size of the box size to 0.025 , yet there is no loss of accuracy. But it will also suffer from low frequency breakdown when the finest level group size is further reduced due to the numerical error. In the future, the FDMA will be applied for both the layered media and half-space electromagnetic problems. ACKNOWLEDGMENT The authors are grateful to Prof. R. Chen for his help.

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[5] N. Yuan, T. S. Yeo, X. C. Nie, and L. W. Li, “A fast analysis of scattering and radiation of large microstrip antenna arrays,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2218–2226, Sep. 2003. [6] K. Z. Zhao, M. N. Vouvakis, and J. F. Lee, “The adaptive cross approximation algorithm for accelerated method of moments computations of EMC problem,” IEEE Trans. Electromagn. Compat., vol. 47, no. 4, pp. 763–773, Nov. 2005. [7] S. Kapur and D. E. Long, “IES : Efficient electrostatic and electromagnetic solution,” IEEE Comput. Sci. Eng., vol. 5, no. 4, pp. 60–67, Oct.–Dec. 1998. [8] L. Tsang and Q. Li, “Wave scattering with UV multilevel partitioning method for volume scattering by discrete scatters,” Microw. Opt. Technol. Lett., vol. 41, no. 5, pp. 354–361, Jun. 2004. [9] A. Breuer, P. Borderies, and J. L. Poirier, “A multilevel implementation of the QR compression for method of moments,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2520–2522, Sep. 2003. [10] E. Michielssen and A. Boag, “A multilevel matrix decomposition algorithm for analyzing scattering from large structures,” IEEE Trans. Antennas Propag., vol. 44, no. 8, pp. 1086–1093, Aug. 1996. [11] J. Parron, J. M. Rius, and J. R. Mosig, “Application of the multilevel matrix decomposition algorithm to the frequency analysis of large microstrip antenna arrays,” IEEE Trans. Magn., vol. 38, no. 2, pp. 721–724, Mar. 2002. [12] J. M. Rius, J. Parron, E. Ubeda, and J. R. Mosig, “Multilevel matrix decomposition for analysis of electrically large electromagnetic problems in 3-D,” Microw. Opt. Tech. Lett., vol. 22, no. 3, pp. 177–182, Aug. 1999. [13] R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: A pedestrian prescription,” IEEE Antennas Propag. Mag., vol. 35, no. 6, pp. 7–12, Jun. 1993. [14] W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast Efficient Algorithms in Computational Electromagnetics. Boston, MA: Artech House, 2001. [15] Y. C. Pan, W. C. Chew, and L. X. Wan, “A fast multipole-methodbased calculation of the capacitance matrix for multiple conductors above stratified dielectric media,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 480–490, Mar. 2001. [16] L. J. Jiang and W. C. Chew, “Modified fast inhomogeneous plane wave algorithm from low frequency to microwave frequency,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jun. 2003, vol. 2, pp. 22–27. [17] B. Engquist and L. X. Ying, “Fast directional multilevel algorithms for oscillatory kernels,” SIAM J. Sci. Comput., vol. 29, no. 4, pp. 1710–1737, 2007. [18] H. Chen, J. Zhu, R. S. Chen, and Z. H. Fan, “Fast direction multilevel algorithm combined with Calderon multiplicative preconditioner for stable electromagnetic scattering analysis,” Microw. Opt. Technol.Let, vol. 52, no. 9, pp. 1963–1969, 2010. [19] O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag., vol. 37, pp. 918–926, 1989. [20] B. Dembart and E. Yip, “The accuracy of fast multipole methods for Maxwell’s equations,” IEEE Comput. Sci. Eng., vol. 5, no. 3, pp. 48–56, Jun.–Sep. 1998. [21] L. J. Jiang and W. C. Chew, “A mixed-form fast multipole algorithm,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 4145–4156, Dec. 2005. [22] Y. Saad, Iterative Methods for Sparse Linear System. New York: PWS Publishing, 1996.

REFERENCES [1] R. F. Harrington, Field Computation by Moment Methods. Malabar, FL: R. E. Krieger, 1968. [2] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 409–418, 1982. [3] 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. [4] J. R. Phillips and J. White, “A precorrected-FFT method for electrostatic analysis of complicated 3-D structures,” IEEE Trans. Comput.Aided Des. Integr. Circuits Syst., vol. 16, pp. 1059–1072, Oct. 1997.

Hua Chen was born in Anhui Province, China. She received the B.S. degree in electronic information engineering from Anhui University, China, in 2005. She 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, antennas and electromagnetic scattering and propagation, electromagnetic modeling of microwave integrated circuits.

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Kwok Wa Leung (S’90–M’93–SM’02–F’11) was born in Hong Kong on April 11, 1967. He received the B.Sc. degree in electronics and 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, Chinese University of Hong Kong. In 1994, he joined the Department of Electronic Engineering, City University of Hong Kong, as an Assistant Professor and is currently a Professor. From Jan. to June 2006, he was a Visiting Professor in the Department of Electrical Engineering, The Pennsylvania State University. 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 2006 and 2007. He was the Chairman of the Technical Program Committee, 2008 Asia-Pacific Microwave Conference, the Co-Chair of the Technical Program Committee, IEEE TENCON, Hong Kong, Nov. 2006, and the Finance Chair of PIERS 1997, Hong Kong. He was as an Editor for the Hong Kong Institution of Engineers (HKIE) Transactions from 2006 to 2010. He is currently serving as an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and received a Transactions Commendation Certificate presented by the IEEE Antennas and Propagation Society for his exceptional performance in 2008. He is also an Associate Editor for the IEEE Antennas Wireless Propagation Letters and a Guest Editor of IET Microwaves, Antennas and Propagation. 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 is a Fellow of HKIE.

Edward K. N. Yung was born in Hong Kong. He received the B.S. degree in 1972, the M.S. degree in 1974, and the Ph.D. degree in 1977, all from the University of Mississippi, University. After graduation, he briefly worked in the Electromagnetic Laboratory, University of Illinois at Urbana-Champaign. He returned to Hong Kong in 1978 and began his teaching career at 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 also active in applied research, consulting, and other technology transfers. He is the author of over 450 papers, including 270 in refereed journals. 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 Institution of Engineering and Technology, and the Hong Kong Institution of Engineers. He is also a member of the Electromagnetics Academy. He is listed in Marquis Who’s Who in the World and Who’s Who in the Science and Engineering in the World.

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Efficient Computation of the Off-Diagonal Elements of the Vector-Potential Multilayered Periodic Dyadic Green’s Function Ana L. Fructos, Rafael R. Boix, Member, IEEE, and Francisco Mesa, Member, IEEE

Abstract—The authors focus on the efficient computation of the slowly convergent infinite series that lead to the off-diagonal elements of the vector potential multilayered periodic dyadic Green’s function. Two different approaches based on Kummer’s transformation are applied to the evaluation of these series. The well-known approach that makes use of the generalized pencil of functions (GPoF) and Ewald’s method is the fastest approach, but it does not provide accurate results when the distance between the field point and any of the source points is close to zero. To avoid this problem, we present a novel approach based on the GPoF and the spectral Kummer-Poisson’s method with higher-order asymptotic extraction. This latter approach is slightly slower than the former one, but it is accurate in the whole range of distances between the field point and the sources. Index Terms—Convergence of numerical methods, Green’s functions, multilayered media, periodic structures, series.

I. INTRODUCTION HE analysis of the scattering of electromagnetic waves by periodic structures in multilayered media is essential to the design of frequency selective surfaces [1] and reflectarray antennas [2]. Also, the analysis of multilayered microwave circuits enclosed by rectangular shields [3]–[6] is a related problem that can be handled as an electromagnetic problem of periodic structures in layered media (see [4, Fig. 6] and [4, Eq. (71)]). One of the most popular techniques for the electromagnetic analysis of multilayered periodic structures is the application of the method of moments (MoM) to the solution of mixed potential integral equations (MPIE) [3]–[10]. The solution of these integral equations requires the numerical computation of both the scalar potential multilayered periodic Green’s functions (SPMPGF) and the vector potential multilayered periodic dyadic Green’s functions (VPMPDGF) in the spatial domain [4], [6], [11]. In the particular case of multilayered structures with 2-D periodicity, these spatial domain Green’s functions

T

Manuscript received July 01, 2010; revised November 02, 2010; accepted January 05, 2011. Date of publication May 12, 2011; date of current version July 07, 2011. This work was supported in part by the Spanish Ministerio de Educación y Ciencia and European Union FEDER funds (projects TEC201016948 and Consolider Ingenio 2010 CSD2008-00066) and in part by Junta de Andalucía (project P09-TIC-4595). A. L. Fructos and R. R. Boix are with the Microwaves Group, Department of Electronics and Electromagnetism, College of Physics, University of Seville, 41012 Seville, Spain (e-mail: [email protected]; [email protected]). F. Mesa is with the Microwaves Group, Department of Applied Physics 1, ETS de Ingeniería Informática, University of Seville, 41012 Seville, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2152344

can be expressed as double infinite series involving spectral domain Green’s functions [4], [11]. Unfortunately, the series are slowly convergent and their accurate computation demands a high CPU time consumption [4], [11]. For this reason, a large number of analytical and numerical methods have been proposed for the efficient evaluation of these series. Kummer’s transformation combined with Poisson’s formula [12] has been the most widely used analytical method [3], [10], [11, Sect. II]. Another relevant approach for the efficient computation of multilayered periodic Green’s functions with 2-D periodicity is based on the use of the discrete complex image method (DCIM). In [13] the DCIM is used to approximate the spatial multilayered non-periodic Green’s functions (MNPGF) in terms of spherical waves, and the double infinite series of these spherical waves are computed by means of Kummer’s transformation in the spectral domain. In [9] both spherical and cylindrical waves are included in the approximation of the spatial MNPGF via the DCIM, and the series of spherical waves and cylindrical waves are computed by means of Ewald’s method and the lattice sum method respectively. The problem with the DCIM is that spatial MNPGF do not only consist of spherical and cylindrical waves but also consist of residual waves [14], [15], and obtaining a general expression for the spatial MNPGF may become a formidable task [14], [15]. Therefore, an alternative interesting approach for the computation of multilayered periodic Green’s function is that presented in [16] and [17], where the DCIM is hybridized with Kummer’s transformation. In these two papers, the DCIM is used just to obtain the asymptotic expressions of the spectral multilayered Green’s functions, and the series containing these asymptotic expressions are computed by means of Ewald’s method. The spatial images method presented in [6] is a method that determines multilayered periodic Green’s functions in terms of a finite set of spatial MNPGF with a relatively small number of calculations. Among the strictly numerical methods employed for the efficient computation of multilayered periodic Green’s functions with 2-D periodicity, it is worth mentioning the “summation by parts technique” of [11, Sect. III]. Since the computation of multilayered periodic Green’s functions may be required thousands of times in real MoM problems, several researchers have proposed to compute the values of the periodic Green’s functions in a certain number of points within the unit cell, and then perform interpolation among the pre-computed values. An interpolation algorithm based on neural networks has been proposed in [5], and an interpolation algorithm that uses third-order B splines after singularity extraction has been reported in [18].

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In this work we focus on the efficient computation of the off-diagonal elements of the VPMPDGF. These off-diagonal elements are not required in the analysis of multilayered periodic structures or shielded microwave circuits that are strictly planar [3], [5], [10], but they are necessary when the periodic structures or the shielded circuits contain vertical interconnects such as vias, shorting pins or air bridges [19], [20]. Although the hybrid Kummer-DCIM method of [16] and [17] has proven to be a fast and accurate tool for the computation of the SPMPGF and the diagonal elements of the VPMPDGF, in this paper we show that this method fails to provide accurate results for the off-diagonal elements of the VPMPDGF when the distance between the field point and the sources approaches zero. In order to avoid this inaccuracy problem, we propose a novel method to compute the off-diagonal elements of the VPMPDGF. This novel method is based on Kummer’s transformation, and it uses an asymptotic term [21] plus a modified DCIM term in order to obtain accurate asymptotic expressions for the off-diagonal spectral Green’s functions. In the novel method the asymptotic series are calculated by means of the spectral Kummer-Poisson’s method with asymptotic extraction of higher-order terms [22]. The novel method is typically 50% slower than the hybrid Kummer-DCIM, but it turns out to be accurate in the whole range of distances between the field point and the sources. Concerning the organization of the paper, in Section II we present both the hybrid Kummer-DCIM method [16], [17] and the novel method for the efficient computation of the off-diagonal elements of the VPMPDGF. In Section III we present numerical results that show the pros and cons of the two methods, and in Section IV we summarize the conclusions. II. EFFICIENT COMPUTATION OF THE OFF-DIAGONAL ELEMENTS OF THE VPMPDGF Among the different MPIE formulations [4], in this paper we will strictly follow the formulation C of Michalski and Zheng [7]. Fig. 1(a) shows a lossless multilayered medium layers of permittivity and thickness consisting of . In the frame of formulation C of Michalski , and Zheng, let , and be the off-diagonal elements of the non-periodic modified dyadic Green’s function for the vector potential in the multilayered medium ( is the -component of the “correction term” vector defined in [7, Eq. (13)]), where is the vector pointing at the field point, is , and the vector pointing at the source point, . According to [4, Eq. (41)] and to the properties of Fourier transforms, the two-dimensional Fourier transforms , , and with respect to and of can be written as (1) (2) (3) (4)

Fig. 1. Multilayered medium limited by free-space (" ;  ) at the upper end, and limited either by a PEC or by free-space (" ;  ) at the lower end. A 2-D periodic array of point sources is located inside the multilayered medium, and the plane containing the array of sources is parallel to the interfaces between layers. (a) Side view of the multilayered medium. (b) Top view of the periodic array of source points, where a and a are the primitive non-orthogonal vectors of the 2-D periodic lattice.

and are cartesian Fourier variables, is the radial cylindrical Fourier variable, and . Let be the inverse two-dimensional Fourier transform of ( , 2) [4, Eq. (9)]. The functions can be expressed as a zeroth-order inverse Hankel transform of (also known as Sommerfeld integral) [4, Eqs. (34) and (35)] as shown below where

(5) where is the zeroth-order Bessel function. Coming back to Fig. 1(a) and 1(b), let us concentrate on the periodic array of point sources located inside the multiis the vector layered medium, where pointing at the point sources, , , and . Let , , and be the off-diagonal elements of the periodic modified dyadic Green’s function for the vector potential generated by the periodic array of sources in the multilayered medium. According to (1), (3) and (5), it can be shown that and can be written as

(6) where

,

is the projection on plane of the wavevector of a wave emitted by the the array of sources (positive sign) or a wave incident on the array are the angular spherof sources (negative sign), and and ical coordinates that indicate the propagation direction of these and waves. According to (1)to (5), an expression for both that is analogous to (6) can be obtained if and are substituted by and in (6) respectively.

FRUCTOS et al.: EFFICIENT COMPUTATION OF THE OFF-DIAGONAL ELEMENTS OF THE VPMPDGF

In the following, we will focus on the computation of the since the large degree of similarity between the function and indicates that the computation of , functions and can be carried out in a parallel way. Applying can be Poisson’s formula [12], the expression (6) for rewritten in the spectral domain as

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is available, we apply Kummer’s transformaOnce tion [12] to (7) by choosing as the asymptotic beas . Also, Poisson’s formula havior of is applied to the spectral asymptotic series containing [17]. After all these manipulations, the following approximate is obtained (see [16, Eq. (3)] for a similar expression for expression)

(7) , and . The function of (7) stands for the values of the function at the sampling points . can be obtained in closed form by means of the formulas provided in [4, Eq. of (6) (45), Sect. VI]. However, the function cannot be obtained in closed form and numerical integration of the Sommerfeld integral of (5) is the only existing way for the in the whole range accurate determination of [4], [7]. This means that expression (7) seems to be more convenient than expression (6) for the compu. Unfortunately, the series in (7) presents a slow tation of , , and the disconvergence when tance from the source and field points of Fig. 1(a) to the closest and . dielectric interface is much smaller than One possible procedure to speed up the computation of is to use the hybrid Kummer-DCIM method of [16] and [17]. Although this method was originally proposed for the computation of the SPMPGF and the diagonal elements of the VPMPDGF [16], [17], in principle the method may be also applicable to the computation of the off-diagonal elements of the VPMPDGF. Following [16], for every pair of values of and , we have to approximate the function in the portion plane where the singularities (poles of the real axis of the and branch points) of are not located. In particular, in this paper we approximate in the interval by means of a linear combination of complex exponential functions as shown below (see [16, Eq. in the variable (2)] for a similar expression) where

(8) and of (8) The unknown coefficients are obtained in this paper via the generalized pencil of functions technique (GPoF) [23] using two sampling paths. These two sampling paths are chosen to be the two intervals and of the real axis of (the the complex -plane, where -th layer of the multilayered substrate of Fig. 1(a) is the layer containing the array of point sources). Although other choices of sampling paths are possible which detour around the singu[24], it should be reminded we want to larities of stay far from these singularities in the approximation of (8).

(9)

( ; ). The limits of the first series in (9) have been of (8) truncated because we assume that the function provides a sufficiently accurate approximation for in the interval . In many practical applications the number of terms required for the computation of the truncated first series of (9) is below 100 (see the comments at the end of this Section). Also, the second series of (9) can be split into two series with Gaussian convergence by means of Ewald’s method [25], [26]. Therefore, (9) is substantially faster than (7) concerning the computation . Unfortunately, the truncation of the first series in (9) of . introduces inaccuracies in the computation of In order to understand the origin of the inaccuracies of (9), we have to take into account that the asymptotic behavior of as , , can be written as a linear combination of functions of the type (see [21, Eqs. (4) and (5)] and [27, Eq. (46)] for similar expressions) where

(10) The factor of (10) has been deliberately introduced to avoid the singularity of when [27], and the parameter has been chosen as so that is roughly one in the interval . The exponent of (10) may be zero when and the source and field points of Fig. 1 are located at one of the dielectric interfaces of the multilayered substrate (e.g., see [21, Eq. (5)] and [27, Eq. (38)]). Here, we would like to point as , out that the asymptotic behavior of , is also a linear combination of functions of the type shown in (10) (see [27, Eqs. (40)–(43)]), which justifies why the steps are completely required for the efficient computation of parallel to those required for the efficient computation of (and consequently, for the efficient computation of both ). The inaccuracies of (9) have to do with the fact that the funchas a decay of the type as tion (see (10)), and the function provides a wrong decay of the type for as

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(please remind the exponent may become zero). Since does not correctly capture the asymptotic behavior of as , the truncation carried out in the first series of (9) leads to inaccuracies in the computation of as , as it will be shown in Section III. This latter reas sult is to be expected since the behavior of is closely related to the behavior of as [21], [27]. The aforementioned inaccuracies are not present when the hybrid Kummer-DCIM method is applied to the computation of the SPMPGF and the diagonal elements of the VPMPDGF [16], [17]. And this is because the spectral , , , and of [7, formulation C] have functions an asymptotic decay of the type as , which is exactly the type of decay predicted by the approximation employed in (8). In this paper, we propose a novel alternative for the efficient which solves the inaccuracy problems computation of arising from (9). In this alternative, we first propose to approxin the interval imate the function in terms of complex exponential functions as shown below (11) where

(12) is again introduced to and where the factor avoid the singularity of when . The procedure followed for obtaining the unknown coefficients and of (12) is parallel to that followed for and . Note that unlike , the function obtaining provides for a correct decay of as . Once is available, the type we apply Kummer’s transformation to (7) by choosing as the asymptotic behavior of as . Assuming that gives a sufficiently accurate in the interval approximation of (as in (9)), the application of Kummer’s transformation leads : to the following approximate expression for

(13)

The time required for the computation of the truncated first series of (13) is basically the same as that required for the truncated first series of (9). Also, the second and third series of (13) can be efficiently computed by using the spectral Kummer-Poisson method with higher-order asymptotic extraction [22] that is described in the Appendix. This latter method is slightly slower than Ewald’s method, and therefore (13) demands a slightly larger CPU time than (9). Despite this drawback, (13) is preferable to (9) for the computation of since it does not suffer from inaccuracies as . When (9) and (13) are compared, one may think that the is not needed in both (11) and (13) since the asymptotic term is assumed to provide the correct mathematical term behavior of as . However, numerical simulations presented in Section III will prove that accuracy is lost in the approximation of for when is neglected. The reason for this is that whereas provides the analytically exact asymptotic behavior of as , only provides a numerical approximation of this asymptotic behavior in terms of complex exponentials. It should be pointed out that the number of terms that have to be added in the truncated first series of (9) and (13) grows with . In many practical increasing values of the ratio antenna and microwave applications, the ratio is smaller than 1 in order to avoid grating lobes in the scattering by periodic structures as well as cavity resonances in the analysis of shielded microwave circuits. In fact, the ratio is only larger than 1 in special circumstances such as periodic leaky wave antennas radiating higher order space harmonics [28], [29] or hybrid printed-waveguide filters where cavity resonances are deliberately excited to introduce transmission zeros [30]. When the condition applies, the number of terms required for the computation of the truncated first series of (9) and (13) is below 100 in the worst case scenario, which gives an idea of the numerical efficiency of these two equations in a large number of applications. III. NUMERICAL RESULTS ( , In Fig. 2(a) and (b), we plot the spectral functions 2) defined in (1) to (4) as a function of for fixed values of and . The figures also show the magnitude of the relative differences and . and Since are proportional to the terms appearing in the first series of (9) and (13), the plot of their magnitudes in Fig. 2(a) provides information about both the rate of convergence and the degree of accuracy of those series. In fact, Fig. 2(a) shows that and go to zero very fast in the interval , which implies that the number of terms required in the computation of the first series of (9) and (13) will be small. The range of values is the interval and where the functions are assumed to be zero in (9) and (13) respectively. In Fig. 2(a), we observe that whereas roughly remains within 0.01% when , substantially increases above 1%

FRUCTOS et al.: EFFICIENT COMPUTATION OF THE OFF-DIAGONAL ELEMENTS OF THE VPMPDGF

Q i=1 Q Q Q Q Q Q +Q f = 10 GHz N = 4 h = 0:7 mm h = 0:3 mm h = 0:5 mm h = 0:3mm " = 2:1 " = 12:5 " = 9:8 " = 8:6 N =N =5

Fig. 2. Solid lines stand for the magnitudes of ( , 2). Dotted lines, and dots and dashes stand for the magnitudes of the relative errors between and , between and , and between and . The results are presented for a four-layered medium limited by a PEC at the lower end. Parameters: , , , , , , , , , , .

in that range. Since the high values of in the interval occur when the values of are still relevant, this means that the truncation assumed in the first series of (9) may lead to important errors in the evaluation of . These errors are not expected in the truncated first series of (13) since the values of in Fig. 2(a) typically remain below when , and, therefore, (13) will always provide the values of with an accuracy of roughly four significant figures. In Fig. 2(a) we have also plotted the values obtained for when is obtained via (11) . Note that these and (12) by excluding the asymptotic term values of get close to in the interval . This indicates that (13) would have only provided with an accuracy of two significant figures at most if we had only extracted in that equation when applying Kummer’s method (i.e., if we had excluded the term in (11)). Comparison between Fig. 2(a) and (b) shows that

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@Q =@   i = 1 @Q =@   Q Q +Q

Q

Fig. 3. Magnitude of j 0 j ( , 2) computed in terms of ( , 2) via numerical integration of (5) (solid line). Comparison with the j 0 jj that are obtained when is replaced approximate values of j in (5), and when is replaced by (2 and ). by Relative errors between exact and approximate values (dashed line, and dots and dashes). Parameters of the multilayered medium and the GPoF as in Fig. 2.

i=1 Q

Q

all the conclusions drawn for the behavior of and the computation of can be immediately extended to the behavior and the computation of . of In Fig. 3(a) and (b), we plot the numerical values of ( , 2) as a function of . Manipulating (1) to (5), it can be shown that the values of are proportional to and , are proportional to and that the values of and (by virtue of the chain rule). In order to obtain the numerical values of , we have taken the derivative of (5) with respect to , and we have subsequently used the numerical integration technique described in [31]. In Fig. 3(a) and (b), we also plot the values of and . In order to obtain these by and latter values, we have replaced in (5), we have subsequently carried out the integration in closed form, and we have finally taken the deriva. Since and tive of the result with respect to are assumed to be the asymptotic limits of as

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, the theory of Fourier transforms tells us that the values and should match those of as of . And consequently, the values of and should also match those of as . Fig. 3(a) and (b) clearly shows match those of that the values of as , and that the relative error between the two sets of values is always smaller than 0.5% when . However, the values of do not correctly match those of (this is especially relevant in the case shown in Fig. 3(a), where the relative errors between the two sets of values can be as large as 100% ). This different behavior can be again when correctly provides the attributed to the fact that as whereas does not asymptotic limit of (see the comments about the results of Fig. 2(a) and (b) in the interval ). Note that as in Fig. 3(a), but as in Fig. 3(b) (see also [27, Figs. 10 and 11] for a similar behavior). The explanation for this difference has to do with ( , 2). the asymptotic behavior of the spectral functions Whereas in the case treated in Fig. 3(a) has an algebraic decay of the type as , in the case treated has an exponential decay of the type in Fig. 3(b) as . The relative errors introduced by ( , 2) as seem to be more relevant in Fig. 3(a) than in Fig. 3(b), which is attributed to the fact that shorter distances between the field and source points are involved in the former case. Fig. 3(a) and (b) has shown that the inaccuracies of ( , 2) for large will lead to inaccuracies in the computation of , , and as , i.e., as the field point approaches the source point. This seems to indicate that (9) will also lead to inaccuracies in the computation of the periodic Green’s function as the field point approaches any of the sources of the array in Fig. 1(b). This latter conclusion becomes apparent in Fig. 4(a), and it can be extended to the computation of as shown in Fig. 4(b). In Fig. 4(a) we plot the relative errors of the values of provided by (9) and (13) with respect to the virtually exact values of that are obtained with (7) when a sufficiently large number of terms is retained in this series. It can be observed that, whereas (13) is capable to keep the relative error in the computation of within roughly 0.01%, (9) leads to relative errors that are above 1% when (this latter result is coherent with the results observed in Fig. 3(a) when ). Also, in Fig. 4(a) we compare the CPU times required by (9) and (13) for computing with an accuracy of at least four significant figures whenever this accuracy can be achieved (as commented above, it will not be possible to achieve the required accuracy in the case of (9) as ). Fig. 4(a) shows that the CPU times required by (13) are typically 1.5 times larger than those required by (9). Therefore, (13) turns out to be much more accurate than (9) for the computation of at the expense of being slightly slower. Comparison between Fig. 4(a) and (b) shows that the conclusions drawn for the accurate and efficient

K

Fig. 4. Relative errors provided by (9) and (13) in the computation of (a), and relative errors provided by the equivalent equations in the computation of (b) ( and ). The curves stand for both the ratio between (a), and the ratio the CPU times required by (9) and (13) to compute between the CPU times required by the equivalent equations to compute (b) (2). Parameters of the periodic unit cell, field point and sources: a = (0 6^ x + 0 3^y) , a = (0 2^x + 0 4^y) , ( 0 ) = 0 875( 0 ), = = 4. Parameters of the multilayered medium and the GPoF as in Fig. 2.

K



:

T =T



:  =

computation of computation of

:

K

:  y y

:

K x x

can be straightforwardly extended to the . IV. CONCLUSIONS

The off-diagonal elements of the VPMPDGF can be expressed in terms of double infinite series with a very slow convergence rate. In this work, two different implementations of Kummer’s transformation have been used in order to speed up the convergence of these series. It has been found that the approach based on the combined use of the GPoF and Ewald’s method is the fastest approach, but is not accurate when the distance between the field point and the source points is close to zero. In order to avoid this problem, the authors have developed a novel approach based on the combined used of GPoF and the spectral Kummer-Poisson’s method with higher-order asymptotic extraction. This novel approach is slightly slower

FRUCTOS et al.: EFFICIENT COMPUTATION OF THE OFF-DIAGONAL ELEMENTS OF THE VPMPDGF

than the approach based on the GPoF and Ewald’s method, but it is accurate in the whole range of distances between the field and source points.

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B. Case In this case, the series below

can be efficiently computed as shown

APPENDIX The efficient computation of the second and third series of (13) requires the fast and accurate evaluation of series of the type

(14)

The terms of the series (14) decay like as both and , and therefore, the convergence of the and . In series may be very slow when both these slow convergence situations, we propose to compute the series in an efficient way by using the spectral Kummer-Poisson method with higher-order asymptotic extraction described in and separately. [22]. We will consider the two cases

(16)

are chosen as in the where the coefficients and the coefficients are given by case the solution of the following linear system of equations:

A. Case In this case, the series of the expression

(17)

can be efficiently computed by means The functions be written in the range

that appear in (17) can as (18) (19) (20) (21)

(15)

(22) (23)

where the coefficients

are chosen to be (see [22, Eq. (30)] for

(24)

a similar expression), (see [22, Eq. (27)] for a similar expression), and is the modified Bessel function of the first kind. With and , the terms in the first series of (15) this definition of for large and , and the terms in decay as the second series exponentially decay as a linear combination of for large terms of the type and .

(25) With the proposed choice of and , the terms in the first series of (16) decay as for large and , and the terms in the second series exponentially decay as a linear combination of terms of the type for large and .

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REFERENCES [1] B. A. Munk, Frequency Selective Surfaces. Theory and Design. New York: Wiley, 2000. [2] J. Huang and J. A. Encinar, Reflectarray Antennas. Hoboken, NJ: Wiley, 2008. [3] G. V. Eleftheriades and J. R. Mosig, “A fast integral equation technique for shielded planar circuits defined on nonuniform meshes,” IEEE Trans. Microw. Theory Tech., vol. 44, pp. 2293–2296, Dec. 1996. [4] K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propag., vol. 45, pp. 508–519, Mar. 1997. [5] J. Pascual-García, F. Quesada-Pereira, D. Cañete-Rebenaque, J. L. Gómez-Tornero, and A. Álvarez-Melcón, “A neural-network for the analysis of multilayered shielded microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 309–320, Jan. 2006. [6] J. S. Gómez-Díaz, M. Martínez-Mendoza, F. J. Pérez-Soler, F. Quesada-Pereira, and A. Álvarez-Melcón, “Practical implementation of the spatial images technique for the analysis of shielded multilayered printed circuits,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 131–141, Jan. 2008. [7] K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, part I: Theory,” IEEE Trans. Antennas Propag., vol. 38, pp. 335–344, Mar. 1990. [8] R. A. Kipp and C. H. Chan, “A numerically efficient technique for the method of Moments solution for planar periodic structures in layered media,” IEEE Trans. Microw. Theory Tech., vol. 42, pp. 635–643, Apr. 1994. [9] Y. Yu and C. H. Chan, “Efficient hybrid spatial and spectral techniques in analyzing planar periodic structures with nonuniform discretizations,” IEEE Trans. Microw. Theory Tech., vol. 48, pp. 1623–1627, Oct. 2000. [10] P. Baccarelli, S. Paulotto, and C. Di Nallo, “Full-wave analysis of bound and leaky modes propagating along 2D periodic printed structures with arbitrary metallization in the unit cell,” IET Microw. Antennas Propag., vol. 1, no. 1, pp. 217–225, Feb. 2007. [11] A. Álvarez-Melcón and J. R. Mosig, “Two techniques for the efficient numerical calculation of the Green’s functions for planar shielded circuits and antennas,” IEEE Trans. Microw. Theory Tech., vol. 48, pp. 1492–1504, Sep. 2000. [12] G. Valerio, P. Baccarelli, P. Burghignoli, and A. Galli, “Comparative analysis of acceleration techniques for 2D and 3D Green’s functions in periodic structures along one and two directions,” IEEE Trans. Antennas Propag., vol. 55, pp. 1630–1643, Jun. 2007. [13] R. M. Shubair and Y. L. Chow, “Efficient computation of the periodic Green’s function in layered dielectric media,” IEEE Trans. Microw. Theory Tech., vol. 41, pp. 498–502, Mar. 1993. [14] F. Mesa, R. R. Boix, and F. Medina, “Closed-form expressions of multilayered planar Green’s functions that account for the continuous spectrum in the far field,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 1601–1614, Jul. 2008. [15] R. R. Boix, A. L. Fructos, and F. Mesa, “Closed-form uniform asymptotic expansions of Green’s functions in layered media,” IEEE Trans. Antennas Propag., vol. 58, pp. 2934–2945, Sep. 2010. [16] M. J. Park and S. Nam, “Efficient calculation of the Green’s function for multilayered planar periodic structures,” IEEE Trans. Antennas Propag., vol. 46, pp. 1582–1583, Oct. 1998. [17] D. Wang, E. K. N. Yung, R. S. Chen, D. Z. Ding, and W. C. Tang, “On evaluation of the Green function for periodic structures in layered media,” IEEE Antennas Wireless Propag. Lett., vol. 3, no. 1, pp. 133–136, 2004. [18] G. Valerio, P. Baccarelli, S. Paulotto, F. Frezza, and A. Galli, “Regularization of mixed-potential layered-media Green’s functions for efficient interpolation procedures in planar periodic structures,” IEEE Trans. Antennas Propag., vol. 57, pp. 122–134, Jan. 2009. [19] O. Luukonen, M. G. Silveirinha, A. B. Yakovlev, C. R. Simovski, I. S. Nefedov, and S. A. Tretyakov, “Effects of spatial dispersion on reflection from mushroom-type artificial impedance surfaces,” IEEE Trans. Antennas Propag., vol. 57, pp. 2692–2699, Nov. 2009. [20] A. Hill, J. Burke, and K. Kottapalli, “Three dimensional electromagnetic analysis of shielded microstrip circuits,” Int. . Microw. MillimeterWave Comput.-Aided Eng., vol. 2, no. 4, pp. 286–296, 1992. [21] N. Hojjat, S. Safavi-Naeini, R. Faraji-Dana, and Y. L. Chow, “Fast computation of the nonsymmetrical components of the Green’s function for multilayer media using complex images,” Proc. Inst. Elec. Eng.-Microwave Antennas Propag., vol. 145, no. 4, pp. 285–288, 1998.

[22] A. L. Fructos, R. R. Boix, F. Mesa, and F. Medina, “An efficient approach for the computation of 2D Green’s functions with 1D and 2D periodicities in homogeneous media,” IEEE Trans. Antennas Propag., vol. 56, pp. 3733–3742, Dec. 2008. [23] Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag., vol. 37, pp. 229–234, Feb. 1989. [24] M. I. Aksun, “A robust approach for the derivation of closed-form Green’s functions,” IEEE Trans. Microw. Theory Tech., vol. 44, pp. 651–658, May 1996. [25] K. E. Jordan, G. E. Richter, and P. Sheng, “An efficient numerical evaluation of the Green’s function for the Helmholtz operator in periodic structures,” J. Comp. Phys., vol. 63, pp. 222–235, 1986. [26] S. Oroskar, D. R. Jackson, and D. R. Wilton, “Efficient computation of the 2D periodic Green’s function usign the Ewald method,” J. Comp. Phys., vol. 219, pp. 899–911, 2006. [27] R. R. Boix, F. Mesa, and F. Medina, “Application of total least squares to the derivation of closed-form Green’s functions for planar layered media,” IEEE Trans. Microw. Theory Tech., vol. 55, pp. 268–280, Feb. 2007. [28] A. A. Oliner, “Leaky-wave antennas,” in Antenna Engineering Handbook, R. C. Johnson, Ed., 3rd ed. New York: McGraw-Hill, 1993, ch. 10. [29] J. L. Gómez-Tornero, F. D. Quesada-Pereira, and A. Álvarez-Melcón, “Analysis and design of periodic leaky-wave antennas for the millimeter waveband in hybrid waveguide-planar technology,” IEEE Trans. Antennas Propag., vol. 53, pp. 2834–2842, Sep. 2005. [30] M. Martínez-Mendoza, J. S. Gómez-Díaz, D. Cañete-Rebenaque, J. L. Gómez-Tornero, and A. Álvarez-Melcón, “Design of bandpass transversal filters employing a novel hybrid structure,” IEEE Trans. Microw. Theory Tech., vol. 52, pp. 2670–2678, Dec. 2007. [31] J. R. Mosig and F. E. Gardiol, “Analytical and numerical techniques in the Green’s function treatment of microstrip antennas and scatterers,” Proc. Inst. Elect. Eng., vol. 130, no. 2, pt. H, pp. 175–182, 1983.

Ana L. Fructos received the Licenciado degree in physics and the Ph.D. degree from the University of Seville, Spain, in 2005 and 2010, respectively. In 2006, she joined the Electronics and Electromagnetism Department, University of Seville. Dr. Fructos was the recipient of a Scholarship financed by the Junta de Andalucía.

Rafael R. Boix (M’96) received the Licenciado and Doctor degrees in physics from the University of Seville, Spain, in 1985 and 1990, respectively. Since 1986, he has been with the Electronics and Electromagnetism Department, University of Seville, where he became Associate Professor in 1994. His current research interests are focused on the numerical analysis of periodic planar electromagnetic structures with applications to the design of frequency selective surfaces and electromagnetic bandgap passive circuits.

Francisco Mesa (M’93) was born in Cádiz, Spain, on April 1965. He received the Licenciado degree in June 1989 and the Doctor degree in December 1991, both in physics, from the University of Seville, Spain. He is currently an Associate Professor in the Department of Applied Physic 1, University of Seville, Spain. His research interest focus on electromagnetic propagation/radiation in planar structures.

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A Scalable Parallel Wideband MLFMA for Efficient Electromagnetic Simulations on Large Scale Clusters Vikram Melapudi, Member, IEEE, Balasubramaniam Shanker, Fellow, IEEE, Sudip Seal, Member, IEEE, and Srinivas Aluru, Fellow, IEEE

Abstract—The development of the multilevel fast multipole algorithm (MLFMA) and its multiscale variants have enabled the use of integral equation (IE) based solvers to compute scattering from complicated structures. Development of scalable parallel algorithms, to extend the reach of these solvers, has been a topic of intense research for about a decade. In this paper, we present a new algorithm for parallel implementation of IE solver that is augmented with a wideband MLFMA and scalable on large number of processors. The wideband MLFMA employed here, to handle multiscale problems, is a hybrid combination of the accelerated Cartesian expansion (ACE) and the classical MLFMA. The salient feature of the presented parallel algorithm is that it is implicitly load balanced and exhibits higher performance. This is achieved by developing a strategy to partition the MLFMA tree, and hence the associated computations, in a self-similar fashion among the parallel processors. As detailed in the paper, the algorithm employs both spatial and direction partitioning approaches in a flexible manner to ensure scalable performance. Plethora of results are presented here to exhibit the scalability of this algorithm on 512 and more processors. Index Terms—Accelerated Cartesian expansion (ACE), Cartesian expansions, fast multipole method (FMM), fast solvers, integral equation (IE), multipole methods, parallel multilevel fast multipole algorithm (MLFMA), scattering, self-similar tree, wideband MLFMA.

I. INTRODUCTION

T

HE study of radiation and scattering of electromagnetic (EM) fields aids in understanding the physics of a wide range of applications. These range from radar scattering cross-section of large objects, to finite antenna arrays, to remote sensing to biomedical imaging to signal integrity etc. Complex topologies of the bodies that are under interrogation preclude analytical treatment, and one needs to resort to numerical Manuscript received May 16, 2009; revised November 03, 2010; accepted January 29, 2011. Date of publication May 12, 2011; date of current version July 07, 2011. This work was supported by the National Science Foundation under Grants CCF-0729157 and DMS-0811197. V. Melapudi was with the Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824 USA. He is now with Ansys Inc., Ann Arbor, MI 48108-5213 USA (e-mail: [email protected]). B. Shanker is with the Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824 USA. S. Seal is with the Modeling and Simulations Group, Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6085 USA (e-mail: [email protected]). S. Aluru is with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011 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.2011.2152311

methods. Among several numerical methods, it is well-known that the integral equation (IE) methods are well-suited to model fields propagating in unbounded media as they impose exact radiation boundary condition and, in many cases, requires discretization of the boundary surfaces only [1]. The principle bottleneck to the popularity of IE based solver was its compu, tational complexity; both memory and time scale as is the number of spatial degrees of freedom. In the where past two decades, several fast algorithms have been developed to ameliorate the computational complexity of IE based solvers. Amongst them, the fast multipole method (FMM) [2], [3] and its variants enjoy a widespread popularity [4], [5]. The multilevel variant for the Helmholtz potential is popularly referred to as the multilevel fast multipole algorithm (MLFMA) in the computational electromagnetics community [4]. Many realistic simulations, involving complicated structures, fall under the category of multiscale problems that exhibit multiple scales in length or frequency or both. It is well known that the classical MLFMA suffer from low frequency breakdown [6], [7], and several modifications have been proposed to overcome this limitation [8]–[12]. In this paper, we will use the wideband MLFMA that was introduced by some of the authors in [11]. Development of the above fast methods has increased the size of the problems being analyzed from thousands to millions of unknowns [13]–[15]. As the problem size exceeds few millions of unknowns, the serial implementation of fast algorithms on single processor machines face limitations in terms of computational memory and speed. This, along with inexpensive and widespread availability of distributed or cluster computers serves as the motivation for exploring the parallel implementation of MLFMA [15]–[18]. However, the algorithmic sophistication of fast methods makes the development of efficient parallel algorithms difficult. MLFMA relies on (i) tree data structure to hierarchically partition the computational geometry and (ii) an alternate representation of the Greens function using multipoles. Several different approaches to partitioning the tree-data have been explored and they can be broadly classified into (a) spatial partitioning, (b) direction partitioning and (c) hierarchical partitioning. In spatial partitioning, the nodes of the tree are distributed among the processors and FMM expansions associated with a node is completely contained within the processor it resides in. This approach is effective when the number of terms in the FMM expansions associated with a node is constant, as in the case of the Laplace FMM. While spatial partitioning has been successfully used for the computation of electrostatic interactions in molecular dynamic simulations [19]–[24], it is not efficient when applied to the case of

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Helmholtz FMM, where the number of terms in the FMM expansions associated with a node depend on its level in the tree. This led to the development of direction partitioning strategy [17], [25]. In this approach, spatial partitioning is used up to a particular level and beyond this level the nodes are duplicated in all processors and their FMM expansion data is partitioned among the processors [17]. The level beyond which direction partitioning is used is determined by heuristics or a one-time tuning analysis. Though this approach does not ensure scalability beyond hundreds of processors, it has been well exploited to solve problems with several millions of unknowns on smaller clusters [25], [26]. Hierarchical partitioning was developed recently as a combination of spatial and direction partitioning approaches [27], [28]. Here spatial partitioning is used at the leaf level, and a systematic combination of spatial and direction partitioning is used at all other levels. This approach shows the promise of being provably scalable albeit with strict dependence and [28]. between In this paper, we develop a parallel implementation of the wideband MLFMA [11] that is scalable on large number of processors. This is achieved by developing strategies that ensure self-similar distribution of tree data, which leads to an algorithm that is implicitly load balanced. This work extends the recent developments in parallel algorithms for Laplace FMM [23], [29] to the Helmholtz FMM and the resulting algorithm is a seamless combination of spatial and direction partitioning strategies. It is well known, from Amdahl’s law, that the maximum parallel speed-up achieved is limited by the minimum time spent on serial computations of an algorithm. Hence we address the parallel implementation of every step of the tree computation algorithm and provide their cost analysis. The main contributions of this paper are: we present: 1) a scalable parallel algorithm for hierarchical tree computations; 2) theoretical bounds on the parallel performance of the presented algorithm; 3) a scalable parallel EM solver for wideband-MLFMA. Note, the implementation presented in this paper relies on the operators that are defined in [11] for wideband MLFMA. It was demonstrated that using the operators defined in [11], to traverse from accelerated Cartesian expansions (ACE)-MLFMA (and downwards), or for that matter, purely up and down the MLFMA tree, errors in fields computed are controllable to very if desired). This is aided by using high precision ( spherical harmonic filters defined in [30] or a modifications thereof for vector problems [31]. The rest of the paper is organized as follows: Section II provides a brief introduction to integral equations and the wideband MLFMA and its counterparts, the classical MLFMA for high frequencies and the ACE for low and mid range frequencies. Section III is devoted to the details of the parallel algorithms-parallel MLFMA (P-MLFMA), parallel ACE (P-ACE) and parallel ACE and MLFMA (P-ACE-MLFMA) proposed in this work. Here, we expound on the parallel implementation of every step of the hierarchical tree computation. Section IV presents a plethora of results that demonstrate the scalability and efficiency of the proposed algorithm. Finally, Section V summarizes the work presented in this paper.

II. PRELIMINARIES Consider the electromagnetic scattering from a closed perfect electric conductor (PEC) that is immersed in free space. Let denote the surface of this object that is equipped with a unit outward pointing normal . Electromagnetic fields deare incident upon the object. The incinoted by dent field induces an electric current that radiates scattered . The unknown current can be found fields by solving the combined field integral equation (CFIE), that may be written as (1) where and are the total electric is a surface conformal to and magnetic fields, respectively, and just inside and is an arbitrary scalar constant. through The scattered electromagnetic fields are related to the dyadic Green’s function (2) (3) (4) (5) In above relations, is the wavenumber, is the characteristic impedance of free space, is the identity dyad and is the scalar are repGreen’s function. Typically, the unknown currents [32] and the resented using RWG vector basis functions system of matrix equations obtained by using a Galerkin testing procedure is (6) The fast evaluation of the matrix-vector product relies on the rapid evaluation of the scalar potential. Hence, with no loss of generality, we consider the evaluation of the scalar potential due to sources (7) Variation to the vector case is well established [33] and will not be elucidated here. A. The Fast Multipole Method for the Helmholtz Potential Consider the evaluation of the potential at a point that is well-separated from a cluster of sources that reside within a sphere of radius . The FMM expansions that enables the fast evaluation of this potential is given as [33]

(8a)

MELAPUDI et al.: A SCALABLE PARALLEL WIDEBAND MLFMA FOR EFFICIENT ELECTROMAGNETIC SIMULATIONS

(8b) (8c) (8d) (8e) where is the center of multipole (local) expansion for source (observation) cluster, is the translation operator, and denotes an order spherical Hankel function of second kind and Legendre polynomial, respectively. The translation operator contains a spherical Hankel function which is singular at the origin. Thus, for numerical stability, neither the translation nor the wavenumber can be arbitrarily small [34], distance [35]. Hence, the classical MLFMA is inefficient or numerically unstable when applied to sub-wavelength problems, where the to capture the domain is discretized at a rate higher than geometric details. These limitations have been reported in detail in [36] and several alternatives have been proposed to overcome them [5], [8]–[10]. In this work we employ the accelerated Cartesian expansion (ACE) [11]. Next, we briefly present the ACE algorithm and its integration with MLFMA to create a wideband MLFMA.

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(9e) where and denote the -th order multipole and local is the ACE translation expansion Cartesian harmonics and operator. ACE is an almost kernel independent method as (a) all quantities of the ACE algorithm, except the translation operator, are independent of the form of the kernel and (b) the ACE expansions are rapidly converging for any non-oscillatory function [5], [38]–[40]. Readers interested in the details and other salient features of the ACE algorithm are referred to [38]. In the case of the Helmholtz potential, the ACE translator operator is given as [5]

(10) where

B. Accelerated Cartesian Expansion (ACE) ACE is a hierarchical tree computation algorithm that employs the generalized Taylor’s expansion to derive alternate representation of the Green’s function. The construction of ACE algorithm is similar to MLFMA in that it uses the oct-tree for geometry processing and derives equivalent operators for tree computation. In contrast, ACE employs Cartesian harmonics as multipole and local expansions. Use of Taylor’s series expansion for fast computation has been explored previously albeit with severe limitation on accuracy and performance. ACE provides a generic framework by adopting a tensorial formulation to exploit the full power of Taylor’s series expansion for fast denotes a tensor of rank computation. In rest of the paper, , the polyadic associated with is given by where and , an fold contraction between two tensors and is denoted when ; for more details by on these definitions and operations see [37], [38]. The ACE expansions for computing the potential in (7) can be written as (9a) (9b)

where, represents the modified Hankel function of order and is the floor operation. It is well known that above Taylor’s series expansion is convergent when either the domain size or frequency is small or both. Further, as the frequency tends to zero, it has been rigorously shown that, the above ACE expansion (10) seamlessly transitions to Laplace FMM [11]. C. Hybrid Algorithm for Multiscale Problems Multiscale problems, by definition, contains a mixture of subwavelength and large-wavelength problems. From above discussions it can be seen that, for evaluation of Helmholtz potential, the ACE algorithm is stable and efficient for sub-wavelength problems; while, MLFMA is efficient and optimal for large-wavelength problems. Thus, individually neither of the two algorithm is efficient for multiscale problems. A hybrid approach, where both ACE and FMM expansions are used in an optimal and seamless fashion, is required to achieve full efficiency with multiscale geometries. This implies that one needs to develop transition operators to switch from ACE to FMM expansions and vice versa [11]. These maps are given by

(9c)

(11)

(9d)

(12)

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Fig. 2. The Z-space filling curves or Morton ordering formed by the sorting the nodes of the tree at a particular level.

+

Fig. 1. An example compressed tree used in ACE MLFMA hybrid approach.

where denotes the center of ACE multipole expansion and the mapping operator . The derivation of and the proof of convergence can be found in [11]. The overall wideband MLFMA proceeds as shown in Algorithm 1. The computational geometry is represented using a compressed oct-tree [39], [41]. This is constructed by first embedding a cube enclosing the computational domain and recursively sub-dividing the large parent cubes into eight smaller, non-overlapping children cubes. The boxes at the lowest level of the tree, beyond which no sub-division occurs, are referred to as the leaf boxes. In rest of the paper, box and nodes are used inter-changeably. Interaction list is constructed for all nodes and nearfield list is constructed for all leaf nodes only [39], [42]. Next, as shown in Fig. 1, the tree nodes are classified as ACE or FMM based on the side length of the domain they represent. All nodes representing domains less than a certain pre-determined or , are classified as ACE nodes and rest size, typically of the nodes are labeled as FMM. The ACE-to-FMM multipole transition operator in (12) is used in step 4 and FMM-to-ACE local expansion transition operator in (11) is used in step 6 of Algorithm 1 [11]. Algorithm I Wideband Multilevel Fast Multipole Algorithm (MLFMA) 1: Construct the tree representation for the given geometry (distribution of discrete points). 2: Build interaction list for all tree nodes and the near-field list for leaf nodes only. 3: S2M: compute multipole expansions at each leaf node from sources contained within it. 4: M2M (upward traversal): compute the parent node multipole by combining the multipole expansions at their children node. 5: M2L (translation): for all nodes in the tree convert the multipole expansions to local expansions of the nodes in their interaction list. 6: L2L (downward traversal): update the local expansion information at a child node using the local expansion of their parent node. 7: L2O: use the local expansions about each leaf node to compute the farfield potential at its observation points. 8: NF: use direct method for computation of nearfield potential at observation points in each leaf node from sources contained in its near-field nodes.

III. PARALLEL ALGORITHM FOR MLFMA We present details of the parallel implementation of the wideband MLFMA outlined in the previous section. First, we present a scheme for constructing and partitioning the oct-tree data structure in parallel environment. This is followed by the details on parallel implementation of the individual tree computation steps in Algorithm 1. As mentioned in the introduction, our emphasis is on reducing the latency among processors to ensure the scalability of the algorithm. A. Parallel Construction of the Oct-Tree Although the construction of oct-tree is a one time effort and takes a negligible fraction of the overall parallel run-time, it is important because (a) tree partitioning among the processors directly affects the load balancing of the rest of the algorithm and (b) creation of various interaction lists at this stage are communication-intensive. In our implementation, the tree is stored in postorder traversal order. It will be shown that the resulting ordering of nodes enables load balanced computation of various tree operations, obviating the need for explicit load balancing. denote the total number of points (sources and obLet servers) distributed within a cubical domain of side length and let be the number of processors. Given the smallest side length associated with leaf boxes, the total number of levels . An integer coding or height of the tree is scheme [21] is used to uniquely represent a node in the tree. This has several advantages as (a) the keys encode a wealth of information such as the center position of the box represented by the node, level of the node, its entire ancestral lineage etc., and (b) the sorted keys conform to Morton ordering [43]. Morton ordering of the sorted leaf nodes distributed across processors results in a self-similar structure in each processor [14], [44] as shown in Fig. 2. Self-similarity is critical to parallel processing as it ensures that each processor has a similar local tree. Hence, the number of tree operations is similar for all processors and this leads to an implicitly load balanced scheme. In each processor, the full post-order tree (up to the root node) is constructed using the leaf nodes in that processor only. Thus the actual tree representing the entire computational domain is split among the processors, as illustrated with a binary tree in Fig. 3. In the case of oct-trees, the partitioning scheme is equivalent to distributing volume-regions of equal number of source/observers among the parallel processors. Next, we comment on the resulting distribution of tree-nodes among the processors. In each processor, as shown in Fig. 3, we have a part of the

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B. Distribution of ACE and MLFMA Multipole and Local Expansion Data

Fig. 3. Illustration of the tree partitioning scheme proposed in this paper. The subsequent distribution of nodes and duplicate nodes in each processor is also shown.

tree with at-least one node from every level. It is evident that some nodes can occur in multiple processors. When considering the global postorder traversal tree across processors, each such node is associated with a processor where its occurrence is appropriate (the processor which has the rightmost leaf box in the sub-tree of the node). This processor is referred to as the native processor for that node and every tree-node has an unique native processor. All other occurrences of the node are termed duplicate nodes and the following Lemma 1 provides a bound on number of such nodes. Lemma 1: The number of duplicate nodes in each processor is bounded by the height of the tree, and will appear sequentially at the end of the local postorder traversal tree. Proof: Let denote the height of the tree. A processor can have at most one duplicate node per level in the tree. The rationale for this statement is as follows: assume that a processor has at least two duplicate nodes at the same level in the tree. Let and be two such nodes, with occurring to the right of in the tree. A processor has a node in its local tree only if at least one of the leaf boxes in the subtree under the node falls in the same processor. Also, all the leaf boxes in a processor are consecutive in Morton ordering. Taken together, these two observations imply that the rightmost leaf box under must reside in the same processor. Thus, is native to this processor and cannot be a duplicated node. This argument demonstrates that a processor can have at most one duplicate node per level, shared with the next processor. Similarly, one can show that the number of multiply occurring nodes that are native to a processor are limited to one per level. The proof that the duplicate nodes will appear sequentially at the end of local postorder traversal tree follows from the fact that the postorder sequencing always places nodes before their parents. The parent of a duplicate node is also a duplicate node in the same processor. Hence all duplicate nodes in a processor appear in sequence at the end of the local postorder traversal tree. All tree-related processing can be implemented in an efficient fashion using a binary search algorithm operating on the postorder traversal tree. Next, we provide details on the parallel implementation of each step of the tree computations shown in Algorithm 1.

The above tree partitioning scheme ensures that the nodes are uniformly partitioned among the processors. If the size of data associated with each node is same, then actual data, not considering those of the duplicate nodes, is also uniformly distributed amongst the processors. This is true only for the ACE portion of the tree; in MLFMA, the number of expansions depends on the level of the node. Thus the FMM expansion data contained in the native processors can be considerably high, leading to severe load imbalance during tree computations especially during multipole-to-local translation operations. This undermines the scalability of the algorithm. To achieve uniform distribution of FMM expansion data, we use direction partition for duplicate nodes only. The FMM expansions data of a duplicate node is partitioned such that each copy contains an equal and distinct portion. This and the selfsimilar distribution of the tree ensures that each processor has an equal amount of FMM expansion data. Note that the nodes to be partitioned and number of partition are the result of the tree partitioning algorithm. It should be emphasized that not all nodes at a particular level are partitioned by the same amount. Evidently, the number of partition for a node depends on the number of its duplicate copies, which directly depends on the distribution of the sources/observers. A parent node with more number of sources/observers will have more duplicate copies as its children nodes, across levels, will be distributed among more processors. Henceforth this scheme is referred to as adaptive direction partition. This approach bears some similarity to the recently introduced hierarchical partitioning approach, where the multipole data of all nodes except the leafs are partitioned in a systematic manner [27], [28]. This imposes a strict dependence between and for scalability. In contrast, the proposed adaptive direction partition scheme is flexible and differs in the following manner: (a) it combines the spatial and direction partitioning in a seamless manner, i.e., not all nodes at a particular level are partitioned equally; (b) direction partitioning is used only when it is necessary; and (c) it provides a means of preserving the self similarity of tree computations. C. Construction of Interaction Lists Tree computation requires the construction of interaction and nearfield lists. Interaction lists are built for all the nodes in the local tree except duplicate nodes. This operation is split into serial and parallel portions. In the serial portion, the interaction list of each node is built assuming that the full tree is constructed [14]. Given a node’s key code, straightforward bit manipulation yields its parent node, the parent’s neighbor nodes and their children. This information is used to construct the interaction list of each local node. Due to locality there is no communication cost associated with this operation. In the parallel portion, the non-existent nodes are eliminated using one time communication. At this stage, we also construct different communication maps that will be used later for information exchange during tree traversal. A similar procedure is used to construct the nearfield list of leaf nodes.

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D. Multipole and Local Expansion Computation In each processor, the multipole expansions are computed at every node in the local postorder traversal tree up-to the root node. The postorder traversal order ensures that a parent node appears only after its children nodes (in case of duplicate nodes, all children that reside in the same processor). Thus, when a parent node occurs the necessary children multipoles are already computed. Multipole expansions are computed for all the local nodes, including the duplicate nodes. Note that the multipole expansions at the duplicate nodes are only partially filled as they account for sources in that processor only. Thus, after the local computation, all processors with duplicate nodes send their multipole expansions to the appropriate native processors of the duplicate nodes they host. The native processor of a node simply adds the received multipole expansion data to the appropriate local node. This algorithm is a one step update process with the following bound on communication overhead. Lemma 2: Total number of nodes received by a processor . during multipole computation is bounded by Proof: This follows from the fact that the number of du(see Lemma 1). plicate nodes in a processor is bounded by Since only the duplicate nodes are exchanged during multipole computations, the maximum number of nodes received by any . processor will be no more than The computation of local expansion is the reverse analogue of multipole computation. In the downward tree traversal, the child node local expansions are updated with the local expansion of their parent node. First, the processors with the duplicate nodes obtain their local expansion from the respective native processors. Then, the downward tree traversal is performed locally in each processor as we traverse the local postorder tree from right to left. Cost Analysis: Each processor has at least one node from every level of the tree and their multipole expansions are computed in every processor as we traverse the local postorder traversal tree. Thus, this part of the process is load balanced if every processor has the same number of leaf nodes. This is true even in the case of FMM where the number of multipoles increases as the level increases. Since the number of duplicate nodes per processor is bounded, the communication overhead involved in exchange of their multipole information is also bounded (and, as shown in the results section, the associated computational cost is negligible). Hence the overall process is load balanced. E. Translation Operation At each node in the global postorder traversal tree, we compute local expansions using multipole expansions of the nodes in its interaction list. This process is divided into parallel and serial portions. In the initial parallel portion, we exchange multipole information between processors. While building the interaction lists, as described in Section III.C, for each node in the local postorder traversal tree we identify the set of processors that require their multipole expansions. At every processor, we traverse through this list and send the requisite information to the appropriate processors. In our implementation, we exchange this data in blocks whose size is defined by the

user. This serves two purposes (a) the number of communication calls can be greatly reduced when compared to a scheme where the multipole data is exchanged one node at a time, and (b) the block size can be adjusted according to the communication architecture of the distributed environment to ensure maximum performance. There are several means of communicating data in blocks and the choice greatly depends on the system architecture and programming convenience. In our implementation, we observed that MPI_PACK/MPI_UNPACK were slow on the IBM-Bluegene and hence we used a complex buffer array for communicating data in blocks. The block size was chosen based on the available RAM after the geometry- and MLFMA pre-processing stages. Once the required multipole expansion data is received, the actual translation operation is performed in a serial manner to compute the local expansion of nodes in the local tree. In case of FMM translations, the duplicate nodes exchange and compute only part of their FMM expansion data in accordance with the adaptive direction partitioning strategy. This is possible due to the diagonal translation operation of the Helmholtz MLFMA. In actual implementation, the parallel communication and serial computation parts are performed in an intertwined fashion such that the translation operation is performed as and when the data is received. Communication of data by blocks facilitates this communication-computation overlap and asynchronous communication primitives like MPI_ISEND/MPI_IRECV were used to minimize latency. Cost Analysis: The translation operation is reciprocal. Thus, if two interacting nodes are in different processors, then both processors need to exchange same amount of information. This process would be load balanced if all processors receive and process the same amount of multipole data. In case of ACE computations, where the number of terms in the expansions are constant for all nodes, an uniform partitioning of tree nodes automatically ensures that the data is also uniformly partitioned. The same argument is not true in the case of MLFMA computation, where the number of terms in multipole expansions is a function of the level of the node. However, the use of adaptive direction partition ensures that the data is distributed uniformly across the processors. Hence, this part of the algorithm is also load balanced. F. Evaluation of Potential The farfield potential at the observation points are evaluated from the leaf node local expansion they reside in. However, the evaluation of the potential is completed only after accounting for the nearfield interactions. These are interactions only among leaf boxes, as specified by the nearfield list. Similar to translation operation, for each leaf node a list of processors that require its information is created and then sorted by processor-ID. At every processor we traverse through this list and communicate the leaf box information to appropriate processors in blocks. The nearfield potential is computed in a serial manner from the received data. This completes the evaluation of potential at every point across all processors. As in the case of translation operation, the communication and computation parts are intertwined and asynchronous communication is used to minimize wait time.

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TABLE I AVERAGE TIME SPENT BY AN INDIVIDUAL PROCESSOR AT DIFFERENT STAGES OF HIERARCHICAL TREE COMPUTATION MILLION POINTS UNIFORMLY DISTRIBUTED IN A CUBE USING THE P-ACE ALGORITHM, FOR

N = 40

G. Parallel Electromagnetic (EM) Solver Next, we briefly describe the use of the wideband, P-ACEMLFMA within the framework of EM solvers. When the discretization size is in the orders of millions, the geometry processing to create basis functions, etc. becomes computationally intensive. Though this is a one time process, an efficient parallel implementation of these steps is necessary to complement the P-ACE-MLFMA and reduce the overall solution time. We assume that the input to the parallel EM solver is a simple position of each node and the elemesh file with a list of ment-to-node connectivity table. Each of the processors reads nodes and elements from the input an equal share of the mesh file. Using the local element-to-node connectivity, a list of edges is created in each of the processors. Each edge is represented by the two global node numbers that make the edge and this two element integer array is sorted in parallel. Thus, every processor has approximately equal number of edges. The data of global node numbers allows one to gather the the nodes as they are sequentially distributed across the processors. This allows us to compute the centers of each edge which is then used to construct the oct-tree as described in Section III.A. Based on the distribution of leaf boxes the edge data are exdata changed among the processors and the necessary of related nodes are gathered from their global node number. IV. RESULTS In this Section, we present plethora of results that exhibit scalability of the parallel algorithm presented here. All the results, unless specified otherwise, were obtained on a IBM Blue Gene/L cluster with 1024 nodes, each with two dual core 700 MHz PPC440 processors and 512 MB RAM (http://bluegene. ece.iastate.edu/). The message passing interface (MPI) was used for communication between the processors. In our implementation of the classical MLFMA, we employ spherical harmonic filters [30] for interpolation and anterpolation during upward and downward tree traversal, respectively. With 512 MB RAM per node, pre-computation of these filter coefficients places memory constraints and this restricts the maximum number of FMM levels to 10 or the overall domain size to 128 . Furthermore, the computation of these filter coefficients on each processor also implies unfavorable memory scalability of our current implementation. In-spite of memory limitations, our choice of spherical harmonic filters was motivated by the fact that large scale levels) and simulations involves trees with high levels ( this technique guarantees accuracy and numerical stability. The first set of results correspond to kernel-only evaluations and this helps us to conduct in-depth study on the various aspects of the

parallel algorithm. This is followed by the use of these algorithms, augmented with an IE solver, for electromagnetic scattering problems. In all cases, the timings are reported in secis comonds and the parallel efficiency of the algorithm puted using (13) where and respectively denote the average time taken for evaluation of potential a processor and number of processors in the -th processor set, and ref is the smallest size processor set for a given . A. Kernel Evaluation The P-ACE algorithm and P-MLFMA were separately emrandom, uniployed to evaluate the scalar potential (7) at formly distributed source/observer pairs within a volume and surface. For volume distribution, we fill a cube of side-length and in case of surface distribution the points were placed on a sphere of radius . In evaluating the P-ACE algorithm the overall size of the domain was fixed at and the leaf box size was chosen such that the average number of sources/observers per leaf box is approxwas chosen so imately 60. The order of ACE harmonic as to evaluate the potential to an accuracy of [11]. The number of points was varied from 1 million to 80 million and in each case the number of processors was varied from 32 to 512. Table I shows, as a representative sample, the time spent at different stages of the tree computation for the case of million on different processor sets. In Table I, the columns titled parallel-multipole and parallel-local represents the time spent in communication of duplicate node data during upward and downward tree-traversal operations. These are negligible when compared to the computation times in columns Local-Multipole and Local Local-exp. This is in accordance with the theoretical design of the algorithm as Lemma 2 ensures that the volume of communications at this stage is bounded and small. Notice that the timing data for rest of the process scales proportionally with the number of processors. This is a direct consequence of the self-similar tree partitioning algorithm that ensures load balanced tree computation. The parallel efficiency of P-ACE algorithm for different cases is shown in the Fig. 4. In all cases ref million where was chosen to be 32 except for was used. The presented algorithm exhibits efficiency as high as 90% on 512 processors. Fig. 6 shows the time spent by indimillion at vidual processors of a 128 processor set for different steps of tree computation. This exhibits the excellent load balance of the prescribed algorithm as all the processors

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Fig. 4. Efficiency of the P-ACE algorithm for computation of Helmholtz potential between uniformly distributed random point within a cubical volume.

Fig. 6. Time spent by individual processors of a 128 processor set at different steps of tree computation for million using the P-ACE algorithm.

Fig. 5. Computational complexity of the P-ACE algorithm for the case of uniformly distributed random points in a cubical volume. The slope of linear line fits, shown by dotted lines, are close to unity and indicates the linear complexity of P-ACE algorithm.

Fig. 7. Efficiency of the P-ACE algorithm for computation of Helmholtz potential between uniformly distributed random points placed on the surface of a sphere.

N

spend almost the same amount at every step of the tree comas function of , for each putation. Next, in Fig. 5, we plot processor set, to measure the cost complexity of our parallel implementation. The slope of the linear line fits are close to unity scaling of our parallel implementawhich indicates the tion. Fig. 7 plots the efficiency of our P-ACE algorithm for the case of spherical distribution. Here again the efficiency is as high as 90% on 512 processors. This indicates the scalability of our parallel algorithm on large number of processors. In evaluating the P-MLFMA, the side-length of the leaf box and overall size of the domain was chosen such was fixed at that the average number of points per leaf box was approximately 60. The size of the domain for different number of points and distribution is presented in Table III. Table II shows, as a representative sample, the average time taken by one processor at different stages of hierarchical tree computation for million on different processor sets. As in the case of P-ACE algorithm, the parallel part of upward and downward tree traversal time are negligible and this is attributed to the bounded number of communications. Fig. 8 shows the time taken by the individual processors in a 128 processor set during translation with and without adaptive direction partition strategy. Without the adaptive direction partition, the native processors that host the

N = 40

N

duplicate nodes spend more time in communication and computation than others. The load balance among the processors improves with adaptive direction partition strategy and helps to improve the scalability of the P-MLFMA. The efficiency in case of volume distribution of points is shown in Fig. 10. The P-MLFMA offers efficiency as high as 90% on 512 processors. Fig. 9 shows the time taken by individual processors of a 128 processor set at different stages of tree computation. The negligible variations in time taken by different processors indicate the excellent load balance of the algorithm; which stems from the self-similar partitioning of tree nodes. In Fig. 11, the average is plotted as a function of time taken for potential evaluation for different processor sets . The slope of the linear line fits are close to unity which indicates the linear complexity of the proposed PFMM algorithm. The efficiency of the P-MLFMA for surface distribution of points is in shown in Fig. 12. The algorithm offers high efficiency on 512 processors even for relmillion. atively small number of points B. EM Simulations We employ the CFIE formulation to solve for electromagnetic scattering from closed PEC objects. The numerical systems were solved by a parallel GMRES solver with diagonal preconditioner.

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TABLE II AVERAGE TIME SPENT BY AN INDIVIDUAL PROCESSOR AT DIFFERENT STAGES OF HIERARCHICAL TREE COMPUTATION USING MILLION POINTS UNIFORMLY DISTRIBUTION WITHIN A CUBE OF SIDE-LENGTH 20 THE P-MLFMA, FOR

N = 20

TABLE III SIZE OF THE DOMAIN FOR P-ACE-MLFMA KERNEL-ONLY EVALUATION FOR VOLUME AND SURFACE DISTRIBUTIONS

Fig. 9. Time spent by individual processors of a 128 processor set at different million using the P-MLFMA. steps of the tree computation for N

= 40

Fig. 8. Time taken by individual processors, of a 128 processor set, in MLFMA multipole-to-local translation operation with and without adaptive direction partitioning.

First we validate our parallel solver by computing the plane wave scattering from a PEC sphere and comparing the RCS with analytical results from Mie series. In the examples presented here, the distribution of unknowns is rather uniform. As a result, we choose a leaf box of size of 0.25 that translates to approximately 20–30 unknowns per leaf node. Fig. 13 shows the comparison of RCS from a PEC sphere of radius 64 computed using the parallel solver and Mie series. As is evident, there is a excellent agreement between the two solutions and validates our parallel implementation. Fig. 14 shows the comparison of RCS of a PEC sphere of radius 128 , discretized with 14 million unknowns and both the solutions exhibit excellent agreement. Next, we present the scalability of the parallel solver on small number of processors and for this we consider the example of plane wave scattering from a 24 PEC sphere discretized with 600 000 unknowns [45] on the AMD cluster with infiniband connection at the High Performance Computing Center

Fig. 10. Efficiency of the P-MLFMA for evaluation of Helmholtz potential between N uniformly distributed random points within a cubical volume.

(HPCC), Michigan State University (http://www.hpc.msu.edu) to avail its larger memory—8 GB per node. Table IV shows and the time taken by the parallel solver for it shows excellent scalability on small number of processors and also exhibits the portability of our implementation. Next, we compute scattering from PEC spheres of different radius (and different number of unknowns) on the Bluegene system for large number of processors. We consider three spheres of rawith thousand unknowns. dius These simulations were performed on 32, 64, 128, 256 and 512 processors. When computing the efficiency using (13), the time denotes the solution time averaged across the processors and 32 . As shown in Fig. 15, the processor set as reference,

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Fig. 11. Computational complexity of P-MLFMA as a function of number of unknown for different processor sets P . The slope of linear line fits are shown by dotted lines.

N

Fig. 12. Efficiency of the P-MLFMA for evaluation of Helmholtz potential between uniformly distributed random points on the surface of a sphere.

parallel solver exhibits efficiency as high as 87% on 512 processors with 3.24 million unknowns. Next, we apply the parallel algorithm to two realistic geometries. In these two examples, we use the wideband FMM presented in [11] to avoid having a high concentration of unknowns cost-scaling. To do this, we in any box and the resulting choose the leaf size such that the number of basis functions at the smallest level lies between 20–30. This necessitates using both P-ACE at lower levels and P-MLFMA at higher levels and operators to communicate between the two at the transition level. Given the distribution of unknowns, the tree is inherently non-uniform, i.e., some branches are longer than the other. Further, the parameters for ACE and MLFMA were chosen such in the that they would guarantee an error in the field of norm, for details refer [11]. The first geometry is a PEC toy-aircraft with fine edges that is densely discretized with 1.75 million unknowns. At 3 GHz the principal dimension of the geometry was 64 and the Fig. 17(a) shows the induced surface currents. The time taken for one matrix-vector multiplication on a 256 processor cluster was approximately 50 seconds. The second geometry is a PEC sharp arrow discretized with 3.24 million unknowns. The principal dimension was 64 long at 3 GHz and the induced surface currents are shown in Fig. 17(b). The time taken for one matrix-vector product on a 256 processor cluster was approximately 30 seconds. The different matvec time for the two objects with different number of unknowns indicates that the distribution of unknowns, viz., the height of the tree, the number of ACE levels and transitions thereof play a key role in

Fig. 13. Comparison of RCS due to plane wave scattering from a 64 PEC sphere computed using the parallel EM solver and Mie series solution: (a) shows the full  range and (b) shows only a portion of the RCS for clarity.

Fig. 14. Comparison of RCS due to plane wave scattering from a 128 PEC sphere, discretized with 14 million unknowns computed using the parallel EM solver and Mie series solution.

the overall time. To validate that the timings for the 1.75 million unknown was a not an algorithmic issue, we performed the following numerical experiment: we recorded times for a matvec,

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TABLE IV TIME FOR ONE MATRIX-VECTOR CALCULATION FOR A 24 PEC SPHERE DISCRETIZED WITH 600 000 UNKNOWNS ON THE HPC, MSU

Fig. 15. Efficiency of the parallel EM solver, using the P-ACE-MLFMA, for the scattering from PEC sphere with different number of unknowns N and as number of processors was varied from 64 to 512.

Fig. 17. The resulting induced surface currents distribution, computed using the P-ACE-MLFMA augmented EM solver, on two multiscale geometries. (a) Toy-aircraft with sharp edges and (b) pyramid shaped arrow.

Fig. 16. Scalability of the upward and downward tree traversal computations in the parallel-EM solver when considering scattering from spheres of different size.

on IBM Bluegene, for a 1.75 million sphere ( in diameter) with a leaf box size of 0.25 which translates to approximately 30 unknowns/leaf node. The overall height of the tree is 9-levels. A single matvec for this configuration is approximately 7 on 128 processors. Reducing the height of the tree to 8-levels ). These results are in reduces the time to 2.8 s (7-levels keeping with the fact that distribution of unknowns and operations required to effect wideband MLFMA contribute to overall matvec times. Reference timings for other parallel MLFMA implementation can be obtained from [27]. Note, when comparing the data, it is important to compare uniformity of distribution in addition to clock speed and machine specific characteristics. V. CONCLUSION In this paper, we have developed a scalable parallel implementation of a wideband-MLFMA. Efficient EM simulation of multiscale geometries necessitates the use of the wideband-MLFMA—a hybrid and seamless combination of ACE

algorithm and MLFMA. In parallel implementation of these algorithms we lay emphasis on improving the scalability such that they can be efficiently executed on large scale clusters with thousands of processors. In this regard, the major contributions of this work can be listed as development of (1) an algorithmic framework for parallel implementation of every step of the tree computation algorithm, including the processing of mesh geometry, (2) a new strategy for distribution of tree nodes that results in self-similar trees in each processor, (3) theoretical bounds on communication during tree computations and (4) adaptive direction partition strategy to ensure equal distribution of FMM expansion data among the processors. The self-similar tree distribution and adaptive direction partition strategy ensures load balanced evaluation of different steps of the tree computation. Plethora of results are presented to demonstrate the different aspects of the parallel algorithms (P-ACE,P-MLFMA,P-ACE-MLFMA); particularly their scalability on large number of processors. ACKNOWLEDGMENT The authors are grateful to the High Performance Computing Center at Michigan State University, the ECE Department at Iowa State University and the NCSA Teragrid at the University of Illinois at Urbana-Champaign for providing computational resources. They also thank the anonymous reviewers for their detailed comments as it led to this revised paper that is both clear and complete. REFERENCES [1] A. Peterson, S. Ray, and R. Mittra, Computational Methods for Electromagnetics. Piscataway, NJ: IEEE Press, 1997.

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[2] V. Rokhlin, “Rapid solution of the integral equations of classical potential theory,” J. Comput. Phys., vol. 60, pp. 187–207, 1985. [3] L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems. Cambridge, MA: MIT Press, 1988. [4] , W. C. Chew, J.-M. Jin, E. Michielssen, and J. Song, Eds., Fast and Efficient Algorithms in Computational Electromagnetics. Norwood, MA: Artech House, 2001. [5] M. Vikram and B. Shanker, “An incomplete review of fast multipole methods—From static to wideband—As applied to problems in computational electromagnetics,” presented at the ACES Conf., 2008. [6] L. Greengard, J. Huang, V. Rokhlin, and S. Wandzura, “Accelerating fast multipole methods for the Helmholtz equation at low frequencies,” IEEE Comput. Sci. Engrg., vol. 5, pp. 32–38, 1998. [7] J. S. Zhao and W. C. Chew, “MLFMA for solving boundary equations of 2D electromagnetic scattering from static to electrodynamic,” Microw. Opt. Technol. Lett., vol. 20, pp. 306–311, 1999. [8] J. S. Zhao and W. C. Chew, “Three-dimensional multilevel fast multipole algorithm from static to electrodynamic,” Microw. Optical Technol. Lett., vol. 26, pp. 43–48, 2000. [9] E. Darve and P. Have, “Efficient fast multipole method for low-frequency scattering,” J. Comput. Phys., vol. 197, pp. 341–363, Jan. 2004. [10] H. Cheng, W. Y. Crutchfield, Z. Gimbutas, L. F. Greengard, J. F. Ethridge, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, “A wideband fast multipole method for the Helmholtz equation in three dimensions,” J. Comput. Phys., vol. 216, pp. 300–325, 2006. [11] M. Vikram, H. Huang, B. Shanker, and T. van, “A novel wideband FMM for fast integral equation solution of multiscale problems in electromagnetics,” IEEE Trans. Antennas Propag., vol. 57, pp. 2094–2104, 2009. [12] I. Bogaert, J. Peeters, J. Fostier, and F. Olyslager, “NSPWMLFMA: A low frequency stable formulation of the MLFMA in three dimensions,” presented at the IEEE Antennas and Propagation Society Int. Symp., Jul. 2008. [13] S. Velamparambil, J. M. Song, W. C. Chew, and G. K. , “ScaleME: A portable scalable multipole engine for electromagnetic and acoustic integral equation solvers,” in Proc. IEEE AP-S Int. Symp., 1998, vol. 3, pp. 1774–1777. [14] B. Hariharan, S. Aluru, and B. Shanker, “A scalable parallel fast multipole method for analysis of scattering from perfect electrically conducting surfaces,” in Proc. ACM/IEEE Conf. on Supercomputing, Los Alamitos, CA, 2002, pp. 1–17. [15] L. Gurel and O. Ergul, “Fast and accurate solutions of extremely large integral-equation problems discretised with tens of millions of unknowns,” Electron. Lett., vol. 43, pp. 499–500, Apr. 2007. [16] K. Donepudi, J. M. Jin, S. Velamparambil, J. M. Song, and W. C. Chew, “A higher-order parallelized fast multipole algorithm for 3D scattering,” IEEE Trans. Antennas Propag., vol. 49, pp. 1069–1078, 2001. [17] S. Velamparambil, W. C. Chew, and J. Song, “10 million unknowns: Is it that big?,” IEEE Trans. Antennas Propag., vol. 45, pp. 43–58, 2003. [18] C. Waltz, K. Sertel, M. Carr, B. Usner, and J. Volakis, “Massively parallel fast multipole method solutions of large electromagnetic scattering problems,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1810–1816, Jun. 2007. [19] P. B. Callahan and S. R. Kosaraju, “A decomposition of multidimensional point sets with applications to k-nearest neighbors and n-body potential fields,” J. ACM, vol. 42, pp. 67–90, 1995. [20] M. S. Warren and J. K. Salmon, “Astrophysical n-body simulations using hierarchical tree data structures,” in Proc. Supercomputing, 1992, pp. 570–576. [21] M. Warren and J. Salmon, “A parallel hashed oct-tree n-body algorithm,” in Proc. Super-Computing, 1993, pp. 1–12. [22] P. Liu and S. Bhatt, “Experiences with parallel n-body simulation,” IEEE Trans. Parallel Distrib. Syst., vol. 11, pp. 1306–1323, 2000. [23] F. Sevilgen, S. Aluru, and N. Futamura, “A provably optimal, distribution-independent parallel fast multipole method,” in Proc. IEEE Int. Parallel and Distributed Processing Symp., 2000, pp. 77–84. [24] S. H. Teng, “Provably good partitioning and load balancing algorithms for parallel adaptive n-body simulation,” SIAM J. Sci. Comput., vol. 19, pp. 635–656, 1998. [25] O. Ergul and L. Gurel, “Efficient parallelization of the multilevel fast multipole algorithm for the solution of large-scale scattering problems,” IEEE Trans. Antennas Propag., vol. 56, pp. 2335–2345, 2008. [26] S. Velamparambil and W. C. Chew, “Analysis and performance of a distributed memory multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag., vol. 8, pp. 2719–2727, 2005.

[27] O. Ergul and L. Gurel, “A hierarchical partitioning strategy for an efficient parallelization of the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag., vol. 57, pp. 1740–1750, Jun. 2009. [28] J. Fostier and F. Olyslager, “A provably scalable parallel multilevel fast multipole algorithm,” Electron. Lett., vol. 44, pp. 1111–1113, Sep. 2008. [29] M. Vikram, A. Baczewski, B. Shanker, and S. Aluru, “Parallel accelerated Cartesian expansions for particle dynamics simulations,” presented at the IEEE Int. Parallel and Distributed Processing Symp., 2009. [30] R. J. Chien and B. K. Alpert, “A fast spherical filter with uniform resolution,” J. Comput. Phys., vol. 136, pp. 580–584, 1997. [31] B. Shanker, A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag., vol. 51, no. 3, pp. 628–641, Mar. 2003. [32] S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, pp. 409–418, May 1982. [33] 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. [34] S. Koc, J. Song, and W. C. Chew, “Error analysis for the numerical evaluation of the diagonal forms of the scalar spherical addition theorem,” SIAM J. Numer. Anal., vol. 36, no. 3, pp. 906–921, 1999. [35] E. Darve, “The fast multipole method I: Error analysis and asymptotic complexity,” SIAM J. Numer. Anal., vol. 38, pp. 98–128, 2000. [36] M. Nilsson, “Stability of the high frequency fast multipole method for Helmholtz equation in three dimensions,” BIT Numer. Math., vol. 44, pp. 773–791, 2004. [37] J. Applequist, “Traceless Cartesian tensor forms for spherical harmonic functions: New theorems and applications to electrostatics of dielectric media,” J. Phys. A: Math. Gen., vol. 22, pp. 4303–4330, 1989. [38] B. Shanker and H. Huang, “Accelerated Cartesian expansions—A fast method for computing of potentials of the form R for all real  ,” J. Comput. Phys., vol. 226, pp. 732–753, 2007. [39] M. Vikram and B. Shanker, “Fast evaluation of time domain fields in sub-wavelength source/observer distributions using accelerated Cartesian expansions (ACE),” J. Comput. Phys., vol. 227, pp. 1007–1023, 2007. [40] M. Vikram, A. Baczewski, B. Shanker, and L. Kempel, “Accelerated Cartesian expansion (ACE) unified framework for the rapid evaluation of potentials associated with the diffusion, lossy wave and KleinGordon equations,” J. Comput. Phys., 2008. [41] H. Cheng, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm in three dimensions,” J. Comput. Phys., vol. 155, pp. 468–498, 1999. [42] L. Greengard and V. Rokhlin, “A new version of the fast multipole method for the Laplace equation in three dimensions,” Acta Numerica, vol. 6, pp. 229–269, 1997. [43] G. M. Morton, “A Computer Oriented Geodetic Data Base and a New Technique in File Sequencing,” IBM, Tech. Rep.. [44] M. Griebel, S. Knapek, and G. Zumbusch, Numerical Simulation in Molecular Dynamics. Berlin: Springer-Verlag, 2007. [45] S. Velamparambil and W. C. Chew, “Parallelization of MLFMA on distributed memory computers,” in Proc. Int. Con. on Electromagnetics in Advanced Applications (ICEAA01), Sep. 2001, pp. 141–144. Vikram Melapudi (M’09) received the B.Tech. degree from the Indian Institute of Technology, Madras, India, in 2003 and the Ph.D. degree in electrical engineering from Michigan State University, East Lansing, in 2009. He continued as a Visiting Postdoctoral Research Associate at Michigan State University, until March 2010. Currently, he works at Ansys Inc. (former Ansoft), Ann Arbor, MI, as a Senior Research and Development Engineer for signal integrity products. His primary research interest is in the broad area of computational science and its applications; especially, in the development of fast and parallel algorithms for electromagnetics and multiphysics simulations. Among his other research interests are signal and image processing, applied analysis and non destructive evaluation. Dr. Melapudi was the recipient of the third prize award in the student paper competition of the IEEE AP-S International Symposium and USNC/URSI National Radio Science Meeting, San Diego, CA, 2008.

MELAPUDI et al.: A SCALABLE PARALLEL WIDEBAND MLFMA FOR EFFICIENT ELECTROMAGNETIC SIMULATIONS

Balasubramaniam Shanker (F’09) received the B.Tech. degree from the Indian Institute of Technology, Madras, India, in 1989, and the M.S. and Ph.D. degrees from the Pennsylvania State University, University Park, in 1992 and 1993, respectively. From 1993 to 1996, he was a Research Associate in the Department of Biochemistry and Biophysics, Iowa State University, where he worked on the molecular theory of optical activity. From 1996 to 1999, he was with the Center for Computational Electromagnetics, University of Illinois at Urbana-Champaign, as a Visiting Assistant Professor, and, from 1999 to 2002, he was with the Department of Electrical and Computer Engineering, Iowa State University, as an Assistant Professor. Currently, he is a Professor in the Department of Electrical and Computer Engineering, Michigan State University, East Lansing. He has authored/coauthored over 300 journal and conferences papers and presented a number of invited talks. His research interests include all aspects of computational electromagnetics (frequency and time domain integral equation based methods, multi-scale fast multipole methods, fast transient methods, higher order finite element and integral equation methods), propagation in complex media, mesoscale electromagnetics, and particle and molecular dynamics as applied to multiphysics and multiscale problems. Dr. Shanker is a full member of the USNC-URSI Commission B and is an elected Fellow of the IEEE, for his contributions to computational electromagnetics. He has also been awarded the 2003 Withrow Distinguished Junior Scholar, the 2010 Withrow Distinguished Senior Scholar, and the 2007 Withrow Teaching Award. He was an Associate Editor for the IEEE Antennas and Wireless Propagation Letters (AWPL) and is currently an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

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Sudip Seal (M’10) received the Ph.D. degree in theoretical physics from New Mexico State University, Las Cruces, in 2002 and the Ph.D. degree in computer engineering from Iowa State University, Ames, in 2007. From 2007 to 2009, he was a Postdoc in the Modeling and Simulations Group, Oak Ridge National Laboratory, Oak Ridge, TN, where, in 2009, he became a research staff member in the Computational Sciences and Engineering Division. His primary research is in the design and development of parallel algorithms for science and engineering applications. At present, his research focusses on scalability of parallel discrete event simulations on very large parallel platforms.

Srinivas Aluru (F’10) received the B.Tech. degree in computer science from the Indian Institute of Technology, Chennai, India, in 1989 and the M.S. and Ph.D. degrees in computer science from Iowa State University, Ames, in 1991 and 1994, respectively. Currently, he is the Ross Martin Mehl and Marylyne Munas Mehl Professor of Computer Engineering at Iowa State University, Ames, and the Bajaj Chair Professor of Computer Science and Engineering, Indian Institute of Technology, Bombay, India. Previously, he held faculty positions at Syracuse University (1994–96) and New Mexico State University (1996–99). He conducts research in parallel algorithms and applications, bioinformatics and systems biology, and scientific computing. Dr. Aluru is an elected Fellow of the IEEE and a Fellow of the American Association for the Advancement of Science (AAAS). He is a recipient of the NSF Career and IBM faculty awards. He is an Associate Editor of the IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS and served as program Vice Chair for algorithms or applications in IEEE Conferences in parallel computing including Supercomputing (2008 and 2003), and the International Parallel and Distributed Processing Symposium (2007).

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A Quasi-3D Thin-Stratified Medium Fast-Multipole Algorithm for Microstrip Structures Jie L. Xiong, Yongpin Chen, and Weng Cho Chew, Fellow, IEEE

Abstract—An accurate and efficient full-wave simulation method is proposed for general microstrip structures. It is called the quasi-3D thin-stratified medium fast-multipole algorithm (TSM-FMA). Different from the 2D TSM-FMA method, it is constructed on a vector wave function based dyadic Green’s function for layered medium (DGLM) instead of the symmetric form DGLM. This new form of DGLM is represented in terms of only two Sommerfeld integrals and is suitable for developing fast algorithm in quasi-3D cases. Similar to the 2D TSM-FMA, the path deformation technique and the multipole-based acceleration is used to transform the Sommerfeld integral and expedite the matrix-vector multiplication. The computational time per iteration and the memory requirement is ( log ) in the quasi-3D TSM-FMA. It is suitable to perform a full-wave analysis of a large microstrip array with less computational resource. Index Terms—Fast-algorithm, Green’s function, layeredmedium.

I. INTRODUCTION

M

ICROSTRIP structures have a large number of varieties in their geometry configuration and efficient full-wave solvers for microstrip structures have been studied extensively. One of the popular methods is using integral equation method with dyadic Green’s function for layered medium (DGLM). The advantage of this method is that only the microstrip part needs to be discretized. The influence of the dielectric substrate and the ground plane is taken into account by the DGLM. This results in much fewer unknowns and leads to a matrix equation that is smaller in size. However, there are two major challenges in this approach. First, the matrix element involves Sommerfeld integrals, whose integrand is highly oscillatory and difficult to evaluate. Second, in the case of large microstrip arrays, although the unknowns are only associated with the patches, the full-wave analysis needs the solution of a very large system of linear equations. So the matrix-solving process needs to be accelerated. In an earlier work, Zhao et al. [1], have proposed a thin-stratified medium fast-multipole algorithm (TSM-FMA) to meet the challenge. It could reduce both the memory requirement and the cost Manuscript received February 10, 2010; revised November 04, 2010; accepted January 29, 2011. Date of publication May 10, 2011; date of current version July 07, 2011. J. L. Xiong was with the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign. She is now with the Schulumberger-Doll Research Center, Cambridge, MA 02139-1579 USA. Y. Chen is with the Department of Electrical and Electronic Engineering, University of Hong Kong, Hong Kong. W. C. Chew is with the Faculty of Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 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.2011.2152324

or of matrix-vector multiplication per iteration to less. However, it was a 2D algorithm and could not handle a large class of quasi-3D microstrip structures, such as microstrip arrays fed by probes. In this paper, we use similar ideas as in the 2D TSM-FMA and propose a quasi-3D TSM-FMA. Quasi-3D means that it is most efficient for 3D objects whose horizontal dimension is much larger than their vertical dimension, which is applicable to almost all microstrip structures. To accelerate the numerical integration of Sommerfeld integrals, there are three popular approaches: one is to pre-compute the integrals on a grid of points in the solution domain and to use interpolation techniques. For a strictly planar microstrip structures, only 1D interpolation on is required. If the object is confined in a single layer, the DGLM could be split into two parts, depending on , and , respectively. So 2D interpolation is needed. For an arbitrarily shaped object, the interpolation should be done with three variables: , and [2]. Another vastly popular approach is the discrete complex image method (DCIM) [3]–[10]. The basic idea of the method is to extract the quasi-static image and guided waves from the spectral kernel, and approximate the remaining term by the sum of complex exponentials. Then the Sommerfeld integral could be evaluated in closed form via the Sommerfeld identity [11] for each term, which is known as a complex image. This method is very efficient since it obviates numerical integration. The drawback is that it has no built-in convergence measure, and its accuracy can only be checked against conventional Sommerfeld integrations. In this paper, we would adopt the third approach, the path deformation technique [1], [11]–[14] as in the 2D TSM-FMA. The integrand decays exponentially along the steepest descent path (SDP) and the deformed integral can be evaluated efficiently. Mathematically, this technique is robust and the numerical error could be controlled by the truncation point and the order of Gaussian quadratures used in the numerical integration along the SDP. Usually it is difficult to find an SDP exactly, but a convenient approximate path exists for thin-stratified media. The conjugate gradient fast Fourier transform (CG-FFT) [15]–[20] method and the multilevel fast-multipole algorithm (MLFMA) based methods [21]–[25] are two commonly used methods for solving large microstrip problems. Both methods could significantly speed up the iterative solver by reducing the computation complexity for the matrix-vector multiplication to . Here, CG-FFT is more efficient than MLFMA based method when dealing with regular grid geometries (dense structure with rectilinear meshes), which are suitable for 2D-FFT. Since the FFT algorithm has been studied for several decades, it is highly mature. For irregular meshes or arbitrarily shaped structures, the meshes should be projected

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XIONG et al.: A QUASI-3D THIN-STRATIFIED MEDIUM FAST-MULTIPOLE ALGORITHM FOR MICROSTRIP STRUCTURES

onto a regular grid to apply CG-FFT. When the structure is sparse (e.g., much empty space between microstrip lines), the efficiency of CG-FFT will deteriorate because the empty space has to be padded with zeros in 2D-FFT. On the other hand, the MLFMA based methods can handle triangle meshes and sparse structures easily without additional work or loss of efficiency. In addition, tabulating and interpolating techniques could be used in conjunction with existing FMA methods to effectively increase the computational speed of the Green’s function [26]. The quasi-3D TSM-FMA method is an MLFMA based method as in the 2D TSM-FMA. Although the basic acceleration techniques in the quasi-3D TSM-FMA and the 2D TSM-FMA are similar, they are based on completely different formulation of the DGLM. The 2D TSM-FMA uses a symmetric form of the DGLM. Due to the complication of the propagation factors, a more succinct DGLM is desired in the quasi-3D case. So we choose the vector wave function based DGLM [27]. This form has been proven to be more friendly for matrix element computation when we use the method of moments [28]. It has been extended to simulate the electrically small structures situated in a layered medium [29] as well. In fast algorithm application, this vector wave based DGLM also offers significant advantage over other forms of DGLM. The advantage will be fully revealed in the formulation part.

II. FORMULATION The typical geometry under consideration is shown in Fig. 1, which is an antenna array fed by probes. In this section, the key points of the quasi-3D TSM-FMA are discussed. We start with the vector wave function based DGLM. First, the evaluation of Sommerfeld integrals is expedited by deforming the integration path from the Sommerfeld integration path (SIP) to the vertical steepest descent path. Then the deformed integral is approximated by using standard Gauss-Legendre quadratures. For each quadrature point, two-dimensional multilevel fast-multipole algorithm (2D MLFMA) is applied to accelerate the matrix vector multiplication process. The direct interaction term is treated similarly after some manipulation. A. Review of 2D MLFMA and the New Form of DGLM In the original 2D TSM-FMA, the dyadic Green’s function for layered medium is developed in a symmetrical form. The Sommerfeld integrals contained in the dyadic Green’s function are expressed as the discrete summation on a few Gaussian integration points along the vertical branch cut and the poles. The Hankel function in each term of the summation could be decomposed and the matrix-vector multiplication is accelerated similarly as in 2D MLFMA. We would first briefly review 2D MLFMA and then discuss the choice of the expression of the DGLM in the quasi-3D application. The 2D MLFMA is based on the decomposition of the integral representation of Hankel function using the translational addition theorem [30], [31]. If the elements of an impedance matrix are proportional to , then 2D MLFMA could reduce the matrix-vector multiplication cost to . The

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Fig. 1. Geometry of the probe-fed microstrip antenna arrays.

single level algorithm could be illustrated by a three-stage information transmission from source points to field points: aggregate from each source point to the center of its group, translate the information of each group to other groups who are not their near neighbors, and finally disaggregate the field information from the group centers to each field point. This technique is first applied in the two-dimensional scattering problem, since the 2D scalar Green’s function is . But it is not limited to that simple case. Consider a Green’s function of the following form in the spectral domain:

(1) The matrix-vector product could be calculated efficiently by following the same process described earlier. Only the radiation pattern and receiving pattern have been changed. If we want to apply 2D MLFMA to accelerate the Green’s function in the quasi-3D case, we would like to express the DGLM as a discrete summation of functions that have the form of (1). The discrete form should contain as few terms as possible to minimize the cost. The vector wave function based formulation of DGLM [28] is more suitable for this application than the symmetric form used in the original TSM-FMA [1]. In the matrix-friendly formulation, the DGLM is expressed in terms of only two scalar potential integrals , , corresponding to TE and TM waves respectively. There are only two fundamental Sommerfeld integrals involved, much less than that in [1]

(2) The expressions for

and

are

(3)

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Fig. 2. Deformation of the Sommerfeld integration path to the vertical branch cut.

Here we have assumed that the source point is in layer while the field point is in layer . The poles at in (3) disappear when they are substituted into (2), because physically, they are fictitious poles. In the following part, we would discuss the path deformation technique and use it to write the Sommerfeld integral as a discrete summation of functions with the form of (1). B. Path Deformation of Sommerfeld Integration Since the integrand in (3) is usually highly oscillatory and decays slowly along the Sommerfeld integration path (SIP), it is difficulty to evaluate it efficiently. By virtue of Jordan’s lemma and Cauchy’s theorem, it is desirable to deform the integration path from the original Sommerfeld integration path (SIP) to the steepest descent path (SDP), where the integrand is decaying rapidly so that numerical integration is more efficient. However, the analytical expression of the SDP is not always available and it changes with source and observation points. It would be inefficient if we need to spend resources to find the SDP for each pair of and . Thus, a simple and geometry independent path where the integrand also decays exponentially would be more preferable. Moreover, in the region enclosed by SIP and the new path, the integrand should be analytic except for the pole singularities. For the thin-stratified medium problems, such as microstrip structures, the horizontal dimension is usually much larger than the vertical dimension. In this case, the path around the vertical branch cut (Fig. 2) would be an optimal choice [1], [21], [22]. Now the evaluation of integral in (3) is done in two parts: the detoured path integral around the vertical branch cut and the contribution from the poles enclosed by the new path and SIP. The total integral is expressed as:

Fig. 3. Sign choice for k

and k

in complex k plane.

we would encounter two branch points. For general application, we would discuss the most complicated case where two branch and are multivalue functions assopoints exist. Since , it is very important to choose their signs ciated with and correctly to ensure that the integrand is continuous along the inand should tegration path. Both the imaginary parts of or , so that the intebe positive when . To satisfy the continuity condigrand vanishes at should change whention, the sign of the imaginary part of ever the integration path goes across the solid line representing . Similarly, the sign of the real part of changes whenever the integration path goes across the dash line repre. The same thing is true for . Since senting region I is split by the and lines, it and based on is not efficient to determine the signs of and their imaginary parts. But the signs of the real parts of should be consistent throughout region I and opposite to that in upper part of region III. , we can start from To find the sign of the real parts of the upper part of region III. In this paper, we use prime and double prime to denote the real and imaginary part of a com, we plex variable respectively. If we assume in the upper part of region III (above the have . Thus, from (5) we conclude dash line). In addition, that in this part. This suggests that both and is positive region I

(4)

(5)

We would discuss each of them separately. 1) Integration Around the Vertical Branch Cut: The vertical branch cuts always start from the branch point in the complex plane. For a general planar layered media, there are at most two branch points associated with the outermost layers on the top or bottom respectively (section 2.7 [11]). In microstrip structures, the dielectric substrate always is backed by a ground plane and only one branch point exists. But for other applications, for example, interconnects on top of a silicon substrate,

Similarly, in region III, and is consistently posshould be positive and be itive. In region II, and in each repositive. The rule of choosing signs for gion is summarized in Fig. 3. The deformed integral is smooth and decays fast, so that it can be easily evaluated using Gauss-Legendre quadrature. However, when poles exist near the branch point, it would cause the integrand to vary rapidly in the neighborhood of the pole. Hence, the integration range is divided into two ranges [1]. Starting from

XIONG et al.: A QUASI-3D THIN-STRATIFIED MEDIUM FAST-MULTIPOLE ALGORITHM FOR MICROSTRIP STRUCTURES

the branch point, the first range is short and more points are used in this range. The second range is integrated adaptively depending on the value of . The discretized integral is expressed as

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Substituting (8) into (6) and (7) and applying the 2D FMA decould be expressed as composition [31],

(9)

(6) where is the number of quadrature points, and are the quadrature points along the two vertical branch cuts, and are the weights of the quadrature rule. 2) Pole Singularity Contribution: The factor and might have poles plane. Those poles correspond to the guided in complex modes in the layered slab and their contribution should be considered. Those modes are determined by the structure of the layered media and the working frequency. To locate the poles, we have adopted the root solver from [12], which is efficient and accurate. The basic idea for this solver is that from the Cauchy theorem, the closed contour integral of any analytical function is zero. We take the entire complex plane as a rectangle box and perform a contour integral of a generalized reflection coefficient around it. If it is non-zero, which suggests poles exist, the big box is divided into four smaller boxes, and contour integrals are performed again. Each time, we ignore the boxes that contain no poles and further divide the boxes that contain poles. In this manner, the locations of the poles can be estimated after a few more subdivisions and they are refined within the finest box. Consequently,

where and is the angular variation in 2D MLFMA. The summation runs over three indexes , where is the index in , is the index for the decomposition of propagation factors, and is the index of . The function contains the rest of the factor which only , The term inside the first bracket depends on depends on only, and the term inside the second bracket depends on only. The factor is included in the second bracket, because is the index of the layer where the source point resides. Substituting (9) into the TE part of the DGLM

(10) where

is the ensemble of , , , and . Now the dyadic Green’s function’s dependence on and are completely decomposed, and this is the prerequisite for the implementation of the fast-multipole algorithm. For the TM part of the dyadic Green’s function, it could be expressed similarly

(11) (7) are the poles and is the number of poles. where Next, we will discuss how to realize the fast matrix-vector multiplication with this discrete form. C. Decomposition of DGLM by 2D MLFMA Comparing the terms in (6) and (7) with (3), we observe that the fast algorithm could be applied if the propagation factor can be decomposed into the summation of products of functions depending solely on or (8)

The inner products of the basis function and the factor inside the brackets in (10) and (11) become the radiation patterns and receiving patterns for this quasi-3D TSM-FMA. The decomposition of the propagation factor and the explicit expression for the radiation and receiving patterns are given in detail in the appendix for different cases. In summary, there are three sets of radiation patterns for every source in each mode for each : , , , 2, 3, where is the index of the layer where the source point resides. The detailed expressions of these patterns are presented in the appendix. For each observation point, it needs sets of receiving patterns to receive the field ra, , , diated from layer 1 to including , , . Here, is the and number of layers which contain the simulation object. Although

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Fig. 4. Geometrical configurations of the circular patch. a : ,d : .

0 94 cm = 0 16 cm

= 1:88 cm, b =

Fig. 6. Geometrical configurations of a single row of Yagi-Uda patch array.

B. Radiation Pattern of Microstrip Patch Antennas

Fig. 5. Input impedance of a circular patch.

the propagation factor is different for every pair of , the aggregation and translation process only needs to be performed for three radiation patterns for each mode by adopting this decomposition scheme. This is the advantage of choosing this form of decomposition. III. NUMERICAL RESULTS In this section, we present a few numerical examples and compare some of them with published results. The quasi-3D TSM-FMA has a better modeling capability, while it retains the computational complexity. A. Input Impedance of Probe-Fed Microstrip Patches The new fast algorithm is designed to handle microstrip structures that have vertical variations, for example, patch antennas fed by probes. A probe is modeled by a vertical via connecting the patch and the ground with a delta-gap source on it. The first example is a probe-fed circular patch as shown in Fig. 4. The radius of the patch is 1.88 cm. The vertical probe is 0.94 cm away from the center of the patch. The substrate has a thickness of 0.16 cm and a relative permittivity of 2.6. The input impedance at the probe versus frequency is presented in Fig. 5. The results obtained from the quasi-3D TSM-FMA agrees with those from direct calculation. The calculation shows that the resonance frequency of the patch antenna is at 2.8 GHz. It agrees with the theoretical analysis given in [32].

In this part, the radiation patterns of several types of microstrip patch antennas and arrays with vertical probe feed are simulated. The first example is a planar Yagi-Uda patch array [33], [34]. Similar to the Yagi-Uda dipole array [35], [36], the array is fed at the driven element, and there is a reflector patch on its left, and two director patches on its right. The structure directs the main beam of the radiation pattern at 30 away from the broadside direction. The geometrical configuration and dimensions of a single row Yagi-Uda patch array are given in Fig. 6. To increase the gain of the array, four rows of Yagi-Uda antennas could be aligned on the microstrip substrate. The voltage source at the two probes of the driven element has the same magnitude but a phase difference of 90 , so that a circularly-polarized field could be generated. The working frequency is 1.552 GHz. The number of unknowns for the four-row case is 6,332. If we use the full impedance matrix of MOM, it needs at least 320 MB of memory. However, with TSM-FMA, the total memory usage is reduced to 36 MB. The radiation pattern of one row and four rows of Yagi-Uda arrays are shown and compared in Fig. 7. In the elevation plane, the main beam has been tilted by about 30 degrees for both cases. By deploying a four-row array, the gain has increased by almost 10 dB. In the azimuth plane, the radiation pattern of the four-row array is better focused because of mutual coupling. The second example is a two-layer patch antenna resides in a three-layer substrate. The geometry of the antenna is similar to the one used in [37]. The difference is that here the third layer is an infinitely thick substrate. Without a ground plane, plane and the fast algonow we have two branch points in rithm should be applied to both vertical branch cuts (Fig. 3). This configuration represents the cases where we have quasi-3D metallic structures embedded in layered medium supported by a thick substrate instead of a ground plane. The radar cross section (RCS) of the antenna is calculated by both the method of moments (MOM) and our fast algorithm at 1.9 GHz. The results from both methods agree well as shown in Fig. 8. The order of the computational complexity of the algorithm is not affected by the existence of two vertical branch cuts since the computation cost increases only by a factor of 2.

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Fig. 9. Nearly square diagonal fed antennas. Unit is in cm.

Fig. 7. The radiation pattern of planar Yagi-Uda patch arrays: in the (a) elevation plane ( = 0 ) and (b) in the azimuth plane ( = 50 ).

2

Fig. 10. The radiation pattern of 12 12 microstrip patch array of nearly square patch elements. Radiation pattern in (a)  = 0 plane and (b)  = 90 plane.

Fig. 8. Radiation pattern of a two-layer patch antenna with no ground plane. The unit of the dimension is millimeter. ( ;  ) = (60 ; 180 ),   polarization, frequency = 1:9 GHz.

0

The third example is an array of nearly square diagonal-fed patch antennas. The nearly square patch antenna has two resonance modes close to each other. If it is properly excited at a fre-

quency between the two resonant frequencies, it could produce a circularly polarized field [38]. The dimension of the patch and the position of the feed is given in Fig. 9 with a unit of centimeter [39]. The microstrip substrate has a thickness of 0.3175 cm and a relative permittivity of 2.52. The feeding voltage source is working at 3.15 GHz. A 12 12 patch array made up of this type of patch is simulated and the radiation patterns are given in Fig. 10. The sidelobes are due to the mutual coupling of the patches. In this ex-

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ample, the total number of unknowns is 80,316. The memory used for setup is 358 MB and the additional memory needed for one matrix-vector multiplication is 33 MB. Time per iteration is 61 seconds on an Intel 3.0 GHz CPU. C. Computational Complexity and Memory Usage We increase the number of elements in a patch array, and record the memory usage and time usage versus the number of unknowns. They have an almost linear relationship. It confirms computational comthat the algorithm has at least plexity in both time and memory as the 2D TSM-FMA [1]. IV. CONCLUSION We have developed a quasi-3D thin-stratified medium fastmultipole algorithm (quasi-3D TSM-FMA). It can be applied in the full-wave analysis of electrically large conducting objects in thin-stratified media when the horizontal dimension of the object is much larger than the vertical dimension, for example, probe-fed microstrip structures. We worked with a vector wave function based matrix-friendly form of DGLM instead of the symmetric form DGLM used in the 2D TSM-FMA. The comas putational complexity and memory cost remains in the 2D TSM-FMA. APPENDIX RADIATION AND RECEIVING PATTERNS In the appendix, we will give the detailed expression of the radiation and receiving patterns for different cases. Patterns for Transmitted Field: When the source point and field point are located in different layers, the observed field is a transmitted field from the source. Assuming that the source point is located in layer and the field point in layer , the propagation factor can be naturally decomposed into the product of a dependent factor and a dependent factor. For , i.e., observation layer is above the source example, if layer, the propagation factor is

Fig. 11. (a) Memory versus number of unknowns. (b) Time versus number of unknowns.

coefficient. As a result, there is only one set of radiation pattern and receiving pattern needed for each mode for the transmitted field. For the TE mode, when the field point is above the source , the detailed expression of radiation and repoint ceiving patterns is given below:

(12) where (13) (16) (14) (15) Here, and are the Fresnel reflection coefficient and is the generalized reflection transmission coefficient, and

(17)

XIONG et al.: A QUASI-3D THIN-STRATIFIED MEDIUM FAST-MULTIPOLE ALGORITHM FOR MICROSTRIP STRUCTURES

Here is the first radiation pattern of the TE wave associated with a source in layer , and it represents the upgoing wave generated by the source within this layer. It contains a direct upgoing wave and a upgoing wave after being reflected by is the receiving pattern of an obthe lower interface. And servation point in layer when the source radiates from layer . , the When the field point is below the source point propagation factor is

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has two parts: the direct field which propagates directly from the source to field point and the reflected field, which reaches the field point after being reflected by other layers. The terms represent the reflected field could be rewritten as the summation of decomposed functions of and

(25) (18) where

Since the factors contains is the same as those in the propagation factors for the transmitted field case, there is no need to calculate new radiation patterns. Only new receiving pattern is needed

(19) (20)

(26)

Now the patterns become (27)

(21)

is the receiving pattern in layer associated with the radiation pattern of layer . is associated with the radiation pattern . Similarly, the receiving pattern for the reflected field of TM mode is given by

(22) is used to denote the second TE radiation patSimilarly, tern associated with a source in layer , and it represents the downgoing wave in this layer. operator makes the expression of For TM mode, the the patterns more complicated. Note that , for a plane wave. The rathe operator becomes a vector diation and receiving patterns for TM waves are summarized below. When

(28)

(29) Patterns for Direct Field: The propagation factor for the direct field is and the modulus sign makes it impossible to decompose it as two independent functions of and respectively. The Sommerfeld integral needs to be changed to . The TE part of a dyadic remove the modulus sign over Green’s function in homogenous space is given by

(30) (23) By deforming the integration path and using Gaussian quadrature rule, the term could be expressed as a discrete summation

(24) where . Patterns for Reflected Field: When the observation point and the source point belong to the same layer, the received field

(31)

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 7, JULY 2011

,

right side of the vertical branch cut side of the vertical branch cut

is the branch point.

on the

and the on the left are of opposite signs

(32) , is the value of on Here, . From the factor in the brackets contains and , it is obvious that is an even function of , we can remove and get the same result. The radithe modulus sign over ation and receiving pattern for the TE part of the direct field in the th layer is

(33) Now we look at the TM part of the dyadic Green’s function in homogenous media. Its discrete form is Fig. 12. Physical meaning of radiation and receiving patterns. (a) Upgoing waves. (b) Downgoing waves. (c) Direct waves.

The physical meaning of the radiation and receiving patterns are summarized in Fig. 12. REFERENCES (34) We expand the operator

as

(35) Substituting (35) into (34), we found the first and last terms are while the second and third terms are not. even functions of holds However, the first term dominates because for the significant quadrature points near the branch point. As moves away from the branch point along the vertical path, the integrand decays exponentially and becomes insignificant. Thus the modulus sign here could be removed as well. Similarly, the radiation pattern and receiving pattern of the TM part of the direct field is

(36)

[1] J. S. Zhao, W. C. Chew, C. C. Lu, E. Michielssen, and J. M. Song, “Thin-stratified medium fast-multipole algorithm for solving microstrip structures,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 4, pp. 395–403, Apr. 1998. [2] P. E. Wannamaker, G. W. Hohmann, and W. A. SanFilipo, “Electromagnetic modeling of three-dimensional bodies in layered earths using integral equations,” Geophysics, vol. 49, pp. 60–60, Jan. 1984. [3] D. G. Fang, J. J. Yang, and G. Y. Delisle, “Discrete image theory for horizontal electric dipoles in a multilayered medium,” IEE Proc. Microwaves, Antennas Propag., H, vol. 135, no. 5, pp. 297–303, Oct. 1988. [4] Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closed-form spatial Green’s function for the thick microstrip substrate,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 3, pp. 588–592, Mar. 1991. [5] 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, no. 11, pp. 2055–2062, Nov. 1992. [6] R. A. Kipp and C. H. Chan, “Complex image method for sources in bounded regions of multilayer structures,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 5, pp. 860–865, May 1994. [7] G. Dural and M. I. Aksun, “Closed-form Green’s functions for general sources and stratified media,” IEEE Trans. Microw. Theory Tech., vol. 43, pp. 1545–1552, Jul. 1995. [8] K. A. Michalski and J. R. Mosig, “Discrete complex image mixedpotential integral equation analysis of microstrip patch antennas with vertical probe feeds,” Electromagnetics, vol. 15, no. 4, pp. 377–392, Jul. 1995. [9] F. Ling and J. M. Jin, “Discrete complex image method for Green’s functions of general multilayer media,” IEEE Microw. Guided Wave Lett., vol. 10, no. 10, pp. 400–402, 2000. [10] T. M. Grzegorczyk and J. R. Mosig, “Full-wave analysis of antennas containing horizontal and vertical metallizations embedded in planar multilayered media,” IEEE Trans. Antennas Propag., vol. 51, no. 11, pp. 3047–3054, Nov. 2003.

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[11] W. C. Chew, Waves and Fields in Inhomogeneous Media, ser. IEEE Electromagnetic Series, 2nd ed. New York: IEEE Press, 1995. [12] B. Hu, “Fast inhomogeneous plane wave algorithm form electromagnetic scattering problems,” Ph.D. dissertation, University of Illinois, Urbana-Champaign, 2001. [13] T. J. Cui and W. C. Chew, “Efficient evaluation of Sommerfeld integrals for Tm wave scattering by buried objects,” J. Electromagn. Waves Applicat., vol. 12, no. 5, pp. 607–657, 1998. [14] T. J. Cui and W. C. Chew, “Fast evaluation of Sommerfield integrals for EM scattering and radiation by three-dimensional buried objects,” IEEE Trans. Geosci. Remote Sensing, vol. 37, no. 2, pp. 887–900, Mar. 1999. [15] T. Sarkar, E. Arvas, and S. Rao, “Application of FFT and the conjugate gradient method for the solution of electromagnetic radiation from electrically large and small conducting bodies,” IEEE Trans. Antennas Propag., vol. 34, no. 5, pp. 635–640, May 1986. [16] Y. Zhang, K. Wu, C. Wu, and J. Litva, “A combined full-wave CGFFT method for rigorous analysis of large microstrip antenna array [J],” IEEE Trans. Antennas Propag., vol. 44, no. 1, pp. 102–109, Jan. 1996. [17] C. F. Wang, F. Ling, and J. M. Jin, “A fast full-wave analysis of scattering and radiation from large finite arrays of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 46, no. 10, pp. 1467–1474, 1998. [18] X. Hu, “Full-wave analysis of large conductor systems over substrate,” Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, 2006. [19] F. Ling, V. I. Okhmatovski, W. Harris, S. McCracken, and A. Dengi, “Large-scale broad-band parasitic extraction for fast layout verification of 3-D RF and mixed-signal on-chip structures,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 264–273, Jan. 2005. [20] Y. Shi and C.-H. Liang, “Application of the spatial-spectral CG-FFT method for the solution of electromagnetic scattering by buried flat metallic objects,” IEEE Geosci. Remote Sensing Lett., vol. 4, no. 1, pp. 37–40, Jan. 2007. [21] W. C. Chew and C. C. Lu, “A fast algorithm to compute the wavescattering solution of a large strip,” J. Comput. Phys., vol. 107, no. 2, pp. 378–387, 1993. [22] C. C. Lu and W. C. Chew, “Electromagnetic scattering of finite strip array on a dielectric slab,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 1, pp. 97–100, Jan. 1993. [23] L. Gurel and M. I. Aksun, “Electromagnetic scattering solution of conducting strips in layered media using the fast multipole method,” IEEE Microw. Guided Wave Lett., vol. 6, no. 8, Aug. 1996. [24] M. A. Saville and W. C. Chew, “Multipole-free fast inhomogeneous plane wave algorithm,” Radio Sci., vol. 42, pp. RS5002–RS5002, 2007. [25] V. Jandhyala, E. Michielssen, and R. Mittra, “Multipole-accelerated capacitance computation for 3-D structures in a stratified dielectric medium using a closed-form Green’s function,” Int. J. RF Microw. Comput.-Aided Engrg., vol. 5, no. 2, pp. 68–78, May 2007. [26] P. Atkins and W. C. Chew, “Fast computation of the dyadic Green’s function for layered media via interpolation,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 493–496, 2010. [27] W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix friendly formulation of layered medium Green’s function,” IEEE Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 490–494, Dec. 2006. [28] J. L. Xiong and W. C. Chew, “A newly developed matrix-friendly formulation of layered medium Green’s function,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 868–875, Mar. 2010. [29] Y. P. Chen, J. L. Xiong, W. C. Chew, and Z. P. Nie, “Numerical analysis of electrically small structures embedded in a layered medium,” Microw. Opt. Technol. Lett., vol. 51, no. 5, pp. 1304–1308, May 2009. [30] V. Rokhlin, “Diagonal forms of translation operators for the Helmholtz equation in three dimensions,” Appl. Comput. Harmon. Analy., vol. 1, no. 1, pp. 82–93, 1993. [31] W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics. Norwood, MA, Artech: , 2001. [32] W. C. Chew, J. A. Kong, and L. C. Shen, “Radiation characteristics of a circular microstrip antenna,” J. Appl. Phys., vol. 51, no. 7, pp. 3907–3915, 1980. [33] J. Huang, “Planar microstrip yagi array antenna,” in Proc. IEEE Antennas Propag. Soc./URSI Symp. Dig., Jun. 1989, vol. 2, pp. 894–897. [34] J. Huang and A. C. Densmore, “Microstrip Yagi array antenna for mobile satellite vehicle application,” IEEE Trans. Antennas Propag., vol. 39, pp. 1024–1030, Jul. 1991. [35] S. Uda, “Wireless beam of short electric waves,” J. Inst. Elec. Eng., no. 452, pp. 273–282, Mar. 1926. [36] H. Yagi, “Beam transmission of ultra short waves,” Proc. IRE, vol. 16, pp. 715–741, Jun. 1928.

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[37] Y. Shi and C.-H. Liang, “Three-dimensional planar radiating structures in stratified media,” Int. J. Microw. Millimeter-Wave Comput.-Aided Engrg., vol. 7, no. 5, pp. 330–343, Sep. 1997. [38] W. Richards and Y. T. Lo, “Design and theory of circularly polarized microstrip antennas,” IEEE Trans. Antennas Propag., vol. 17, pp. 117–120, 1979. [39] P. C. Sharma and K. C. Gupta, “Analysis and optimized design of single feed circularly polarized microstrip antennas,” IEEE Trans. Antennas Propag., vol. 31, no. 6, pp. 949–955, Nov. 1983.

Jie L. Xiong was born in Hubei, Chin, in 1981. She received the B.S. degree from Tsinghua University, Beijing, China, in 2002, the M.S. degree from the University of Massachusetts, Amherst, in 2004, and the Ph.D. degree from the University of Illinois at Urbana-Champaign, in 2010. From 2007 to 2009, she worked as a Research Associate with Professor Chew, in the Faculty of Engineering, University of Hong Kong. Currently she is working in the Schulumberger-Doll Research Center, Cambridge, MA, as a postdoctoral Research Scientist. Her research interests include the fast algorithms for layered medium Green’s function, the numerical evaluation of the Casimir force, and the inversion algorithm for acoustic and elastic waves.

Yongpin Chen was born in Zhejiang, China, in 1981. He received the B.S. and the M.S. degrees in microwave engineering from the University of Electronic Science and Technology of China, Chengdu, in 2003 and 2006, respectively. He is currently working towards the Ph.D. degree at the University of Hong Kong, Hong Kong, China. His research interests include waves and fields in inhomogeneous media, integral equation methods and fast algorithms in computational electromagnetics.

Weng Cho Chew (S’79–M’80–SM’86–F’93) was born in Malaysia, on June 9, 1953. He received the B.E., M.S. Engineer, and Ph.D. degrees, all in electrical engineering, from the Massachusetts Institute of Technology, Cambridge, in 1976, 1978, and 1980, respectively. Previously, he was a Department Manager and a Program Leader at Schlumberger-Doll Research, Ridgefield, CT. Currently, he is a Professor at the University of Illinois at Urbana-Champaign, where he is also the Director of the Center for Computational Electromagnetics and Electromagnetics Laboratory (CCEML). His research interest is in the areas of waves in inhomogeneous media for various sensing applications, integrated circuits, microstrip antenna applications, and fast algorithms for solving wave scattering and radiation problems. He is the originator of several fast algorithms for solving electromagnetic scattering and inverse problems. He has led a research group that has developed parallel codes solving tens of millions of unknowns for integral equations of scattering. He authored Waves and Fields in Inhomogeneous Media (New York: Van Nostrand Reinhold, 1990), coauthored Fast and Efficient Methods in Computational Electromagnetics (Norwood, MA: Artech House, 2001), and authored and coauthored over 300 journal publications, over 400 conference publications, and over ten book chapters. Dr. Chew is an OSA Fellow, an IOP Fellow, and was an NSF Presidential Young Investigator. He received the Schelkunoff Best Paper Award from the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, the IEEE Graduate Teaching Award, Campus Wide Teaching Award, and was a Founder Professor of the College of Engineering, and currently, a Y. T. Lo Endowed Chair Professor in the Department of Electrical and Computer Engineering at the University of Illinois. Since 2005, he serves as an IEEE Distinguished Lecturer. He served on the IEEE Adcom for Antennas and Propagation Society as well as Geoscience and Remote Sensing Society. He has been active with various journals and societies. Recently, ISI Citation elected him to the category of Most-Highly Cited Authors (top 0.5%).

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Analytic Fields With Higher-Order Compensations for 3-D FDTD TF/SF Formulation With Application to Beam Excitations Gurpreet Singh, Eng Leong Tan, Senior Member, IEEE, and Zhi Ning Chen, Fellow, IEEE

Abstract—The total-field/scattered-field (TF/SF) formulation is widely used to initiate incident waves for scattering problems in finite-difference time-domain (FDTD) simulations. An important aspect of the TF/SF formulation is the calculation of incident fields at the TF/SF boundary by methods that account for the numerical nature of the incident wave in the FDTD grid. Failure to do so can lead to high levels of field leakage errors across the TF/SF boundary. This paper presents an improved analytic time-domain method for accurately computing incident fields in the TF/SF formulation. Using analytic field expressions with higher-order dispersion and polarization compensations, the proposed method compensates for 1) the FDTD numerical dispersion; and 2) the lack of orthogonality between the frequency-dependent field polarizations and the wavevector, which was not accounted for in existing analytic time-domain methods. The higher-order compensations result in further suppression of field leakage errors in the SF region. In addition to exciting a plane wave, the proposed method can be employed to excite an incident beam. To demonstrate this, numerical experiments that source both 3-D plane wave and focused beam into a free space FDTD grid are presented and compared with existing methods. Index Terms—Beam synthesis, dispersion compensation, finitedifference time-domain (FDTD) methods and total-field/scatteredfield (TF/SF) formulation.

I. INTRODUCTION

T

HE total-field/scattered-field (TF/SF) formulation is widely used to initiate an incident wave for scattering problems in finite-difference time-domain (FDTD) simulations [1]. The TF/SF formulation zones the entire FDTD lattice into two distinct regions, namely, the total-field region where total-fields, that include incident and scattered fields, exist and the scattered-field region where only scattered-fields exist. The boundary separating the total-field and scattered-field regions is defined as the TF/SF boundary. To properly connect the TF and SF regions, field corrections (or consistency conditions) are applied to the field components tangential to the TF/SF Manuscript received March 29, 2010; revised October 25, 2010; accepted November 02, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. G. Singh and E. L. Tan are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: [email protected]; [email protected]). Z. N. Chen is with the Institute for Infocomm Research, Singapore 138632, Singapore. 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.2011.2152328

boundary. These corrections are functions of incident fields at the TF/SF boundary, which have to be calculated by methods that account for the numerical nature of the incident wave in the FDTD grid. Failure to do so can lead to significant field leakage across the TF/SF boundary and into the SF region, which can result in analysis errors. The TF/SF formulation can be described by the equivalence principle [2] in which the Huygens’ sources are related to the field corrections. For a plane wave source excitation, an auxiliary 1-D FDTD grid was employed in [1] to provide the necessary incident fields at the TF/SF boundary. Despite its efficiency, the accuracy of the 1-D grid method was limited for non-grid aligned propagations due to numerical mismatch between the 1-D and main grids in terms of phase velocities and field locations as well as polarizations. To ameliorate this numerical mismatch, several developments of the 1-D grid method have been proposed, namely in the form of higher-order interpolations [3], finer 1-D grid sampling as well as signal-processing techniques [4] and spatial step size optimizations for phase velocity matching over a wide frequency band [5]. Recently, a new 1-D grid method termed as 1-D Multipoint Auxiliary Source Propagator (MAP) was proposed in [6] to source the TF/SF boundary in a 2-D main grid for a discrete set of propagation angles called the rational angles. The 1-D MAP approach produces a dispersion relation identical to that of the main grid at the rational angles and maps the field values to the TF/SF boundary with no interpolation. dB. This method results in a low field leakage of less than However, the computational overhead of the 1-D MAP can depend significantly on the desired resolution of the rational angle [6]. The 1-D MAP method has been extended in [7] as a hybrid technique to source a 3-D plane wave. An alternate method for plane wave source excitation was presented in [8], in which the incident fields at the TF/SF boundary are directly calculated by analytical means that account for the FDTD dispersion and the non-orthogonality between the frequency-dependent field polarizations and the wavevector [9]. This method (Analytic Field Propagator (AFP)) first computes the incident fields in frequency-domain for each point at the TF/SF boundary given the field at a reference point, the direction of propagation and the dispersion relation. An inverse Fourier transform is then performed to obtain the time sequences of the incident fields at the TF/SF boundary. By using the AFP method, the field leakage can be reduced to levels dB. However, a drawback of the AFP method as low as is that it uses inverse Fourier transforms that require significant amount of memory to store the incident fields for every time

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SINGH et al.: ANALYTIC FIELDS WITH HIGHER-ORDER COMPENSATIONS FOR 3-D FDTD TF/SF FORMULATION

and point at the TF/SF boundary, as noted in [7] and [10]. This computational requirement becomes a barrier especially for 3-D problems. It can be avoided by using the time-domain method presented in [11], in which the incident fields are calculated at every time step (on-the-fly) using analytic field expressions that compensate for the FDTD dispersion only. There are several instances where instead of a plane wave, an incident beam (e.g., Gaussian-like focused beam) may be a more preferred choice of excitation. For example, a focused beam can be used to provide a localized illumination to study the scattering effects within a confined area such as in the case of individual scatterers on a larger target [12]–[14]. Focused beams can also be used to minimize illuminating the edges of a finite-sized scatterer that otherwise, can result in interferences between the diffracted waves, that radiate from the edges of the finite-sized scatterer, and the desired reflected and transmitted waves [15]–[18]. Hence, more accurate scattering results such as S parameters can be achieved. An incident beam may also be of interest in modeling part of an optical illumination system such as, the focal area of a converging lens [19]. In general, such beams can be expressed analytically as a finite summation of plane waves, with each plane wave constituent having its own magnitude, phase, polarization and propagation direction [12], [18], [19]. In [18], an approach was outlined to incorporate the FDTD numerical effects into each plane wave constituent of a Maxwellian beam. However, like the AFP method, this approach synthesizes the beam before the FDTD simulation, which may result in considerable memory requirements especially for 3-D problems. In [19], a time-domain method was presented for exciting a focused light beam, in which the FDTD anisotropic grid-velocities were accounted for in the propagation delay/advance times of each plane wave constituent. Although this approach can calculate incident fields on-the-fly, it compensates for the FDTD grid-velocities at the centre frequency of a source pulse and does not account for the non-orthogonality between the frequency-dependent field polarizations and the wavevector. In this paper, we present an improved analytic time-domain method for computing incident fields in the TF/SF formulation and demonstrate its ability in sourcing a focused beam. The proposed method improves on the accuracy of [11] in two ways. Firstly, it enhances the FDTD dispersion compensation with a higher-order approximation of the numerical wavenumber and secondly, it accounts for the lack of orthogonality between the frequency-dependent field polarizations and the wavevector, which was not accounted for in [11] and [19]. The higher-order dispersion and polarization compensations are included into a set of analytic field expressions that can then be used to practically compute the required incident fields at every time step, unlike the AFP method which requires the incident fields to be synthesized and stored before the start of the FDTD simulation. The organization of this paper is as follows: Section II provides the necessary background to develop the proposed method. Sections III and IV present the higher-order numerical wavenumber and frequency-dependent field polarization compensations, respectively. Section V presents the proposed analytic expressions that include both higher-order dispersion and polarization compensations. This section also provides

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formulations to facilitate their implementation. Numerical experiments of both 3-D plane wave and focused beam are presented in Section VI to demonstrate the performance of the proposed method. Section VII finally provides the conclusions. II. BACKGROUND Let the electric and magnetic fields at a position from a reference point be represented in Fourier domain as (1) (2) for a plane wave propagating in the FDTD grid. The numerical wavevector is given as with . The vector components of are for . The angle is relative to the axis and the angle is relative to the axis when projected in the plane. The and electric and magnetic fields at the reference point, , are related through the following vector equations that can be deduced by Fourier analysis of FDTD update equations [8]: (3) (4) where

(5)

A. Numerical Wavenumber Approximation From (3) and (4), the FDTD dispersion relation can be oband upon expansion, it can be reextained as pressed as (6) for . where Let be related to the physical wavenumber by a factor , that is assumed real. By subinto stituting the vector components of (6) and then performing the series expansion of the terms in (6), one can obtain the following expression:

(7)

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where

and Using the Fourier transform identities , (15) is converted to time-domain as

An approximation of correct to can be extracted terms that are greater than from (7) by first truncating the to yield (8) Then, by solving for

in (8)

(16) . The part in (15) is where that compenconverted into a time-domain operator sates the FDTD dispersion and reduces the errors at the TF/SF boundary. C. Frequency-Dependent Field Polarizations

(9) and utilizing the approximation correct to can be arrived at as

and be represented Let the polarization of and respectively. Let be the by the unit vectors that can be expressed, using from unit vector of (5) and , as

(10) By substituting correct to

into

, an approximation of can be obtained as (11)

where

(17) ( for ) Note that the components of vector in [7], [20]. From (6), it correspond to the projection terms can be seen that these components also satisfy the projection . equality From (3) and (4), a set of equations relating and can be arrived at as

(12) The expression (12) has been obtained in [11] using a similar approximation procedure as (8)–(10). It can be viewed as an approximate deviation of the numerical wavenumber from the physical wavenumber . This deviation is both frequency and propagation direction dependent. B. Time-Domain Expression With Dispersion Compensation The time-domain electric field expression with dispersion compensation has been obtained in [11] and its derivation is as follows. First, expression (1) is rewritten as

(18) (19) Note that relations and were used to obtain (18) and (19) with the latter relation obtained from (3) and (4) [8]. Equations (18) and (19) indicate and form a mutually orthogonal set of vecthat tors. can be specified as [20] To satisfy (18) and (19), (20)

(13) Assuming that with mating

is a small quantity and approxi, (13) is then written as

in which the FDTD numerical angles and are the direction that can be determined from the following angles of vector trigonometric relations with the components of vector :

(14) (21)

in (12) is then substituted into (14) to yield

(15)

The polarization angle is specified with reference to the di. can be obtained by replacing and rection in (20) with and respectively. As noted in

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(17), the components of vector are generally frequency-dependent implying that the trigonometric functions of and in and , are fre(21), and consequently the unit vectors quency-dependent [20]. III. HIGHER-ORDER NUMERICAL WAVENUMBER APPROXIMATION In this section, we improve the order of approximation of from 3rd order, as presented in (11), to 5th order. By increasing the order of approximation of , the dispersion compensation can be improved. Let the approximate correct to be . To determine it is possible to follow a denoted as similar approximation procedure in Section II.A, where the lefthand side of expression (7) is first truncated by keeping terms , then the correct to is solved up to is finally multiplied with to from a cubic equation and . However, attaining via the cubic equation in obtain can be a demanding task. In a simplified form similar to what follows, an alternate approach is presented to extract in a form similar to . be written as Let the approximated correct to (22) The corresponding as

correct to

can then be expressed

(23) where

can be written as (24)

Substituting (22) into (7) and keeping terms up to to

leads

(25) To equate the left and right-hand sides of (25) at every real , the polynomial coefficients of should reduce to and in (25) zero. By setting the coefficients of to zero, two equations with the two unknowns and can be formed and are solved as

Fig. 1. Plot of log gation direction of 

k

k

jError = ~ 0 ~ = 70

and 

= 30

j

versus log

!1t) for propa-

(

.

Notice that is equivalent to the coefficient in (10). and are finally deduced by subThe approximations stituting and into (24) and (23). can be verified to reduce to The approximation when the FDTD discretizathe physical wavenumber tion parameters approach zero or when the propagation direction satisfies the relation with a corresponding time step size set to the Courant limit time step. is consistent with that of its analytic This behavior of counterpart [6], [21]. by meaWe next analyze the approximation error from suring the difference with its analytic counterpart over a range of frequencies. The analytic can be determined from (6). In this analysis, and are computed using the following pamm rameters: Uniform cell size and time step size of respectively with being the speed of wave and propagation in free space, frequency range of rad/s that corresponds to cells per wavelength of about 60–10 and propagation direction angles of and . Fig. 1 plots the approximation error from in logarithmic scale. Plotted together, for comparison, is the approximation error from in (11). From the figure, it is obis generally lower served that the approximation error from . This verifies that is a better apcompared to that from . Furthermore, the slope of the logarithmic proximation than error plots corresponding to and are 5 and 7 respectively. This verifies their respective 3rd and 5th order accuracy.

IV. FREQUENCY-DEPENDENT FIELD POLARIZATION COMPENSATION

(26)

As highlighted in Section II.C, the incident fields in the FDTD grid are polarized in a direction orthogonal to . The field in (20), in polarizations can be specified for example by

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which the trigonometric functions of and are in general frequency-dependent. In this section, we present polynomial approximations of the trigonometric functions of and that can be subsequently used to estimate the incident field polarizations, . The inverse Fourier transform of the estimated for example polarization vector then yields a time-domain operator that can compensate for the frequency-dependent field polarizations in the FDTD grid. is first estimated as Each component of vector from the series expansion a polynomial correct to with for of . The estimated components of vector are then substituted into (21) and using the approximations and , the polynomial approximations of trigonometric functions of and , correct to , are attained as Fig. 2. Plot of log jErrorj versus log (! 1t). Dash- and dot-lines represent errors from trigonometric function approximations correct to (! 1t) and ~ cos ; ~ (! 1t) respectively. “o,” “.,” “/” and “+” represent errors from sin ; ~ approximations respectively. Slope of dash- and dot-lines are 2 ~ and sin  cos  and 6 respectively.

(27) with the and coefficients listed in the Appendix. The approximated numerical angles and in (27) can be verified to reduce to the physical angles and when , or when the propagation direction is either grid-aligned or satisfies the relation . This behaviour of and is consistent with that of its analytic counterparts and respectively [20]. Next, we analyze the approximation errors from the approximated trigonometic functions of and by computing their differences with their respective analytic counterparts in (21) across a range of frequencies. For example, the error from the is denoted as frequency-independent approximation of . These errors can be expressed as

(28)

In this analysis, the following parameters are specified: Unimm and time step size of form cell size of , frequency range of rad/s and propagation direction angles of and . Fig. 2 plots, in logarithmic scale, the absolute errors produced from each approximated function in (27) and the errors produced from the frequency-independent approximations of (21). From the figure, we observe that the approximation errors from (27) are generally lower compared to that produced from the frequency-independent approximations. Specifically, rad/s , the maximum error proat duced from the frequency-independent approximations is aptimes more than that produced from (27). From proximately the figure, we further observe that the slope of the logarithmic error plots from the frequency-independent approximations and the approximations in (27) are 2 and 6 respectively. This verifies accuracy of the approximated functions in (27). the By substituting the FDTD trigonometric functions in (20) with their respective approximated functions in (27), the elec, tric field polarization vector in the FDTD grid, for example as can now be estimated correct to (29) and are listed in the The vector coefficients Appendix. is the frequency-independent polarization vector in physical space, obtained simply by replacing and in (20) with and respectively. with and the magnitude Upon substitutions of with the function , the electric field at the refer, can now be estimated as ence point,

(30)

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Using the Fourier transform identity can be converted to time-domain as

, (30)

(31)

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deduced in [11] when . To include the frequency-dependent polarization in (33), the reference electric can be substituted from (31) into (33) to finally field yield the electric field with both compensations as

where

is the inverse Fourier transform of . The operators in (31) can be used to compensate in and consetime-domain the frequency dependence of quently minimize the errors at the TF/SF boundary. A similar . approximation can be established for

(35) The magnetic field expression with higher-order dispersion and polarization compensations can be similarly attained as

V. ANALYTIC FIELD EXPRESSIONS WITH HIGHER-ORDER DISPERSION AND POLARIZATION COMPENSATIONS Following the derivations in Sections III and IV, we next present the time-domain analytic field expressions that include both higher-order FDTD dispersion and frequency-dependent field polarization compensations. The proposed analytic expressions can be used to calculate the incident fields at the TF/SF boundary with minimal field leakage errors, as will be demonstrated in Section VI. Formulations that facilitate the implementation of the proposed analytic expressions are also presented in this section. The electric field expression with higher-order dispersion compensation can be attained by first approximating (13) as

(36) where

is the impedance of the wave in the medium and and are vector coefficients that can be obtained and in and with by replacing and respectively. Expressions (35) and (36) are the proposed analytic field expressions that compensate for the FDTD dispersion and frequency-dependent field polarizations. They can be used to generate the required incident fields at the TF/SF boundary with minimal errors. A. Computing Derivatives of Gaussian Source Functions

(32) where the first two real and imaginary terms of the series expansion of are kept and is estimated by the . Upon substitution of in (24), higher-order term (32) can be expanded as a polynomial function of and converted to time-domain using the Fourier identities and to arrive at (33) is defined as a higher-order dispersion compensation factor that can be expressed in compact form as

The evaluation of (35) and (36) requires the derivatives of the source at a specific time. For commonly used Gaussian source, the value of the th derivative can be easily computed from the following product between the th order Hermite polyand the Gaussian [22], [23]: nomial (37) The explicit expression of is not needed for its evaluation since it can be determined via the recurrence relation (38) and the initial conditions and . For a modulated Gaussian source with variance , similar formulations to compute its th derivative can be obtained, as follows: First, using the chain rule, (37) can be modified for a Gaussian as source with variance (39)

(34)

Next, let the modulated Gaussian source be with and . The th derivative can be written, from Leibnitz’ product rule [23], as of

with and . Note that (34) reduces to the lower-order dispersion compensation factor

(40) By noting that the even and odd th derivatives of are and respectively

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and substituting pressed as

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 7, JULY 2011

with (39), (40) can be ex-

(41) where

With the aid of identities and

, the terms

and

can be

deduced, in terms of recurrence relations, as

2 02 j for fixed N = 20 and (b) phase N values.

Fig. 3. Plot of (a) phase errors j error for different

(42) (43) with the initial conditions and . Expressions (41), (42) and (43) can be easily programmed to compute the value of the th derivative of a modulated Gaussian source at a given time. The th derivative value can then be utilized in (35) and (36). The above procedures can also be used to obtain similar implementations for other forms of Gaussian source functions such as differentiated and cosine-modulated Gaussian.

j2 0 2 j

where coefficients number can be determined from

and depend on the Courant and cells per wavelength and

B. Phase Error Analysis In this part, we analyze the phase errors that arise from four types of series approximation of , including the approximation used in (32). The approximated phase can be determined from the polar form of the series approximation . The reference phase is defined as of . To facilitate this analysis, let the approximated where the indices and phase be denoted in general as represent the order of approximations of and the series exrespectively. The phase error is then pansion of . We consider the direction denoted as and to account for of propagation largest dispersion errors [11], [24] and let . The corresponding phase errors from four types of series approxican then be expressed as follows: mation of

In Fig. 3(a), the absolute values of these phase errors are . The maximum frequency is assumed plotted with respect to where . Uniform cell and as time step sizes of mm and are set respectively. From the figure, we observe that the phase error , from the approximation proposed in (32), is produced by generally the lowest compared to that produced by the other approximated phases and this comparison becomes significant as increases. Fig. 3(b) further plots the phase error produced for different values. From the figure, we observe by decreases. that the phase error generally increases as

VI. NUMERICAL RESULTS

(44)

In this section, numerical results are presented to validate the proposed analytic field expressions (35) and (36) that calculate the incident fields at the TF/SF boundary. For this validation, two experiments are considered: one that sources a plane wave and another that sources a focused beam into the TF region of a free space FDTD grid.

SINGH et al.: ANALYTIC FIELDS WITH HIGHER-ORDER COMPENSATIONS FOR 3-D FDTD TF/SF FORMULATION

Fig. 4. Field leakage in the scattered-field region using modulated Gaussian  rad/s N . The field leakage is below source with ! dB using the proposed analytic expressions.

= 5 2 10

(

= 10)

Fig. 5. Field leakage in the scattered-field region using modulated Gaussian  rad/s N . The field leakage is below source with ! : dB using the proposed analytic expressions.

086 0137 3

A. Plane Wave Source In the first experiment, we initiate a 3-D plane wave into the TF region without any scattering objects. The plane wave propand with the polarizaagates in the direction . The entire computational volume is of free tion angle space and is meshed with 120 120 120 uniform grid cells of size 1 cm each. The TF region has dimensions 100 100 100 uniform grid cells and the Mur absorbing boundary condition (ABC) [25] is placed 10 grids away from the TF/SF boundary to terminate the computational volume. The simulation is pereach. formed over 1200 time steps of size The reference point is specified at the centre of the computamodeled by a modtional volume with the source function ulated Gaussian pulse in time: with and two values of rad/s rad/s. The upper dB magnitude of the source and function spectrum corresponds to and 20 when is rad/s and rad/s respectively. The incident fields at the TF/SF boundary are obtained at every time step are comusing (35) and (36) in which the derivatives of puted using (41), (42) and (43). To measure the performance of the proposed analytic expressions, the field leakage error at the point (112, 112, 112) in the SF region is recorded over the entire simulation and is normalized by the peak incident field value. The field leakage error at this point is plotted in Figs. 4 and 5 for the case of a source rad/s and function with centre frequencies rad/s respectively. Plotted together, for comparison, are the field leakage errors using the expressions presented in [11] that utilizes a lower-order dispersion compensation and does not include compensations for frequency-dependent polarization. From the plots, it can be seen that the proposed analytic dB and expressions can attain field leakage errors below dB when the modulated Gaussian source is discretized values of 10 and 20 respectively. These error levels with are suppressed by amounts of 42.6 dB and 77.6 dB respectively

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= 2 2 10

(

= 20)

from the corresponding error levels obtained using the expressions in [11]. This verifies the improved performance of the proposed analytic expressions. B. Focused Beam Source In the second experiment, we demonstrate the proposed analytic expressions to initiate a 3-D focused beam into the TF region without any scattering objects. The focused beam is constructed from a finite sum of plane waves that appears in the 2-D integral representation of the field in the image space of an aplanatic converging lens [19], originally represented in frequency-domain by Richards and Wolf [26], [27]. The geometry of the experiment is similar to that presented in [19], where the incident beam is propagating towards the axis. The 2-D integral representation of the fields on the TF/SF boundary is numerically computed using the extended midpoint quadrature scheme [19] and considers a total number of 324 plane waves (accurately sourcing such a large number of plane waves by the 1-D grid based methods can be a challenging task). The dispersion and polarization compensations are included into the incident fields of each plane wave constituent through expressions such as (35) and (36). The focal length of . the lens is 10 mm and the numerical aperture is The reference point is specified at the centre of the computational volume, that is also the focal point of the lens, and is modeled by a modulated Gaussian the source function with pulse in time: and rad/s. The upper dB magnitude of the source function spectrum corresponds to . The source function corresponds to the incident electric field denoted as in [19]. The cell dimensions of the entire computational volume and TF regions are same as those in the first experiment with a cell size of 20 nm. The simulation is performed over 1200 time steps of size each. field component (in dB) on the Fig. 6(a)–(f) plots the plane at different time instants with the colorplot dB. The field is normalized by its bar restricted to

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therefore has the potential to be utilized in focused beam implementations to achieve lower levels of field leakage. VII. CONCLUSION

= 0 (xz) plane at different 040 dB.

Fig. 6. Grayscale plots of E field (in dB) on the y time instants (a)–(f). The colorplot bar is restricted to

This paper has presented an improved analytic time-domain method for accurately computing incident fields at the TF/SF boundary. Using analytic field expressions with higher-order dispersion and polarization compensations, the proposed method compensates for 1) the FDTD numerical dispersion; and 2) the lack of orthogonality between the frequency-dependent field polarizations and the wavevector, which was not accounted for in existing analytic time-domain methods. Formulations that facilitate the implementation of the proposed analytic expressions have also been presented. Numerical experiments have been presented to demonstrate the proposed method’s ability to source both 3-D plane wave and focused beam. From the experiments, it was shown that the proposed method is more accurate and can attain field leakage errors much lower than existing analytic time-domain methods. The proposed method is also practical as it can be used to compute the require incident fields at every time step, unlike the AFP method that requires additional memory to store the incident fields prior to the FDTD simulation. These advantages make the proposed method a suitable analytic approach especially for introducing beam excitations (nonplane waves) into the TF/SF formulation. Using the proposed method, we aim to attain in future accurate time-domain representations of other forms of incident beams such as Gaussian beams for the TF/SF formulation. APPENDIX

Fig. 7. Field leakage in scattered-field region is below proposed analytic expressions.

0115:9 dB using the

maximum at the focal point. The white dash-line represents the TF/SF boundary. Fig. 6(a)–(c) first illustrate the focused beam entering the TF region and propagating towards the focal point. Fig. 6(d) shows the centre of the beam at the focal point at time . Fig. 6(e)–(f) illustrate the beam propagating away from the focal point and leaving the TF region. In all plots, the focused beam is well created and isolated in the TF region. To measure the performance of the proposed analytic expresfield leakage error at the point (61, 61, sions, the normalized 10) in the SF region is recorded over the entire simulation and plotted in Fig. 7. Plotted together, for comparison, is the field leakage error from the grid-velocity correction technique presented in [19] that utilizes the FDTD anisotropic grid-velocity at the centre frequency (instead of ) to compute the propagation in Section V. From Fig. 7, it is observed delays, for example that the proposed analytic expressions can attain field leakage dB that is a level reduced by 67.8 dB from error below the corresponding error level obtained using the grid-velocity correction technique in [19]. The proposed analytic expressions

where

SINGH et al.: ANALYTIC FIELDS WITH HIGHER-ORDER COMPENSATIONS FOR 3-D FDTD TF/SF FORMULATION

with

and

Note that

and

are given in Section II.A. REFERENCES

[1] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Boston, MA: Artech House, 2005. [2] D. E. Merewether, R. Fisher, and F. W. Smith, “On implementing a numeric Huygen’s source scheme in a finite difference program to illuminate scattering bodies,” IEEE Trans. Nucl. Sci., vol. 27, no. 6, pp. 1829–1833, Dec. 1980. [3] U. Oguz and L. Gurel, “Interpolation techniques to improve the accuracy of the plane wave excitations in the finite difference time domain method,” Radio Sci., vol. 32, no. 6, pp. 2189–2199, Nov.–Dec. 1997. [4] U. Oguz, L. Gurel, and O. Arikan, “An efficient and accurate technique for the incident-wave excitations in the FDTD method,” IEEE Trans. Microwave Theory Tech., vol. 46, no. 6, pp. 869–882, Jun. 1998. [5] C. Guiffaut and K. Mahdjoubi, “Perfect wideband plane wave injector for FDTD method,” in Proc. IEEE Antennas Propagat. Soc. Int. Symp., Jul. 16–21, 2000, vol. 1, pp. 236–239. [6] T. Tan and M. Potter, “1-D multipoint auxiliary source propagator for the total-field/scattered-field FDTD formulation,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 144–148, 2007. [7] T. Tan and M. Potter, “Optimized analytic field propagator (O-AFP) for plane wave injection in FDTD simulations,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 824–831, Mar. 2010. [8] J. B. Schneider, “Plane waves in FDTD simulations and a nearly perfect total-field/scattered-field boundary,” IEEE Trans. Antennas Propag., vol. 52, no. 12, pp. 3280–3287, Dec. 2004. [9] M. Celuch-Marcysiak and W. K. Gwarek, “On the nature of solutions produced by finite difference schemes in time domain,” Int. J. Numer. Modeling: Electron. Networ., Devices, Fields, vol. 12, no. 1–2, pp. 23–40, Jan.–Apr. 1999. [10] C. D. Moss, F. L. Teixeira, and J. A. Kong, “Analysis and compensation of numerical dispersion in the FDTD method for layered, anisotropic media,” IEEE Trans. Antennas Propag., vol. 50, no. 9, pp. 1174–1184, Sep. 2002. [11] T. Martin and L. Pettersson, “Dispersion compensation for Huygens’ sources and far-zone transformation in FDTD,” IEEE Trans. Antennas Propag., vol. 48, no. 4, pp. 494–501, Apr. 2000. [12] J. W. Schultz, E. J. Hopkins, and E. J. Kuster, “Near-field probe measurements of microwave scattering from discontinuities in planar surfaces,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2361–2368, Sep. 2003.

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[13] J. W. Schultz, E. J. Hopkins, J. G. Maloney, E. J. Kuster, and M. P. Kesler, “A focused-beam methodology for measuring microwave backscatter,” Microwave Opt. Technol. Lett., vol. 42, no. 3, pp. 201–205, Aug. 2004. [14] L.-C. Ma and R. Mittra, “Implementation of Gaussian beam sources in FDTD for scattering problems,” in Proc. IEEE Antennas Propagat. Soc. Int. Symp., 2007, pp. 1665–1668. [15] V. Galdi, P. Kosmas, C. M. Rappaport, L. B. Felsen, and D. A. Castanon, “Short-pulse three-dimensional scattering from moderately rough surfaces: A comparison between narrow-waisted Gaussian beam algorithms and FDTD,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 157–167, Jan. 2006. [16] J. E. Roy, “A numerical technique to compute the plane-wave scattering coefficients of a general slab,” in Proc. IEEE Antennas Propagat. Soc. Int. Symp., 2006, pp. 3487–3490. [17] D. R. Reid and G. S. Smith, “A comparison of the focusing properties of a Fresnel zone plate with a double-hyperbolic lens for application in a free-space, focused-beam measurement system,” IEEE Trans. Antennas Propag., vol. 57, no. 2, pp. 499–507, Feb. 2009. [18] J. E. Roy, “A numerical technique for computing the values of plane wave scattering coefficients of a general scatterer,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3868–3881, Dec. 2009. [19] I. R. Capoglu, A. Taflove, and V. Backman, “Generation of an incident focused light pulse in FDTD,” Opt. Expr., vol. 16, no. 23, pp. 19208–19220, Nov. 10, 2008. [20] T. Tan and M. Potter, “On the nature of numerical plane waves in FDTD,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 505–508, 2009. [21] A. P. Zhao, “Determination of the direction that has maximum phase velocity for the 2-D and 3-D FDTD methods based on Yee’s algorithm,” IEEE Microw. Wireless Comp. Lett., vol. 13, no. 6, pp. 226–228, Jun. 2003. [22] W. M. Heigl, “Computing Gaussian derivative waveforms of any order,” Geophysics, vol. 72, no. 4, pp. H39–H42, Jul.–Aug. 2007. [23] K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering. Cambridge, U.K.: Cambridge Univer. Press, 2006. [24] J. B. Schneider and R. J. Kruhlak, “Dispersion of homogeneous and inhomogeneous waves in the Yee finite-difference time-domain grid,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 2, pp. 280–287, Feb. 2001. [25] G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat., vol. 23, no. 4, pp. 377–382, Nov. 1981. [26] E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. Roy. Soc. A., vol. 253, no. 1274, pp. 349–357, Dec. 15, 1959. [27] B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A., vol. 253, no. 1274, pp. 358–379, Dec. 15, 1959. Gurpreet Singh received the B.Eng. (Electrical) degree (with first class honors) from Nanyang Technological University, Singapore, in 2007, where he is currently working toward the Ph.D. degree. His research interests include computational electromagnetics, RF and microwave circuit design.

Eng Leong Tan (SM’02) received the B.Eng. (electrical) degree (with first class honors) from the University of Malaya, Malaysia, and the Ph.D. degree in electrical engineering from Nanyang Technological University, Singapore. From 1991 to 1992, he was a Research Assistant at the University of Malaya. From 1991 to 1994, he worked part time at Commercial Network Corporations Sdn. Bhd., Malaysia. From 1999 to 2002, he was a Member of Technical Staff at the Institute for Infocomm Research, Singapore. Currently, he is

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an Associate Professor at the School of Electrical & Electronic Engineering, Nanyang Technological University. His research interests include electromagnetic and acoustic simulations, RF and microwave circuit design.

Zhi Ning Chen (F’08) received the B.Eng., M.Eng., Ph.D. and DoE degrees from Institute of Communications Engineering, China and University of Tsukuba, Japan, all in electrical engineering. During 1988–1997, he worked at Institute of Communications Engineering, Southeast University, and City University of Hong Kong, China with teaching and research appointments. In 1997, he was awarded a JSPS Fellowship to join the University of Tsukuba, Japan. In 2001 and 2004, he visited the University of Tsukuba under a JSPS Fellowship Program (senior level). In 2004, he worked at IBM T. J. Watson Research Center, New York, as an Academic Visitor. Since 1999, he has worked with the Institute for Infocomm Research and his current appointments are Principal Scientist and Depart-

ment Head for RF & Optical. He is concurrently holding Adjunct/Guest Professors at Southeast University, Nanjing University, Shanghai Jiao Tong University, Tongji University, and National University of Singapore. He has published 280 journal and conference papers as well as authored and edited the books Broadband Planar Antennas, UWB Wireless Communication, Antennas for Portable Devices, and Antennas for Base Station in Wireless Communications. He also contributed to the books UWB Antennas and Propagation for Communications, Radar, and Imaging as well as the Antenna Engineering Handbook. He holds 28 granted and filed patents with 17 licensed deals with industry. His current research interest includes applied electromagnetic engineering, RF transmission over bio-channel, and antennas for wireless systems, in particular at mmW, submmW, and THz for medical and healthcare applications. Dr. Chen is a Fellow of the IEEE for his contribution to small and broadband antennas for wireless and an IEEE AP-S Distinguished Lecturer (2008–2010). He has organized many international technical events as key organizer. He is the founder of the International Workshop on Antenna Technology (iWAT). He is the recipient of the CST University Publication Award 2008, IEEE AP-S Honorable Mention Student Paper Contest 2008, IES Prestigious Engineering Achievement Award 2006, I2R Quarterly Best Paper Award 2004, and IEEE iWAT 2005 Best Poster Award.

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Scattering by a Two-Dimensional Periodic Array of Vertically Placed Microstrip Lines Amir Khurrum Rashid, Student Member, IEEE, and Zhongxiang Shen, Senior Member, IEEE

Abstract—Scattering by a two-dimensional (2-D) array of vertically placed microstrip lines is important for a number of recently proposed interesting applications. We investigate this scattering problem using an efficient full-wave mode-matching method. Our analysis comprises two parts. First, the propagation characteristics of the 2-D array are studied by solving a unit-cell of this structure, which is a microstrip line sandwiched between two periodic boundaries. In the second part, an air-to-array discontinuity is solved using Floquet modes in the air-region. Based on our full-wave analysis, we then present two frequency selective surfaces using this array, which exhibit stable quasi-elliptic filtering response under a large variation of the angle of incidence. Results of our full-wave analysis are in excellent agreement with those obtained from measurements and CST Microwave Studio. Index Terms—Frequency selective surface, mode-matching method, periodic boundary condition.

I. INTRODUCTION two-dimensional (2-D) periodic array of vertical microstrip lines was recently used to realize a number of interesting applications, such as high-performance frequency selective surface (FSS) [1], wideband microwave absorber [2], [3], and spatial power combiner [4], etc. Unlike the planar microstrip line geometries used in traditional 2-D FSS or leaky-wave antennas, this array is subjected to an incident plane wave whose polarization is mainly perpendicular to the microstrip lines. These vertically placed microstrip lines can support two quasi-TEM modes [2], which is an attractive feature for applications in wideband and angular-stable designs [1]–[3]. A comprehensive understating of scattering by this array is important for improving the existing designs and proposing its future potential applications. Traditionally, electromagnetic scattering from 2-D surfaces of periodic elements has been widely studied primarily for its application in FSS [5], [6] and shielding structures [7]. These periodic elements generally consist of apertures in a conducting plate or conductive patches on a dielectric layer. A formulation based on Floquet’s theorem is well understood for solving this class of problems [8], [9]. Similarly, a microstrip line grating has also been studied extensively with a focus on its leakywave modes [10]–[12]. Based on a similar formulation using

A

Manuscript received May 03, 2010; revised November 05, 2010; accepted December 10, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (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.2011.2152332

Fig. 1. Geometry of a 2-D periodic array of vertically placed microstrip lines, (a) perspective view, (b) top view, (c) cross-sectional view of a unit-cell.

Floquet modes, we develop an accurate and efficient approach for the analysis of the recently proposed periodic array of vertically placed microstrip lines. The problem under consideration is shown in Fig. 1. Periods along the - and -axes are denoted by and , respectively. Width of the microstrip lines is represented by , and they are printed on a substrate material of dielectric constant and height . Thickness of the strips is assumed to be negligibly small. This array can be constructed by a periodic placement of a number of small printed circuit boards (PCBs), where each PCB consists of one-dimensional (1-D) periodic array of microstrip lines. Section II of this paper studies the propagation characteristics of a 2-D array of vertically placed microstrip lines. Our re-

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sults for the periodic boundary condition (PBC) are also compared with the case when a microstrip line is shielded with side-walls of perfect electric conductor (PEC) or perfect magnetic conductor (PMC). Section III solves the air-to-microstrip line discontinuity using Floquet modes in the air region. Based on that, Section III demonstrates its application as angular stable FSSs. Unlike traditional FSSs [6], these structures do not strictly require an additional dielectric layer. The stable angular performance is primarily due to a smaller size of their unit-cells [6], [13]–[15], which follows from the existence of quasi-TEM modes of this 2-D array of vertically placed microstrip lines. An excellent agreement between our analysis and measurement results validates our formulation and design concepts.

and

Fields in Region II are also derived from their -directed potentials:

II. PROPAGATION CHARACTERISTICS Fig. 1 shows a 2-D periodic array of vertically placed microstrip lines. It is seen that the unit-cell of this array is basically a microstrip line sandwiched between two periodic boundaries. Under an oblique incidence, the periodic boundary can be characterized by a phase shift , which is a function of the angle of incidence. The top and bottom PEC planes refer to the ground of a printed circuit board. Since the problem of a 2-D array can be reduced to that of a unit-cell based on Floquet theorem [16], we can therefore study the propagation characteristics of the unit-cell using an efficient full-wave mode-matching method.

(2) where

A. Formulation The unit-cell can be divided into three distinct regions, as shown in Fig. 1(c). Region I consists of an inhomogeneous waveguide that supports LSE and LSM modes [16]. This inhomogeneous region is further divided into two sub-regions (i), (ii). We use the following y-directed Hertzian potentials: Expressions for Region III potentials can be obtained from ” and “ ” with and , respectively. (2) by replacing “ The following boundary conditions are enforced on the tangential field components at the regional interface and the periodic boundaries

(1) where

where and denote the tangential electric and magnetic is the phase shift between two adjacent fields in region . periodic boundaries, which can be related to the incident angle. Solving these boundary conditions through suitable orthogonal relations leads to a system of linear equations. It is then transformed to a matrix equation whose non-trivial solution yields the propagation constant .

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b

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TABLE I CONVERGENCE BEHAVIOR OF THE FIRST TWO MODES (t : : ,f ,h ,d ,"

= 10 mm = 3 mm = 1 27 mm

= 4 mm, = 2 2 = 10 GHz, 8 = 0)

Fig. 3. Variation of the propagation constant as a function of periodic phase shift 8 for first three modes of a periodic array of microstrip lines, (t , : ). b ,h ,d : ,"

= 5 mm = 5 mm = 1 524 mm

Fig. 2. Dispersion diagram of the first six modes of a periodic array of mi,h : , ,d : ," ,b crostrip lines (t 8 ).

=0

= 2 mm = 5 mm = 5 mm = 1 524mm

= 3 38

B. Convergence In order to ensure a fast convergence, the number of modes in each region is dynamically chosen for a simulation frequency point. For every simulation wave number , we define a corre(approximately ten times ), sponding upper bound factor and then calculate the number of modes as follows:

where rounds its argument to the nearest larger integer. The modes considered for Regions II and III are restricted by the same upper bound factor , which ensures the convergence of results for any arbitrary heights ratio and dielectric constant . The number of modes in Region I is taken as an algebraic and . Using this scheme, we obtain conversum of is chosen as ten times the operating wave gent results when number , as given in Table I. C. Dispersion Diagram Fig. 2 shows a comparison of the dispersion diagram generated by our method with the one obtained from CST Microwave Studio (MWS). An excellent agreement can be observed. Considering a small phase shift, we can observe the existence of two , these two modes are simquasi-TEM modes. When ilar to those of a microstrip line shielded with PMC side walls. Based on their field patterns [2], these modes were termed as “substrate-mode,” and “air-mode” in [1], [2]. Most of the power

= 3 38

= 2 mm

Fig. 4. Cross-section of a microstrip line shielded with (a) PEC, (b) PMC side walls.

carried by the first mode is concentrated in the substrate region, while the second propagates dominantly in the air region above the substrate [2]. Interestingly, the existence of dual-mode propagation was previously also observed for an open strip grating, and these were named as grating and surface wave modes in [11]. Fig. 3 presents the variation of propagation constants of the is varied from 0 to 120 . In the refirst three modes when gion of lower frequencies, propagation constants of these modes decrease with an increase in the periodic phase shift. The substrate mode remains almost stable while the air-mode becomes evanescent at lower frequencies when the periodic phase shift is increased, as shown in Fig. 3. Basically, when the periodic , virtual magnetic walls may be considphase shift is ered between the strips of the periodic array of microstrip lines. The unit-cell of this geometry then takes the form of a microstrip line shielded with PMC sidewalls as shown in Fig. 4(b). Hence, the existence of two quasi-TEM modes can be attributed to the presence of three isolated conductors in this case. On the other , a virtual PEC wall may be considextreme, when ered between the strips and the unit-cell of the resulting periodic array is given in Fig. 4(a). Since this geometry is known to support the propagation of single quasi-TEM mode, it may explain the variation of the air-mode in Fig. 3, which becomes evanescent with an increase of the periodic boundary phase shift. While using equivalent PMC/PEC walls at can explain the behavior of the dominant modes of a periodic array of microstrip lines, it may not be thought of as a substitute for the periodic boundary. This comparison is more clearly discussed in the next sub-section.

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Fig. 5. A schematic diagram of the 2-D periodic array of microstrip lines, indicating the direction of propagation of a mode inside the periodic structure.

Alternatively, the trend of Fig. 3 can also be understood by considering the fact that there exists a preferred direction of the group velocity for every propagating mode of the periodic array of microstrip lines [11], as indicated in Fig. 5. If is the periodicity of this array, then the periodic boundary phase shift can be written as

(3) where is the propagation constant along the direction of propagation making an angle with the -axis, as shown in Fig. 5. Since is the -directed projection of [11], we can write

(4)

In the range of lower frequencies when the periodicity of the array is much smaller than the operating wavelength, can be assumed to be independent of , which means that the group velocity of a mode remains the same in every direction since the layer of strips acts as a homogenous surface. This assumption seems reasonable especially for the quasi-TEM modes at lower frequencies where the field distribution is quite uniform within the substrate or the air-region [2]. Differentiating both sides of (3) and (4) with respect to , and after a few simple algebraic manipulations we can obtain

(5) The above result is clearly in agreement with the trend of Fig. 3 at lower frequencies, where we note a reduction in the propagation constant of a mode with an increase in the periodic boundary phase shift. Furthermore, the air-mode changes more rapidly than the substrate-mode, as expected from (5), since the rate of variation is inversely proportional to the propagation constant of a mode. However, our relation breaks down at higher frequencies, which implies that the magnitude of the group velocity of a mode is not independent of its direction of propagation at higher frequencies, as assumed in the derivation of (5). This is understandable because the periodicity of the array becomes comparable with its operating guided wavelength at higher frequencies.

Fig. 6. Comparison of dispersion diagrams of a microstrip line with PEC side (t walls, and PBC side walls when the periodic phase shift 8 ,b ,h : ," : ). ,d

2 mm = 5 mm = 5 mm = 1 524 mm

= 3 38

= 180

=

D. Comparison of PBC With PMC and PEC Walls In many electromagnetic problems of periodic structures, especially those involving microstrip line arrays, a periodic boundary is often replaced by a PEC or a PMC wall [1], [10]. It can be intuitively argued that when the strips are excited in phase, virtual PMC walls shall be formed between the adjacent strips. Similarly, if the two adjacent strips are excited out of phase, a virtual PEC wall may be considered between them. Based on that, we find it an interesting investigation to seek a comparison of PMC and PEC walls with PBC when the and 180 , periodic boundary phase shift is set to be respectively. We use an efficient mode-matching method to calculate the dispersion diagram of a microstrip line shielded with either PEC or PMC side wall, as indicated in Fig. 4. This figure shows the classification of four distinct regions used in our formulation. Tangential field components are equated at the regional interfaces represented by the dotted lines in Fig. 4. The major steps of our procedure are similar to yet more general than those given in [17]. Unlike [17], since we solve the two discontinuity interfaces, our mode-matching approach is also applicable when the microstrip line is placed asymmetrically. However, for the present comparison, it is sufficient to consider the case of a symmetrically placed microstrip line shielded with PMC or PEC side walls, and compare its dispersion diagram with that of a periodic array of microstrip lines when periodic phase shift or 180 . Fig. 6 shows dispersion diagrams of a microstrip line shielded with PEC sidewalls, and that of a periodic array of microstrip lines when its adjacent strips are fed out of phase. As expected, the dominant modes of the two cases are identical. However, it is interesting to note that certain higher order modes are also similar, for instance, the third and fifth modes in Fig. 6. Fig. 7 presents the dispersion diagram of a microstrip line shielded with PMC side walls compared with that of a periodic array of microstrip lines when the strips are excited in phase. The first two modes of these cases are identical, as expected. However, the higher order modes are significantly different in this case. Based on the above, it is seen that a periodic boundary is significantly different from a PEC or a PMC wall. A PEC or

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and are the coefficients of incident and rewhere denotes the propagation conflected waves, respectively. stant of a mode. Expressions for transverse modal field and admittance are given in [8]. Region 2 supports hybrid modes as follows:

(7)

Fig. 7. Comparison of dispersion diagrams of a microstrip line with PEC side (t , walls, and PBC side walls when the periodic phase shift 8 b ,h : ," : ). ,d

= 5 mm = 5 mm = 1 524 mm

= 3 38

=0

= 2 mm

where and are the coefficients of the forward and backward waves, respectively. Expressions of the transverse modal are derived from the previous section. Modal fields fields in the two regions are normalized according to the following orthogonality relations:

Enforcing the continuity of tangential field components at the air-to-microstrip line interface results in two sets of equations: Fig. 8. Cross-sectional view of the unit-cell of a finite length array illustrating a cascaded junction of two air-to-microstrip line discontinuities.

PMC wall requires the tangential electric or magnetic field components to vanish on the wall surface, while a periodic boundary condition requires the phase shifted replicas of field pattern based on Floquet theorem [16]. However, under certain excitation conditions, some modes of a microstrip line with PBC become identical to those of a microstrip line with PEC/PMC side-wall. For those modes, PEC/PMC side-wall may be treated as special cases of periodic boundaries.

(8) (9) where “ ” is a unified index for “

” modes in Region 1, and

III. AIR-TO-ARRAY DISCONTINUITY A. Formulation Fig. 8 shows a unit-cell of the proposed array of thickness , which is illuminated by a plane wave incident from air. Regions 1 and 2 denote the air region and the microstrip line region, respectively. Based on Floquet’s theorem, the problem of air-toarray discontinuity is also reduced to that of an air-to-microstrip line discontinuity. Tangential field components in Region 1 can be written as:

The generalized scattering matrix of the air-to-array discontinuity is calculated as:

(6)

(11)

(10) and are the identity matrices of sizes and , respectively. An array of finite thickness can be modeled as a cascaded junction of two air-to-array discontinuities, and its S-parameters can be calculated as [18] where

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TABLE II CONVERGENCE BEHAVIOR OF S-PARAMTERS WITH THE NUMBER OF MODES ,b : ,h , ,d IN THE AIR REGION (t " : ,f ,8 ,N )

= 1 mm = 5 mm = 5 mm = 1 524 mm = 3 38 = 10 GHz = 0 = 10

TABLE III CONVERGENCE BEHAVIOR OF S-PARAMTERS WITH THE NUMBER OF MODES IN THE ARRAY REGION (t ,b ,h ,d : , : ,f ,P Q ) ,8 "

= 1 mm = 5 mm = 5 mm = 1 524 mm = 3 38 = 10 GHz = 0 = = 2

Fig. 9. S-parameters of a cascaded junction of two air-to-array discontinu: ,b ,h : ,d ities under TE incidence (t : ,L : ,' ). ," ,

1 524 mm

where is an diagonal matrix based on propagation constants of modes in Region 2, as calculated in the Section II of this paper

..

.

B. Results For the given angles of incidence and , we first express the periodic boundary shift as , calculate the modes of Region 2, and then apply the above procedure to obtain scattering parameters. Similar to the previous section, the number of modes considered in our analysis is a function of the frequency, angle of incidence, and the geometrical parameters of the array. We define an upper bound wave number , which is approximately ten times the simulation wave number , and then calculate the number of propagating Floquet modes in the air region corresponding to , for a known angle of incidence, and geometrical dimensions of the array. These values of and are then used in the actual simulation using our code for the operating point . The number of propagating modes in Region 2 cannot be found analytically for a given upper bound . For this case, we fix the number of modes of Region 2 to a small finite value . It is seen that our mode-matching approach actually results in a rapidly converging results and hence a few modes are sufficient for Region 2. Table II shows the convergence behavior of is varied to change the number of modes S-parameters when leads to sufficiently conin Region 1. It is noted that vergent results. Table III shows convergence behavior of S-parameters when the number of modes in Region 2 is varied. Fig. 9 compares our results with those obtained from CST MWS, whereby a very good agreement is noted. Fig. 10 shows the reflection coefficient of a single air-to-array discontinuity

= 0 2 mm = 5 mm = 3 = 9 5 mm = 0 = 40

= 3 124 mm =

for and Floquet modes. It also presents the transmission coefficients of the first two modes of the microstrip line and refer to the substrate-mode and region, where the air-mode, respectively. It may be noted that most of the inFloquet mode is coupled to the cident energy carried by air-mode (second mode) of microstrip lines. This is understandable from the field distribution of the air-mode [2]. On the other hand, the first two modes of the array are not excited by the Floquet mode. Therefore, it results in strong reflection, as seen in Fig. 10(b). Based on this result, the applications of this 2-D array are actually limited to TE incidence only [1], [2]. Reference [1] includes a comprehensive discussion on the parametric study and operation of this array under normal incidence when virtual PMC walls may be assumed between the strips. As shown in Section II of the present paper, a consideration of PMC walls is actually a valid assumption for the first two dominant modes of a periodic array. Since the applications of this array are based on these quasi-TEM modes, discussion and design guidelines given in [1] are directly applicable here. However, as shown is Fig. 3 and justified by (5), an oblique incidence basically changes the propagation constants of the modes of this array, and the amount of change is related to the phase shift between two adjacent periodic boundaries. With a selection of smaller unit-cell size, this phase shift can be reduced, and hence it is possible to obtain an angular stable design while still following the guidelines of [1]. The next section of this paper presents two such examples of FSSs, where excellent performance stability under a large variation of incidence angle has been achieved. IV. EXPERIMENTAL VERIFICATION Since a 2-D periodic array of vertical microstrip lines supports two quasi-TEM modes, in principle, it is possible to choose an arbitrarily smaller size of the unit-cell, which in turn may result in a highly stable filtering response under different angles of incidence. Based on that, this array can practically be considered as a promising candidate for designing angular stable FSSs. We follow the general guidelines in [1], and design two examples to demonstrate the application and validity of our results. The intuitive guidelines in [1] are useful for designing

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Fig. 10. Reflection coefficient of air-to-microstrip line discontinuity, and transmission coefficients of the first two propagating modes under TE and TM indi,b ,h ,d : ," : , dence (t ' , ). (a) TE incidence; (b) TM incidence.

Fig. 11. S-parameters of a narrow-band design example exhibiting a stable : , filtering performance under different angles of TE incidence (t ,L : ,' ). b ,h : ,d : ," (a) Reflection; (b) transmission.

simple FSSs exhibiting one or two transmission zeros. However, more complicated designs of this geometry may also be taken up on the basis of a coupling matrix based synthesis [19]. Fig. 10 shows that S-parameters of an air-to-array discontinuity do not vary rapidly, and remain fairly constant for a wide range of frequencies. This implies that a coupling matrix based approach may become applicable even for a wideband design. This is unlike the waveguide structures where discontinuity parameters are highly frequency dependent, and hence coupling matrix based synthesis is applicable for very narrow-band designs only. Fig. 11 shows a narrow-band example, where two reflection zeros corresponding to the two modes of the array can be identified. The first reflection zero corresponding to the substrate mode remains very stable under a large variation of angle of incidence. This is in line with (5) and the trend of Fig. 3, as discussed in Section II. The second reflection zero corresponding to the air-mode changes significantly, yet the over-all frequency response of the FSS is stable under even a large variation of the angle of incidence. Fig. 12 presents results of a wideband example. The two reflection zeros in this case correspond to the two resonances of the substrate mode over the finite thickness of this array. An

excellent stability of frequency performance has been obtained under oblique incidence, without requiring any additional dielectric layer. These figures also show a very good agreement between measurement results and those obtained through our efficient full-wave mode-matching method.

= 2 mm = 5 mm = 5 mm = 1 524 mm =0 =0

= 3 38

= 5 mm = 3 124 mm = 1 524 mm

= 0 2 mm = 3 = 9 5 mm = 0

V. CONCLUDING REMARKS A periodic array of vertically placed microstrip lines has been analyzed for its performance under oblique incidence. It supports two quasi-TEM modes when excited by a TE wave. An efficient full-wave mode-matching method has been employed to investigate both the propagation characteristics and the air-toarray discontinuity for this 2-D array of microstrip lines. It has been experimentally demonstrated that this array is very suitable for designing a pseudo-elliptic FSS, with highly stable response under a large variation of angle of incidence. Since this 2-D array of microstrip lines is also an integral part of many recent structures like microwave absorber [2], [3], spatial power combiner [4], etc., we believe that our mode-matching formulation presented here may also be extended to those and other future applications involving this array of vertically placed microstrip lines.

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[9] C.-C Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microw. Theory Tech., vol. 21, no. 1, pp. 1–6, 1973. [10] P. Baccarelli, P. Burghignoli, C. Di Nallo, F. Frezza, A. Galli, P. Lampariello, and G. Ruggieri, “Full-wave analysis of printed leaky-wave phased arrays,” Int. J. RF Microw. Comput.-Aided Engrg., vol. 12, no. 3, pp. 272–287, 2002. [11] F. H. Bellamine and E. F. Kuester, “Guided waves along a metal grating on the surface of a grounded dielectric slab,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 7, pp. 1190–1197, 1994. [12] J. A. Encinar, “Mode-matching and point-matching techniques applied to the analysis of metal-strip-loaded dielectric antennas,” IEEE Trans. Antennas Propag., vol. 38, no. 9, pp. 1405–1412, 1990. [13] H. L. Liu, K. L. Ford, and R. J. Langley, “Design methodology for a miniaturized frequency selective surface using lumped reactive components,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2732–2738, 2009. [14] K. Sarabandi and N. Behdad, “A frequency selective surface with miniaturized elements,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1239–1245, 2007. [15] F. Bayatpur and K. Sarabandi, “Miniaturized FSS and patch antenna array coupling for angle-independent, high-order spatial filtering,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. 2, pp. 79–81, 2010. [16] R. E. Collin, Field Theory of Guided Waves. New York: McGraw Hill, 1960. [17] H.-W. Yao, A. Abdelmonem, J.-F. Liang, and K. A. Zaki, “Analysis and design of microstrip-to-waveguide transitions,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 12, pp. 2371–2380, 1994. [18] J. Uher, J. Bornemann, and U. Rosenberg, Waveguide Components for Antenna Feed Systems: Theory and CAD. Norwood, MA: Artech House, 1993. [19] S. Amari and U. Rosenberg, “Direct synthesis of a new class of bandstop filters,” IEEE Trans. Microw. Theory and Tech., vol. 52, no. 2, pp. 607–616, 2004.

Fig. 12. S-parameters of a wideband design example exhibiting a stable fil: ,b tering performance under different angles of TE incidence (t ,h ,d : ," : ,L ,' ). (a) : Reflection; (b) transmission.

1 5 mm = 2 mm = 1 27 mm

= 1 3 mm = = 10 2 = 7 mm = 0

Amir Khurrum Rashid (S’10) was born in Layyah, Pakistan, in 1982. He received the B.S. degree in electronic engineering from Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Topi, Pakistan, in 2003. Currently, he is working towards the Ph.D. degree at Nanyang Technological University, Singapore. He worked as Assistant Manager (Technical) at the National Engineering and Scientific Commission, Pakistan, from 2003 to 2006. His research interests include RF/microwave antennas and absorbers.

REFERENCES [1] A. K. Rashid and Z. Shen, “A novel band-reject frequency selective surface with pseudo-elliptic response,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1220–1226, 2010. [2] A. K. Rashid, Z. Shen, and S. Aditya, “Wideband microwave absorber based on a two-dimensional periodic array of microstrip lines,” IEEE Trans. Antennas Propag., vol. 58, no. 12, pp. 3913–3922, 2010. [3] B. Zheng and Z. Shen, “Wideband radar absorbing material combining high-impedance transmission line and circuit analogue screen,” Electron. Lett., vol. 44, no. 4, pp. 318–319, 2008. [4] M. P. DeLisio and R. A. York, “Quasi-optical and spatial power combining,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 929–936, 2002. [5] R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces—A review,” Proc. IEEE, vol. 76, no. 12, pp. 1593–1615, 1988. [6] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New Yor: Wiley, 2000. [7] D. C. Love and E. J. Rothwell, “A mode-matching approach to determine the shielding properties of a doubly periodic array of rectangular apertures in a thick conducting screen,” IEEE Trans. Electromagn. Compat., vol. 48, no. 1, pp. 121–133, 2006. [8] C.-C. Chen, “Scattering by a two-dimensional periodic array of conducting plates,” IEEE Trans. Antennas Propag., vol. 18, no. 5, pp. 660–665, 1970.

Zhongxiang Shen (M’98–SM’04) received the B.Eng. degree from the University of Electronic Science and Technology of China, Chengdu, in 1987, the M.S. degree from Southeast University, Nanjing, China, in 1990, and the Ph.D. degree from the University of Waterloo, Waterloo, ON, Canada, in 1997, all in electrical engineering. From 1990 to 1994, he was with Nanjing University of Aeronautics and Astronautics, China. He was with Com Dev Ltd., Cambridge, ON, as an Advanced Member of Technical Staff in 1997. He spent six months each in 1998, first with the Gordon McKay Laboratory, Harvard University, Cambridge, MA, and then with the Radiation Laboratory, the University of Michigan, Ann Arbor, MI, as a Postdoctoral Fellow. He is presently an Associate Professor in the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. His research interests are in microwave/millimeter-wave passive devices and circuits, small and planar antennas for wireless communications, and numerical modeling of various RF/microwave components and antennas. He has authored or co-authored over 100 journal articles and more than 100 conference papers. Dr. Shen served as Vice-Chair and Chair of the IEEE MTT/AP Singapore Chapter in 2008 and 2009, respectively. He currently serves as the AP-S Chapter Activities Coordinator.

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On the Co-Polarized Phase Difference of Rough Layered Surfaces: Formulae Derived From the Small Perturbation Method Saddek Afifi and Richard Dusséaux

Abstract—We determine the statistical distribution of the co-polarized phase difference of fields scattered from a stack of two two-dimensional rough interfaces in the incidence plane. The electromagnetic fields are represented by Rayleigh expansions and a perturbation method is used to solve the boundary value problem and to determine the first-order scattering amplitudes. For slightly rough interfaces with infinite length and Gaussian height distributions, we show that the probability density function is only a function of two parameters. For a sand layer on a granite surface in backscattering configurations, we study the influence of the incidence angle, the layer thickness, the cross-spectral density and the wave frequency upon both parameters of the probability law. Index Terms—Co-polarized phase difference, density probability function, random surfaces, scattering amplitudes, small perturbation method.

I. INTRODUCTION EVERAL approximate models were implemented for the analysis of the scattering of electromagnetic waves by rough layered interfaces: the small perturbation method [1]–[9], the full-wave method [10], [11], the Kirchhoff method and the geometrical optics approximation [12], [13] and the small slope approximation [14]. These models give analytical solutions and thus cannot be compared in terms of computing time with the exact methods that are associated with numerical techniques [15]–[23]. Within their domains of validity, these approximate models allow a fast analysis of the multilayered structures by means of analytical formulae giving the scattered field and intensities. The work presented in this paper is based on the first-order small perturbation method. Elson was one of the first authors to develop a vector theory of scattering from a stack of two-dimensional slightly rough interfaces. This vector theory allows the angular distribution of scattered light to be determined [1] and can be used with fully or partially correlated or uncorrelated surface roughness. Many authors have studied the role

S

Manuscript received August 08, 2010; revised November 06, 2010; accepted December 10, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. S. Afifi is with the Laboratoire de Physique des Lasers, de Spectroscopie Optique et d’Opto-électronique (LAPLASO), University Badji Mokhtar Annaba, 23000 Annaba, Algeria (e-mail: [email protected]). R. Dusséaux is with Laboratoire Atmosphères, Milieux, Observations Spatiales (LATMOS), Université de Versailles Saint-Quentin en Yvelines, Quartier des Garennes, 78280 Guyancourt, France (e-mail: richard.dusseaux@latmos. ipsl.fr). Digital Object Identifier 10.1109/TAP.2011.2152347

of correlation lengths and roughness cross-correlation properties on the angular scattering [1], [2], [5] and studied the relationship of the scattering from multilayered structure to angle of incidence and polarisation [2], [4]. References [1]–[5] deal with light scattering from multilayer optical coatings. Many authors have developed a vector perturbative theory for analyzing remote sensing problems [6]–[9]. In [1]–[9], the perturbation method has been used to determine the first-order complex amplitude of the far-field and the average radiation intensity. But, these works do not give the statistical distribution of the scattered field in modulus and phase [25], [26] and the statistical distributions of the co-polarized and cross-polarized phase differences [27]–[30]. In [31], [49], we studied the scattering of an incident plane wave by a stack of two one-dimensional rough interfaces. Within the framework of the small perturbation method and by assuming Gaussian height distributions with zero mean values, we showed that for a finite extension interfaces, the real and imaginary parts of the scattering amplitude are random centered and correlated Gaussian variables, with unequal variances. The modulus of the scattering amplitude thus follows a Hoyt law and the phase law is not uniformly distributed between and [31]–[33], [49]. For an infinite length, we have shown that the real and imaginary parts are uncorrelated and have the same variance. For fully or partially correlated or uncorrelated interfaces, the modulus then obeys a Rayleigh law and the phase is uniform [31]–[35], [49]. In [32], we have obtained the same results for the field scattered by a single rough surface separating two media. Consequently, the forms of probability density functions do not allow to differentiate the case of a single surface from the case of a layered medium. For a single surface separating two media, the small perturbation method shows that the scattering amplitude is proportional to the Fourier transform of the surface [32], [36]. Consequently, the co-polarized phase difference is not a random variable. For rough layered interfaces, the two co-polarized scattering amplitudes are not proportional and their phase difference obeys a probability law. In this paper, we present a statistical study of the co-polarized phase difference of far-fields scattered from a stack of two rough two-dimensional interfaces illuminated by a plane wave. Such a study is important in the characterization of both interfaces by polarimetry radar, for the detection of wet sub-surface [37], [38], for oil spill observation by a phase signature [39] or for the classification of polarimetric SAR images [40]. The polarimetric signature can be represented by the scattering matrix which binds the scattered far-field to the incident

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field. In [27], Sarabandi described the components of this matrix by Wiener processes [24], [25], [30]. When a radar illuminates a random surface, many point scatterers contribute to the total scattered field. Each element of the scattering matrix elementary waves can be represented by a superposition of produced from scattering elements of the rough surface. When is large enough, the application of the central limit theorem shows that the scattering matrix components are represented by independent Gaussian processes. Based on these assumptions, the covariance matrix depends on four parameters which can be obtained from the Mueller matrix. The probability density function (pdf) of the co-polarized phase difference is defined by only two parameters and the cross-polarized phase difference is uniform. These assumptions and the distributions are confronted with experimental results for rough surfaces illuminated in C-band, X-band and L-band [27], [41], [42]. In [28], the authors investigated the phase statistics of interferograms of distributed scatterers. They used the co-polarized phase difference distribution presented in [27] and analyzed the interferograms of interferometric Synthetic Aperture Radar by taking into account the effects of thermal noise and the SAR processor aberrations on the phase. To reduce statistical variations, polarimetric and interferometric radars use the multi-look processing by averaging spatially in the complex representation of both co-polarized backscattered signals. The statistical properties of both signals are different from those of the single-look processing [27]. In [29], the authors derived the multi-look phase difference pdf from the complex Wishart distribution. In [30], the authors established results under the K-distribution model for a multi-look processing [24]. The models presented in [26]–[30] are based above all on signal theory methods. The connection with fundamental equations of the electromagnetism does not appear. But, for physicists, it is also important to derive the statistical distributions of scattered fields from an electromagnetic model, either an analytical approach or Monte-Carlo simulations. In the present paper, our step is not based on the a priori assumption that the scattered field is described by a random walk. We study the scattering of an incident plane wave by a stack of two rough interfaces. Assuming Gaussian height distributions with zero means, we derive from the first-order small perturbation method the phase difference pdf of 1-look signals in the incidence plane. For infinite extension interfaces, we show that the covariance matrix depends on four parameters and that the pdf of the co-polarized phase difference is only a function of two parameters. The paper is organised as follows. Section II presents the statistical properties of the rough interfaces and Section III recalls the scattered field expressions within the framework of the Rayleigh hypothesis [43]. In Section IV, the analytical solution to scattering by stratified medium is presented. A perturbation method is used to solve the boundary value problem and to determine the first-order scattering amplitudes [7], [31], [49]. In Section V, the analytical formulae giving the probability density function and the distribution function of the co-polarized phase difference are established [27], [28]. Section V is also devoted to the study of the probability law according to the dielectric permittivity vertical profile and the statistical properties of the layered medium. For a sand layer on a granite surface and backscatter configurations, the influence of the incidence angle,

Fig. 1. Structure with two nonparallel interfaces.

the layer thickness, the cross-spectral density and the wave frequency upon the probability law is analyzed in Section VI. II. CHARACTERIZATION OF THE ROUGH INTERFACES The geometry of the problem is described in Fig. 1. The structure is a stack of two two-dimensional rough interfaces. Both . interfaces are randomly deformed over a surface area is the layer thickness. and describe the upper and The functions lower interfaces, respectively. Both interfaces are random processes, centered, isotropic and stationary to the second order. They can be correlated or non-correlated. The random functions can be synthesized using a filtering process [31], [44], [49]. For numerical applications, we consider Gaussian correlaand the tion functions. The autocorrelation functions are defined as follow: intercorrelation function

(1) are the rms heights and are the correlation radius. is the mixing parameter with a value between 1 and +1 [31], [49]. The correlation coefficient between both interfaces depends on mixing parameter: (2) If , the interfaces are not correlated . The surfaces are fully correlated when (i.e., for and ). The spectra and the inter-spectrum are Gaussian:

(3) denotes the 2D Fourier transform of

.

III. INCIDENT FIELD AND SCATTERED FIELDS The structure is illuminated by a monochromatic plane wave is defined by with a wavelength . The incident wave vector the zenith angle and the azimuth angle (4)

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with

(5) , and are the unit vectors of the Cartesian coordinate system. We consider a harmonic temporal dependence . Medium 1 is assimilated to the vacuum. We note as the vacuum impedance and as , , and indicate the wave number. Henceforth, the optical indices, the relative permittivity, the impedances and the wave numbers of layers 2 or 3. The incident electric and magnetic field vectors are defined as follows:

(14) In medium 3, the Rayleigh expansion uses only downwards propagating component

(6) (15) (7) where

and, (16) (8)

is the wave vector of an elementary plane wave and is given by

(9)

) polarization, the electric field vector For horizontal (or plane and we take and is parallel to the . For vertical (or ) polarization, the magnetic plane and we take field vector is parallel to the and . The factor makes the incident power independent on the deformation length : (10) The scattered field is expressed by a Rayleigh expansion. In the vacuum, the electric and magnetic components are given by:

(11)

(12) In medium 2, it is necessary to consider upwards and downwards propagating waves.

(17) and

designate the polarization vectors (18) (19)

The constants of propagation or zero imaginary part

(

, 2, 3) present a negative

(20) associated with The constants of propagation are denoted with . The unknowns of the problem are the scattering amplitudes ( , 2, 3). The Rayleigh hypothesis is compatible with the Rayleigh expansions at any point of the space including the interfaces. We determined the scattering amplitudes from a first-order perturbation method applied to the boundary value problem. In Section IV, we give the zeroth- and first-order solutions. In the far-field zone, the angular dependence of the field scattered within the upper medium at the observation point is given by the product where and [45]–[47]. is the zenith angle and , the azimuth angle. In Section V, we will establish the analytical expression of the co-polarized phase difference pdf. IV. ZEROTH-AND FIRST-ORDER SCATTERING AMPLITUDES

(13)

The zeroth-order problem gives the field diffracted by perfectly smooth interfaces and the specular amplitude

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[7]. After some mathematical manipulations, we obtain for the horizontal polarization

(21)

(31)

where (22) and for the vertical polarization (32)

(23) where

(33) (24)

After some tedious mathematical manipulations, the first-order perturbation theory applied to the boundary value problem gives the direct and cross components as follows [7]: (25) is defined from the Fourier transform of

by

(26) The factors and depend on the polarization. For a h-polarized incident wave, we obtain the complex amplitudes and

(34) where . If the optical indices and are are equal to zero and we obtain, as equal, the factors expected, the scattering amplitudes of a single interface [36]. , there is no depolarIn the incidence plane where ization and the cross scattering amplitudes and are equal to zero. Moreover, the direct factors and become identified with the factors obtained for a stack of two one-dimensional rough interfaces [31], [49]. Henceforth, we study configurations in the incidence plane. for the other Expressions of the complex amplitudes layers are given in the reference [31], [49]. V. STATISTICAL CHARACTERISTICS OF THE CO-POLARIZED PHASE DIFFERENCE A. Joint Probability Density Function of the Co-Polarized Scattering Amplitudes

(27) (28)

(29)

We assume that the height distributions of both interfaces are Gaussian. Since the Gaussian character is preserved by linear operation, the Fourier transforms of both functions as well as the first-order scattering amplitudes are Gaussian processes. For rough layered interfaces, the two co-polarized first-order scattering amplitudes are not proportional and their phase difference changes from one realization to another. We propose to determine the properties of this stochastic process. We write

(30) where complex amplitudes

. For a v-polarized incident wave, the and are given by

(35) where the letter designates the polarization ( or ). According to the relation (25), the real and imaginary parts of the first-order scattering amplitudes are given in the following form:

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(42) (36)

(43) (44)

are centered both random functions , both functions and are centered. The brackets stand for a statistical average. For and , the random variables , given values of , and follow a Gaussian joint pdf:

, the covariance matrix depends only on four When , , and ). Consequently, the parameters ( inverse of the covariance matrix is defined as follows:

(37)

(45)

Insofar

The symbol

denotes the transpose.

is the covariance matrix where (46)

(38) and For a finite length , the covariance matrix is full. In this paper, we consider infinite extension interfaces in order to determine the analytical expression of the probability density function of the co-polarized phase difference. We show from (25), (26) , the variances and and (36) that when of the real and imaginary parts are equal with

(47) The coefficient is by definition equal to or greater than zero. , the variances and and the According to (47), if covariances and satisfy the condition (48) After some mathematical operations, we show from (39), (41) and (42) that the relation (48) is equivalent to

(39) In addition, we have established that for a given polarization, the real and imaginary parts are uncorrelated (40) , We found from (25), (26) and (36) the covariances , and and prove that for an infinite length the random variables and are correlated with and with the variables

(41)

(49) According to (3) and knowing that (49) is valid for any value for , , .

, the relation and . Consequently,

B. Probability Density Function of the Co-Polarized Phase Difference Using the polar coordinates, we obtain from (37) and (38) the , joint probability density function of the random variables , and that represent the modulus and the phase of and , respectively

(50)

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The joint probability density function of and can be deby a double integration shown in (51) duced from at the bottom of the page, where

(52)

The distribution function is defined by (58) After some mathematical operations, we obtain

The probability density function of the co-polarized phase difference is obtained from by the following relation: (53) Taking into account that

(59)

, we obtain We can check that

and

.

C. Study of Limit Cases

(54)

, the difference phase pdf According to (54), if is uniform between and . According to (47), occurs if the covariances and are the case null (60)

where (55) The probability density function of the co-polarized phase difference (54) is entirely determined by the coefficient (47) and the phase (55). The parameter is expressed as a function of the covariances and and of the variances and [27]. The angle is only a function of both covariances [27]. After some mathematical manipulations, we show that the statistical average is given by

According to (54), the phase difference is no longer a random . It is equal to the angle variable if (61) occurs when the variances and and The case and satisfy the equality in (48). the covariances Taking into account expressions (39), (41) and (42), there are two configuration types leading to a constant phase difference. The first case occurs when the incidence angle, the observation angle, the optical indexes and the layer thickness satisfy the following condition: (62)

(56) is not equal to . We did not establish the analytical for. This variance is calculated numerimula of the variance cally by (57)

The condition (62) does not use the statistical properties of the surfaces and can be verified for the correlated or . When the connon-correlated surfaces with dition (62) is satisfied, we can show from (25) that for each realization of rough two-dimensional layers, the scattering and are proportional amplitudes . Consequently, the phase difference is not a random variable and . The

(51)

AFIFI AND DUSSÉAUX: ON THE CO-POLARIZED PHASE DIFFERENCE OF ROUGH LAYERED SURFACES

second case occurs when the incidence and observation angles satisfy the condition (63)

(63) In particular, we find from the expressions of the spectra and the inter-spectrum (3) that the condition (63) is satisfied when . For a single surface separating two media, the co-polarized . Within the phase difference is not a random variable and framework of the first-order small perturbation method, we can . only infer the presence of several layers when D. An Example Configuration Where Varies From 0 to 1 There exist several configurations that satisfy the condition (60) and lead to . For example, in the case of correlated interfaces having the same correlation length and taking into account the relations (3), (41) and (42), we find that the covariand are null when ances

(64a)

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The mixture parameter which is negative according to (67) (68) depend on the incidence angle and on and the ratio the refractive index but do not depend on the wavelength (On a given frequency range, we can assume that the refractive index is constant). For an incidence angle and a refractive index satisfying (67) and (68), when the wavelength and the thicksatisfy (65), we get in backscattering. ness In addition, for a configuration where , we show from (27), (28), (31) and (32) that the condition (62) is satisfied . The values of in backscattering when leading to are given by (69)

where is an integer. This condition does not use the root mean square heights and the correlation lengths. The selected configuration is in backscattering and is char, , an incidence angle acterized by and a real refractive index satisfying the conditions (67) and (68). When the thickness , the wavelength and the refractive index satisfy the relation (65), the phase difference follows a uniform law because . When they satisfy the relation (65), and the phase difference is no longer a random varithen able. For a value given of , when the layer thickness or the wavelength vary, the parameter oscillates between 0 and 1. E. On the Cross-Polarized Phase Difference in the Incidence Plane

(64b) and Let us consider a configuration where where medium 2 is lossless. Let us suppose moreover that and (i.e., and ). We can write (65)

where

is an integer. For such a configuration in backscattering , the terms are purely imaginary. The condition (64b) thus is satisfied and the condition (64a) leads to

In the incidence plane, the cross scattering amplitudes and are equal to zero. The cross-polarized phase difference is reduced to the . For one-dimensional rough interfaces, we showed in phase , the phase is uniformly distributed [31], [49] that when between and . This result remains true for two-dimensional rough layers. Consequently, the cross-polarized phase difference in the incidence plane is also distributed between and and does not contain information about the medium [27], [41], [42]. VI. NUMERICAL RESULTS A. Configuration Where the Coefficient Varies From 0 to 1

(66) where the mixture parameter

satisfies the relation

We study the configuration of Section V-D and determine the for two incidence behavior of as a function of the ratio and . The study is angles undertaken in backscattering . The refractive index is equal to 3/2. The mixture parameter (67) and the ratio (68) depend on the incidence angle with

(67) (70a) (70b)

The single solution of the (66) is given by the ratio (68)

satisfies the relation (65), the coefficient When the ratio is equal to zero and the phase difference follows a uniform law.

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Fig. 2. (a) Coefficient r versus ratio u = for the configuration of Section V-D. (b) Phase versus ratio u = for the configuration of Section V-D. Fig. 3. (a) Coefficient r versus ratio u = for a sand layer on a granite surface. (b) Phase versus ratio u = for a sand layer on a granite surface.

For the chosen incidence angles, we obtain the following values of : (71a) (71b) satisfies the relation (69), the coefficient When the ratio is equal to 1 and the phase difference is not a random variable. For both incidence angles, we obtain the following values: (72a) (72b) Curves of Fig. 2(a) and (b) confirm that for the structure under study, the coefficient oscillates between 0 and 1. For values satisfying (68), the phase presents a discontinuity and of to according to expression (55). swings from B. Study of a Granite Surface Covered With a Sand Layer We consider a sand layer on a granite surface where the relative permittivities and are fixed at and 8 [16], [17]. The two interfaces are characterized by the root mean

square heights and , and by the correlation radius . For each simulation, we modify the layer thickness , the mixture parameter and the wave frequency . Three values of the zenith angle are considered: 10 , 40 and 70 . Azimuth angles are fixed to 0 . Fig. 3(a) and (b) give coefficient and phase versus thickness . The mixture parameter is equal to 0, and the wavelength is equal to 20 cm. The oscillation amplitude decreases slightly due to the low dielectric loss angle of the sand. The oscillation amplitude increases with the incidence angle . For an incidence angle of 10 , the coefficient is quasi-constant and close to 1 whatever the layer thickness and the phase presents , presents a varivariations lower than 1 degree. For ation of 10% between a consecutive maximum-minimum pair and the variations of are higher than 110 . The phase is than the pamore sensitive to the variations of the ratio rameter . Fig. 4(a) gives the coefficient as a function of the mixture parameter . Fig. 4(b) shows the phase . The layer thickness is equal to and the wavelength is equal to 20 cm. We notice that the sensitivity to the mixture parameter increases with the

AFIFI AND DUSSÉAUX: ON THE CO-POLARIZED PHASE DIFFERENCE OF ROUGH LAYERED SURFACES

Fig. 4. (a) Coefficient r versus mixture coefficient q for a sand layer on a granite surface. (b) Phase versus mixture coefficient q for a sand layer on a granite surface.

incidence angle . When , the value of increases to reach the limit value 1. For , the value of depends on the incidence angle. Fig. 5(a) and (b) give the parameter and the phase on the is equal to range 0.5 GHz to 1.5 GHz. The layer thickness 40 cm and the mixture parameter is equal to 0. Curves are depends on the incidence angle. As periodic and the period for the curves of Fig. 3(a) and (b), the oscillation amplitude increases with the incidence angle. The parameters and are given by (47) and (55). According to (39), (41), and (42), these (27), (28), (31), are functions of the complex amplitudes (32). The sand dielectric loss angle is weak and the granite surface is assumed to be a lossless material. In a first approach, . We dethe oscillations are related to the function and the layer thickness satisfy the duce that the period following relation: (73)

is the speed of light in vacuum. We relate from curves Fig. 5(b) the periods as follows: with and with . From (73), we and which are deduce the values very close to the values of the simulation parameters.

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Fig. 5. (a) Coefficient r versus frequency for a sand layer on a granite surface. (b) Phase versus frequency for a sand layer on a granite surface.

Fig. 6(a) and (b) show the probability density function and the distribution function of the phase difference. The incidence angle is equal to 70 and the wavelength is equal to 20 cm. . Two values of the Both interfaces are non-correlated layer thickness are chosen: and . As shown in Fig. 3(a) and (b), these thicknesses give the extrema and and lead to the of with and values of close to 0 with . The probability density functions are centered on the . The average are close to zero with values and . Insofar as and , the width . Although the of the pdf is more important when is close to 1, we notice that the angular width at value of half-height reaches 20 degrees. The probability density function (54) is a bell-shaped curve but as shown in Fig. 6(a) and (b), the theoretical law of the co-polarized phase difference should not be confused with a Gaussian distribution (the Gaussian diswith tribution is given for and ). Fig. 7 shows the theoretical probability density function of co-polarized phase difference and normalized histogram obtained from Monte-Carlo simulations. We consider a stack

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first-order solution. Nevertheless, the comparison is conclusive. This comparison confirms the validity of the theoretical formulae and show that for the slightly rough surfaces, the first-order small perturbation model gives the averaged bistatic coefficient [1]–[9] but also the pdf of the co-polarized phase difference. VII. CONCLUSION

Fig. 6. (a) Probability density function of the co-polarized phase difference. (b) Distribution function of the co-polarized phase difference.

Fig. 7. Theoretical pdf and normalized histogram. The histogram is estimated from 5000 results obtained with the C-method. The length used with the C method is 30.

of two one-dimensional rough interfaces. The incidence angle is equal to 40 , the wavelength to 20 cm and the layer . thickness to . The two interfaces are uncorrelated The statistical distributions are given by (54). The histogram is derived from 5000 results obtained with the exact C-method . The numer[48]. The length used with the C method is ical results take into account higher-order terms beyond the

In this article, we determined the analytical expression of the probability density function of the co-polarized phase difference for a stack of two rough two-dimensional interfaces illuminated by a plane wave. The electromagnetic fields are expressed as Rayleigh expansions and a perturbation method is used to solve the boundary value problem and to determine the first-order scattering amplitudes. These amplitudes are expressed as linear combinations of the Fourier transforms of the height profiles. Assuming Gaussian height distributions with zero means, we and and the imagshowed that the real parts and of the first-order scattering inary parts amplitudes are Gaussian random variables and we derived the probability density function of the co-polarized phase difference in the incidence plane from the joint probability law of these four random variables. For infinite extension interfaces, the pdf of the co-polarized phase difference is only a function of two parameters and ( can vary from to and we proved that can vary from 0 to 1). The parameter is expressed as a function of the covariances and and of the variances and . The angle is only a function of the covariance and . and depend on the spectra of both interfaces and their inter-spec, trum that is defined by the mixture parameter . When the phase difference is uniform. This configuration occurs when and are null. When , the the covariances phase difference is no longer a random variable but is equal to . There exist two types of configurations for which the angle . The first case occurs if the incidence and observation angles, the refractive indices and the layer thickness satisfy the condition (62). This condition does not use the statistical properties of both surfaces. The second case occurs when the squared modulus of the inter-spectrum is equal to the product of spectra , we showed that of both height profiles. For whatever the incidence and observation angles. For a single surface separating two homogeneous media, the co-polarized phase difference is not a random variable and . The relation thus does not indicate the presence of several layers. Within the framework of the first-order small perturbation method, we can only infer the presence of several layers . when For a granite surface covered by a sand layer and backscattering configurations, we studied the influence of the layer thickness upon both parameters and and showed that the variations increases with the incidence angle and that the phase is more sensitive to the variations of the ratio than the parameter . We also studied the influence of the wave frequency. The variations also increase with the incidence angle. We showed that a frequential analysis under two incidence angles permits to determine the thickness of the sand layer and the real part of its refractive index.

AFIFI AND DUSSÉAUX: ON THE CO-POLARIZED PHASE DIFFERENCE OF ROUGH LAYERED SURFACES

The proposed method provides an efficient analytical tool for analyzing the rough layers by means of the probability density function of the phase difference. Although the analytical formulae are valid for slightly rough interfaces only, it is well suited for the remote sensing of layered structure using low-frequency radars [7]. This model cannot be compared in terms of computational cost with the exact methods that are associated with numerical techniques and that require the solutions for many realizations of rough two-dimensional layers. As shown in [31], [32], [49], the statistical distribution depends on the surface sizes. The exact methods based on Monte Carlo simulations require illumination finite areas. Therefore, from a numerical point of view, one should be wise and choose large enough sizes. Within its domains of validity, the proposed method is a practical and realistic alternative with negligible computational cost. REFERENCES [1] J. M. Elson, “Infrared light scattering from surfaces covered with multiple dielectric overlayers,” Appl. Opt., vol. 16, no. 11, pp. 2873–2881, Nov. 1977. [2] J. M. Elson, J. P. Rahn, and J. M. Bennett, “Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation length, and roughness cross-correlation properties,” Appl. Opt., vol. 22, no. 20, pp. 3207–3219, Oct. 1983. [3] P. Bousquet, F. Flory, and P. Roche, “Scattering from multilayer thin films: Theory and experiment,” J. Opt. Soc. Am., vol. 71, no. 9, pp. 1115–1123, 1981. [4] C. Amra, C. Grèzes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt., vol. 32, no. 28, pp. 5492–5503, Oct. 1993. [5] C. Amra, J. H. Apfel, and E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt., vol. 31, no. 16, pp. 3134–3151, Jun. 1992. [6] I. M. Fuks, “Wave diffraction by a rough boundary of an arbitrary plane-layered medium,” IEEE Trans. Antennas Propag., vol. 49, no. 4, pp. 630–639, Apr. 2001. [7] A. Tabatabaeenejad and M. Moghaddam, “Bistatic scattering from three-dimensional layered rough surfaces,” IEEE Trans. Geosci. Remote Sens., vol. 44, no. 8, pp. 2102–2114, Aug. 2006. [8] P. Imperatore, A. Iodice, and D. Riccio, “Electromagnetic wave scattering from layered structures with an arbitrary number of rough interfaces,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 4, pp. 1056–1072, Apr. 2009. [9] P. Imperatore, A. Iodice, and D. Riccio, “Transmission through layered media with rough boundaries: First-order perturbative solution,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1481–1494, May 2009. [10] E. Bahar and Y. Zhang, “A new unified full wave approach to evaluate the scatter cross sections of composite random rough surfaces,” IEEE Trans. Geosci. Remote Sensing, vol. 34, no. 4, pp. 973–980, Jul. 1996. [11] E. Bahar and Y. Zhang, “Diffuse like and cross-polarized fields scattered from irregular layered structures-full-wave analysis,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 941–948, May 1999. [12] N. Pinel, N. Déchamps, C. Bourlier, and J. Saillard, “Bistatic scattering from one-dimensional random rough homogeneous layers in the high-frequency limit with shadowing effect,” Waves Random Complex Media, vol. 17, no. 3, pp. 283–303, Aug. 2007. [13] N. Pinel and C. Bourlier, “Scattering from very rough layers under the geometric optics approximation: Further investigation,” J. Opt. Soc. Am. A., vol. 25, no. 6, pp. 1293–1306, May 2008. [14] G. Berginc and C. Bourrely, “The small-slope approximation method applied to a three-dimensional slab with rough boundaries,” Progr. Electromagn. Res., vol. PIER 73, pp. 131–211, 2007. [15] Z. H. Gu, J. Q. Lu, and A. A. Maradudin, “Enhanced backscattering from a rough dielectric film on a glass substrate,” J. Opt. Soc. Am. A., vol. 10, no. 8, pp. 1753–1764, Aug. 1993. [16] M. Saillard and G. Toso, “Electromagnetic scattering from bounded or infinite subsurface bodies,” Radio Sci., vol. 32, no. 4, pp. 1347–1359, Jul.–Aug. 1997.

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[17] M. El-Shenawee, “Polarimetric scattering from two-layered two dimensional random rough surfaces with and without buried objects,” IEEE Trans.Geosci. Remote Sens., vol. 42, no. 1, pp. 67–76, Jan. 2004. [18] V. Ciarletti, B. Martinat, A. Reineix, J. J. Berthelier, and R. Ney, “Numerical simulation of the operation of the GPR experiment on NETLANDER,” J. Geophys. Res., vol. 108, no. E4, p. 8028, Feb. 2003. [19] Q. Di and M. Wang, “Migration of ground-penetrating radar data with a finite-element method that considers attenuation and dispersion,” Geophysics, vol. 69, no. 2, pp. 472–477, Mar.–Apr. 2004. [20] N. Déchamps, N. de Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: Propagation-inside-layer expansion method,” J. Opt. Soc. Am. A., vol. 23, no. 2, pp. 359–369, Feb. 2006. [21] C.-H. Kuo and M. Moghaddam, “Scattering from multilayer rough surfaces based on the extented boundary condition method and truncated singular value decomposition,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2917–2929, Oct. 2006. [22] N. Déchamps and C. Bourlier, “Electromagnetic scattering from a rough layer: Propagation-inside-layer expansion method combined to an updated BMIA/CAG approach,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2790–2802, Oct. 2007. [23] N. Déchamps and C. Bourlier, “Electromagnetic scattering from a rough layer: Propagation-inside-layer expansion method combined to the forward-backward novel-spectral acceleration,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3576–3586, Dec. 2007. [24] E. Jakeman and K. D. Ridley, Modeling Fluctuations in Scattered Waves. New York: Taylor and Francis, 2006. [25] A. Abdi, H. Hashemi, and S. Nader-Esfahani, “On the PDF of the sum of random vectors,” IEEE Trans. Commun., vol. 48, no. 1, pp. 7–12, Jan. 2000. [26] H. J. Eom and W.-M. Boerner, “Statistical properties of the phase difference between two orthogonally polarized SAR signals,” IEEE Trans. Geosci. Remote Sensing, vol. 29, no. 1, pp. 182–184, Jan. 1991. [27] K. Sarabandi, “Derivation of phase statistics from the Mueller matrix,” Radio Sci., vol. 27, no. 5, pp. 553–560, Sep.–Oct. 1992. [28] D. Just and R. Bamler, “Phase statistics of interferograms with applications to synthetic aperture radar,” Appl. Opt., vol. 33, no. 20, pp. 4362–4368, Jul. 1994. [29] J.-S. Lee, A. R. Miller, and K. W. Hoppel, “Statistics of phase difference and product magnitude of multi-look processed Gaussian signals,” Waves Random Complex Media, vol. 4, no. 3, pp. 307–319, Jul. 1994. [30] R.-J.-A. Tough, D. Blacknell, and S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. Royal Society: Math. Phys. Sci., vol. 449, no. 1937, pp. 567–589, Jun. 1995. [31] S. Afifi, R. Dusséaux, and R. de Oliveira, “Statistical distribution of the field scattered by rough layered interfaces: Formulae derived from the small perturbation method,” Waves Random Complex Media, vol. 20, no. 1, pp. 1–22, Feb. 2010. [32] R. Dusséaux and R. de Oliveira, “Effect of the illumination length on the statistical distribution of the field scattered from one-dimensional random rough surfaces: Analytical formulae derived from the small perturbation method,” Waves Random Complex Media, vol. 17, no. 3, pp. 305–320, Aug. 2007. [33] R. de Oliveira, R. Dusséaux, and S. Afifi, “Analytical derivation of the stationnarity and the ergodicity of the field scattered by a slightly rough random surface,” Waves Random Complex Media, vol. 20, no. 3, pp. 396–418, Aug. 2010. [34] S. Afifi and R. Dusséaux, “Statistical study of the radiation losses due to surface roughness for a dielectric slab deposited on a metal substrate,” Opt. Commun., vol. 281, pp. 4663–4670, 2008. [35] S. Afifi and R. Dusséaux, “Statistical study of radiation loss from planar optical waveguides: The curvilinear coordinate method and the small perturbation method,” J. Opt. Soc. Am. A, vol. 27, no. 5, pp. 1171–1184, May 2010. [36] L. Tsang and J. A. Kong, Scattering of electromagnetic waves—Advanced Topics. New York: Wiley-Interscience, 2001. [37] Y. Lasne, P. Paillou, T. August-Bernex, G. Ruffié, and G. Grandjean, “A phase signature for detecting wet subsurface structures using polarimetric L-band SAR,” IEEE Trans. Geosci. Remote Sensing, vol. 42, no. 8, pp. 1683–1694, Aug. 2004. [38] Y. Lasne, P. Paillou, G. Ruffié, and G. Grandjean, “Effect of multiple scattering on phase signature of buried wet subsurface structures: Applications to polarimetric L and C-band SAR,” IEEE Trans. Geosci. Remote Sensing, vol. 43, no. 8, pp. 1716–1726, Aug. 2005.

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[39] M. Migliaccio, F. Nunziata, and A. Gambardella, “On the co-polarized phase difference for oil spill observation,” Int. J. Remote Sensing, vol. 30, no. 6, pp. 1587–1602, Mar. 2009. [40] G. Chang and Y. Oh, “Polarimetric SAR image classification based on the degree of polarization and co-polarized phase-difference statistics,” in Proc. Asia-Pacific Microwave Confe., APMC2007, pp. 2282–2285. [41] Y. Oh, K. Sarabandi, and F. T. Ulaby, “An empirical model for phase difference statistics of rough,” in Proc. IEEE Geosci. Remote Sensing Symp., Tokyo, Japan, Aug. 1993, vol. III, pp. 1003–1005. [42] Y. Oh, K. Sarabandi, and F. T. Ulaby, “Semi-empirical model of the ensemble-averaged differential Mueller matrix for microwave backscattering from bare soil surfaces,” IEEE Trans. Geosci. Remote Sensing, vol. 40, no. 6, pp. 1348–1355, Jun. 2002. [43] L. Rayleigh, Theory of Sound. New York: Dover, 1945, vol. II. [44] L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of electromagnetic waves—Numerical simulations. New York: Wiley-Interscience, 2001. [45] C. Baudier and R. Dusséaux, “Scattering of an E//—Polarized plane wave by one-dimensional rough surfaces: Numerical applicability domain of a Rayleigh method in the far-field zone,” Progr. Electromagn. Res., vol. PIER 34, pp. 1–27, 2001. [46] R. Dusséaux and C. Baudier, “Scattering of a plane wave by 1-dimensional dielectric rough surfaces—Study of the field in a nonorthogonal coordinate system,” Progr. Electromagn. Res., vol. PIER 37, pp. 289–317, 2002. [47] K. Aït Braham, R. Dusséaux, and G. Granet, “Scattering of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces—Study with the curvilinear coordinate method,” Waves Random Complex Media, vol. 18, no. 2, pp. 255–274, May 2008. [48] R. Dusséaux, G. Granet, K. S. Edee, and S. Afifi, “Electromagnetic scattering from rough layered interfaces Analysis with the curvilinear coordinate method,” in Proc. Applied Computational Electromagnetics Society Conf., ACES, 2010, pp. 508–513.

[49] S. Afifi, R. Dusséaux, and R. de Oliveira, Erratum, Waves Random Complex Media, vol. 20, no. 2, p. 332, May 2010.

Saddek Afifi was born in Asfour, El Tarf, Algeria, on April 30, 1959. He received the Engineering degree in electronics from the Polytechnique Algiers in 1983, the Ph.D. degree (Docteur Ingénieur) in electronics from Blaise Pascal University, Clermont-Ferrand, France, in 1986, and the Doctorat ès-sciences in physics from the University of Badji Mokhtar Annaba, Algeria, in 2006. In 1986, he joined the University of Annaba where he worked as an Assistant Professor in the Electronics Department and was a member of the “Laboratoire de Physique des Lasers, de Spectroscopie Optique et d’Opto-électronique (LAPLASO).” He leads and teaches modules at both B.Sc. and M.Sc. levels in signal processing, electromagnetics and electronics. His main research interests are the scattering and propagation of electromagnetic waves in random rough surfaces and remote sensing.

Richard Dusséaux received the Ph.D. degree in electronics and systems from Blaise Pascal University, Clermont-Ferrand, France, in 1993. In 1994, he joined the University of Versailles—Saint-Quentin. He is a Professor of electrical engineering and a member of the “Laboratoire Atmosphères, Milieux, Observations Spatiales.” His research interests include waveguides, scattering by gratings and rough surfaces.

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Parametric Design of Compact Dual-Frequency Antennas for Wireless Sensor Networks Simone Genovesi, Member, IEEE, Sergio Saponara, and Agostino Monorchio, Senior Member, IEEE

Abstract—A parametric study for the design of a planar compact dual-frequency antenna is presented. The proposed geometry and the suggested tailoring procedure provide a useful template for generating a single antenna able to serve as bidirectional node for a generic Wireless Sensor Network within the UHF and microwave frequency bands. The constructive parameters can be set to adapt the radiating device for different unlicensed bands and to comply with the international radiation power regulations. A careful design procedure will be described to tune the antenna template for working at the required frequencies. In order to prove the effectiveness of the suggested approach as well as the reliability of the adopted technique, comparisons between simulations and measurements of realized prototypes will be reported. Index Terms—Dual-frequency antenna, loop antenna, printed antenna, wireless sensor network.

I. INTRODUCTION

W

IRELESS SENSOR networks (WSN) are receiving a growing interest for several applications such as logistic, home automation, healthcare, structural monitoring, security/safety systems, intelligent transport systems [1]–[12]. They are typically [6] constituted by several client modules, distributed in the environment to be monitored. They are realized as compact printed circuit board (PCB) hosting one or more sensors, a battery or energy harvesting unit, a programmable IC for signal conditioning [12], [13] and a RF transmitter for communicating towards a server module. In many cases, the client’s RF unit can be configured so to support different modulations schemes (ASK, OOK, BPSK, ), frequencies (usually ISM unlicensed bands in the UHF range) and transmitted power. The configurability of the RF part is important also to cope with different national regulations. As an example, we mention tire pressure monitoring systems (TPMS) [1]–[3] where a wireless client hosting pressure and temperature sensors is mounted on each wheel’s rim transmitting tire status to a centralized receiver mounted on the car chassis. Usually, the network covers a local range and the maximum client RF transmitted power amounts to few tens of mW. A key issue for sensor networks is the availability of compact antennas at the wireless client side, possibly requiring only conventional PCB technologies for their production. Manuscript received November 15, 2009; revised November 12, 2010; accepted December 02, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. The authors are with the Dipartimento di Ingegneria dell’Informazione, University of Pisa, 56122 Pisa, Italy (e-mail: [email protected]; sergio. [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.2011.2152313

In WSN a bidirectional communication scheme has to be preferred between client and server units. In addition to the wireless transmission of data from a client sensor to a server unit, a wireless link is needed from the server towards the clients for a wide range of tasks. For example, it can be necessary to acknowledge the safe receipt of data or to request their re-transmission in case of errors. Moreover, it may be useful to wake-up the clients and enable data transmission in wireless networks where the clients are normally in idle mode for power optimization or to configure the clients. Finally, two-way communication can be requested to exchange security codes for identification or for encrypted data transfer. To this aim, most systems known in literature adopt two separate antennas at the client side [1]–[3], [14] operating at distinct frequencies, thus increasing module size and cost. Typical solutions adopt a printed antenna at UHF together with a LF coil and its relevant demodulator (this solution is adopted for the TPMS in [3] or for safe entry system in [11]) or two separate antennas printed on opposite sides of the board and operating at distinct UHF frequencies (e.g., 433 MHz and 868 MHz for the TPMS in [2]). To exploit all added features of bidirectional communication schemes in WSN while avoiding the use of two separate antennas, this work proposes the design of a novel and compact double-loop antenna, patent filed [15], resonating at multiple frequencies in the UHF range, whose realization complies with standard low-cost PCB technologies. Another distinct characteristic of this work is that the antenna design is parametric; while the external loop is fixed to limit the overall size, the other elements (inner loop, tuner) are parameters that can be configured to allow the synthesis of different antennas. Therefore the antenna design is not fixed and customized only for a specific application, as in [7], [8]. Different antennas can be easily generated from the same architectural template having the same topology and the same general characteristics but resonating at different specific frequencies that can be tuned to meet the typical requirements of different national authorities. In fact, it is worth noting that 433 MHz and 868 MHz are the typical frequencies used in Western Europe, 315 MHz, 450 MHz and 915 MHz in US, 315 MHz and 915 MHz in South America, 433 MHz and 915 MHz in Australia [16]–[18]. Hereafter in Section II the parametric antenna design is proposed and the effects of the sizing of its building elements (the tuning element, the inner and outer loops, the matching network) on the resonating frequencies are discussed. Some tuning heuristic rules are derived in Section III allowing to generate and realize the desired antenna configuration for a given target of dual-band working frequencies, starting from the parametric antenna template. The antenna design and the configuration process are described in Section III, while Section IV

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Fig. 1. Top view of the antenna template.

Fig. 2. Comparison between the antenna input impedance without the inner loop (continuous and dotted lines with triangles) and with the inner loop (continuous and dotted lines). The real part is a dotted line while the imaginary part is continuous.

illustrates the generation, realization and measurements of distinct multi-frequencies antennas presented as case studies. Section V proves that the achieved performances of the novel proposed antennas are useful to realize wireless networking systems compliant with the radiation power regulations (US FCC [19] and ETSI [20]). Finally, conclusions are drawn in Section 6. II. ANTENNA TEMPLATE DESCRIPTION The configuration of the antenna comprises an outer loop, an inner loop and a tuning element (Fig. 1). The outer loop is connected to the source, or the electronic circuitry, by using a simple matching network. In order to show the effect of each single basic element of the antenna template, we consider the investigated radiating device printed on a commercially availmm) and able dielectric substrate ( a trace width equal to 1.25 mm. It is important to point out that we are interested in the impedance of the antenna determined at the open ends of the outer loop. This feeding configuration allows the antenna to be easily connected to electronic circuitry and RFIC input/output and, at the same time, prevents the use of balun or via-hole. The width of the trace equal to 1.25 mm has been chosen according to the solutions commonly adopted in commercial devices but a slightly different width can be employed to perform the design process. As an example, an outer cm has been loop of fixed dimension chosen to perform our parametric analysis but different sizes can be adopted depending on the available space on the PCB. These dimensions refer to the solution proposed in [3]; however, as additional advantage, we now propose an antenna design that occupies the same area but supports multiple frequencies. It is important to highlight that the perimeter of the outer loop is related to the value of the higher resonance, as it will be deeply discussed in the following. Therefore, a good starting guess for the size is represented by an outer loop with a perimeter close to the free-space wavelength at the higher frequency (we can find the same resonance frequency by using, for instance, dimensions of 4.2 cm 4.2 cm instead of 5.3 cm 3.2 cm since

Fig. 3. The increase of the inner loop length lowers both the resonance frequencies F (continuous line) and F (dashed line). The ratio (dotted line) between F and F reveals a faster decrease of F with the increase of the inner loop length.

the perimeter is almost the same). Then, if the antenna dimension does not match the imposed space requirement, a proper value of the matching network and the tuner element allows to shrink it providing a device with the same resonance frequency but with a smaller size. A. Effect of the Inner Loop The most important effect of the presence of the inner element , while is the occurrence of the lower resonance frequency is only slightly changed. In the higher resonance frequency Fig. 2 we show the input impedance of the same antenna with and without the inner loop. If the length of the investigated element is increased by and decreases while folding the inner loop, the value of increases (Fig. 3). Therefore, the inner loop the ratio while its length determines the first resonance frequency has an effect on both the resonance frequencies, although decreases more rapidly than .

GENOVESI et al.: PARAMETRIC DESIGN OF COMPACT DUAL-FREQUENCY ANTENNAS FOR WIRELESS SENSOR NETWORKS

Fig. 4. Changes in the inner loop position (a), determine an effect of the inner loop position on both resonance frequencies (b).

Fig. 5. Comparison between the antenna input impedance without the tuning element (continuous and dotted lines with triangles) and with the inner loop (continuous and dotted lines). The real part is a dotted line while the imaginary part is continuous.

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Fig. 6. The increase of the tuner length slightly changes F (continuous line) while F (dashed line) shifts toward lower values. The ratio between F and F (dotted line) decreases in the investigated range (no tuner-4.6 cm).

Fig. 7. Higher values of the inductance in the matching network determine smaller values of both the resonance frequencies although F (dashed line) decrease more rapidly than F (continuous line), as the ratio between F and F (dotted line) highlights.

C. Effect of the Matching Network The position of the inner loop influences the resonance frequencies. In Fig. 4 it is possible to observe that increasing the offset (Fig. 4(a)) between the tuner and the inner loop raises F2 while F1 slightly decreases (Fig. 4(b)). In particular, the dB bandwidth for the first resonance remains value of the stable around 1.0% whereas for the second one we obtain 1.5%. These values are typical for wireless sensor network applications where the amount of information to be transferred requires low bit-rates and they also agree with the ones presented in [9]. Therefore, although the resonance frequencies shift as shown in Fig. 4, the impedance bandwidth is not affected and remains stable for all offsets. B. Effect of the Tuning Element The length of the tuning element placed between the outer and produces a small and the inner loop affects the value of (Fig. 5). variation of More in detail, if we increase the length of the tuning element, decreases as well as the ratio (Fig. 6).

The matching network considered for the study comprises only an inductance connected to the outer loop. The considered inductance is in the range of tens of nH. For these values surface mount inductors exist that can be easily soldered to the PCB trace just before the source or the electronic circuitry. As shown in Fig. 7, the increase of the inductance value determines a shift toward lower resonance frequencies whereas the real part of the input impedance at the resonance varies within a set of reasonable values for the realization of a standard matching network (Fig. 8). III. TUNING OF THE ANTENNA TEMPLATE Each element of the antenna template has to be carefully dimensioned and tuned in order to obtain the desired resonance frequencies. In particular, as pointed out in the previous section, even a single change in one element affects both the resonance and and their ratio . This section will frequencies provide guidelines to exploit the antenna framework for the design of an antenna with two resonances within the ISM band at and . the desired frequencies

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Fig. 8. The real part of the input impedance is within an acceptable range of values for both the frequencies F (continuous line) and F (dashed line).

As well known, the design can be subject to a series of limitations such as the allowed space for the radiating device or the discrete values assumed by the lumped loads of the matching network. As a case study, we consider again the dimensions of the outer loop equal to 5.3 cm 3.2 cm in order to design an antenna with two resonance frequencies satisfying the Australia MHz and MHz). requirements ( The first step is to obtain an antenna with a resonance and, at the same time, a resonance . Basically, this means that we have to tune the framework in order to achieve the first resonance by choosing a proper matching network and the inner loop length. To accomplish this goal, we set the tuner element as long as possible at this stage since its function will in order to make the antenna resbe to increase the value of onating also at . The aforementioned rules will suggest the changes that have to be made to the length or the position of the elements. For example, in order to increase the first resonance we have to decrease the inner loop size or the value of the inductance of the matching network. Suppose we start our tuning procedure with a matching network comprising an inductance of 33 nH and an inner loop with a length of 25 cm. In this case, we have a value of around 380 MHz and a second resonance frequency around 870 MHz. To allow our framework to satisfy the requirements of the first step we need to decrease the inner loop until its length is equal to with MHz. Next, since 19.5 cm. and hence , we have still to tune the second resonance frequency and therefore we reduce the tuner element from 2.3 cm to 2.15 . All the presented results have been valcm to obtain idated by using Ansoft HFSS [21] and are reported in Fig. 9. IV. MEASUREMENTS The procedure described in the previous section has been successfully applied for different sets of resonance frequencies, also including all the resonance frequencies mentioned in Section I. Moreover, in order to validate the simulated results, two prototypes have been realized and measured. The former is an antenna with two resonances in accordance with the

Fig. 9. Tuning of the antenna framework. The dotted lines are the real part of input impedances while the continuous lines illustrate the imaginary parts. The MHz and lines with triangles refers to the starting configuration with F MHz, the lines with circles to the antenna with F F and the F remaining lines to the final antenna satisfying the imposed requirements.

= 870

= 380 =

Fig. 10. Front view of the prototype for the double loop antenna satisfying the requirements of Australia resonance frequencies.

Australia requirements whereas the latter complies with the South America standard. An half-loop version of both of the antennas was fabricated (Figs. 10 and 11) and its measured input impedance was compared to the simulated results. We recurred to an half-loop version for the fabrication because it allows to validate the antenna performance while preventing the need to design and build a balun for the antenna. The dimensions of the half-loop antennas were determined by using HFSS, following the aforementioned rules for the tuning. More in detail, the tuner length is equal to 4 cm and the inner element is 21.5 cm long for the Australia configuration while the tuner is 4.6 cm and the inner element is 39 cm length for the South America antenna. Each device was printed on a commercially available dielectric substrate mm) with a trace width equals to ( 1.25 mm. The antenna is excited via a SMA connector with the outer conductor connected directly to the ground plane and the center conductor connected to the matching network of outer conductor of the half-loop. According to images’ theorem, the impedance of an half-loop above an infinite ground plane is

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Fig. 11. Front view of the prototype for the double loop antenna satisfying the requirements of South America resonance frequencies.

Fig. 13. Comparison between simulation (lines) and measurements (dots) of the input impedance related to the second resonance (915 MHz) of Australia configuration: real part (continuous line) and imaginary part (dashed line) are shown.

Fig. 12. Comparison between simulation (lines) and measurements (dots) of the input impedance related to the first resonance (433 MHz) of Australia configuration: real part (continuous line) and imaginary part (dashed line) are shown.

equal to one-half of the corresponding full loop. If the ground plane is sufficiently large, this statement still holds and we can use it to estimate the impedance of the half-loop. The input impedances of the fabricated half-loops were used to estimate the performance of the radiating device and comparisons with the numerical results have been reported in order to assess the reliability of the proposed design procedure. A Wiltron 37311A network analyzer was used to measure the input impedance versus frequency of the fabricated prototypes. In Fig. 12 we compare the real and imaginary part of the simulated and measured input impedance at the first and second resonance of the Australia configuration, respectively. The real part of the measured input impedance for the first resonance is in good agreement with the simulated one and also the measured imaginary part is close to the computed one since there is only a 4% off-set discrepancy with respect to the expected resonance frequency of 433 MHz (mainly due to technology and component spreading for the adopted manufacturing process). The real part of the measured input impedance for the second resonance (Fig. 13) exhibits a behavior similar to the simulated curve whereas the imaginary part is quite close to the expected one of 915 MHz (0.14% of discrepancy). The comparison between the real and imaginary part of the input impedance for the South America configuration is shown in Fig. 14 for the first resonance. The measured real part of the

Fig. 14. Comparison between simulation (lines) and measurements (dots) of the input impedance related to the first resonance of South America configuration: real part (continuous line) and imaginary part (dashed line) are shown.

impedance at the first resonance is in accordance with the simulated results and also the imaginary part of the prototype is close to the expected 315 MHz (4% of discrepancy). At the second resonance, the real part of the input impedance (Fig. 15) has a similar trend to the simulated one and assumes acceptable values whereas the imaginary part is quite close to the estimated one (2.29% of discrepancy). The simulated results are in good agreement with the measured one, therefore the proposed procedure for the design can be considered reliable. The small discrepancy (always below 4%) could be determined by a series of different causes such as the tolerance of the components (5% for the inductance of the matching network), the effect of the welding in the half-loop prototypes, a non ideal insertion of the SMA connector to the antenna and the ground plane. However, a further tuning can be easily done by trimming the printed antenna. V. RADIATION PATTERN AND LINK BUDGET In order to characterize the performance of the proposed antenna configuration, radiation patterns for the Australian and

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Fig. 15. Comparison between simulation (lines) and measurements (dots) of the input impedance related to the second resonance of South America configuration: real part (continuous line) and imaginary part (dashed line) are shown.

South American configuration are reported in Figs. 16 and 17, respectively. As it can be noticed, the pattern retains an omnidirectional behavior on the horizontal plane for both resonance frequencies for the Australian as well as the South American configuration, thus providing a good connection among the nodes of the sensing network. The gain values of the Australia configuration are equal to dB and dB while those of the South America condB and dB respectively at the first figuration are and second resonance. The values of the gain are comparable to those find in [9] and [10] which are recent works in literature targeting similar applications of that considered in our paper. Since the proposed antenna can be employed in the sensing node of a wireless system, it has to be compliant with the radiation power regulations (US FCC and ETSI). Let us consider our antenna on each sensing node and a standard patch antenna (7.0 dB of gain), working at the same frequencies, at the master node. The MCU (Micro Controller Unit) in the master node generally hosts an RF transceiver able to receive signals with a sensitivity dBm, therefore the sensing node has to deliver a below very low power level. At this purpose, currents of a few mA are sufficient [2], [3] for the sensing node to transmit an intelligible signal to the master node. On the other side, particularly in passively-powered [22], [23] or power-optimized [2] sensor networks, the sensing node has an RF rectifier, which delivers outputs compliant with standard microcontroller voltage levels. The RF rectifiers can work properly if the received power at the dBm to dBm, depending on antenna is in the range of the implementation technology [22], [23]. For the case of Australian configuration, the sensing unit communicates with the master node at 433 MHz and the master node uses an RF link at 915 MHz to send data to the sensing node. By using the well known Friis formula in the maximum radiation condition and by supposing a distance between transmitter and receiver equals to 3.0 m, the required power for the 915 MHz transmitter master node ranges from tens of mW up to 105 mW (Fig. 18), well below the allowed transmitter power

Fig. 16. Normalized radiation pattern for Australia configuration at (a) first resonance of 433 MHz; (b) second resonance 915 MHz. The line with dot markers cut, the triangular markers to the  cut and the refers to the  square markers to  plane.

=0

= 90

= 90

limit (500 mW). The aforementioned sensitivity of the MCU RF transceiver guarantees the link at 433 MHZ to work properly with much less power. In order to assess the maximum possible communication distance and hence the covered range of the system, we have studied the necessary level of transmitted power with the increment of the distance between the RF transceiver of the master node and the receiver at the sensing node. As already mentioned, if we consider a distance of 3.0 m we need to transmit few tens of mWs if we employ a performing RF dBm) or around 100 mW if we use an ordinary rectifier (

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Fig. 18. Relation between transmitted and received power at distance equal to 3.0 m between transmitting and receiving antennas for the case of Australian configuration. The dotted line refers to the received power at 433 MHz whereas the continuous one to the received power at 915 MHz.

Fig. 17. Normalized radiation pattern for South America configuration at (a) first resonance of 315 MHz; (b) second resonance 915 MHz. The line with dot cut, the triangular markers to the  cut and markers refers to the  the square markers to  plane.

=0 = 90

Fig. 19. Relation between transmitted and received power at 915 MHz for three different distances antennas for the case of Australian configuration. The continuous line refers to the received power at a distance equal to 3.0 m, the dashed line to 6.0 m and the dotted line to 10.0 m. The grey zone indicates the interval of received power necessary to a correct behavior of the RF rectifier.

= 90

one ( dBm). As shown in Fig. 19, if we double the distance (6.0 m), we have to transmit slightly less than 100 mW if we exploit a good RF rectifier whereas it is necessary to increase the power to 400 mW for the less sensitive one. Even in the worst case we are still under the limit of 500 mW of transmitted power. If we go farther and reach 10.0 m from the transmitter, we are still able to have a reliable communication between the master node and the sensing node if we use an RF rectifier with dBm (Fig. 19). a good sensitivity, at least For the case of South America configuration, the sensing unit communicates with the master MCU at 315 MHz and the master

MCU uses an RF link at 915 MHz to send data to the node. If we set again the distance between transmitter and receiver to 3.0 m, the maximum necessary level of transmitted power is required for the MCU working at 915 MHz, ranging from tens of milliwatts up to 125 mW (Fig. 20). From the analysis of the link budget (minimum transmitted power versus the distance between the nodes), a result similar to the previous case emerges. In fact as shown in Fig. 21, the communication is possible for a distance equal to 6.0 m with the respect of the 500 mW limit even with an ordinary RF rectifier. If the distance is set equal to 10 m, the link is possible only by employing an RF rectifier dBm. with a sensitivity at least equal to

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allowed frequencies by following the described tailoring procedure. Thus, it is possible to realize a bidirectional node for a generic Wireless Sensor Network simply by using a single low-cost antenna. Moreover, careful attention has been paid to the fulfillment of the international radiation power norms but, at the same time, a communication range up to 10.0 m has been guaranteed. The good agreement between simulations and measurements on the realized prototypes has proved the reliability of the design approach.

REFERENCES

Fig. 20. Relation between transmitted and received power at distance equal to 3.0 m between transmitting and receiving antennas for the case of South America configuration. The dotted black line refers to the received power at 315 MHz whereas the continuous black one to the received power at 915 MHz.

Fig. 21. Relation between transmitted and received power at 915 MHz for three different distances antennas for the case of South America configuration. The grey continuous line refers to the received power at a distance equal to 3.0 m, the black dotted one to 6.0 m and the continuous black one to 10.0 m. The grey zone indicates the interval of received power necessary to a correct behavior of the RF rectifier.

As a result of the examples shown, we can affirm that we are able to successfully employ the proposed antenna in a wireless sensor network, also with passively-powered sensing nodes, which complies with the US FCC and ETSI regulations and, at the same time, guarantees a communication range of 6.0 m or even 10.0 m with a performing RF rectifier.

VI. CONCLUSION A configuration of a dual-frequency planar compact antenna has been proposed. This antenna template provides a radiating device with two resonance frequencies within the UHF and microwave band, which can be adapted to the different worldwide

[1] S. Ergen, A. Sangiovanni-Vincentelli, X. Sun, R. Tebano, S. Alalusi, G. Audisio, and M. Sabatini, “The tire as an intelligent sensor,” IEEE Trans. Comput.-Aided Design Integrat. Circuits Syst., vol. 28, no. 7, pp. 941–955, Jul. 2009. [2] F. Iacopetti, S. Saponara, and L. Fanucci, “Improving power efficiency and reliability in RF tire pressure monitoring modules,” in Proc. IEEE ICECS, 2007, pp. 878–881. [3] J. Burgess, “Freescale Tire Pressure Monitor System Demo,” AN1951, Jun. 2005, rev. 2. [4] A. Sabata and S. Brossia, “Remote Monitoring of Pipelines Using Wireless Sensor Network,” U.S. patent 7526944, May 2009. [5] G. Marrocco, “The art of UHF RFID antenna design: Impedancematching and size-reduction techniques,” IEEE Antennas Propag. Maga., vol. 50, no. 1, pp. 66–79, Feb. 2008. [6] J. P. Lynch and K. J. Loh, “Summary review of wireless sensors and sensor networks for structural health monitoring,” Shock Vibr. Digest, pp. 38–91, 2006. [7] S. Genovesi, A. Monorchio, and S. Saponara, “Double-loop antenna for wireless tyre pressure monitoring,” Electron. Lett., vol. 44, no. 24, pp. 1385–1386, Nov. 2008. [8] K. S. Leong, M. L. Ng, and P. H. Cole, “Dual-frequency antenna design for RFID application,” presented at the 21st Int. Technical Conf. on Circuits/Systems, Computers and Communications (ITC-CSCC 2006), Chiang Mai, Thailand, Jul. 2006. [9] K. Tanoshita, K. Nakatani, and Y. Yamada, “Electric field simulations around a car of the tire pressure monitoring system,” IEICE Trans. Commun., vol. E90-B, no. 9, pp. 2416–2421, Sept. 2007. [10] M. Brzeska, J. Pontes, G.-A. Chakam, and W. Wiesbeck, “RF-design characterization and modelling of tire pressure sensors,” in Proc. EuCAP, 2007, pp. 1–5. [11] M. Brzeska and G.-A. Chakam, “Modelling of the coverage range for modern vehicle access systems at low frequencies,” in Proc. 37th Eur. Microwave Conf., Oct. 2007, pp. 771–774. [12] S. Saponara, E. Petri, L. Fanucci, and P. Terreni, “Sensor modeling, low-complexity fusion algorithms and mixed-signal IC prototyping for gas measures in low-emission vehicles,” IEEE Trans. Instr. Meas., vol. 60, no. 2, 2011. [13] S. Saponara, L. Fanucci, and P. Terreni, “Architectural-level power optimization of microcontroller cores in embedded systems,” IEEE Trans. Ind. Electron., vol. 54, no. 1, pp. 680–683, 2007. [14] ATMEL, “LF—Wake Up Demonstrator ATAK5276,” 2007. [15] “Antenna Multirisonante Compatta a Doppia Spira Con Elemento Parassita (“Compact Multiresonant Double Loop Antenna With Parasitic Element”),” Italian Patent P.I. 2009-A000131, Oct. 22, 2009. [16] [Online]. Available: http://www.atmel.com/products/SmartRF/ [17] A. Harney, “Design, simulate, and document proprietary wireless systems,” Analog Dialogue, vol. 42, no. 10, pp. 15–17, Oct. 2008. [18] Freescale Semiconductor, “MC33596 data sheet,” Mar. 2009, rev. 4. [19] FCC Codes of Regulation, pt.15. Available online: [Online]. Available: http://www.access.gpo.gov/nara/cfr/waisidx_08/47cfr15_08.html [20] ETSI EN 301 489-3,1.4.1 (2002–08) [Online]. Available: http://www. etsi.org/WebSite/Technologies/ShortRangeDevices.aspx [21] [Online]. Available: http://www.ansoft.com/products/hf/hfss/ [22] T. Le, K. Mayaram, and T. Fiez, “Efficient far-field radio frequency energy harvesting for passively powered sensor networks,” IEEE J. Solid State Circuits, vol. 43, no. 5, pp. 1287–302, May 2008. [23] G. De Vita and G. Iannaccone, “Design criteria for the RF section of UHF and microwave passive RFID transponders,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2978–2990, Sep. 2005.

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Simone Genovesi (S’99–M’07) received the Laurea degree in telecommunication engineering and the Ph.D. degree in information engineering from the University of Pisa, Pisa, Italy, in 2003 and 2007, respectively. Since 2003, he has been collaborating with the Electromagnetic Communication Laboratory, Pennsylvania State University, University Park. From 2004 to 2006, he was a Research Associate at the ISTI institute of the National Research Council of Italy (ISTI-CNR), Pisa. He is currently a Research Associate at the University of Pisa. His research is focused on metamaterials, antenna optimization and evolutionary algorithms.

Sergio Saponara received the Laurea degree (cum laude) and the Ph.D. degree in electronic engineering from the University of Pisa, Pisa, Italy, in 1999 and 2003, respectively. In 2002, he was with IMEC, Leuven (B), Belgium, as a Marie Curie Research Fellow. Since 2001, he has been collaborating with the Consorzio Pisa Ricerche, Pisa. He holds the Chair of Electronic Systems for Automotive and Automation at the Faculty of Engineering, University of Pisa, where he is a Senior Researcher at in the field of electronic circuits and systems for telecom, multimedia, space and automotive applications. He coauthored more than 100 scientific publications and holds four patents. Dr. Saponara is also Research Associate of CNIT and INFN and served as Guest Editor of special issues on international journals and as a program committee member of international conferences.

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Agostino Monorchio (S’89–M’96–SM’04) received the Laurea degree in electronics engineering and the Ph.D. degree in methods and technologies for environmental monitoring from the University of Pisa, Pisa, Italy, in 1991 and 1994, respectively. During 1995, he joined the Radio Astronomy Group, Arcetri Astrophysical Observatory, Florence, Italy, as a Postdoctoral Research Fellow, in the area of antennas and microwave systems. He has been collaborating with the Electromagnetic Communication Laboratory, Pennsylvania State University (Penn State), University Park, and he is an Affiliate of the Computational Electromagnetics and Antennas Research Laboratory. He has been a Visiting Scientist at the University of Granada, Spain, and at the Communication University of China in Beijing. He is currently an Associate Professor in the School of Engineering, University of Pisa, and Adjunct Professor at the Italian Naval Academy of Livorno. He is also an Adjunct Professor in the Department of Electrical Engineering, Penn State. He is on the Teaching Board of the Ph.D. course in “Remote Sensing” and on the council of the Ph.D. School of Engineering “Leonardo da Vinci” at the University of Pisa. His research interests include the development of novel numerical and asymptotic methods in applied electromagnetics, both in frequency and time domains, with applications to the design of antennas, microwave systems and RCS calculation, the analysis and design of frequency-selective surfaces and novel materials, and the definition of electromagnetic scattering models from complex objects and random surfaces for remote sensing applications. He has been a reviewer for many scientific journals and he has been supervising various research projects related to applied electromagnetic, commissioned and supported by national companies and public institutions. Dr. Monorchio has served as Associate Editor of the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS. He received a Summa Foundation Fellowship and a NATO Senior Fellowship.

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Integrated Wide-Narrowband Antenna for Multi-Standard Radio Elham Ebrahimi, Student Member, IEEE, James R. Kelly, Member, IEEE, and Peter S. Hall, Fellow, IEEE

Abstract—An integration concept for multi-standard antennas is described. This technique is based on utilizing a relatively large antenna that is printed on the top side of a substrate, acting as a ground for a smaller antenna. The smaller antenna is printed onto the bottom side of the substrate. To validate this concept, an integrated wide-narrowband demonstrator antenna is presented. This antenna is composed of a shorted microstrip patch integrated with a coplanar waveguide (CPW) fed ultrawideband (UWB) antenna. A prototype of the integrated antenna was fabricated and its performance was verified. This arrangement is a promising candidate for applications where some level of reconfigurability is required. For this reason, a set of external tuning circuits were designed to demonstrate the potential of the proposed configuration for such applications. In order to improve the isolation between the wideband and narrowband ports several modified arrangements were presented and investigated. Index Terms—Integrated antenna, multi-band antenna, reconfigurable antenna, shorted patch antenna, ultrawideband antenna.

I. INTRODUCTION NTENNA design is becoming a bottleneck for wireless mobile devices where multiple communication services are integrated in the same platform. Today, a typical wireless device might support multiple services such as GSM, DVB-H, and GPS that all rely on a multi-band antenna [1]–[9]. Implementing multi-band antennas in a small terminal is challenging due to the limited space and the need for low fabrication costs. Accordingly, antennas used for such devices must also follow the downsizing trend of the terminal unit. In particular, physically smaller built-in antennas are required for next generation devices. Different techniques are documented to design multiband antennas [8], [9]. One method, which is widely used for handset antennas, is the use of meandered metal segments. The meandering is achieved by cutting slots into non-radiating edges of the antenna. This effectively elongates the surface current path on the antenna and increases the reactive loading, resulting in another mode of operation at lower frequencies. However, for non-resonating antennas that radiate over a wide frequency range (e.g., ultrawideband (UWB) antennas) this technique is

A

not applicable. Another approach for designing multi-band antennas is to use a separate antenna for each operating mode [10]. In this method the antenna can either be formed of one radiating structure with multiple ports, or a few separate antennas. This technique often requires more space. Therefore, integrating several antennas into a limited space is practically challenging. In order to efficiently use the space allocated for the antenna in the wireless devices, a novel antenna integration technique is introduced in this paper. We integrated a narrowband antenna with a wideband antenna. The proposed structure is composed of a shorted microstrip patch which is integrated with a coplanar waveguide (CPW) fed UWB monopole antenna. A prototype of the antenna has been fabricated and verified. Tackling the challenge of limited space in small devices is indeed the main motivation of this approach. The concept which is introduced here could be applied to a wide range of different antennas. However, considering the interest in dynamic and open access spectrum in future networks is also a motivation for this research. In recent years, the underutilization of the frequency spectrum has inspired the use of reconfigurable radios concepts such as cognitive radio (CR) [11]. A CR is capable of sensing the spectrum and changing system parameters such as frequency, transmitted power, or standard if required. Various antenna scenarios have been reported in the literature [12]–[15]. In one scenario a wideband antenna might be used for spectrum sensing whilst a reconfigurable narrowband antenna is employed for communication [13]. Tuning circuits can easily be integrated with this antenna making it useful for both fixed and reconfigurable radio applications [15]. This paper is organized as follows. In Section II the integration method and design details are presented. Simulation and measurement results are also reported and discussed. Three sets of matching circuits have been designed to tune the antenna to 4 GHz, 8 GHz and 10 GHz. The matching circuit design and results are discussed in Section III. The isolation between antenna ports is important in closely spaced antennas. Hence, in Section IV, the antenna configuration is modified to reduce the level of mutual coupling between the two ports. II. THE INTEGRATED WIDE-NARROWBAND ANTENNA

Manuscript received October 28, 2010; accepted January 15, 2011. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported by EPSRC Grant EP/F017502/1. E. Ebrahimi and P. S. Hall are with the Department of Electronics, Electrical and Computer Engineering, The University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K. (e-mail: [email protected]). J. R. Kelly is with the Department of Electronic and Electrical Engineering, The University of Sheffield, Sheffield S1 3JD, U.K. 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.2011.2152353

A. Integration Concept We propose to use a printed antenna with a relatively large metallization area as a ground plane for an additional antenna. Depending on the application and the amount of available space, it may be possible to use the first antenna as the coplanar ground for the second antenna or the second antenna can be printed on the reverse side of the substrate and, where necessary, it can be galvanically connected to ground with a via. Taking into account

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that small portable devices require low profile antennas, printed antennas can be good candidates for the proposed integration technique. Therefore, we used a printed UWB monopole antenna and a shorted patch antenna. In general, printed UWB monopoles are composed of two major printed parts, namely the radiator and the ground plane. The feeding system is either CPW or microstrip line [16], [17]. The impedance matching is optimized by the shaping of the radiator and controlling the height of the gap between radiator and the ground plane. To improve the bandwidth of the antenna, various configurations of the radiator and ground plane have been documented [16]–[19]. These include circular [16], square [18], and elliptical shapes [19]. In comparison with microstrip designs CPW-fed UWB antennas are better candidates due to their simple configuration, manufacturing advantages, repeatability, and low cost. Printed UWB antennas, in general, have a relatively large metalized layer that can be used as the ground plane for an additional relatively small antenna. This means that the space of one antenna can be used for two antennas. Importantly, the second antenna should be positioned on part of the first antenna where there is less concentration of surface currents. This ensures less interaction between the two antennas. Hence, by carefully choosing the right place to integrate the second antenna, the mutual coupling may be minimized. There is a lower concentration of the surface current on the ground plane of the UWB antenna, which might indicate that the second antenna can be placed at that region. However, in a typical mobile phone or a USB dongle, the ground plane is used for mounting the RF front-end and baseband circuitry. This area is normally tightly packed with elements and chipsets. Thus, it is not only unhelpful but impractical to use that area of the ground plane for mounting the additional antenna. B. Antenna Design A CPW fed UWB monopole and a shorted microstrip patch antenna or planar inverted F antenna (PIFA) were chosen as demonstrators for the concept of integration, presented here. The topology of the antenna is presented in Fig. 1. The structure consists of two printed antennas namely a wide- and a narrowband antenna. The wideband antenna is an hour glass shaped CPW-fed UWB monopole, printed on one side of the substrate. Fig. 1(a) shows the geometry and dimensions of the UWB antenna. The structure is printed on a Taconic TLC laminate with a relative permittivity of and a thickness of 0.79 mm. Since the ground plane is part of the matching network, in this class of antennas, the dimension of the ground plane and the feed gap are optimized to provide the optimum impedance matching [16], [17]. The narrowband antenna is printed on the reverse side of the substrate in order to use the UWB antenna as its ground plane. Consequently, less space required for this antenna even though it provides two states of operation. The shorting pin of the microstrip patch antenna is used to connect the patch to the wideband antenna. The patch is fed through a microstrip line. The microstrip feed line is printed above a defected ground plane, which is formed by a tapered slot on the reverse side of the substrate.

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Fig. 1. Topology of the proposed antenna, (a) top view and (b) bottom view.

The dimensions of the patch and its lateral position relative to the wideband antenna and the shorting pin position, all influence the resonant frequency of the narrowband antenna. This is due to changes in the current distribution which are caused by the close proximity of the wideband antenna. The feed line for the patch antenna is a 50 transmission line connected to a linearly tapered microstrip line. The taper forms a smooth transition between the 50 and 100 microstrip lines. The positions of the via pin and the patch, relative to the UWB antenna, are illustrated in Fig. 1(b). The computer simulation results, reported in the following section, were obtained using CST Design Studio. C. Fabrication and Measurement The antenna has been simulated, fabricated and verified. Its geometry is etched on the top and bottom side of the substrate. The narrowband antenna is then connected to its ground plane, that is the wideband antenna, through a via hole. The geometrical parameters were optimized to achieve wideband operation from 3.1 GHz to 10.6 GHz. The measured and simulated reflection coefficient for the wideband antenna is depicted in Fig. 2(a). The wideband antenna provides reasonable matching for the whole UWB spectrum. There is a good agreement between simulated and measured results. Fig. 2(b) compares the measured and simulated reflection coefficient for the narrow band antenna. The impedance bandwidth 10 dB) is 0.45 GHz (4.9 GHz–5.35 (reflection coefficient GHz). It is slightly wider than that of the simulated result. This difference may be attributed to the actual substrate having a larger loss tangent than that used in simulation and the tolerance in manufacturing. In this structure, the antennas are not completely isolated from each other. In Fig. 3 the measured and simulated transmission coefficient shows that the coupling is less than 10 dB for the whole band except in the range of 4.7 GHz to 7.3 GHz. The coupling peaks ( 4 dB) at 5.15 GHz. Our investigations showed that the high level of coupling is due to the location of the narrowband antenna. Fig. 4 shows the current distribution at 5.15 GHz supporting our argument. High current density can be observed at the lower edge of the UWB radiator. The narrowband feed line crosses this region on the bottom side of substrate, leading to high mutual coupling. The antenna and feeding

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Fig. 4. Simulated current distribution. (a) Top view and (b) bottom view.

Fig. 2. Simulated and measured reflection coefficient (a) UWB antenna and (b) narrowband antenna.

Fig. 5. Simulated and measured UWB antenna radiation patterns at 4.86 GHz, (a) xy plane and (b) zy plane.

Fig. 3. Simulated and measured transmission coefficient.

mechanism arrangement can be modified to enhance the isolation between the ports. This issue has been investigated and detailed discussions are presented in Section IV. Fig. 5 and Fig. 6 illustrate measured and simulated theta component of the radiation patterns for the UWB antenna at 4.86 GHz and 10 GHz, respectively. Fig. 7 depicts the measured

and simulated radiation patterns for the narrowband antenna at 5.15 GHz. The measurement results are in good agreement with simulated. Placing two antennas in close proximity changes the current distributions on each and consequently causes some changes in the far field radiation patterns. However, in practice due to the demand for smaller devices with more services, fitting multiple antennas in a small available space is the only solution. Thus, some degree of pattern degradation is inevitable. The challenge is to minimize the impact of one antenna on the radiation pattern of the other. The peak realized gain of UWB antenna is demonstrated in Fig. 8. III. MATCHING CIRCUIT DESIGN AND RESULTS At low frequencies such as GSM or Bluetooth frequencies, the antenna dimensions are significant comparing to the UWB

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Fig. 6. Simulated and measured UWB antenna radiation patterns at 10 GHz, (a) xy plane and (b) zy plane.

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to design. Most continuously reconfigurable matching circuits are based around a non-linear tuning element such as varactor diodes. This leads to intermodulation products, which are highly undesirable. It would also be necessary to design a bias circuit for varactor diode and that adds to the cost and complexity of the system. Although a bank of switched matching networks may require more space it would be much easier to design. In a mass produced commercial system these matching circuits could be housed within a microwave integrated circuit (MIC), which could be optimized in order to enhance performance and reduce the space requirement. A microprocessor would be used to switch between each of the different circuits. It would be important to investigate the effect of component and switch losses, on the total efficiency of the antenna. If this is a problem it may be necessary to use a low loss component, such as a MEMS switch. This section investigates the possibility for tuning the narrowband antenna while retaining the ultra-wideband performance on the second port. For this reason three tuning circuits are designed, based around fixed elements. The simulated and measured reflection coefficients for the antenna, with various matching circuits, are provided. A. Matching Circuit Design

Fig. 7. Simulated and measured narrowband antenna radiation patterns at 5.15 GHz, (a) xy plane and (b) zy plane.

Fig. 8. UWB antenna realized gain.

antenna. Thus, the UWB antenna is not sufficiently large to act as the ground plane for the narrowband antenna. In this case, the narrowband antenna is designed at higher frequencies together with a matching circuit. Furthermore, in cognitive radio a wideband antenna can be utilized for wideband spectrum sensing while a tunable narrowband antenna would be used for communications. Thus, the combination of the wideband antenna and a narrowband antenna together with either a bank of matching circuits, or several reconfigurable matching circuits, can be a promising candidate. A continuously reconfigurable matching network would probably require less space. However it would be more complicated

In this paper two types of matching circuit have been used to tune the narrowband antenna to three different frequencies. A pair of L-networks was used to tune the narrowband antenna to 4 GHz and 10 GHz, whilst a Pi-network was employed to achieve operation at 8 GHz. These three frequencies have been chosen to demonstrate possibility to tune the antenna across the UWB band. All three circuits are composed of just inductors and capacitors. In order to reduce the complexity of the manufacturing process, the matching circuit is designed to be fabricated on the same substrate as the antenna. The overall size of the antenna, along with the matching circuit, is 49 mm by 54 mm. The original structure has been elongated by 10 mm to accommodate the matching circuit. Using the CPW ground as the ground for the circuits, the elements can be placed around the narrowband feed line on the reverse side of the substrate. Fig. 9 demonstrates the integration of the matching circuit with the antenna. The reactances on the vertical branches of the network (i.e., X2 and X3) are connected to the ground plane by means of two vias. The reactance in the horizontal branch (i.e., X1) is connected across a gap made in the feed line of the antenna. The three different matching circuits are shown in Fig. 10. The elements were chosen from a range of commercially available surface mount device (SMD) inductors and capacitors. The lumped element model for each of the components accounted for the effect of parasitics, and was obtained from the manufacture’s literature. The simulated and measured scattering parameters of the three circuits are depicted in Figs. 11, 12 and 13 for the 4 GHz, 8 GHz and 10 GHz circuits, respectively. Measured results for the 4 GHz and 8 GHz circuits are in agreement with those obtained through simulation. For the 10 GHz circuit there is a 900 MHz difference between measurement and simulation. This might be due to other unknown parasitics of

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Fig. 9. The topology of the antenna with an integrated matching circuit.

Fig. 12. Simulated and measured reflection coefficient of the antenna with a 4 GHz matching circuit.

Fig. 10. Matching circuit for (a) 4 GHz, (b) 8 GHz, (c) 10 GHz.

Fig. 13. Simulated and measured reflection coefficient of the antenna with a 10 GHz matching circuit. Fig. 11. Simulated and measured reflection coefficient of the antenna with a 4 GHz matching circuit.

the elements which are not considered in the simulation. Since the elements are not placed with a pick and place machine on the PCB, positional errors might give rise to the discrepancy. The hand-made grounding vias can also increase the loss and change the reactive values in each branch. Investigations show that for low frequency matching, a Pi-network or L-network with an inductor in the horizontal branch and capacitors in the vertical branches is the most suitable choice, as it filters out the high frequency resonances. Table I compares the simulated total efficiency and realized gain of the tuning circuit when integrated with the original antenna. The total efficiency is defined as the ratio of radiated to stimulated power. The difference between the input power and the stimulated power is that the latter considers any reflections which occur at the feeding location. The realized gain includes , this impedance mismatch loss and is defined by G. where G is the IEEE gain, defined in [20]. The antenna becomes electrically larger as the frequency increases, and so the efficiency and gain increase. However, at lower frequencies the

TABLE I EFFICIENCY AND GAIN OF VARIOUS NARROWBAND ANTENNAS

Realized gain = Gain (1 S ) 1

0

Original antenna

structure is electrically small and the level of total efficiency and realized gain is lower, which agrees with the Chu-Harrington expression. IV. COUPLING One of the main issues in the design of multi-port antennas is achieving adequate isolation between the ports. However controlling the power which is coupled between the ports is very

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Fig. 14. Geometry of configuration A, (a) top view, (b) bottom view. Fig. 15. Transmission coefficient for all three configurations and the reference antenna.

challenging in closely spaced antennas. Although the patch antenna is positioned in a region of the UWB antenna with low current concentration, the coupling between the ports is still relatively high. From Fig. 3, it is clear that the coupling is less than 10 dB throughout the whole band, except in the range 4.7 GHz to 7.3 GHz, where the peak value is 4 dB at 5.15 GHz. Investigations show that when the UWB antenna is excited the intensity of the current flow along the lower edge of the UWB ellipse and the top edge of the CPW ground plane is high. In effect a slot to microstrip transition is created at the point where the narrowband feed line crosses over this region. Therefore, a significant portion of the input power is coupled into the microstrip feed line, and returns to the feed port of the narrowband antenna. Similar discussion applies when the narrowband antenna is fed. In this case, it creates a microstrip to slotline transition and the fields around the microstrip line couple into the tapered slot of the wideband antenna. Fig. 4 illustrates this argument. Hence, to increase the isolation between the ports one solution is to ensure that the feed line does not cross this region. This section presents three modifications to the structure in order to reduce the level of coupling. A. Configuration A Fig. 14 demonstrates the geometry of the first configuration (Configuration A). The modification involves shifting the patch feed line away from the tapered slot at the edge of the wideband antenna and bending it at right angles. Using this method, the peak value of mutual coupling drops from 4 dB to just below 7 dB (see Fig. 15). The impedance bandwidth of the wideband antenna is not affected by this modification, i.e., the 10 dB bandwidth still covers the whole of the FCC UWB band. The narrowband antenna exhibits a resonance at 4.81 GHz. At the resonant frequency the reflection coefficient is just below 20 dB. B. Configuration B Investigating the current distribution reveals that there are less currents on the upper part of the monopole radiator representing a good location for the second antenna. Fig. 16 illustrates the second modified antenna (denoted Configuration B). The monopole radiator is elongated so that it extends to the edge

Fig. 16. Geometry of configuration B, (a) top view, (b) bottom view.

of the board. The narrowband patch is printed on the reverse side of the substrate. The isolation between the two ports is demonstrated in Fig. 15. In this configuration, the peak transmission coefficient is 18 dB. It remains below 25 dB throughout the majority of the FCC UWB band. The reflection coefficient for the wideband antenna, in this configuration, is well below 10 dB from 3–11 GHz. The narrowband antenna operates at 5.26 GHz where the reflection coefficient is 7.5 dB. C. Configuration C For some applications, it may not be physically affordable for the designer to have ports on opposite sides of the board. Thus in the third arrangement of this structure the goal is to have two ports on the same side, while maintaining a good level of isolation. The geometry of the third antenna (Configuration C) is presented in Fig. 17. In this structure the wideband monopole has been moved along the -axis, from middle of the substrate to one side. The UWB radiator and ground plane are printed onto the upper surface of the substrate. The UWB monopole is fed through a bent section of microstrip line which is printed onto the back of the substrate. The feed line is connected to the monopole radiator through a via hole. The narrowband antenna is located above the rectangular part of the monopole. This configuration provides approximately 18 dB of isolation between

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to make minor changes to the geometry in order to accommodate the matching circuits. Additionally the overall dimensions of the antenna are quite small. It is concluded that the specific arrangement of the narrowband antenna’s feed line in the original antenna design creates a microstrip to slot transition suggesting that approximately 40% of the power is coupled from one antenna into the other. The final section of this paper evaluated three different approaches for reducing the mutual coupling between the two antennas. Each of these configurations yielded to a reduction in the isolation while maintaining the wideband and narrowband functionality. In the third configuration the coupling level has dropped dramatically from 4 dB to below 20 dB. Fig. 17. Geometry of configuration C, (a) top view, (b) bottom view.

ACKNOWLEDGMENT the two ports (see Fig. 15). The 10 dB impedance bandwidth of the wideband antenna has not distorted. The narrowband antenna operates at 5.35 GHz.

V. CONCLUSION This paper introduced a novel concept for integrating two planar antennas suitable for multi-band applications. The integration approach was based on using a relatively large printed antenna as the ground plane for an additional antenna. To avoid high interaction between the two antennas, it is preferable to place the second antenna above a region of the larger antenna where there is a relatively low concentration of surface current. The ground plane of the large antenna appeared to be the most suitable location. However, after taking some industrial concerns into account it was clear that the extra antenna could not be placed above the main ground plane. In practice, the ground plane of the antenna is used for mounting the RF and digital circuits. The concept was validated by integrating a narrowband antenna with a UWB antenna. The structure consisted of a wideband CPW-fed monopole and a microstrip shorted patch antenna. The wideband antenna was printed onto the top side of the substrate and the shorted patch was printed above the monopole radiator, on the bottom side of the substrate. Other UWB antennas with different radiator shapes and feeding mechanisms may also be customized and designed by applying the same concept. Depending on the frequency of operation there are more possibilities to choose from when selecting the second antenna. The structure was fabricated and verified. By providing two separate ports for two services this antenna makes it possible for the user to switch the services on and off, one at any time. In circumstances when the size of the ground plane is not large enough for the second antenna a smaller antenna can be designed at higher frequency and then tuned to the required frequency by means of a tuning circuit. By employing a bank of matching circuit, or several reconfigurable matching networks it is possible to use this arrangement to create reconfigurable radio systems as well. Three matching circuits were designed to demonstrate the possibility of tuning the narrowband antenna across the wide range of frequencies. An advantage of this integrated antenna concept is that one only needs

The authors would like to thank Taconic Advanced Dielectric Division for kindly donating the microwave substrate material used in this investigation. REFERENCES [1] H. W. Hsieh, Y. C. Lee, K. K. Tiong, and J. S. Sun, “Design of a multiband antenna for mobile handset operations,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 200–203, 2009. [2] H. W. Hsieh, Y. C. Lee, K. K. Tiong, and J. S. Sun, “Multiband printed monopole slot antenna for WWAN operation in the laptop computer,” IEEE Trans. Antennas Propag., vol. 57, pp. 324–330, Feb. 2009. [3] M. Martínez-Vázquez, O. Litschke, M. Geissler, D. Heberling, A. M. Martínez-González, and D. Sánchez-Hernández, “Integrated planar multiband antennas for personal communication handsets,” IEEE Trans. Antennas Propag., vol. 54, pp. 384–391, Feb. 2006. [4] L. Lizzi, F. Viani, E. Zeni, and A. Massa, “A DVBH/GSM/UMTS planar antenna for multimode wireless devices,” IEEE Antenna Wireless Propag. Lett., vol. 8, pp. 568–571, May 2009. [5] Y. X. Guo, M. Y. W. Chia, and Z. N. Chen, “Miniature built-in multiband antennas for mobile handsets,” IEEE Trans. Antennas Propag., vol. 52, pp. 1936–1944, Aug. 2004. [6] Z. Li and Y. Rahmat-Samii, “Optimization of PIFA-IFA combination in handset antenna designs,” IEEE Trans. Antennas Propag., vol. 53, pp. 1770–1778, May 2005. [7] J. Holopainen, O. Kivekäs, C. Icheln, and P. Vainikainen, “Internal broadband antennas for digital television receiver in mobile terminals,” IEEE Trans. Antennas Propag., vol. 58, p. 3363, Oct. 2010. [8] Z. D. Liu, P. S. Hall, and D. Wake, “Dual-frequency planar inverted-F antenna,” IEEE Trans. Antennas Propag., vol. 45, pp. 1451–1458, Oct. 1997. [9] C. T. P. Song, P. S. Hall, H. Ghafouri-Shiraz, and D. Wake, “Triple band planar inverted F antennas for handheld devices,” IET Electron Lett., vol. 36, pp. 112–114, Jan. 2000. [10] S. Yoon, C. Jung, Y. Kim, and F. De Flaviis, “Multi-port multi-band small antenna design,” presented at the Asia-Pacific Microwave Conf., Dec. 2007. [11] J. Mitola and G. Q. Maguire, “Cognitive radio: Making software radios more personal,” IEEE Personal Commun., vol. 6, pp. 13–18, Aug. 1999. [12] P. S. Hall, P. Gardner, J. Kelly, E. Ebrahimi, M. R. Hamid, and F. Ghanem, “Antenna challenges in cognitive radio,” presented at the Antennas Propag. Int Symp., Taiwan, Oct. 2008. [13] Y. Hur, J. Park, W. Woo, K. Lim, C.-H. Lee, H. S. Kim, and J. Laskar, “A wideband analog multi-resolution spectrum sensing (MRSS) technique for cognitive radio (CR) systems,” presented at the IEEE Circuits and Systems Int. Symp., 2006. [14] H. Harada, “A software defined cognitive radio prototype,” presented at the IEEE 18th Personal, Indoor Mobile Radio Comms. Int. Symp., Sep. 2007. [15] E. Ebrahimi, J. Kelly, and P. S. Hall, “A reconfigurable narrowband antenna integrated with wideband monopole for cognitive radio applications,” presented at the IEEE Int. APS/URSI/AMEREM, 2009.

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[16] C. J. Liang, C. C. Chiau, X. Chen, and C. G. Parini, “Study of a printed circular disc monopole antenna for UWB systems,” IEEE Trans. Antennas Propag., vol. 53, pp. 3500–3504, Nov. 2005. [17] J. Liang, L. Guo, C. C. Chiau, X. Chen, and C. G. Parini, “Study of CPW-fed circular disc monopole antenna for ultra wideband applications,” Proc. IEE Microw. Antennas Propag., vol. 152, pp. 520–526, Dec. 2005. [18] S. W. Su, J. H. Chou, and K. L. Wong, “Internal ultrawideband monopole antenna for wireless USB dongle applications,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1180–1183, Apr. 2007. [19] K. Bahadori and Y. R. Samii, “A miniaturized elliptic-card UWB antenna with WLAN band rejection for wireless communications,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3326–3332, Nov. 2007. [20] IEEE Standard Definitions of Terms for Antennas, 145–1983, Jun. 1983.

as a Research Associate. His research topic is reconfigurable antennas for body-worn wireless networks. He has published more than 50 papers in peer reviewed international journals and conference proceedings, and frequently acts as a reviewer for several technical publications. During the past two years he has filed three patents on new developments in small antenna technology. His research interests include: reconfigurable antennas, ultrawideband antennas, small antennas, dielectric constant measurement, metamaterial structures, and microwave filters. Dr. Kelly is a member of the IET and EurAAP. In October 2009, he was keynote speaker at the IET seminar on Adaptable and Tunable Antenna Technology for Handsets and Mobile Computing Products. He served on the TPC for the 2010 Mediterranean Microwave Conference. He also Co-Chaired a session on adaptive antennas for software radio during the 2009 IEEE International Symposium on Antennas and Propagation, in Toronto, Canada.

Elham Ebrahimi (S’09) received the B.Sc. degree in electrical engineering from KN Toosi University, of Technology, Tehran, Iran, in 2004 and the M.Sc. degree in electrical engineering from Tarbiat Modarres University, Tehran, in 2006. She is currently working toward the Ph.D. degree at the University of Birmingham, Birmingham, U.K. Her current research interests include small ultrawideband and reconfigurable antenna design and analysis.

Peter S. Hall (M’88–SM’93–F’01) received the Ph.D. degree in antenna measurements from Sheffield University, Sheffield, U.K., in 1973. He spent three years with Marconi Space and Defence Systems, Stanmore, U.K., largely working on a European Communications satellite project. He then joined The Royal Military College of Science as a Senior Research Scientist, progressing to Reader in Electromagnetics. He joined The University of Birmingham in 1994. Currently, he is Professor of Communications Engineering, Leader of the Antennas and Applied Electromagnetics Laboratory, and Head of the Devices and Systems Research Centre in the Department of Electronic, Electrical and Computer Engineering at The University of Birmingham, Birmingham, U.K. He has researched extensively in the areas of antennas, propagation and antenna measurements. He has published five books, over 350 learned papers and taken various patents. These publications have earned many awards, including the 1990 IEE Rayleigh Book Award for the Handbook of Microstrip Antennas. Professor Hall is a Fellow of the IET and the IEEE and a past IEEE Distinguished Lecturer. He recently received the LAPC IET James Roderick James Lifetime Achievement Award. He is a past Chairman of the IEE Antennas and Propagation Professional Group and past Coordinator for Premium Awards for IEE Proceedings on Microwave, Antennas and Propagation. He is a member of the IEEE AP-S Fellow Evaluation Committee. He Chaired the 1997 IEE ICAP conference, was Vice Chair of EuCAP 2008 and has been associated with the organization of many other international conferences. He was Honorary Editor of IEE Proceedings Part H from 1991 to 1995 and currently on the editorial board of Microwave and Optical Technology Letters. He is a past member of the Executive Board of the EC Antenna Network of Excellence.

James R. Kelly (M’10) was born in Derbyshire, England. He received the Masters degree in electronic and electrical engineering and the Doctorate in microwave filters from Loughborough University, Leicestershire, England, in 2002 and 2006, respectively. From 1999 to 2000, he worked for Interfleet Technology (international rail vehicle consultancy). During summer 2001, he was on placement within the Rolls-Royce Strategic Research Centre. After completing his Ph.D. he began working as a Research Associate/Fellow. He has worked at Loughborough University (2006–2007) as well as The University of Birmingham (2008–2010). Those projects focused on the design of 2D leaky-wave antennas and reconfigurable antennas for Cognitive Radio. He is currently with the Communications Research Group, University of Sheffield, Sheffield, U.K., where he is employed

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Multiband Inverted-F Antenna With Independent Bands for Small and Slim Cellular Mobile Handsets Hattan F. AbuTarboush, Member, IEEE, R. Nilavalan, Senior Member, IEEE, Thomas Peter, and S. W. Cheung, Senior Member, IEEE

Abstract—The design of a small ultra-thin printed inverted-F antenna (PIFA) with independent control on the resonant frequency bands is proposed. The antenna consists of a slotted radiator supported by shorting walls and a small ground plane. The structure is designed and optimized to operate at 2.09, 3.74 and 5 GHz with achievable bandwidths of 11%, 8.84% and 10%, respectively. These three bands cover the existing wireless communication frequency bands from 1.5–6.8 GHz. Each of the three bands can be controlled independently without affecting the other two bands. The 2.09 GHz band can be controlled to operate between 1.5–2.09 GHz (33.33%), the 3.74 GHz band can be controlled over the range of 3.57–4.18 GHz (15.76%) and the 5 GHz band can be controlled to cover the band from 5.00–6.80 GHz (30.50%). Results of intensive investigations using computer simulations and measurements show that the ground plane and the feed locations of the antenna have marginal effects on the performance of the antenna. The effects of the user’s hand and mobile phone housing on the return loss, radiation patterns, gains and efficiency are characterized. The measured peak gains of the prototype antenna at 2.09, 3.74 and 5 GHz are 2.05, 2.32 and 3.47 dBi, respectively. The measured radiation efficiencies for the corresponding three bands are 70.12, 60.29 and 66.24% respectively. Index Terms—Antenna for mobile phone, independent control, printed inverted-F antenna (PIFA), PIFA ground plane, small PIFA, the effect of user’s hand, thin PIFA.

I. INTRODUCTION

LANAR Inverted-F Antennas are widely used in a variety of communication systems especially in mobile phone handsets due to low profile, light weight, easy integration and manufacturability [1]–[3]. In recent years, there have been a number of PIFA designs with different configurations to achieve single and multiple operations by using different shapes of slots [2]–[11]. Truncated corner technique [12], meandered strips [13] and meandered shapes [14], [15] have been used to create multiple band operations. Branch line slit has been used to achieve dual-band operations [16]. Broadband

P

Manuscript received August 25, 2010; revised November 07, 2010; accepted December 02, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. H. F. AbuTarboush, R. Nilavalan and T. Peter are with the Wireless Networks and Communications Centre (WNCC), School of Engineering and Design, Brunel University, West London UB8 3PH, U.K. (e-mail: [email protected]). S. W. Cheung is with the Department of Electrical and Electronic Engineering, University of Hong Kong, Pokfulam Road, Hong Kong. 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.2011.2152350

multi-resonant antennas utilizing capacitive coupling between multiple conductive plates was claimed in a patent [17]. These antennas are generally designed to cover one or more wireless communications bands such as the Global System for Mobile Communications (GSM900 and 800), Personal Communication System (PCS 1800 and 1900), Digital Communication Systems (DCS), Global Position System (GPS), Universal Mobile Telecommunications System (UMTS), Wireless Local Area Networks (WLAN) and Worldwide Interoperability for Microwave Access (WiMAX), etc. The ground plane of a PIFA can play an important role to enhance the performance of the antenna [18]. For example, for low frequency operation such as for the GSM 900/800 bands, the ground plane has to be used as a radiating part. However, if the ground plane also acts as a radiating part, the effect of the user’s hand is likely to degrade the antenna performance when the antenna is fitted inside the mobile phone. This causes several practical engineering problems [19]–[22]. In some designs, the location of the antenna on the substrate is also an important factor to be considered as it can enhance the bandwidth of the antenna by few more percentages [23]. Some work has been done to achieve frequency independent control for a small-size and thin antenna. For example in [24], a Planer Inverted-F Antenna (PIFA) was thoroughly studied to control three resonant frequencies for GSM/DCS/DMB with an . However, the structure of overall size of the antenna could not provide a wide-independent control for the three resonant frequency bands and the large ground plane also affected the frequency bands. In [25], a switchable design for dual bands at (1.9 GHz, 5.2 GHz) and (1.9 GHz, 3.5 GHz) was presented with some tuning capability for a reconfigurable system. In [26], a double U-Slot patch antenna was proposed to independently control three WiMAX bands. However, the control ranges of the three bands were limited to only few percentages. In the patent reported in [27], [36], a multi-frequency band antenna, capable of tuning the low-band portion to a low frequency band and the first high-band portion to a first high frequency band, was introduced for use in mobile handsets applications. In this paper, the design of a relatively small and ultra-thin PIFA that can support three frequency bands at 2.09 GHz, 3.74 GHz and 5 GHz with achievable bandwidths of 10%, 8.8% and 11%, respectively is proposed and presented. The effects of different ground plane dimensions, locations of the antenna on the substrate and physical heights of the antenna from the ground plane are studied. The three bands can be independently designed over a wide range and also re-designed to any other bands between 1.5 GHz to 6.8 GHz. The proposed antenna satisfies the

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Fig. 1. The layout of the proposed antenna (a) 3D view and (b) detailed dimensions.

TABLE I DETAILED DIMENSIONS OF THE PROPOSED ANTENNA (IN MILLIMETER)

return loss, bandwidth, gain and efficiency requirements for applications within the frequency range from 1.5 GHz to 6.8 GHz. The measured reflection coefficient, radiation pattern, gain and radiation efficiency are characterized. The effects of a user’s hand model and the mobile phone housing model on the return loss, gain, radiation pattern and efficiency are also studied. II. DESIGN OF SMALL AND THIN PIFA A. Antenna Configuration and Measurements Fig. 1(a) shows the structure of the proposed antenna with detailed dimensions given in Fig. 1(b) and Table I. The proposed antenna consists of a main radiator with an irregular shape, a rectangular slot, shorting walls, and a ground plane. The material used is FR-4 substrate with a dielectric constant of 4.4, a loss tangent of 0.02 and a substrate height of 1.57 mm. The proposed antenna has a very small size and is physically thin. The total , volume of the radiator with feed point is while the overall volume of the antenna including the ground . The EM software, High Frequency plan is Structure Simulator (HFSS) V.11.4 package, is used for full wave analysis of the antenna and material losses is taken into account in the simulation studies. To validate the simulated results, the proposed antenna is also fabricated on a FR-4 substrate with the same characteristics used in simulation. The thickness of the copper used in the prototype is 0.15 mm. The simulated and measured reflection coefficient of the proposed antenna is presented in Fig. 2(a) and the

prototype is shown in Fig. 2(b). It can be seen that the simulated and measured results are in good agreements. The little discrepancies might be due to many factors such as the soldering proficiency and accuracy of cutting the edges of the copper. The results in Fig. 2(a) show three distinct bands are generated at 2.09 GHz, 3.74 GHz and 5 GHz. The corresponding bandwidths defined by 6 dB for the three bands are 11% (1.978–2.2 GHz) for the 2.09 GHz band, 8.84% (3.571–3.9 GHz) for the 3.74 GHz band and 10% (4.887–5.391 GHz) for the 5 GHz band. These bandwidths satisfy the requirements for most of the wireless applications. The antenna achieves a wider bandwidth, smaller ground plane size and thinner structure than the designs reported in [24] and [25]. B. Radiation Mechanism and Current Distributions Further understanding of the antenna behaviour can be observed from the current distribution plots shown in Fig. 3(a)–(c). These current distribution plots can be used to identify the electrical lengths for the first, second and third resonant frequencies, , and , at 2.09 GHz, 3.74 GHz and 5 GHz, respectively. It can be seen in Fig. 3(a) that there are two major current paths on the radiator generating the 2.09 GHz band. The first current path and whereas the second current path is along is along and of Fig. 1(b). Both paths have an electrical length of about a quarter wavelength at 2.09 GHz. At 3.74 GHz, Fig. 3(b) shows that there is only one major current path concentrated and on the radiator. This path has an electrical along

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Fig. 2. (a) Simulated and measured S

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for the proposed antenna (b) prototype antenna.

Fig. 3. Currents distribution for the proposed antenna at (a) 2.09 GHz, (b) 3.74 GHz, and (c) 5 GHz.

length of about a quarter wavelength at 3.74 GHz. In Fig. 3(c), there are two major current paths on the radiator. The first path and whereas the second path is formed around is formed around and . The electrical lengths for both paths are about a quarter wavelength at 5 GHz. Thus by varying these parameters, the current paths for the first, second and third resonances can be independently controlled over a wide range, which is further elaborated in the next section. C. Parametric Analysis and Independent Control Over a Wide Range To design an antenna with multiple band operation, it is desirable to have an independent frequency control on two or more separate frequencies. Achieving this option is very challenging. Very often, when one parameter is changed, all the other frequencies are affected [28], [29] and the antenna needs to be completely redesigned for other bands. The idea proposed in this paper to achieve an independent frequency control on different frequencies of a single antenna is to find out the radiation elements of the antenna responsible for individual bands. From the current distribution discussions in Section II-B, we can identify the key radiation elements by observing the current paths for each resonant frequency, so we

can control each band independently. For example, the current and is for the 2.09 GHz band. Increasing the path along in the X-direction moves the lower order mode reslength of onance (at 2.09 GHz) toward the lower frequencies as shown in Fig. 4(a) and Table II. The 2.09 GHz band can be controlled over 33.33% between 1.5–2.09 GHz. Similarly, the current path for and , so by changing the size of the 3.74 GHz is along width of in the X-direction (without changing the parame)), the 3.74 GHz band can be tuned to a lower or ters ( and higher frequency, as shown in Fig. 4(b) and Table II. Here, we can tune the 3.74 GHz band over 15.75% between 3.57–4.188 GHz. For the 5 GHz band, the current path is formed along and . By varying the length and the width simultaneously, we can tune the 5 GHz band over 30.50% between 5–6.8 GHz without affecting the 2.09 GHz band and the 3.74 GHz bands, as shown in Fig. 4(c) and Table II. It should be noted that the 2.09 GHz and 5 GHz bands have one common current path around and , so by changing the length or the width of , these two bands can be controlled without affecting the 3.74 GHz band. Since these three bands can be controlled independently over wide frequency ranges (compared with the design reported in [24] and [26]), the antenna can be designed easily for other applications.

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Fig. 4. Parametric studies showing independent control for each band over wide range (a) 2.09 GHz band, (b) 3.74 GHz band, and (c) 5 GHz band.

TABLE II INDEPENDENT CONTROL RANGE IN THREE BANDS

III. SIGNIFICANCE OF SOME PARAMETERS ON ANTENNA PERFORMANCE The effects of the ground plane size, the antenna location and the height of the PIFA on the performance of the antenna are examined and further elaborated in this section. A. Ground Plane Effect The side and geometry of a ground plane in a PIFA are known to affect the antenna performance. In [30], slots were added to the ground plane to significantly improve the bandwidth performance of the antenna. In [24], it was shown that varying the . An antenna can be designed ground plane size would affect to couple more energy to the ground plane, making the ground plane a radiating part and resulting in a wider impedance bandwidth. However, this makes the ground plane quite sensitive. Since our proposed antenna is designed for use in the mobile

phone systems which require a relatively narrow width, there is no need to use the ground plane to increase the bandwidth. Moreover, there are many advantages of having a less sensitive ground plane. For example, with an insensitive ground plane, the antenna performance will not be affected by other electronic components and circuits nearby. When multiple antennas are integrated together, there will be strong isolation between antennas, allowing easy optimization of antennas positions. The antenna can be used in mobile phones with different ground plane sizes without changing the performance. The user’s hand will not affect the matching of the bands and also the radiation efficiency [31]. For these reasons, we should design the antenna to have the ground plane as less sensitive as possible so that the performance mainly depends on the structure alone and not the surrounding elements [19], [22]. For our proposed antenna, results in Fig. 5(a) show that varying the length of the ground plane from 40 40 mm to 40 100 mm does not affect the

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Fig. 5. The effects of (a) ground plane size, (b) antenna location, and (c) physical height of the PIFA on S

matching or the bandwidth of the antenna, indicating that the ground plane effect is quite small. Similar observation was also reported in [23]. However, this will not be the case if the ground plane is used as part of the radiating part as in [24] and [30]. When the size of the ground plane changes the current distribution on the main radiator and the ground plane for the three bands do not significantly change. Also the gain and the radiation efficiency have been observed when changing the size of the ground plane. No significant changes in the gain and the radiation efficiency have been noticed at the three bands. B. Antenna Location The location of the antenna can also affect the performances [23]. However, in our proposed antenna, results in Fig. 5(b) shows that changing the location of the proposed antenna along the substrate does not affect the matching or bandwidth of the bands. More results have also shown that this would not change the gain and the shape of the radiation pattern. C. Height of the PIFA Fig. 5(c) shows the effects of the height of the PIFA above the ground plane on the bandwidth of the antenna. It can , the reflection be seen that, at a smaller value of coefficient is larger at the high frequency band and significantly lower at only 5 dB in the lower frequency band. This is because the radiator was too close to the substrate base to resonate at

performance.

, 6 and 9 mm, the relow frequency. At the heights of in the low freturn loss is larger than 10 dB quency band, but less than 5 dB in the high frequency bands. , the return losses of the three bands are larger With than 10 dB which satisfies many applications. Since the objective of this research is to design a small antenna with a thin strucfrom the subture, we have selected the heights strate and 3.57 mm from the ground plane for further studies. With these dimensions, the proposed antenna can operate in the UMTS, m-WiMAX and 5 GH WLAN bands with a bandwidth wide enough to cater for these applications. More results have does not alter the radiation effialso shown that changing ciency and gain of the antenna significantly enough to affect the , where the effiperformance, except at 5 GHz for ciency changes slightly. IV. SIMULATION AND MEASUREMENTS A. Measurements Setup The antenna is measured using the antenna measurement equipment, StarLab, manufactured by Satimo [31]. Before any measurement is done, calibration is carried out by using the standard antennas provided. For radiation pattern and gain measurements, it is just like other antenna measurement equipment. For power efficiency measurement, the equipment first measures the gain, radiation intensity and reflection coefficient of the antenna and computes the directivity using the radiation

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Fig. 6. Simulated and measured Co and X-pol radiation patterns in E and H planes (a) 2.09 GHz, (b) 3.74 GHz, and (c) 5 GHz.

intensity. The efficiency of the antenna is then computed using the equation:

where

is the voltage reflection coefficient, and are the gain and directivity, respectively, of the antenna and are functions of spherical coordinate angles and . The directivity is calculated by using the radiation intensity [32]. B. Simulated and Measured Radiation Patterns and Its Relationship With Current Distribution The simulated and measured radiation patterns for co- and cross- polarizations in the E-plane and H-plane at the frequencies of 2.09, 3.74 and 5 GHz are shown in Fig. 6(b)–(d). It can be observed the radiation patterns are quite stable throughout the UMATS, m-WiMAX and WLAN bands. To relate the X-Y-Z orientation of the antenna in Fig. 1 to the E- and H-planes in Fig. 6, we use the current directions of Fig. 3(a)–(c) on the radiator in the individual frequency bands. Fig. 3(a) shows that the current direction for the first band at 2.09 GHz is in the X-direction, so the E- and H-planes in Fig. 6 are the X-Z and Y-Z planes, respectively, in Fig. 1. The current direction for the second band at 3.74 GHz is in the Y-direction as shown in Fig. 3(b), indicating that the E- and H-planes are the Y-Z and X-Z planes, respectively. Here, a high cross-polarization level is found at 3.74 GHz. This might be due to high current concentration around the feed and the end of , as shown in Fig. 3(b). Finally, the

Fig. 7. Measured S , peak gain and radiation efficiency.

current direction for the 5 GHz band in Fig. 3(c) indicates that the E- and H-planes are the Y-Z and X-Z planes, respectively. To conclude these, at 3.74 GHz and 5 GHz, the currents have the same direction and their E-planes are the Y-Z plane, whereas at 2.09 GHz, the current has a different direction and the E-plane is the X-Z plane. C. Measured Gain and Radiation Efficiency Simulations and measurements on the peak gain and radiation efficiency of the antenna have been carried out. Results have shown that, in the 2.09 GHz, 3.74 GHz and 5 GHz bands, the simulated peak gains are 2.14 dBi, 2.4 dBi and 5 dBi, respectively, and the corresponding measured peak gains 2.05 dBi,

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TABLE III SIMULATED AND MEASURED GAIN AND EFFICIENCY WITH AND WITHOUT THE MATERIAL LOSSES

Fig. 8. The proposed antenna with (a) hand and (b) mobile phone housing and hand.

Fig. 9. The effect of the human hand and the plastic housing of the mobile phone on S .

2.32 dBi and 4.42 dBi, as shown in Fig. 7. The measured radiation efficiencies for the three bands is 70.12%, 60.29% and 66.24%, respectively, as shown in Fig. 7. These results have taken into account the loss of the FR-4 substrate. If the loss is neglected, the gains and the radiation efficiencies in the three-band are higher, as shown in Table III. V. EFFECTS OF MOBILE PHONE HOUSING AND USER’S HAND ON ANTENNA’S GROUND PLANE The effects of a human hand model and a mobile phone , radiation housing model on the reflection coefficient

Fig. 10. Normalized Co-Pol and X-Pol radiation patterns at 2.09 GHz band in the presence of human hand model and the mobile phone housing in (a) X-Z plane and (b) Y-Z plane.

patterns, gain and efficiency of the antenna have also been investigated. Fig. 8(a) shows the simulation model of the antenna with the human hand model. The fingers and the palm are attached directly to the ground plane and the main substrate, respectively. When the mobile phone housing model is used as well, the simulation model is shown in Fig. 8(b), where the mobile phone housing model is in direct contact with the

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Fig. 11. Results of the proposed antenna with human hand model and mobile phone housing (a) peak gain and (b) radiation efficiency.

Fig. 12. Simulation model for different positions of user’s hand (a) user’s hand covering radiator, (b) user’s hand partly covering radiator, and (c) user’s hand not covering radiator of antenna.

antenna. The relative permittivity and conductivity of 3.5 and 0.02 S/m, respectively, for the mobile phone housing model [33] and of 54 and 1.45 S/m, respectively, for human hand model [34] have been used in simulations. The results in Fig. 9 show that the human hand and mobile phone housing increase the return losses in the two lower frequency bands and slightly increase it in the higher frequency band. With the human hand and mobile phone housing in place, the radiation patterns at 2 GHz are shown in Fig. 10. It can be seen that the shape of the radiation patterns do not change much. The simulated peak gain and radiation efficiency are shown in Fig. 11. At 2 GHz, Fig. 11(a) shows that the mobile phone housing and the hand increase the peak gain by 2 dB. But if only the human hand is attached directly to the ground plane, the peak gain is decreased by almost 1 dB. This can also be observed in the radiation pattern of Fig. 10 where the antenna loses some energy in the direction of the ground plane and gain from its maximum value. At 3.74 GHz and 5 GHz, the gains drop by approximately 1 dBi, yet maintaining the shape of the radiation patterns. There is no significant change in radiation efficiency when both the human hand model and mobile phone housing are present, as shown in Fig. 11(b).

These results indicate that the ground plane of the proposed design is not too sensitive to the hand and mobile phone housing, thus the antenna has a low ground plane effect. VI. EFFECT OF USER’S HAND AT DIFFERENT POSITIONS ON ANTENNA It is also essential to examine the effects of the hand at different positions on the return loss, gain and efficiency of the antenna. In [35], results of studies showed a dual-band of the antenna was significantly affected by a hand placing on the top of the radiator with 1 cm gap between them. Here, the performances of the antenna with the hand in three different positions, positions 1, 2 and 3, as shown in Fig. 12(a)–(c), respectively, are studied. In position 1, the hand (palm) is placed 1 mm above the top of the radiator and the fingers are touching the ground plane on the other side. The results in Fig. 13(a) show that the first and the third resonances at 2.09 and 5 GHz remain about the same. The second resonance at 3.74 GHz slightly moves to 3.69 GHz. The return losses of the three bands are still greater , which is different from the results than 6 dB reported in [35]. In position 2, where the hand is at the centre of the antenna and the palm of the hand partially covering the

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Fig. 13. Effect of different positions of user’s hand on (a) the reflection coefficient (S ), (b) gain, (c) radiation efficiency.

radiator with 1 mm gap between them as shown in Fig. 12(b), the results in Fig. 13(a) show that the three resonant frequencies again remain about the same. In position 3, where the hand model is relatively far from the radiator as shown in Fig. 12(c), the three resonances again remain about the same, as indicated in Fig. 13(a). The simulated results on the gain and radiation efficiency for the three different positions are shown in Fig. 13(b)–(c). It can be seen that, when the hand moves closer to the radiator, the gain drops by almost 0.7 dBi compared with that when the antenna in free space. The radiation efficiency drops to 63% when the hand moves closer to the radiator. These results indicate that, in these 3 positions, the performance of the antenna is not very much sensitive to the user’s hand. The best position, in terms of maximum efficiency and gain, is when the user is holding the mobile phone from the bottom, i.e., position 3. When the user’s hand gets closer to the radiator, the gain and the radiation efficiency drop slightly compared with the case when the antenna is in free space. Even for the worst case scenario where the user’s hand is completely or partly covering the radiator with 1 mm gap between them, as in positions 1 and 2, the simulation results show that it still can attain above 60% efficiency which is considered quite acceptable for mobile phone applications unlike the design reported in [33] where the radiation efficiency is about 28% at 1795 MHz when the hand is covering the radiator. The gain only drops by 2.2 dBi in the first band of 2.09 GHz and by smaller amounts in the other bands. VII. CONCLUSIONS The design of a compact multiband PIFA having independent controls of the resonant bands for UMTS, m-WiMAX and 5 GHz WLAN over a wide range have been presented and proposed. The key controlling parameters of the antenna have been studied using the current distributions on the radiator. The antenna has a small size and is thin, making it a good choice for modern mobile phones. Simulation and measurement results have shown good performances in terms of return loss, gain, radiation efficiency and radiation. The ground plane of the antenna has minimal effects on the antenna performance and the performance is not too sensitive to the human hand and the mobile phone housing used in the studies. This feature allows the nearby electronic components to be placed closed to the antenna, making the overall size the mobile phones even more compact and thin.

REFERENCES [1] Z. Ying, “Some important antenna innovations in the mobile terminal industry in the last decade,” in Proc. 1st Eur. Conf. on Antennas and Propagation EuCAP, Nov. 6–10, 2006, pp. 1–5. [2] P. S. Hall, E. Lee, and C. T. P. Song, “Planar inverted-F antennas,” in UPrinted Antennas for Wireless Communications, R. Waterhouse, Ed. Hoboken, NJ: Wiley, 2007, ch. 7. [3] T. G. Moore, “Multiband PIFA Antenna for Portable Devices,” U.S. patent application number: 9/814,171 publication number: US 2002/ 0135521-A1, Mar. 21, 2001. [4] L. Duixian and B. Gaucher, “A branched inverted-F antenna for dual band WLAN applications,” in Proc. IEEE Int. Symp. Antennas and Propagation Society, Jun. 2004, vol. 3, pp. 2623–2626. [5] J. Byun, J. Jo, and B. Lee, “Compact dual-band diversity antenna for mobile handset applications,” Microw. Opt. Technol. Lett., vol. 50, no. 10, pp. 2600–2604, 2008. [6] Y. Ge, K. P. Esselle, and T. S. Bird, “Compact diversity antenna for wireless devices,” Electron Lett., vol. 41, pp. 52–53, 2005. [7] C. H. See, R. A. Abd-Alhameed, D. Zhou, and P. S. Excell, “Dualfrequency Planar Inverted F-L-Antenna (PIFLA) for WLAN and short range communication systems,” IEEE Trans. Antennas Propag., vol. 56, no. 10, pp. 3318–3320, Oct. 2008. [8] A. Azremi, N. Saidatul, R. Ahmad, and P. Soh, “A parametric study of broadband planar inverted F antenna (PIFA) for WLAN application,” in Proc. Int. Conf. on Electronic Design ICED, Dec. 1–3, 2008, pp. 1–6. [9] D. Nashaat, H. Elsadek, and H. Ghali, “Dual-band reduced size PIFA antenna with U-slot for Bluetooth and WLAN applications,” in Proc. IEEE Int. Symp. Antennas and Propagation Society, June 22–27, 2003, vol. 2, pp. 962–965, vol.2. [10] Y. J. Cho, Y. S. Shin, and S. O. Park, “Internal PIFA for 2.4/5 GHz WLAN applications,” Electron. Lett., vol. 42, pp. 8–10, 2006. [11] P. Salonen, M. Keskilammi, and M. Kivikoski, “New slot configurations for dual-band planar inverted-F antenna,” Microw. Opt. Technol. Lett., vol. 28, no. 5, pp. 293–298, Mar. 5, 2001. [12] C.-Y.-D. Sim, “Dual and triple-band PIFA design for WLAN applications,” Microw. Opt. Technol. Lett., vol. 49, no. 9, pp. 2159–2162, 2007. [13] P. W. Chan, H. Wong, and E. Yung, “Wideband planar inverted-F antenna with meandering shorting strip,” Electron. Lett., vol. 44, no. 6, pp. 395–396, March 13, 2008. [14] P. W. Chan, H. Wong, and E. Yung, “Dual-band printed inverted-F antenna for DCS, 2.4 GHz WLAN applications,” in Proc. Loughborough Antennas and Propagation Conf., LAPC, Mar. 17–18, 2008, pp. 185–188. [15] W. P. Dou and Y. W. M. Chia, “Novel meandered planar inverted-F antenna for triple frequency operation,” Microw. Opt. Technol. Lett., vol. 27, pp. 58–60, 2000. [16] F.-R. Hsiao, H.-T. Chen, T.-W. Chio1, G.-Y. Lee, and K.-L. Wong, “A dual-band planar inverted-F patch antenna with a branch-line slit,” Microw. Opt. Technol. Lett., vol. 32, no. 4, pp. 310–312, Feb. 20, 2002. [17] Z. Ying and A. Dahlstroem, “Compact Broadband Antenna,” U.S. patent 6650294 B2, Nov. 18, 2003. [18] M. C. Huynh and W. Stutzman, “Ground plane effects on planar inverted-F antenna (PIFA) performance,” lEE Proc. Microw. Antennas Propag., vol. 150, no. 4, pp. 209–213, Aug. 2003. [19] Y. Chi and K. Wong, “Internal compact dual-band printed loop antenna for mobile phone application,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1457–1462, May 2007. [20] Y. Huang and K. Boyle, Antennas: From Theory to Practice. Hoboken, NJ: Wiley, 2008.

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[21] K. Wong, S. Su, C. Tang, and S. Yeh, “Internal shorted patch antenna for a UMTS folder-type mobile phone,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3391–3394, Oct. 2005. [22] Z. Chen, T. S. See, and X. Qing, “Small printed ultrawideband antenna with reduced ground plane effect,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 383–388, Feb. 2007. [23] S. R. Best, “The significance of ground-plane size and antenna location in establishing the performance of ground-plane-dependent antennas,” IEEE Antennas Propag. Mag., vol. 51, no. 6, pp. 29–43, Dec. 2009. [24] D. Kim, J. Lee, C. Cho, and T. Lee, “Design of a compact tri-band PIFA based on independent control of the resonant frequencies,” IEEE Trans. Antennas Propag., vol. 56, pp. 1428–1436, 2008. [25] J. H. Lim, G. T. Back, Y. I. Ko, C. W. Song, and T. Y. Yun, “A reconfigurable PIFA using a switchable PIN-diode and a fine-tuning varactor for USPCS/WCDMA/m-WiMAX/WLAN,” IEEE Trans. Antennas Propag., vol. 58, no. 7, pp. 2404–2411, Jul. 2010. [26] H. F. AbuTarboush, R. Nilavalan, D. Budimir, and H. S. Al-Raweshidy, “Double U-slots patch antenna for tri-band wireless systems,” Int. J. RF Microw. Comput.-Aided Engineering, vol. 20, no. 3, pp. 279–285, 2010. [27] Z. Ying, “Multi Frequency-Band Antenna,” Singapore Patent WO01/91 233, May 2001. [28] R. Sujith, V. Deepu, D. Laila, C. Aanandan, K. Vasudevan, and P. Mohanan, “A compact dual-band modified T-shaped CPW-fed monopole antenna,” Microw. Opt. Technol. Lett., vol. 51, no. 4, pp. 937–939, 2009. [29] S. Lee, H. Park, S. Hong, and J. Choi, “Design of a multiband antenna using a planner inverted-F structure,” in Proc. 9th Int. Conf. on Advanced Communication Technol., Feb. 2007, vol. 3, pp. 1665–1668. [30] M. Abedin and M. Ali, “Modifying the ground plane and its effect on planar inverted-F antennas (PIFAs) for mobile phone handsets,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 226–229, 2003. [31] [Online]. Available: http://www.satimo.com/ [32] C. A. Balanis, Antenna Theory—Analysis and Design, 3rd ed. Hoboken, NJ: Wiley, 2005. [33] Y.-W. Chi and K.-L. Wong, “Internal compact dual-band printed loop antenna for mobile phone application,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1457–1462, 2007. [34] D. Zhou, R. A. Abd-Alhameed, C. H. See, A. G. Alhaddad, and P. S. Excell, “Compact wideband balanced antenna for mobile handsets,” IET Microw. Antennas Propag. J., vol. 4, pp. 600–608, 2010. [35] Z. D. Liu, P. S. Hall, and D. Wake, “Dual-frequency planar inverted-F antenna,” IEEE Trans. Antennas Propag., vol. 45, no. 10, pp. 1451–1458, Oct. 1997. [36] Z. Ying and A. Dahlstroem, “Multi Frequency Band Antenna for Mobile Telephone,” Singapore patent No. WO200191233-A; EP1168491-A; WO200191233-A1.

Hattan F. AbuTarboush (M’07) received the B.Sc. (Eng) degree in electrical communications and electronics engineering from Greenwich and MSA University, London, U.K., in 2005 and the M.Sc. degree in mobile personal and satellite communication from Westminster University, London, in 2007. His master’s thesis was about the design of integrated antenna-filter for mobile WiMAX applications. Currently, he is pursuing the Ph.D. degree at Brunel University, West London, U.K. He has published several journal articles and conference papers. His current research interests lie in the design of reconfigurable antennas, antennas for mobile phones, miniaturized antennas, multiple antennas, smart antennas, antenna arrays, EBG and RF/microwave circuit design. He is a member of the IET and a reviewer for several journals and international conferences.

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R. Nilavalan (M’05–SM’10) received the B.Sc. Eng. degree in electrical and electronics engineering from University of Peradeniya, Sri Lanka, in 1995 and the Ph.D. degree in radio frequency systems from the University of Bristol, Bristol, U.K., in 2001. From 1999 to 2005, he was a Researcher at the Centre for Communications Research (CCR), University of Bristol, where his research involved theoretical and practical analyses of post reception synthetic focussing concepts for near-field imaging and research on numerical FDTD techniques. Since 2005, he has been with the Electronics and Computer Engineering Department, Brunel University, West London, U.K., where he is currently a Lecturer in wireless communications. His main research interests include antennas and propagation, microwave circuit designs, numerical electromagnetic modelling and digital video broadcast techniques. He has published over 70 papers and articles in international conferences and journals in his research area. Dr. Nilavalan was a member of the European commission, Network of Excellence on Antennas (2002–2005) and a member of the IET.

Thomas Peter received the M. Eng. degree in electrical engineering from the University of Technology Malaysia (UTM), in 2007. He is currently pursuing the Ph.D. degree at Brunel University, West London, U.K. His current research interest include UWB antennas and communications, semi-planar antennas for high gain, body centric antennas, antennas for green technology, and low detection antennas for stealth. From January to March 2011, he was a Visiting Researcher at the University of Hong Kong developing transparent green antennas for UWB applications as part of a collaborative research effort. Mr. Thomas was awarded a VC’s travel prize by the Graduate School of Brunel University in November 2010 to present his research paper on the development of a Green UWB antenna at the ISAP2010 conference in Macau.

S. W. Cheung (SM’98) received the B.Sc. degree (with First Class Honours) in electrical and electronic engineering from Middlesex University, U.K., in 1982 and the Ph.D. degree from Loughborough University of Technology, U.K., in 1986. From 1982 to 1986, he was a Research Assistant in the Department of Electronic and Electrical Engineering, Loughborough University of Technology, where he collaborated with Rutherford Appleton Laboratory and many U.K. universities to develop a project for new generations of satellite systems. From 1986 and 1988, he was a Postdoctoral Research Assistant with the Communications Research Group, King’s College, London University, working on research for future generations of satellite systems. In 1988, he joined the Radio and Satellite Communications Division, British Telecom Research Laboratories (now British Telecom Laboratories), as an Assistant Executive Engineer. He is an Associate Professor at the University of Hong Kong. His current research interests include 2G, 3G and 4G mobile communications systems, antenna designs, MIMO systems and satellite communications systems, predistortion of high power amplifiers and e-learning. He has published over 120 technical papers in international journals and conferences in these areas. Dr. Cheung has served as reviewer for different international journals and conferences in the areas of mobile communications, antennas and propagation. Currently, he is Chairman of the IEEE Hong Kong Joint Chapter on Circuits and Systems and Communications.

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Rectenna Application of Miniaturized Implantable Antenna Design for Triple-Band Biotelemetry Communication Fu-Jhuan Huang, Chien-Ming Lee, Chia-Lin Chang, Liang-Kai Chen, Tzong-Chee Yo, and Ching-Hsing Luo

Abstract—A novel antenna design that effectively covers three bands (the medical implant communications service (MICS) band at 402 MHz, and the industrial, scientific, and medical (ISM) band at 433 MHz and 2.45 GHz) using a -shaped radiator with a stacked and spiral structure is described. The antenna has compact size of 254 mm 3 (10 mm by 10 mm by 2.54 mm). The proposed design is effective for triple-band biotelemetry with data telemetry (402 MHz), wireless powering transmission (433 MHz), and wake-up controller (2.45 GHz). An experimental prototype of the compact stacked rectenna was fabricated on a Roger 3210 substrate. This antenna was used in a rectenna (rectifying antenna) for 433 MHz wireless powering transmission, and provided a conversion efficiency of 86% when 11 dBm microwave power was received at 433 MHz with a 5 k load. The optimal antenna was fabricated and tested in a minced front leg of pork. The simulated and measured bandwidths were 86 MHz and 113 MHz in the MICS band, and 60 MHz and 70 MHz in the ISM band, respectively. Index Terms—Biomedical telemetry, implantable antenna, rectifying antenna (rectenna), triple-band biotelemetry.

I. INTRODUCTION

B

IOTELEMETRY provides wireless communication from outside the body to inside it, or vice versa. Higher-frequency telemetry links are being developed for medical implants. For cardiac telemetry, dipole antenna [1] and spiral or serpentine micro-strips antenna [2], [3], as well as circumferential antenna [4], and waffle-type antenna [5] have been designed for implantation into the shoulder. Implantable antennas for biomedical implants have advanced rapidly recently [6]–[9]. The miniaturization of biomedical implants is limited by the size of the antenna. Power conservation is critical to the long-term continuous monitoring of implants. To improve and extend the lifetime of an implantable device, Zarlink developed a transceiver with dual-band operation (MICS and ISM band), an ultra-low power sleep mode [10], and a dual-band implantable antenna [11]. In the normal mode, when fully powered, the device transmits and receives data in

Manuscript received November 30, 2009; revised November 01, 2010; accepted April 04, 2011. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported in part by the NCKU-Delta Cooperation Research Project (D04) and in part by the Taiwan National Science Council (Contract 97-2220-E-006-008). The authors are with the Department of Electrical Engineering, National Cheng Kung University, Tainan 701, Taiwan (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.2011.2152317

the MICS band; in sleep mode, it listens for a wake-up signal at the ISM receiver with a power consumption of 1 uW. In order to develop a test environment for the dual band operation, DGBE (diethylene glycol butyl ether) and rat skin were proposed for [12]–[14] for mimicking skin tissue. This dual-mode operation in separate bands greatly improves the lifetime of the battery. To increase its lifetime further, in this investigation, a rectifying circuit is integrated into the proposed antenna to form a rectenna for wireless power transmission for biomedical implant applications. A smaller antenna with a stacked and spiral structure, which supports triple-band operations, was developed. The details of the antenna design will be discussed in Section II. For wireless power transmission from external wireless stations, a double rectifier was integrated into the back side of the antenna as a rectenna that provides a high output voltage and a very high RF-to-DC conversion efficiency to supply electrical power to implants [15]. To confirm the performance of the proposed antenna, the antenna was implanted in a test tissue (a minced front leg of pork). This approximate approach significantly reduced the number of steps required to prepare skin or muscle mimicking gels and saved time. Measurements revealed that this antenna has a wide impedance bandwidth (113 MHz at MICS covering 433 MHz, 70 MHz at ISM) at a return loss of 10 dB. It has a maximum gain of 7 dB at 402 MHz, 11 dB at 433 MHz and 15 dB at 2.45 GHz. The simulated 1-g averaged specific absorption rate (SAR) satisfies the limits set by the American National Standards Institute (ANSI/IEEE) in each band for triple-band implantable biotelemetry [16]. II. MINIATURIZED IMPLANTABLE ANTENNA DESIGN AND RECTIFIER The proposed antenna was constructed and implanted into and conductivity a minced pork with permittivity at low frequency (402 MHz and 433 MHz), and conductivity and permittivity at high frequency (2450 MHz). Its geometry and characteristics are discussed below. A. Antenna Design Fig. 1 displays the geometry of the proposed implantable antenna, which has dimensions of 254 mm 3 (10 mm by 10 mm by 2.54 mm) and consists four layers. Layer 1 is a ground plane;

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Fig. 1. The geometry of the proposed triple-band implantable antenna.

layer 2, layer 3, and layer 4 are the radiating elements of the antenna. Using the -shape with two meandered strips, the PIFA structure can achieve dual-resonance over a large bandwidth [17], [18]. Furthermore, in order to reduce the size of antenna, it was designed with stacked and spiral structures. To investigate the characteristics of the proposed antenna, the detailed current distributions are plotted in Fig. 2. In Fig. 2(a) and (b), the proposed antenna has two fundamental resonant frequencies that correspond to the two meandering and stacked strips. The first and second resonant frequencies are determined in the open loop mode (Fig. 2(a)) and PIFA mode (Fig. 2(b)), respectively. The comb structure in layer 4 was added to increase capacitive coupling of the open loop mode and reduce the effective length of the current path. In addition to these two low-frequency bands (402 MHz and 433 MHz), here, a third resonant frequency, which is used to trigger the wake-up circuit in the 2.45 GHz ISM band, is required. Since ignoring the resonant structure of the third resonant frequency enables the size of antenna to be effectively reduced, the ISM band (2.45 GHz) was realized by exciting a harmonic mode in a meander-strips structure, whose current distribution is shown in Fig. 2(c). Fig. 2(a), (b), and (c) also plot the current distributions on the ground plane at different frequencies. Notably, the currents on the radiator induce inverse currents on the ground plane, so the currents on the radiator and the ground plane are anti-phase. In MICS band operation, the open loop and the PIFA modes are controlled to excite dual resonance frequencies of 400 MHz and 433 MHz, respectively. The third resonant frequency (harmonic mode) is controlled at 2.45 GHz by varying the widths of each meander strip to increase or reduce the effective length of the current path. The parameters were optimized using Ansoft’s

Fig. 2. The current distributions of the proposed antenna at 402 MHz, 433 MHz and 2450 MHz.

high-frequency structure simulator (HFSS) software to yield a broad bandwidth of 86 MHz (381–467 MHz) in the MICS band

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Fig. 3. Configurations for integrating the doubler rectifier as a rectenna.

Fig. 4. Minced pork and measurement setup.

and a bandwidth of 60 MHz (2.42–2.48 GHz) in the 2.45 GHz ISM band, during the simulation. A prototype of this antenna was fabricated on a Roger 3210 substrate with a thickness of 0.635 mm and a dielectric constant of 10.2.

B. Rectifier Once received, the RF signal is converted to dc power by means of a high-frequency rectifier circuit. Power conversion efficiency (PCE) was the figure of merit used in evaluating the rectifier designs. Multiple stages were implemented to further increase the output voltage to maximize the charge delivered to the load resistor. Although increasing the number of stages increases the output voltage, it decreases efficiency. The rectifier was composed of dual microwave Schottky detector (HSMS286c) diodes, a DC block capacitor C1, a DC filter capacitor C2, and a 50 impedance matching element, L1, for powering the DC load RL, shown in Fig. 3. The rectifier was placed on the back side of the proposed antenna. Schottky diodes were used for the RF rectifier because of their excellent high frequency performance and low forward-bias voltage. This low forward voltage drop (150 mV–300 mV) supports rectification at low input voltage biases, which favors low power operation. The rectifier was designed with an ADS EM/Circuit co-simulation feature that was connected with micro-strip line components. Notably, the high output dc voltage, generated by the doubler, reduced the junction capacitance of the diode, further improving the RF-to-DC conversion efficiency [19].

Fig. 5. The comparison of permittivity and conductivity versus frequency of the test tissues (pork) with the reference skin and muscle. (a) Permittivity; (b) conductivity.

III. THE ELECTRICAL PROPERTIES OF MINCED PORK Since the implantable antenna must be embedded into test tissue to verify its practical effectiveness, the characteristics of simulated human fluid must be mapped accurately to imitate standard human tissue. In a recent study, simulated skin fluids were developed by simply varying the concentrations of alcohol and salt [7]. Although this approach can easily yield the permittivity and conductivity of skin, it is not suitable for preparing test tissue for multi-band application. Here minced pork was adopted to provide an easy approach to mimicking the environment of multi-band applications and to verify the characteristics of the proposed antenna. In this case, the test tissue was a minced front leg of pork. Agilent’s 85070E dielectric probe kit and an 8753E network analyzer were used to make dielectric measurements, and measurements were made of the test tissue between 100 MHz and 3 GHz. The dimensions of the test tissue were 65 mm by 92 mm by 50 mm. Fig. 4 shows the minced pork used in these experiments for dielectric measurement. Fig. 5 compares the permittivity and conductivity of the test tissue (pork) with that of reference skin and muscle [20]–[22]. The properties of the minced pork correspond to that of human skin and muscle between 100 MHz and 3 GHz, therefore it is suitable for verifying the multi-band implantable antenna design. Considering the differences between the tissues we also simulated the proposed antenna in

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Fig. 6. The fabricated triple-band miniaturized antenna.

Fig. 7. The implantable antenna embedded in minced pork.

skin, muscle, and pork tissues. We discuss the results in next section. IV. RESULTS AND DISCUSSION Fig. 6 shows the fabricated triple-band miniaturized antenna, and its layers. To test the antenna in vitro, the antenna was first placed on the base of a plastic container, to which the musclemimicking material was then added. Fig. 7 shows the antenna embedded in the minced pork and the measurement setup. Fig. 8 plots the simulated and measured S11 of the proposed antenna implanted into the different tissues (skin, muscle and pork). Clearly, even though the proposed antenna was implanted into skin and muscle, the S11 was still met the 10 dB requirement for the MICS and ISM bands. Thus, the characteristics of the proposed antenna remained consistent even in severe biological environments, such as various tissues with complex structures. Hence, the resonant frequency of proposed antenna at 402 MHz and 433 MHz was controlled using the open loop and PIFA modes, respectively. In Fig. 9, changing the gap “g” of the comb structure changes the coupling capacitance. Accordingly, the resonant frequency that is excited in open loop mode can be adjusted. Good agreement exists between measurements taken in the MICS band with a 113 MHz (402–515 MHz) bandwidth and those taken in the ISM band with a 70 MHz (2.43–2.5 GHz) bandwidth. Fig. 10(a), (b), and (c) present the simulated three-dimensional far-field gain patterns and 1-g average SAR distribution

Fig. 8. (a) low frequency band (402 MHz and 433 MHz), (b) high frequency band (2.45 GHz), (c) measured and simulated S using minced pork.

at 402 MHz, 433 MHz, and 2.4 GHz for the proposed antenna implanted into muscle tissue, respectively. When the proposed antenna was assumed to deliver 1 W, the maximum SAR values at 402, 433 and 2450 MHz were 341, 320 and 382 W/kg, respectively. Therefore, the maximum power must be decreased to a suitable level (4.7 mW, 5 mW and 4.2 mW, respectively) to satisfy SAR regulations (1.6 W/kg) of ANSI/IEEE [16]. Table I compares the proposed antenna to those from previous studies [7]–[9], [11], [14], [17], [18]. This table demonstrates that the

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Fig. 9. Simulation S

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for parameter of “g”.

proposed antenna features a high gain, wide bandwidth, and small size compared to previous antenna designs. Fig. 11(a) displays the fabricated rectifier, which was integrated on the back of the antenna as a rectenna, as shown in Fig. 11(b). Actually, the rectenna will be coated with thin film gel (shown in Fig. 11(c)) for insulating it from the tissue. To evaluate the performance of the rectifier as a rectenna, the output DC power was obtained by measuring the output voltage drop across the load resistor. The RF-to-DC conversion efficiency is defined as

(1) is the output DC power; is the power received by where the rectifier; is the output DC voltage, and is the load resistance. The performance of the rectifier was measured as 0 dBm input power at 433 MHz at various loads, as plotted in Fig. 12(a). The input of the rectifier was connected to a microwave signal generator and its load was tuned to measure the output DC voltage. Owing to the trade off between the output DC voltage and the RF-to-DC conversion efficiency, the 5 k resistor load was selected as the load for conversion efficiency optimization. In the same manner, the power was directly fed to the rectifier; it was , as presented in Fig. 12(b). The swept with the load set to 5 power conversion efficiency for both implementations is plotted for input powers from 15 dBm to 15 dBm. As estimated, the optimal RF-DC conversion efficiency of 86% was obtained with an 11 dBm input RF power. At the maximum available power, a DC voltage of higher than 7 V was generated at 15 dBm and efficiency fading at input powers greater than 11 dBm was caused by the fact that the output DC voltage exceeded the breakdown voltage of the diodes. To achieve a comparison and quantify the effects of implantation, the performance across free space was first evaluated by placing the rectenna into the minced pork and positioning the transmitting antenna at distances of 50 and 100 cm in front of the device. The block diagram and experiment setup are shown in Fig. 13. The setup consisted of the transmitting antenna (Moxon antenna) with high directivity, RF and dc testing instruments,

Fig. 10. Simulated three-dimension gain patterns and the 1-g average SAR distribution for the proposed implantable rectenna were delivered power = 1 W . (a) 402 MHz, SAR calibration line was positioned at Y = 0 mm. (b) 433 MHz, SAR calibration line was positioned at Y = 0 mm. (c) 2.45 GHz, SAR calibration line was positioned at Y = 0 mm.

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TABLE I PERFORMANCES COMPARISON OF THE TRIPLE-BAND IMPLANTABLE ANTENNA DESIGN

Fig. 11. (a) Fabricated rectifier and (b) the top and bottom view of the rectenna. (c) With thin film gel coating.

and the minced pork. The power transmission efficiency (PTE) is defined as

(2) is the output DC power; is the transmitting where power; and are the output DC voltage and the load resistance of rectenna, respectively. The results plotted in Fig. 14 summarize the comparison of varying distances. The measured results show that in an implanted setting with the Moxon antenna placed 50 cm and 100 cm away from the device, the rectenna can produce 7.37 V and resistor using a large trans3.4 V, respectively, cross a 5 mitted power of 44 dBm. The power transmission efficiency is quit poor due to the free space loss and low antenna gain. V. CONCLUSION This study describes a miniaturized implantable rectenna with a spiral and stacked structure and a 113 MHz bandwidth (402–515 MHz) at frequencies of 402 MHz and 433 MHz, and a 70 MHz bandwidth (2.43–2.5 GHz) at a frequency of 2.45 GHz. By varying the width of each meander strip and the gap “g”, the antenna can be made to excite three resonant frequencies and broad bandwidth. The proposed antenna structure is very suitable for use in implantable devices used in multi-band biotelemetry. Moreover, minced pork was used as a test tissue to confirm the properties of the multi-band implantable antenna

Fig. 12. (a) Measured DC voltage and conversion efficiency with different load values at 0 dBm input power. (b) Measured DC voltage and conversion efficiency by sweeping input power at 5 k load resistance.

bypassing the need for the complicated formulation of simulated tissue liquid for multi-band application. The results also reveal that the proposed antenna satisfies the 1-g average SAR regulation set by ANSI/IEEE in the MICS and ISM bands. The rectifier also exhibits high output DC and RF-DC conversion efficiency at various loads and power values. The optimal RF-DC conversion efficiency is 86% at 11 dBm. The rectifier

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Fig. 13. (a) Block diagram and (b) experiment setup with a transmit antenna fed by a signal generator through a power amplifier, and a digital multi-meter to measure the DC output voltage.

Fig. 14. Comparisons of the output dc voltages of the rectenna and power transmission efficiencies at distance 50 cm and 100 cm.

[4] J. Johansson, “Wireless Communication With Medical Implants: Antennas and Propagations,” Ph.D. dissertation, Lund Univ., Switzerland, 2004, 1402-8662. [5] K. Gosalia, J. Weiland, M. Humayun, and G. Lazzi, “Thermal elevation in the human eye and head due to the operation of a retinal prosthesis,” IEEE Trans. Biomed.l Eng., vol. 51, no. 8, Aug. 2004. [6] P. Soontornpipit, “Design of Implantable Antennas for Communication With Medical Implants,” M.S. thesis, Dept. Elect. Comput. Eng., Utah State Univ, Logan, 2002. [7] C. M. Lee, T. C. Yo, C.-H. Luo, C. H. Tu, and T. Z. Juang, “Compact broadband stacked implantable antenna for biotelemetry with medical devices,” Electron Lett., vol. 43, pp. 660–662, 2007. [8] W. C. Liu, F. M. Yeh, and M. Ghavami, “Miniaturized implantable broadband antenna for biotelemetry communication,” Microwave Opt Technol Lett., vol. 50, pp. 2407–2409, 2008. [9] W. C. Liu, S. H. Chen, and C. M. Wu, “Bandwidth enhancement and size reduction of an implantable PIFA antenna for biotelemetry devices,” Microwave Opt Technol Lett., vol. 51, no. 3, pp. 755–757, 2009. [10] Medical Implantable RF Transceiver ZL70101 Data Sheet Zarlink Semiconduct.. Ottawa, ON, Canada, Oct. 2006. [11] T. Karacolak, A. Z. Hood, and E. Topsakal, “Design of a dual band implantable antenna and development of skin mimicking gels for continuous glucose monitoring,” IEEE Trans. MTT, vol. 56, no. 4, pp. 1001–1008, Apr. 2008. [12] T. Yilmaz, T. Karacolak, and E. Topsakal, “Characterization and testing of a skin mimicking material for implantable antennas operating at ISM band (2.4 GHz–2.48 GHz),” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 418–420, 2008. [13] T. Karacolak and E. Topsakal, “Electrical properties of nude rat skin and design of implantable antennas for wireless data telemetry,” IEEE MTT-S Int., pp. 907–910, 2008. [14] T. Karacolak, R. Cooper, and E. Topsakal, “Electrical properties of rat skin and design of implantable antennas for medical wireless telemetry,” IEEE Trans. AP, vol. 57, no. 9, pp. 2806–2812, Sep. 2009. [15] T. C. Yo, C. M. Lee, C. M. Hsu, and C. H. Luo, “Compact circularly polarized rectenna with unbalanced circular slots,” IEEE Trans. AP, vol. 56, no. 3, pp. 882–886, Mar. 2008. [16] IEEE Standard for Safety Levels With Respect to Human Exposure to Radio Frequency Electromagnetic Fields, 3 KHz to 300 GHz, IEEE Standard C95.1-1999, 1999. [17] C. M. Lee, T. C. Yo, F. J. Huang, and C. H. Luo, “Dual-resonant  -shape with double L-strips PIFA for implantable biotelemetry,” Electron. Lett., vol. 44, pp. 837–838, 2008. [18] C. M. Lee, T. C. Yo, F. J. Huang, and C. H. Luo, “Bandwidth enhancement of planar inverted-F antenna for implantable biotelemetry,” Microwave Opt Technol Lett., vol. 51, pp. 749–752, 2009. [19] J. O. Mcspadden, T. Yoo, and K. Chang, “Theoretical and experimental investigation of rectenna element for microwave power transmission,” IEEE Trans. MTT, vol. 40, no. 12, pp. 2359–2366, Dec. 1992. [20] C. Gabriel, S. Gabriel, and E. Corthout, “The dielectric properties of biological tissues: I. Literature survey,” Phys. Med. Biol., vol. 41, pp. 2231–2249, 1996. [21] 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. [22] S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues,” Phys. Med. Biol., vol. 41, pp. 2271–2293, 1996.

exhibits favorable impedance matching with the proposed antenna with the optimal frequency response at 433 MHz.

REFERENCES [1] L. Griffiths, “Analysis of Wire Antennas for Implantation in the Body,” Master’s thesis, Utah State Univ., Logan, UT, 2002. [2] P. Soontornpipit, C. M. Furse, and Y. C. Chung, “Design of implantable microstrip antennas for communication with medical implants,” IEEE Trans. MTT, vol. 52, no. 8, pp. 1944–1951, Aug. 2004. [3] J. Kim and Y. Rahmat-Samii, “Implanted antennas inside a human body: Simulations, designs, and characterizations,” IEEE Trans. MTT, vol. 52, no. 8, pp. 1934–1943, Aug. 2004.

Fu-Jhuan Huang was born in Kaohsiung, Taiwan, R.O.C., in 1984. He received the B.S. degree in electronic engineering from the National Chin-Yi University of Technology, Taichung, Taiwan, in 2006 and the M.S. degree from National Cheng Kung University, Tainan, Taiwan, in 2008, where he is currently working toward the Ph.D. degree. His research interests include RF circuits, micro-strip antenna, rectenna, and wireless charging system, and also the planar antennas for implantable biotelemetry.

HUANG et al.: RECTENNA APPLICATION OF MINIATURIZED IMPLANTABLE ANTENNA DESIGN

Chien-Ming Lee was born in Taichung, Taiwan, in 1982. He received the B.S. degree in aeronautical engineering from National Formosa University, Yunlin, Taiwan, in 2004, and the Ph.D. degree from National Cheng Kung University, Tainan, Taiwan, in 2008. During 2008, he was a Visiting Researcher with the Radio Frequency Circuits and Systems Research Group, University of Florida, Gainesville. Since 2009, he has been with the Antenna and Wireless Integration Department, HTC Corporation, Xindian, Taiwan, as a Senior Engineer. His research interests include planar antennas for wireless communications, especially in the planar antennas for implantable biotelemetry, mobile phones, wireless local area networks, and ultra-wideband applications, and also in microwave passive components and RF circuit design.

Chia-Lin Chang was born in Tainan, Taiwan, in 1982. He received the B.S. degree in electrical engineering from National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan, in 2005 and the M.S. degree in electrical engineering from National Cheng Kung University, Tainan, Taiwan, in 2007, where he is currently working toward the Ph.D. degree. His research field includes biotelemetry device and chip design, biomedical wireless communication and analog IC design.

Liang-Kai Chen was born in Kaohsiung, Taiwan, R.O.C., in 1982. He received the B.S. degree in electronic engineering from Southern Taiwan University, Tainan, Taiwan, in 2004 and the M.S. degree from Southern Taiwan University, Tainan, Taiwan, in 2006. He is currently working toward the Ph.D. degree at National Cheng Kung University, Tainan, Taiwan, R.O.C. His research interests include antenna for implantable biotelemetry, UHF RFID antenna, micro-strip antenna, and also research including, WWAN, WLAN, WiMAX, UWB band for Notebook wireless system.

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Tzong-Chee Yo received the B.S., M.S., and Ph.D. degrees in electrical engineering from National Cheng Kung University, Tainan, Taiwan, R.O.C., in 2001, 2003 and 2008, respectively. He currently works at National Synchrotron Radiation Research Center, HsinChu, Taiwan, R.O.C. as a Postdoctoral Researcher. His research interests include high power RF amplifier design, RF circuits and devices, microstrip antenna, rectenna, and wireless charging system.

Ching-Hsing Luo received the B.S. degree in electrophysics from National Chaio Tung University, Hsinchu, Taiwan, R.O.C., the M.S. degree in electrical engineering from National Taiwan University, in 1982, the M.S. degree in biomedical engineering from The Johns Hopkins University, Baltimore, MD, in 1987, and the Ph.D. degree in biomedical engineering from Case Western Reserve University, Cleveland, OH, in 1991. He is a Full Professor in the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., since 1996 and was honored as a Distinguished Professor in 2005. His research interests include biomedical instrumentation- on-achip, assistive tool implementation, cell modeling, signal processing, home automata, RFIC, gene chip, and quality engineering.

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Spherical Horn Array for Wideband Propagation Measurements Ondˇrej Franek, Member, IEEE, and Gert Frølund Pedersen

Abstract—A spherical array of horn antennas designed to obtain directional channel information and characteristics is introduced. A dual-polarized quad-ridged horn antenna with open flared boundaries and coaxial feeding for the frequency band 600 MHz–6 GHz is used as the element of the array. Matching properties and coupling between the elements are investigated via measurements and numerical simulations. Radiation patterns and sum beams of the array on selected frequencies throughout the band are also presented. Based on the obtained results it is concluded that the array is a good candidate for a wideband multipath propagation studies. Index Terms—Antenna arrays, antenna array mutual coupling, horn antennas.

I. INTRODUCTION N order to measure the propagation channel characteristics without antenna influence and extensive postprocessing, a spherical array of antennas is needed. The elements of the array should theoretically have medium gain, both polarizations, and cover wide frequency range. Also, in order to directly retrieve the directional channel information and to obtain sufficiently isotropic coverage, we need moderately narrow radiation beam covering the whole sphere. One possibility is to use a directional antenna with the desired characteristics on a pedestal, scanning the space to obtain directional information [1], but this solution cannot accommodate the velocities typically involved in mobile experiments. Fast and simultaneous measurements are, however, feasible with spherical arrays. A spherical array composed of microstrip patch antennas was introduced in [2], but the bandwidth offered by this type of antennas is generally poor. In our previous measurement campaign, we used a spherical array of monopoles [3] with the view of increasing the bandwidth. Although successful in some aspects, it did not allow a simple estimation of directional information because the elements of the array, monopoles, were not directional, instead having maximum radiation around the circumference. This made the determination of signal direction particularly difficult in terms of postprocessing. For the next measurement campaign we therefore decided to use horn antennas to compose the array.

I

Manuscript received August 18, 2010; revised November 08, 2010; accepted November 25, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported by the Danish Center for Scientific Computing. The authors are with the Antennas, Propagation and Radio Networking Section, Department of Electronic Systems, Aalborg University, DK-9220 Aalborg øst, Denmark (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.2011.2152349

Horn antennas are well known for being used as primary reflector feeds, in electromagnetic compatibility measurements and for standard gain calibration purposes [4]. Their main advantages are relatively wide bandwidth, high gain, and the possibility to accommodate both polarizations within one antenna. The bandwidth can be further enlarged by introducing ridges, which, together with symmetrical feeding, substantially extend the waveguide single-mode operation [5]. As our measurement campaigns were expected to require wideband measurements from 600 MHz to 6 GHz (i.e., LTE interfaces span from 700 MHz up to 2.7 GHz) as well as both polarizations, we decided to use the ridged design with dual polarization, i.e., quadrupleridged horn antenna. Double ridged horn antennas have been studied widely in the literature, for examples we refer to [6]–[9]. Publications on quad-ridged horns, on the other hand, are few. An early presentation of quad-ridged design, although very brief, can be found in [10]. A dual-polarized ridged horn with coaxial feeding and bandwidth from 2 to 26.5 GHz was introduced in [11]. Another original design was presented in [12], where the quadridged horn intended for anechoic chamber operation in 2–18 GHz band was stripped off the flare boundaries while maintaining very good radiation and matching parameters. Recently, two novel designs of quad-ridged horn antennas, with full flare boundaries though, were presented in [13] and [14], with respective bandwidths 8–18 GHz and 2–18 GHz and coaxial feeding. However, none of the papers addressed the possibility of using such antenna as an array element. In this paper, we introduce a spherical array of quad-ridged dual-polarized horn antennas with frequency range 600 MHz-6 GHz. Focus is given on aspects relevant to using this array for obtaining channel characteristics and directional information. In , coupling particular, we present reflection coefficient and radiation patterns. The data were obtained from anechoic chamber measurements and from simulation by the finite-difference time-domain (FDTD) method [15]. II. DESCRIPTION A. The Antenna The element of the proposed array is a diagonal horn antenna with four ridges added for simultaneous use of both vertical and horizontal polarizations (see Figs. 1 and 2). The initial design of the horn antenna has been prepared using simulations in our in-house FDTD code. Optimization has been performed by sweeping through many values for the feed position, the gap between the ridges, the thickness of the ridges, the chamfering angle near the feed, and the position of the shorting strip. The

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Fig. 1. The realization of the quad-ridged dual-polarized horn antenna. Fig. 3. Detail of the feed with two ridges present: (a) side-view, (b) cross-section. Arrangement of the perpendicular ridges is similar, only the feed is shifted 2 mm forward.

Fig. 2. Dimensions of the horn antenna.

optimization had strong influence on matching, while the radiation pattern has been found to be largely insensitive to small changes in the antenna geometry. Fine tuning of the parameters has been done using measurements on a prototype. The flare is designed with open boundary, but includes a shorting strip similar to [7] resulting from matching optimization; the strip is 20 mm wide at a distance of 80 mm from the transition waveguide. The antenna is 348 mm long, with an aperture size of 288 mm. The ridges are tapered approximating an exponential transition between the characteristic impedance of the feeding waveguide and free space. Note that the tapering of the ridges runs right from the feed in full length of the antenna. Towards the feed the ridges are chamfered to accommodate the very narrow gap resulting from tapering and to achieve higher bandwidth and lower impedance [5]. The antenna is fed by a coaxial probe—a coaxial cable passes throughout the ridge with outer conductor connected to it and the inner conductor is attached to the opposite ridge (see Fig. 3). The passage is filled with dielectric material and keeps the characteristic impedance at 50 Ohms. There are two probes arranged

Fig. 4. Simulations for various gaps between the ridges.

perpendicularly to each other for excitation of both polarizations. The probes are positioned 2 mm apart, counting for 14 degrees of phase shift at the upper end of the frequency range. The gap between the ridges at the feedpoint is one of the most crucial dimensions of the antenna. While the tapering length was kept constant, we eventually chose 1 mm as the optimum distance between the ridges at the feedpoint with respect to the re. We also found, however, that the tenflection coefficient dency of the reflection coefficient is generally not monotonic with the gap width (see Fig. 4 with FDTD simulations). This is in contrast to [16] and [17] which report that the matching improves with decreasing the gap between the ridges. Exact physical explanation remains unclear to us, but we hypothesize that the size of the feed may be the cause, since the diameter of the inner conductor of the coaxial probe is 1 mm, comparable to the gap itself.

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Fig. 6. Distribution and numbering of the horn antennas on the spherical array; (a) top view, (b) bottom view, (c) theta angles of the element rows. Fig. 5. The quad-ridged horn antennas in the spherical array arrangement.

Thickness of the ridges has been determined by numerical optimization as 8 mm. Reducing the overall thickness of the ridges negatively influences the performance of the antenna and also poses some practical difficulties, e.g., does not allow for the coaxial feeding through the ridges. However, large portion of the ridges has been milled out resulting in 2 mm thickness everywhere except a 10 mm wide frame at the edges—we did not find any significant influence of this modification. The benefit is reduced weight of the antenna, which is especially important when it is used as an array element. Another component of the antenna of which dimensions were numerically investigated was the cavity behind the feed, but again without noticing any considerable impact.

Fig. 7. Reflection coefficient (s ) of the quad-ridged horn antenna. Solid line: measurement, dashed line: simulation.

B. The Array The antenna array presented in this paper has 16 previously described horn antennas arranged in a quasi-spherical pattern (Fig. 5), with emphasis on the coverage of the upper hemisphere. The elements in the array (schematic in Fig. 6) are arranged into 4 groups (rows) with antennas sharing the same theta angle from the vertical axis in each group. The first group contains one element pointing directly upwards, the second group has 4 , the third group has 7 elements with elements with , pointing slightly above the horizon, and the fourth , pointing slightly below group has 4 elements with the horizon. The described constellation has been chosen seeking the highest possible homogeneity in obtaining the directional channel information, but also with respect to practical matters like mounting and mobility. In particular, the lower hemisphere is not covered entirely due to the presence of the mounting rod and also because of generally lower probability of signal incoming from the downward direction. Generally, we were searching for the highest possible distance between any two antenna elements in order to minimize the coupling. The pattern homogeneity was, however, also expected to be improved by

this approach. All the mounting pods for the antennas were arranged in such a way that the antennas are positioned on a sphere, i.e., with approximately equal distance from the geometrical center of the array. The homogeneity of the final array turned out to be satisfactory, as shown below in results. III. RESULTS Fig. 7 shows the parameter of the horn antenna across the intended operation frequency range, obtained by measurement and FDTD simulations. This plot is a result of extensive optimization on the shape of the antenna, as we tried to find dimensions giving maximum possible bandwidth, favorable matching and well-defined radiation beam, all at the same time. The relower than dB in most of the frequency range of sult is dB in few segments. However, there interest and lower than are notable differences between the measurement and the simulation which are caused by subtle mechanical changes on the final prototype (ridge gap, waveguide walls perforation) which could not be reflected in the simulation. Differences appeared also when comparing radiation patterns, therefore we decided

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Fig. 8. Measured mutual coupling (s parameter) between the two nearest neighboring antennas (11–15) with both polarizations (solid: horizontal, dashed: vertical).

Fig. 11. Measured radiation pattern for the antenna cluster in H- (solid line) and E-plane (dashed line), and for single antenna in H- (dash-dotted line) and E-plane (dotted line), at 2.3 GHz.

Fig. 9. Measured mutual coupling (s the same antenna.

Fig. 12. Measured radiation pattern for the antenna cluster in H- (solid line) and E-plane (dashed line), and for single antenna in H- (dash-dotted line) and E-plane (dotted line), at 4.5 GHz.

parameter) between the two ports of

Fig. 10. Measured radiation pattern for the antenna cluster in H- (solid line) and E-plane (dashed line), and for single antenna in H- (dash-dotted line) and E-plane (dotted line), at 776 MHz.

to rely entirely on measurement in obtaining the remaining antenna parameters.

One of the most important parameters of any antenna array is the amount of mutual coupling between the array elements. The results for the worst coupling in the array (approx. 40 apart) are shown in Fig. 8. The maximum coupling in the frequency band of interest occurs at the lower end with magnitude at most dB. Only the ports with the same orientation (H-H, V-V) are significantly coupled as shown in Fig. 8, whereas mixed polardB in all cases (not shown). ization coupling is less than It is the H-plane coupling (V-V) which turned out to be the strongest. The coupling between the two ports of the same antenna is shown in Fig. 9. The radiation pattern is quite insensitive to small changes in the geometry of the antenna. However, its properties change throughout the band as it is demonstrated for four selected frequencies: 776 MHz, 2.3 GHz, 4.5 GHz and 6 GHz (see Figs. 10–13). In this case, the measurement setup consisted of the measured element and four neighboring identical antennas tilted by 40 from the main axis (“antenna cluster”), in order to capture an image of the real-world performance of the antenna as a part of the array. Radiation patterns for the single antenna only are also included in Figs. 10–13 for comparison.

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Fig. 13. Measured radiation pattern for the antenna cluster in H- (solid line) and E-plane (dashed line), and for single antenna in H- (dash-dotted line) and E-plane (dotted line), at 6 GHz.

Fig. 14. Measured directivity versus frequency.

The directivity (directive gain) grows with frequency (Fig. 14) and the single beam becomes narrower (Fig. 15), while the front-to-back ratio improves. The beam is generally wider in the H-plane than in the E-plane. At the upper frequency limit (6 GHz, Fig. 13) the beam is still intact with a wide flat top in H-plane and mild sidelobes in E-plane. Still, some of the sidelobes occur throughout the entire band, but these have generally lower level and they are also pointing around 40 degrees or more from the main beam, suggesting that they are caused by the influence of the four neighboring antennas. Figs. 16–21 show the sum beams in -and -polarization of the whole spherical array composed of 16 horn antennas. The sum beam is obtained by summing power contributions from all of the antennas in the array and both polarization feeds. It shows how homogeneous the array will be in retrieving the directional channel information over the sphere. Each antenna is represented by a full 3D radiation pattern in both polarizations, including the influence of its four neighbors. These patterns are then rotated into the appropriate direction and added together. The plot center in Figs. 16–21 corresponds to the “north pole” of the spherical domain (radiation upwards) while the circumference corresponds to the downwards radiation, or “south

Fig. 15. Measured beamwidth (full-width half-maximum) in H- (solid line) and E-plane (dashed line) versus frequency.

Fig. 16. Sum beam of array (relative power to maximum),  -polarization, 776 MHz.

Fig. 17. Sum beam of array (relative power to maximum), -polarization, 776 MHz.

pole”. The horizontal radiation, or “the equator”, is denoted by a dashed line at half distance from the center to the circumference. From the schematic of the array (Fig. 6) it follows that the coverage in the downwards direction will be minimal, and this is indeed the case. The homogeneity is generally better when

FRANEK AND PEDERSEN: SPHERICAL HORN ARRAY FOR WIDEBAND PROPAGATION MEASUREMENTS

Fig. 18. Sum beam of array (relative power to maximum),  -polarization, 2.3 GHz.

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Fig. 21. Sum beam of array (relative power to maximum), -polarization, 6.0 GHz.

upper hemisphere (inner part of the circle), while at 2.3 GHz we have only 4 dB and for 6 GHz even 6 dB span due to sporadic dips. These numbers apply to a 16-element array. For 32 and 62 elements in the array (uniform distribution around the sphere, not displayed) the homogeneity would be better than 6 dB and 4 dB, respectively, for both hemispheres at 6 GHz, but then the array would have a larger diameter and corresponding weight. IV. CONCLUSION

Fig. 19. Sum beam of array (relative power to maximum), -polarization, 2.3 GHz.

In this paper, a spherical array with dual-polarized quad-ridged horn antenna as an element has been introduced. The 16-element array has an isotropy of max. 6 dB over the sum radiation pattern in the upper hemisphere within frequency range from 600 MHz to 6 GHz. The reflection coefficient of dB across the frequency range the horn antenna is below and the coupling between the elements is typically better than dB. The radiation pattern of a single antenna in the array shows a single well-defined beam with small sidelobes and good front-to-back ratio. The directivity reaches 12 dB for the most part of the band. Although there was a quite significant proximity effect of the neighboring antennas on the radiation pattern, this did not manifest in the coupling. We therefore conclude that the presented array is suitable for wireless propagation studies with potential for accurate estimation of signal direction. ACKNOWLEDGMENT The authors would like to thank the TAP reviewers and Dr. E. de Carvalho for their useful comments on the manuscript.

Fig. 20. Sum beam of array (relative power to maximum),  -polarization, 6.0 GHz.

the antenna beam is wider, as at the lower frequency (776 MHz, Figs. 16, 17), or when it has many secondary sidelobes, which can be observed at the upper end of the band (6 GHz, Figs. 20, 21). However, at higher frequencies there is also a higher probability of sharp dips in the sum radiation pattern. In particular, the homogeneity of the sum beam at 776 MHz is within 3 dB for

REFERENCES [1] M. Knudsen and G. Pedersen, “Spherical outdoor to indoor power spectrum model at the mobile terminal,” IEEE J. Sel. Areas Commun., vol. 20, no. 6, pp. 1156–1169, Aug. 2002. [2] K. Kalliola, H. Laitinen, L. Vaskelainen, and P. Vainikainen, “Realtime 3-D spatial-temporal dual-polarized measurement of wideband radio channel at mobile station,” IEEE Trans. Instrum. Meas., vol. 49, no. 2, pp. 439–448, Apr. 2000. [3] O. Franek, G. Pedersen, and J. Andersen, “Numerical modeling of a spherical array of monopoles using FDTD method,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 1952–1963, July 2006. [4] C. Balanis, Modern Antenna Handbook. New York: Wiley-Interscience, 2008.

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[5] W. Sun and C. Balanis, “Analysis and design of quadruple-ridged waveguides,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 12, pp. 2201–2207, Dec. 1994. [6] J. Kerr, “Short axial length broad-band horns,” IEEE Trans. Antennas Propag., vol. 21, no. 5, pp. 710–714, Sep. 1973. [7] C. Bruns, P. Leuchtmann, and R. Vahldieck, “Analysis and simulation of a 1–18 GHz broadband double-ridged horn antenna,” IEEE Trans. Electromagn. Compat., vol. 45, no. 1, pp. 55–60, Feb. 2003. [8] V. Rodriguez, “New broadband EMC double-ridge guide horn antenna,” RF Design, vol. 27, no. 5, 2004. [9] M. Botello-Perez, H. Jardon-Aguilar, and I. Ruiz, “Design and simulation of a 1 to 14 GHz broadband electromagnetic compatibility DRGH antenna,” in Proc. 2nd Int. Conf. on Electrical and Electronics Engineering, Sep. 2005, pp. 118–121. [10] S. Soroka, “A physically compact quad ridge horn design,” in Proc. Antennas and Propagation Society Int. Symp., Jun. 1986, vol. 24, pp. 903–906. [11] Z. Shen and C. Feng, “A new dual-polarized broadband horn antenna,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 270–273, 2005. [12] V. Rodriguez, “An open-boundary quad-ridged guide horn antenna for use as a source in antenna pattern measurement anechoic chambers,” IEEE Antennas Propag. Mag., vol. 48, no. 2, pp. 157–160, Apr. 2006. [13] R. Dehdasht-Heydari, H. Hassani, and A. Mallahzadeh, “Quad ridged horn antenna for UWB applications,” Progr. Electromagn. Res., vol. 79, pp. 23–38, 2008. [14] R. Dehdasht-Heydari, H. Hassani, and A. Mallahzadeh, “A new 2–18 GHz quad-ridged horn antenna,” Progr. Electromagn. Res., vol. 81, pp. 183–195, 2008. [15] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Boston: Artech House, 2005. [16] M. Abbas-Azimi, F. Arazm, and J. Rashed-Mohassel, “Sensitivity analysis of a 1 to 18 GHz broadband DRGH antenna,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 2006, pp. 3129–3132. [17] M. Kujalowicz, W. Zieniutycz, and M. Mazur, “Double-ridged horn antenna with sinusoidal ridge profile,” in Proc. Int. Conf. on Microwaves, Radar & Wireless Communications, May 2006, pp. 759–762.

Ondˇrej Franek (S’02–M’05) was born in 1977. He received the M.Sc. (Ing., with honors) and Ph.D. degrees in electronics and communication from Brno University of Technology, Czech Republic, in 2001 and 2006, respectively. Currently, he is working in the Department of Electronic Systems, Aalborg University, Denmark, as a Postdoctoral Research Associate. His research interests include computational electromagnetics with focus on fast and efficient numerical methods, especially the finite-difference time-domain method. He is also involved in research on biological effects of non-ionizing electromagnetic radiation, indoor radiowave propagation, and electromagnetic compatibility. Dr. Franek was the recipient of the Seventh Annual SIEMENS Award for outstanding scientific publication.

Gert Frølund Pedersen was born in 1965. He received the B.Sc. E.E. degree (with honour) in electrical engineering from College of Technology in Dublin, Ireland, and the M.Sc. E.E. and Ph.D. degrees from Aalborg University, Denmark, in 1993 and 2003, respectively. He has been employed by Aalborg University since 1993 where he is now Full Professor heading the Antennas, Propagation and Radio Networking Group and is also the Head of the Doctoral School on Wireless which has close to 100 Ph.D. students enrolled. His research has focused on radio communication for mobile terminals, and especially on small antennas, diversity systems, propagation and biological effects, and he has published more than 75 peer reviewed papers and holds 20 patents. He has also worked as consultant for developments of more than 100 antennas for mobile terminals including the first internal antenna for mobile phones in 1994 with lowest SAR, first internal triple-band antenna in 1998 with low SAR and high TRP and TIS, and lately various multi antenna systems rated as the most efficient on the market. He has been one of the pioneers in establishing the over-the-air measurement systems. The measurement technique is now well established for mobile terminals with single antennas and he is now chairing the COST2100 SWG2.2 group with liaison to 3 GPP for over-the-air tests of MIMO terminals.

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Location Specific Coverage With Wireless Platform Integrated 60-GHz Antenna Systems Arnaud L. Amadjikpè, Student Member, IEEE, Debabani Choudhury, Fellow, IEEE, George E. Ponchak, Fellow, IEEE, and John Papapolymerou, Senior Member, IEEE

Abstract—60-GHz antennas are embedded inside a laptop computer chassis to evaluate suitable integration scenarios for effective far-field range coverage. A broad-beam patch and a switched-beam directive quasi-Yagi array are designed and utilized to conduct experimental tests on a real laptop computer. An electromagnetic modeling tool is used to fine tune the antenna’s specific position at different locations in the laptop lid and base. In general, it is found that the platform embedded antennas exhibit satisfactory performance when they illuminate a small area of the chassis in the boresight direction, which prevents unwanted surface waves radiated from the chassis discontinuities (edges, corners, apertures) from interfering with the antenna main beam. In practice, this is simply achievable by keeping the antenna within a wavelength (5 mm) or closer to the frontal cover surface. Improper antenna placement may lead to antenna beamwidth reduction, boresight gain decrease, boresight angle tilt, and shadow regions formation. The derived results are not solely specific to the laptop chassis problem, and can thus be used to design general purpose wireless platform integrated 60-GHz antenna systems. Index Terms—60-GHz, antenna integration, antenna packaging, antenna systems, embedded antenna, internal antenna, platform integrated antenna, switched-beam end-fire array.

I. INTRODUCTION ULTI-GIGABIT short-range wireless communications for the consumer market is becoming a reality with the recent advancements in 60-GHz integrated system technologies. This fast-growing technology with unlicensed broadband frequency range shows tremendous potential for integration with most consumer electronic devices such as smartphones, tablets, e-books, netbooks and laptop computers [1], [2]. 60-GHz wireless network environments are extremely dense, and thus link reliability and robustness in such environments will critically depend on the ability of the antenna systems to provide effective coverage between different nodes. In the consumer electronics industry, aesthetic and packaging reasons have forced designers to embed antennas inside the host chassis, as opposed to the older, external monopole type of antenna approach.

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Manuscript received April 20, 2010; revised September 07, 2010; accepted December 10, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported by Intel Corporation, USA. A. L. Amadjikpè and J. Papapolymerou are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30338 USA (e-mail: [email protected]). D. Choudhury is with Intel Corporation, Hillsboro, OR 97124 USA. G. E. Ponchak is with the NASA Glenn Research Center, Cleveland, OH 44135 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.2011.2152330

Various antenna system integration with the platform chassis has been extensively studied using measurement as well as modeling approaches, starting with AM and FM antennas for automobiles ([3], [4]), followed by VHF antennas for civil aircrafts ([5]–[7]), and more recently GPS, GSM, Bluetooth, UMTS, UWB and Wi-Fi antennas for mobile platforms [8]–[14]. The behavior of internal antennas inside electrically small chassis, resonant chassis, and electrically large chassis has been characterized up to 10 GHz using different modeling approaches such as the geometrical optics (GO), the physical optics (PO), the finite difference time domain (FDTD) technique, the boundary element method (BEM), the finite element method (FEM), or integral equation (IE) formulations of the electric and magnetic fields. Most of these methods are implemented in commercial software packages such as FEKO, NEC, HFSS, and CST [15]–[18]. The chassis integration of 60-GHz antennas can be categorized as an electrically large problem because of the very small wavelength compared to the typical wireless platform sizes. For such problems, there is a common understanding that, at low frequencies, internal antenna characteristics are mainly affected by objects that obstruct the path of the rays emanating from the antenna element. However, this is yet to be verified and demonstrated at 60 GHz. Moreover, some studies recently conducted with 60-GHz antennas in proximity of plastic cover materials showed that small size discontinuities such as thin cover edges can significantly deteriorate the antenna performance due to space wave diffraction [19], [20]. This behavior has been less significant at low frequencies because the same cover edges are electrically small and thus less apparent, if not invisible at those frequencies. This emphasizes the necessity to account for the effects of small size features composing the geometry of platform environments at 60 GHz. For the first time, a comprehensive analysis of internal 60-GHz antenna radiation characteristics in a wireless platform environment is investigated. In this work, a laptop computer is utilized as the host platform for embedded 60-GHz antennas. Section II describes the laptop chassis modeling with the specific antenna locations. Antennas suitable for this type of applications are designed in Section III. Thereafter, the radiation performance of these antennas mounted inside the laptop computer chassis is evaluated in Section IV. The last part of the paper is devoted to discuss the major results from this study and their direct implications in the design of wireless platform integrated 60-GHz antennas. II. WIRELESS PLATFORM CHASSIS MODELING It is necessary to mention that modeling of real-life internal antennas requires the exact model constructed by the platform

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Fig. 1. Antenna integration in the laptop lid: (a) back view with the center of the coordinates system aligned with the iAUT location; (b) lateral zoom on the antenna mounted behind the LCD screen. Large arrows indicate possible directions of radiation. iAUT denotes the internal antenna under test.

chassis manufacturer. Failing to use such a model may affect simulations accuracy. In this work, however, the entire laptop chassis was created from scratch in the HFSS graphics modeler because the exact model from the manufacturer was not available. Differences between the model and the real laptop are carefully emphasized. Later in Section IV, a sensitivity study to the chassis environment is carried out to determine the pertinence of chassis structural details in the design of 60-GHz internal antennas. A. Lid-Mounted Antenna Internal antennas are generally integrated inside the laptop lid, close to the top edge as well as side rims around and away from the LCD screen. For the location shown in Fig. 1, radiation may occur either toward , or in the direction. The first one represents the very familiar scenario where the antenna communicates with a spot located behind the laptop screen. The two other scenarios emulate an antenna communicating with wireless spots located on the front left hand of the laptop base or above the computer (a wireless spot on a ceiling for instance). is here considered to be odd as it would Radiation toward mean that the antenna points toward the laptop user. In the simulation tool, the chassis cover material is defined as a dielectric material (with no metal particle) of relative permittivity 3.45 and loss tangent 0.025, at 60 GHz [19]. Although plastic covers with conductive coatings are a very common practice in the industry for electromagnetic interference (EMI) mitigation, this model excludes these types of covers. Indeed, preliminary measurements with a metal coated plastic cover (from a real laptop computer) show that the radiated power is attenuated by at least 40 to 50 dB at 60 GHz, which clearly prevents the use of such coatings, at least, in a small area facing the radiating element. The experimental laptop also uses a plastic cover with no metal particles. The LCD screen is defined as a glass material. B. Base-Mounted Antenna The backside and lateral sides of the laptop base are other areas where internal antennas can be mounted. Fig. 2(a) shows

an antenna in the front and left corner of the base that may raor direction. For this particular laptop, diate either in the the specific location originally contained the audio speaker which was removed, except for the 1/16” thick plastic obstacle and the very thin slots in the plastic. These discontinuities are included in the simulation model of the antenna placed in the surrounding laptop environment. Fig. 2(b) shows an antenna in the back and left side of the base, from where the internal , or . antenna under test (iAUT) can radiate toward The laptop used for the experiments has some in-line apertures in the cover where VGA, USB and other connectors were located. These connectors were removed, and the apertures covered from the inside with small pieces of the same plastic material. In the model and the experimental laptop, a large (30 mm 12.5 mm) parasitic copper piece is attached 4 mm above the iAUT to investigate the proximity effects with metal obstacles. III. 60-GHZ ANTENNAS FOR WIRELESS PLATFORMS Before presenting the antennas used in this work, a figure of merit based on the average 3 dB gain of the iAUT is introduced as a more effective evaluation quantity compared to the usually calculated or measured antenna peak gain. A. 3 dB Average Gain As will be discussed in Section IV, the iAUT radiation pattern at 60 GHz may be significantly distorted because of constructive and destructive interference from scattered fields throughout the host chassis. In some cases, the resulting patterns have ripples of more than 3 dB magnitude and it becomes complex to define the antenna peak gain and beamwidth in such a case. The method proposed in this work consists of: 1) smoothing rippled patterns using the MATLAB moving average method. The purpose here is to obtain data with less than 1 dB magnitude ripples in the antenna main beam. The is used to generate the smoothed data; function 2) a 3 dB beamwidth for the smoothed patterns can then be determined, in both E and H planes; and

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Fig. 2. Antenna integration in the laptop base: (a) the antenna is mounted in the front left corner; (b) the antenna is mounted in the back left corner. In all cases, the antenna location coincides with the center of the coordinates system. Large arrows indicate possible directions of radiation.

Fig. 3. (a) Rectangular patch antenna; (b) measured magnitude of S

3) the average gain

for the rectangular patch antenna in free space.

of the AUT is finally defined as

(1) and are the number of angular points in a 3 dB where beamwidth span of the E and H planes, respectively, is the E plane gain at angle , and is the H plane gain at angle . The advantage of this method is that it smooths the gain ripples and gives an estimate of the average radiated power within the 3 dB beamwidth of the antenna, hence taking into account the gain decrease due to nulls in the antenna main beam. Ideally, if we keep as the sweep angle, the average gain should be computed over a full 3 dB solid angle (that is, the solid ); but from a angle integrated over all angles and within practical point of view, it is reasonable to limit to the E and H plane cuts. B. Rectangular Patch Antenna Rectangular patch antennas are known to have broad-beam type of radiation at boresight. The broad-beam characteristic is especially desirable for wide spatial coverage. An inset-fed patch is here designed and fabricated on an 8 mil thick liquid crystal polymer (LCP) substrate, whose relative permittivity and loss tangent are defined to be 3.16 and 0.004 respectively, at 60 GHz [21]. Standard lithography patterning of a bare 1/4 oz thick copper (Cu) layer is performed to define the patch antenna geometry and feed lines on the top layer, while the bottom layer is covered with a bare 1/4 oz Cu. Testing of the antenna is done with a GPPO connector edge-mounted to

Fig. 4. Simulated and measured normalized radiation pattern of the rectangular patch antenna in free space, at 60 GHz: (a) E plane; (b) H plane.

the microstrip feed line (Fig. 3(a)). The fabricated patch resonates at 60.6 GHz with about 4.5 GHz frequency bandwidth (Fig. 3(b)). The higher resonance above 65 GHz stems from the launcher over-moding. The simulated and measured radiation patterns are very similar, including the E plane ripples that are attributed to radiation from the microstrip feed line and reflections from the launcher. After smoothing the pattern, the calculated and measured 3 dB average gain values are 3.8 and 5.8 dBi respectively. The measured value is slightly higher because the measured H plane cut level coincides with a higher gain level in direction, owing to a slight pattern tilt in the E plane the (Fig. 4). This pattern tilt could be easily due to the curvature of the flexible LCP substrate. To remain consistent, these discrepancies in gain values are taken into account in the embedded antenna characterization.

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Fig. 5. Schematic of the proposed switched-beam quasi-Yagi array fabricated on LCP and integrated with the HMC-SDD112 SPDT switch. Close view of the die to LCP substrate interconnects is also shown.

C. Switched-Beam Directive Quasi-Yagi Planar Array Directive antennas are highly desirable for 60-GHz applications because they can increase the communication range and are less vulnerable to multipath interference in highly dense environments. However, they usually need to be implemented in phased arrays or switched-beam systems to increase the coverage with the directive antenna. A switched-beam antenna capable of switching between two orthogonal directions ( and ) in the E plane is proposed in this work. The structure of the proposed antenna is shown in Fig. 5. It is composed of two 4 1 quasi-Yagi arrays oriented orthogonally. Switching between each array allows changing the direction of radiation. The operating principle of the single element quasi-Yagi printed antenna may be found in [20], [22]. The switched-beam antenna is fabricated on an 8 mil thick LCP substrate with patterned 1/4 oz thick bare Cu ground plane in the bottom layer. The antenna elements, feed lines and bias lines for the cavity-mounted SPDT switch are patterned on the top gold (Au) layer is first evapolayer in two steps: a thin 1.5 rated and patterned to define the entire top layer metallization; thick Au pads are then selectively electroplated close 10 to the switch pads to enhance bond wire adhesion on the soft LCP substrate and prevent scratching of the thin Au seed layer during the bonding process. The cavity for the GaAs PIN die (Hittite: HMC-SDD112) is opened in the LCP substrate with a UV excimer laser. The cavity size is precisely controlled due to gaps between the accuracy of the excimer laser: 10 to 25 the die edges and the cavity contour are achieved in fabrication. The 1/4 oz thick Cu ground plane is also used as a stop layer while ablating the LCP material. A 4 mil thick conduc-

tive silver epoxy film adhesive (ESP8660-WL) from AI Technology is used to attach the die ground and level the die to the top layer of the LCP substrate. Under this configuration, both the die and the antenna have a common ground plane, and bond wire . Single layer lengths are minimized between 150 and 250 chip RF by-pass capacitors (100 pF) from Presidio Components are also cavity-mounted inside the antenna substrate. Note the T-shaped impedance matching network (shunt capacitive stub with a series inductive t-line) at each RF port that is designed to compensate for the inductance of the bond wires. The switch -parameter files were not available from Hittite, and ideal 50 microstrip lines on a 4 mil thick GaAs substrate were used instead to model the switch. Note that the Hittite switch has a typical 1.5 dB insertion loss, better than 30 dB isolation and 12 dB return loss at 60 GHz. Fig. 6 a shows the simulated and meafor the switched-beam antenna in free sured magnitude of space. The measured operating bandwidth spans from 59.2 to 64.5 GHz versus the simulated 57.3–62 GHz bandwidth. A frequency shift of 1.9 GHz (3%) is noticed between the simulated and measured data: an adjustment of the value from 3.16 to 2.9 would be the major source of discrepancy. In fact other litin the erature has reported some variability relative dielectric constant of LCP [23]. Note that the dominant surface wave excited mode in the quasi-Yagi antenna is a mode that is parallel to the substrate, in which case anisotropy of the relative permittivity might also influence the frequency shift. Recall that the simulated model does not use -parameter files of the switch, which might be an additional source of error in the . Parasitic resonances that are partially attributed simulated to the connector transition can be suppressed once this transition is de-embedded. This was verified by simulating the model

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Fig. 6. (a) Simulated and measured magnitude of S of the switched beam quasi-Yagi antenna array in free space; Simulated and measured normalized radiation pattern of the switched beam quasi-Yagi antenna in free space, at 60 GHz: (b) E plane; (c) H plane.

with a waveguide port feeding instead of the GPPO launcher, in the HFSS environment. Excellent agreement is achieved for the simulated and measured E plane patterns (Fig. 6(b)). The E . The measured plane 3 dB beamwidth is in all cases H plane patterns are narrower than expected and this is essenand tially attributed to nulls formation in the directions. These nulls result from the interference between the antenna main beam and the fields radiated upward or downward from the coaxial aperture of the connector and a 26 mm long (not shown in Fig. 5) microstrip feed line (that is used for clearance between the antenna element and the launcher). Also, the slight curvature of the LCP substrate justifies why the measured beams point a bit downward. After de-embedding the connector loss (0.9 dB) and the 26 mm microstrip line loss (0.5 dB/cm), the simulated and measured 3 dB gain values are 8.7 and 8.4 dBi respectively, at 60 GHz. Note that the calculated and measured peak gain values are 9.9 and 10.1 dBi respectively, at 60 GHz. IV. CHARACTERIZATION OF 60-GHZ PLATFORM EMBEDDED ANTENNAS

Fig. 7. Photograph of measurement setup. This picture was taken during the measurement of the embedded switched-beam antenna. The laptop is re-positioned each time for a new measurement run to align the iAUT with the receiving horn antenna.

A. Measurement Setup Description The platform embedded antennas were characterized in a 60-GHz antenna characterization system that holds the iAUT fixed at the center of the coordinates system, where it acts as the transmitting antenna. The receiving antenna rotates in an arc with a radius of 52 cm around the iAUT. This distance is close enough to the far-field range given that the internal antennas illuminate only a localized area of the platform (and not the entire chassis as is the case at low frequencies), in such a way that the effective aperture of the chassis-mounted antenna is limited to 4 cm (8 at 60 GHz) in average. Section IV-B elaborates more on how this effective area is delimited. Note that this aperture size is specific to the type of antenna used in this study and might slightly increase if the internal antenna is larger and thus illuminate a larger portion of the chassis. An Agilent Vector Network Analyzer, a V-Band low noise amplifier on the receiving port, and a pair of 1.85 mm cables are used for the measurements. The receiving antenna is a V-Band, 25 dBi standard gain horn, and a similar antenna is used for the gain measurements performed by the substitution method. A 1.85 mm coaxial to WR-15 adapter and a 1.85 mm

coaxial to GPPO microstrip launcher are used for interconnects between the cables and the horn, and the iAUT, respectively. The entire system is controlled by a LabView program. Although the standalone antenna can be aligned accurately when it is characterized in free space, it is more difficult when the antenna is placed inside the laptop because the iAUT is hidden from view for most of the measurements. Thus, this must be considered a source of error. The measurement setup is shown in Fig. 7. For all these measurements, the keyboard, lid, and all components of the computer are placed in their proper locations except for the CD-Rom drive and the battery to facilitate the antenna placement with the testing cables. The presence of the laptop battery and CD-Rom drive, in a real case scenario however, may alter the observed results if these metal blocks obstruct the path of the radiated fields because the induced surface currents radiate from obstacles discontinuities and add constructively or destructively to the antenna main beam. In addition, the flexible LCP substrate has a slight curvature that could not be eliminated or corrected for by placing the antenna into a fixture; this also introduced a source of error.

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Fig. 8. (a) Magnitude of electric field distribution on the surface of the laptop lid, showing surface waves excitation on the interface of the plastic cover. This is a view from the back of the laptop with a zoom into the area surrounding the patch; (b) normalized H plane co-pol radiation pattern of the patch antenna; (c) normalized E plane co-pol radiation pattern of the patch antenna. The antenna beam is directed toward x.

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B. Rectangular Patch Inside the Laptop Lid Fig. 8(a) shows the simulated magnitude of the electric field on the surface of the laptop lid (HFSS is used to model the laptop-mounted antennas). Concentric field lines that spread out in a cylindrical fashion around the main radiator (the patch in that are always this case) remind of surface wave modes excited in dielectrics [24]–[26]. Excitation of surface waves on the surface and in the bulk of the chassis cover material is particularly enhanced at 60 GHz because of its large effective thickat 60 GHz). Surface waves usually conness ( tribute to radiation when they reach discontinuities from where reflection and/or diffraction occur. In this particular scenario, the rectangular patch is mounted 2 cm away from the lid cover edges (in the and directions) and the distance between the patch and the frontal cover surface is only 2 mm. Because of the 2 cm clearance distance, surface waves propagating in the cover dielectric vanish before they reach discontinuities at the lid cover edges. Therefore, there is very limited surface wave radiation and this explains why both E and H plane patterns are similar to the free space case, as shown in Fig. 8(b) and (c). There is a good agreement between the simulated and measured data. The H plane pattern also remains symmetric despite the large LCD glass screen behind the patch. Pattern symmetry is much more difficult to achieve at low frequencies because low frequency internal antennas require a much larger physical wide patch ground plane is found ground plane. Here, a to be large enough to hide the antenna from the LCD screen effect. The same radiation characteristics are thus expected to be observed if the patch is moved along the top or side edges of the lid. Because the fields are essentially confined in the vicinity (inside a disk of radius 4 centered on the patch) of the patch antenna, only the lid chassis is incorporated in the simulation model. The calculated and measured power levels (averaged 3 dB gain values) are 2.16 and 3.65 dBi respectively; total gain attenuations of 1.64 dB (simulation) and 2.15 dB (measurement) are extracted from the corresponding free space values (Section III-B). Calculated and measured attenuation levels are within 0.5 dB, which is acceptable for simulation and measurement tolerances. Here, the average gain is attenuated because part of the fields radiated from the patch are reflected at the cover interface and a more important part dissipated inside the plastic

Fig. 9. Simulated and measured normalized radiation pattern of the patch antenna: (a) H plane co-polarization; (b) E plane co-polarization. The antenna beam is directed toward y . The E plane cut could not be measured. Standalone and integration in the front of the base are compared.

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cover. Although not shown here, the cross-polarization levels were also measured and found to be better than 12 dB, same as for the standalone antenna. The limited diffraction of surface waves helps maintaining a reasonable cross-polarization level, in this specific scenario. C. Rectangular Patch in the Front of the Laptop Base The measured radiation pattern of the base-mounted patch (see Fig. 2(a) for mounting scenario) shows significant degradation compared to the standalone case (Fig. 9(a)). To better understand the behavior at this location, the patch is placed 0.5 cm from the base sidewall (closer to the base sidewall-Fig. 10(b)) and 2.5 cm from the inner plastic (away from the base sidewall-Fig. 10(a)). In the former case, the thin slots are also removed to assess their contribution (Fig. 10(c)). In the Fig. 10(a) case, a large area in the corner of the laptop base is illuminated and important surface waves are thus excited on adjacent cover surfaces. Diffracted surface waves from the sharp junction edges, corners and the thin slots radiate on one side, and reflected space waves from the inner plastic piece and surrounding cover faces interfere with the main radiated waves on the other side, which in turn significantly affect the total radiated fields (Fig. 11). In Fig. 10(b), the thin slots are the essential discontinuities that create surface wave diffraction since the fields are

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Fig. 10. Magnitude of electric field distribution on the surface of the laptop base: (a) patch at 2.5 cm from the inner vertical plastic obstacle; (b) patch at 0.5 cm from the base vertical wall; (c) patch at 0.5 cm from the base vertical wall without slots in the cover. The antenna location in the coordinates system is represented by a “ ” in the plots.

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Fig. 11. Simulated normalized radiation pattern of the patch antenna for different configurations: (a) H plane co-polarization; (b) E plane co-polarization. The antenna beam is directed toward y .

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Fig. 13. Simulated normalized radiation pattern of the switched-beam antenna array when located 0.5 or 2.5 cm away from the vertical plastic cover of the base: (a) H plane co-polarization x; (b) E plane co-polarization x; (c) H plane co-polarization y ; (d) E plane co-polarization y .

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excited surface waves. Although not shown here, simulations with smaller than 0.5 cm distances between the antenna and the base sidewall resulted in very consistent pattern shapes. The base-mounting scheme of Fig. 10(c) is therefore recommended for such antennas. D. Switched-Beam Quasi-Yagi Array in the Back of the Laptop Base

Fig. 12. Measured normalized radiation pattern of the switched-beam array mounted in the back left corner of the laptop base: (a) H plane co-polarization x; (b) E plane co-polarization x; (c) H plane co-polarization y ; (d) E plane co-polarization y .

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more confined in that area. This infers that to achieve better performance from patch like antenna at this location, it first needs to be mounted as close as possible to the base sidewall to minimize the illuminated area. Moreover, the sidewall should be free of discontinuities such as apertures or thin slots. Discontinuities in the order of a wavelength significantly diffract the

The measured radiation patterns of the base-mounted switched-beam array (See Fig. 2(b) for mounting scenario) show some minor rippling along with a slight decrease in the average gain, in both and directions (Fig. 12). The 3 dB gain values are 5.9 and 5.7 dBi respectively, which corresponds to an average 2.5 dB gain decrease in both end-fire directions, compared to the standalone antenna (Section III-C). These results show that end-fire antennas mounted in the back of the base have satisfactory radiation characteristics although there are minor ripples that could be avoided as explained below. Recall from Section II-B that the back-left (or right) corner of the laptop base is a very complex environment where connectors or power plug-ins are usually installed. The lid prox-

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Fig. 14. Magnitude of electric field distribution on the surface of the laptop lid and base: (a) switched-beam array “ x” at 0.5 cm from the base vertical wall; (b) switched-beam array “ x” at 2.5 cm from the base vertical wall; (c) switched-beam array “ y ” at 0.5 cm from the base vertical wall; (d) switched-beam array “ y ” at 2.5 cm from the base vertical wall. The antenna location in the coordinates system is represented by a “ ” in the plots.

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imity further makes the antenna integration problem challenging at this location. The impact on the end-fire antenna is studied here by moving the iAUT close to (0.5 cm) or far from (2.5 cm) the base sidewalls. The calculated radiation patterns turn out to significantly degrade as the antenna moves away from the base sidewalls (Fig. 13). In fact, the antenna element illuminates a larger area of the chassis when it is located 2.5 cm away from the base sidewalls. In Fig. 14(b) for instance, surface waves are excited not only on the surface of the base, but also on the surface of the lid; diffraction and/or reflection of these parasitic waves from the lid-base junction and also the lid and base edges and corners dramatically distort the H plane pattern (Fig. 13(a)). The pattern ripples result in an average 2.6 dBi gain level at boresight, that is, 5.8 dB less than the free space gain. Ripples in the patterns can thus significantly reduce the internal antenna gain, and ultimately reduce the antenna range (a 5 dB gain drop at both the transmitter and the receiver reduces the antenna range by 2/3-Friis equation). In Fig. 13(c), it is observed that the main beam direction can also be significantly tilted (20 off the end-fire direction), while the gain in the plane of the end-fire antenna decreases by more than 10 dB; in this case, the plane of the end-fire antenna is just a shadow region, which leaves the antenna completely blind in its end-fire direction. The best performance of the end-fire antenna is however achieved when the antenna is closer to the base sidewalls (0.5 cm antenna edge to base sidewall distance). The radiation patterns in that case are very similar to the free space case. The direction is even better than the direction beam in the because of the piece of copper mounted 4 mm above the an: the copper piece first cancels tenna element radiating in upward radiation in the H plane (side lobe attenuation by more than 10 dB), and second, serves as a reflector that narrows the H plane main beamwidth and increases the directivity of the antenna element in the end-fire direction. For instance, the simuis lated directivity of the internal antenna radiating toward 14.5 dBi, compared to 12.1 dBi in free space. More generally, it can be interpreted that the top and bottom surfaces of the base chassis form a structure similar to a parallel plate waveguide (in the end-fire direction) where space waves are subject to multiple reflections, therefore resulting in a more directive beam. Note that the copper piece is electrically large (30 mm 12.5 mm) enough not to create a parasitic resonance at 60 GHz.

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V. DISCUSSION This work is the first to bring insight into the integration of 60-GHz antennas with wireless platforms for consumer electronics applications. Without loss of generality, this paper investigates the challenges of integrating a 60-GHz planar patch and an end-fire switched-beam quasi-Yagi array inside the lid and the base of a laptop computer chassis. The results and observations derived from this study are not solely specific to the laptop chassis problem and are easily transposable to various wireless platforms such as smartphones, tablets, netbooks and e-books where 60-GHz radios are very likely to be integrated. Electromagnetic analysis of the different integration scenarios has shown that with low-profile antennas, only a very limited area of the chassis surface (disk of radius 4 centered on the antenna) is illuminated and effectively contributes to the entire system radiation performance, in contrary to low frequency laptop-mounted antennas. The first conclusion is that a near-field interaction assessment between the antenna element and the host platform is a useful approach to determine the chassis areas where induced surface currents are relevant to the overall antenna-platform system performance. Moreover, this suggests that platform components such as batteries, speakers, connectors and other electronic parts can be safely mounted in the host platform as long as they are isolated from the illuminated area. The illuminated area may get slightly larger than found in this work, if larger size antennas such as arrays with multiple elements are utilized. The second important point is the necessity to illuminate the smallest area of the chassis (or confine fields in an obstacle-free area of the chassis) in the boresight direction to prevent unwanted surface waves radiated from the chassis discontinuities from interfering with the antenna main beam. Indeed, it is observed that when a larger area of the chassis is illuminated, it is more likely that surface waves excited on the chassis get diffracted or reflected from surrounding chassis edges, corners or apertures. The investigations carried out in this work show that in practice, the illuminated area on the chassis can be enough confined by simply keeping the internal antenna element within a wavelength (5 mm) or closer to the frontal cover surface. Satisfactory radiation characteristics (close to the standalone antenna) along with good cross-polarization levels are observed under these considerations. Otherwise, improper antenna

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placement may lead to antenna beamwidth reduction, boresight gain decrease, boresight angle tilt and shadow regions formation. These distortions of the radiation pattern may directly imby drastically reducing the pact the quality of service antenna range (a 5 dB boresight gain drop at both the transmitter and the receiver reduces the antenna range by approximately 2/3-Friis equation). It is also obvious that the internal antenna should not point toward the chassis edges and corners. Surface waves effects are thus effectively mitigated at 60 GHz with the use of low-profile antennas properly mounted inside the platform chassis. It is worth mentioning that surface waves mitigation is achieved in this work without the need for resistive thin-film coatings that have been suggested for lower frequencies applications [27]. Besides, it is also beneficial to keep the frontal cover surface (portion of cover facing the antenna boresight) as a dielectric with no metal particles because metal coated covers attenuate the 60-GHz signal by more than 40–50 dB. Even with dielectric covers with no metal coating, reflections and dissipation losses from the plastic cover will attenuate the antenna boresight gain by 2 to 4.5 dB [19]. From a location specific point of view, both the lid and base can accommodate 60-GHz internal antennas. Lid-mounted broad-beam patch-like ground planes are antennas with electrically large insensitive to the screen display back reflections and can thus be easily attached anywhere along the top or side edges of the lid, while keeping perfect pattern symmetry, in contrary to low frequency similar location scenarios [10]. Broad-beam and end-fire antennas (including switched-beam arrays for increased versatility) mounted along the laptop base sidewalls show satisfactory radiation characteristics when they are closely mounted to the base sidewalls. It is however expected that end-fire types of antennas may have their beamwidth reduced when their main beam is arranged parallel to the top and bottom surfaces of the base: in fact, the top and bottom cover surfaces form a structure similar to a parallel plate waveguide (in the end-fire direction) where space waves are subject to multiple reflections, therefore resulting in a more directive beam. Because the top and bottom surfaces of the cover are not conductive, space waves may leak through these lateral boundaries with potential boresight angle tilt. Electrically large (that is non-resonant) parasitic metal patches are effectively incorporated with the platform to preserve the desired boresight angle. The main observations derived from this study on laptop-embedded 60-GHz antennas may also be compared to the problem of low frequency antennas flush-mounted on the fuselage of an aircraft and thus analyzed in a similar fashion. Indeed, in terms of wavelengths a 60-GHz antenna to laptop size ratio is very comparable to a VHF antenna to aircraft size ratio. It is thus not surprising to observe that in many aircraft mounted antenna modeling approaches, the analysis starts with an identification of the currents induced on the surface of the aircraft body ([28]) or the different source, reflected and diffracted fields components ([5], [29], [30]) that are used to generate the overall antenna-platform radiation pattern. In both problems, induced surface wave currents that are reflected or diffracted from the platform discontinuities are found to significantly distort the antenna main beam. The illuminated area of the platform also matters in terms of evaluating the pertinent portions of the platform

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geometry that contribute to the system performance. In [6] and [29] for instance, dominant energy regions that are associated to the relevant induced surface currents are used to determine the critical region of the aircraft body in the vicinity of the antenna element. In [31], electrically large obstacles in the near-field region of an aircraft antenna are effectively incorporated to increase the antenna directivity. Finally, in the thick-radome enclosed antenna problem for aircrafts, analogous recommendations are derived: the optimal location to mitigate the resonance effects in the thick-radome corresponds to an antenna located closer to the radome apex, in which case the near-fields interactions are limited to a smaller area on the radome surface [32], [33]. This work has essentially focused on the radiation characteristics of 60-GHz internal antennas, which is certainly the most challenging study on platform integrated antennas. It is however reminded that embedded antennas also need to be characterized with respect to their voltage standing wave ratio (VSWR), and therefore their impedance bandwidth within the respective platform environment. It will be essential to maintain the internal antenna matched in the entire WPAN frequency band (57–66 GHz). Resonant frequency shifts that are often attributed to the loading effect of the chassis cover can be easily adjusted by de-tuning the standalone antenna accordingly. Changes in bandwidth are typically observed at low frequencies (GSM-[13], [14]) when the chassis starts and hence easily couples with resonating the internal antenna. At 60 GHz, the chassis is expected not to affect the antenna bandwidth as much because the chassis resonant frequency is much smaller than the internal antenna resonant frequency. Finally, it is reminded that this work considered the laptop in an open environment; additional studies are currently being carried out with the laptop seating on a conference table with other laptops, books, pens, and coffee cups in a 60-GHz environment [34]. VI. CONCLUSION This paper is the first demonstration of 60-GHz embedded antennas for integration with wireless platforms. Broad-beam patch and switched-beam directive quasi-Yagi arrays have been characterized at different locations inside a laptop lid and base. This work has discussed the far-field behavior and suggested simple and efficient approaches to mitigate the distortions effects observed with 60-GHz chassis-mounted antennas. The derived conclusions are not solely specific to the laptop chassis problem, and have been presented in such a way that they can be easily applied to the design of various wireless platform integrated 60-GHz antenna systems. ACKNOWLEDGMENT The authors thank M. Dalal and M. Pardo from the Integrated MEMS Group at the Georgia Institute of Technology for helping with the wire bonding, and D. Chung from the MircTech group at the Georgia Institute of Technology for assisting on the excimer laser. The authors also thank reviewers for their compelling comments that greatly improved the quality of this paper.

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REFERENCES [1] Wireless Gigabit Alliance Website [Online]. Available: http://wirelessgigabitalliance.org/ [2] WirelessHD Consortium Website [Online]. Available: www.wirelesshd.org/news/press.html [3] R. Abou-Jaoude and E. K. Walton, “Numerical modeling of on-glass conformal automobile antennas,” IEEE Trans. Antennas Propag., vol. 46, no. 6, pp. 845–852, Jun. 1998. [4] L. Low, R. Langley, R. Breden, and P. Callaghan, “Hidden automotive antenna performance and simulation,” IEEE Trans. Antennas Propag., vol. 54, no. 12, pp. 3707–3712, Dec. 2006. [5] B. M. Notaros, M. L. Djordjevic, B. D. Popovic, and Z. Popovic, “Rigorous EM modeling of cars and airplanes,” in Proc. IEEE Radio and Wireless Conf., Denver, CO, Aug. 1999, pp. 167–170. [6] A. Barka and P. Caudrillier, “Domain decomposition method based on generalized scattering matrix for installed performance of antennas on aircraft,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1833–1842, Jun. 2007. [7] F. G. Bogdanov, D. D. Karkashadze, R. G. Jobava, A. L. Gheonjian, E. A. Yavolovskaya, N. G. Bondarenko, and C. Ullrich, “Validation of hybrid MoM scheme with included equivalent glass antenna model for handling automotive EMC problems,” IEEE Trans. Electromagn. Compat., vol. 52, no. 1, pp. 164–172, Feb. 2010. [8] D. Liu, B. P. Gaucher, E. B. Flint, T. W. Studwell, H. Usui, and T. J. Beukema, “Developing integrated antenna subsystems for laptop computers,” IBM J. Res. Devel., vol. 47, no. 2–3, pp. 355–367, Mar.–May 2003. [9] J. Guterman, A. A. Moreira, and C. Peixeiro, “Integration of omnidirectional wrapped microstrip antennas into laptops,” IEEE Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 141–144, Dec. 2006. [10] G. H. Huff, J. Feng, Z. Shenghui, G. Cung, and J. T. Bernhard, “Directional reconfigurable antennas on laptop computers: Simulation, measurement and evaluation of candidate integration positions,” IEEE Trans. Antennas Propag., vol. 52, no. 12, pp. 3220–3227, Dec. 2004. [11] Z. N. Chen, D. Liu, and B. Gaucher, “A planar dualband antenna for 2.4 GHz and UWB laptop applications,” in Proc. IEEE Vehicular Technology Conf., Melbourne, Australia, May 2006, pp. 2652–2655. [12] C. Zhang, S. Yang, S. El-Ghazaly, A. E. Fathy, and V. K. Nair, “A lowprofile branched monopole laptop reconfigurable multiband antenna for wireless applications,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 216–219, Apr. 2009. [13] O. Kivekäs, 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. [14] P. Vainikainen, J. Ollikainen, O. Kivekäs, and I. Kelander, “Resonator-based analysis of the combination of mobile handset antenna and chassis,” IEEE Trans. Antennas Propag., vol. 50, no. 10, pp. 1433–1444, Oct. 2002. [15] FEKO [Online]. Available: www.feko.info [16] NEC [Online]. Available: www.nec2.org [17] Ansoft HFSS v12 [Online]. Available: www.ansoft.com [18] CST Microwave Studio 2010 [Online]. Available: www.cst.com [19] A. L. Amadjikpè, D. Choudhury, G. E. Ponchak, B. Pan, Y. Li, and J. Papapolymerou, “Proximity effects of plastic laptop covers on radiation characteristics of 60-GHz antennas,” IEEE Antennas Wireless Propag. Lett., vol. 8, no. 1, pp. 763–766, Jul. 2009. [20] A. L. Amadjikpè, D. Choudhury, G. E. Ponchak, and J. Papapolymerou, “High-gain quasi-Yagi planar antenna evaluation in chassis material environment for 60 GHz wireless applications,” in Proc. IEEE MTT-S Int. Microwave Symp., Boston, MA, Jun. 2009, pp. 385–388. [21] D. Thompson, O. Tantot, H. Jallageas, G. Ponchak, M. Tentzeris, and J. Papapolymerou, “Characterization of liquid crystal polymer (LCP) material and transmission lines on LCP substrates from 30–110 GHz,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1343–1352, Apr. 2004. [22] N. Kaneda, W. R. Deal, Q. Yongxi, R. Waterhouse, and T. Itoh, “A broadband planar quasi-Yagi antenna,” IEEE Trans. Antennas Propag., vol. 50, no. 8, pp. 1158–1160, Aug. 2002. [23] Y. P. Zhang and D. Liu, “Antenna-on-chip and antenna-in-package solutions to highly integrated millimeter-wave devices for wireless communications,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 2830–2841, Oct. 2009.

[24] R. B. Waterhouse, Microstrip Patch Antennas: A Designer’s Guide. Norwell, MA: Artech House, 2003. [25] B. Roudot, J. R. Mosig, and F. E. Gardiol, “Surface wave fields and efficiency of microstrip antennas,” in Proc. 18th Eur. Microwave Conf., 1988, pp. 1055–1062. [26] E. Bahar, “Excitation of surface waves and the scattered radiation fields by rough surfaces of arbitrary slope,” IEEE Trans. Microw. Theory Tech., vol. 28, no. 9, pp. 999–1006, Sep. 1980. [27] A. Sambell, P. Lowes, and E. Korolkiewicz, “Removal of surface-wave induced radiation nulls for patch antennas integrated with vehicle windscreens,” IEEE Trans. Antennas Propag., vol. 45, no. 1, pp. 176–176, Jan. 1997. [28] F. Obelleiro, L. Landesa, J. M. Taboada, and J. L. Rodriguez, “Synthesis of onboard array antennas including interaction with the mounting platform and mutual coupling effects,” IEEE Antennas Propag. Mag., vol. 43, no. 2, pp. 76–82, Apr. 2001. [29] J. J. Kim and W. D. Burnside, “Simulation and analysis of antennas radiating in a complex environment,” IEEE Trans. Antennas Propag., vol. 34, no. 4, pp. 554–562, Apr. 1986. [30] T. Özdemir, M. W. Nurnberger, J. L. Volakis, R. Kipp, and J. Berrie, “A hybridization of finite-element and high-frequency methods for pattern prediction for antennas on aircraft structures,” IEEE Antennas Propag. Mag., vol. 38, no. 3, pp. 28–38, Jun. 1996. [31] F. Obelleiro, L. Landesa, J. L. Rodriguez, A. G. Pino, and M. R. Pino, “Directivity optimisation of an array antenna with obstacles within its near-field region,” Electron. Lett., vol. 33, no. 25, pp. 2087–2088, Dec. 1997. [32] M. J. Povinelli, “Finite element analysis of large wavelength antenna radome problems for leading edge and radar phased arrays,” IEEE Trans. Magn., vol. 27, no. 5, pp. 4299–4302, Sep. 1991. [33] G. A. E. Crone, A. W. Rudge, and G. N. Taylor, “Design and performance of airborne radomes: A review,” IEE Proc. on Communications, Radar and Signal Processing, vol. 128, no. 7, pp. 451–464, Dec. 1981. [34] G. E. Ponchak, A. L. Amadjikpè, D. Choudhury, and J. Papapolymerou, “Experimental investigation of 60 GHz transmission characteristics for WPAN applications between computers on a conference table,” presented at the IEEE Radio and Wireless Symp., Phoenix, AZ, Jan. 2011. Arnaud L. Amadjikpè (S’08) received the Eng. Dip. degree in electrical engineering from the Polytechnic National Institute of Toulouse ENSEEIHT, Toulouse, France, in 2007, and the MSECE degree from the Georgia Institute of Technology, Atlanta, in 2007, where he is currently working toward the Ph.D. degree. His current research focuses on millimeter-wave integrated antenna module design, packaging and fabrication on organic materials for multi-gigabit applications. He is also interested in the characterization of wireless platform integrated 60-GHz antenna systems. He has also been working on the development of frequency reconfigurable waveguide cavity filters using RF-MEMS technology for S-band radar applications.

Debabani Choudhury (F’11) received the Ph.D. degree in electrical engineering from the Indian Institute of Technology, Bombay, India, in 1991. She is a Senior Technologist at Intel Labs, Hillsboro, OR, and carries out research on RF technologies for small-form factor wireless platform integration. Before joining Intel in 2006, she held senior research staff positions at HRL Labs (former Hughes Research Laboratories) and Millitech Corporation where she developed various mm-wave and THz technologies for imaging, space and defense applications. Prior to that, she worked at NASA Jet Propulsion Laboratory (JPL) on submillimeter wave devices and components for space-based heterodyne receiver applications. Her area of expertise is in the field of RF and millimeter-wave technologies, circuits and subsystem integration. She developed first dual-polarized beam steering arrays at 94 GHz for imaging applications. Her work on multi-gigabit package and wideband active amplifier development holds records. She also co-invented and developed state-of-the-art RF MEM (Micro-Electro-Mechanical) switches, tunable devices and transceiver elements

AMADJIKPÈ et al.: LOCATION SPECIFIC COVERAGE WITH WIRELESS PLATFORM INTEGRATED 60-GHz ANTENNA SYSTEMS

for MM-wave IC integration with ultra low-loss, harmonic-free switching and tuning. She has 20+ patents/patent applications and numerous publications. She received several NASA Recognition awards for her work on heterodyne receivers, devices, multipliers and guiding structures/modules developed for space and defense applications. Dr. Choudhury is an IEEE Fellow. She serves on several Technical Program Committees (TPC) for IEEE and SPIE conferences including IEEE-IMS (International Microwave Symposium), IEEE AP-S (Antenna and Propagation Symposium), IEEE RWS (Radio and Wireless Symposium), and APMC (Asia Pacific Microwave Conference). She is a member of two IEEE Microwave Theory and Technique Society (MTT-S) Technical Co-ordination Committees: MTT-6 on Microwave and Millimeter Wave Integrated Circuits and MTT-20 on Wireless Communications. She was elected Vice-Chair of the MTT-20 Committee. She served on different IMS and RWS steering committees and also served as Publications Chair for several IEEE MTT-S conferences. She served as the Guest Editor of IEEE Microwave Magazine Special Issue on RWW. She is serving as a Co-Editor for the IEEE Proceedings Special Issue on Wireless Power Applications. She has presented invited talks at various conferences, workshops, industries and universities worldwide and was invited to participate in DARPA workshops. She has presented and organized several IEEE workshops, panel sessions and focused sessions.

George E. Ponchak (S’82–M’83–SM’97–F’08) received the B.E.E. degree from Cleveland State University, Cleveland, OH, in 1983, the M.S.E.E. degree from Case Western Reserve University, Cleveland, in 1987, and the Ph.D. in electrical engineering from the University of Michigan, Ann Arbor, in 1997. He joined the staff of the Communications, Instrumentation, and Controls Division at NASA Glenn Research Center, Cleveland, in 1983 where he is now a Senior Research Engineer. In 1997–1998 and in 2000–2001, he was a Visiting Professor at Case Western Reserve University. He has authored and coauthored over 150 papers in refereed journals and symposia proceedings. His research interests include the development and characterization of microwave and millimeter-wave printed transmission lines and passive circuits, multilayer interconnects, uniplanar circuits, Si and SiC Radio Frequency Integrated Circuits, and microwave packaging. Dr. Ponchak is a Fellow of the IEEE and an Associate Member of the European Microwave Association. He is Editor-in-Chief of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and was Editor-in-Chief of the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS from 2006–2010, and Guest Editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Special Issue on Si MMICs. He founded the IEEE Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems and served as its Chair in 1998, 2001, and 2006. He is the General Chair of the 2011 IEEE Radio and Wireless Symposium and was the Technical Program Chair of the 2010 IEEE Radio and Wireless Symposium. He served as Chair of the Cleveland MTT-S/AP-S Chapter (2004–2006), and has chaired many symposium workshops and special sessions. He is a member of the IEEE International Microwave Symposium Technical Program Committee on Transmission Line Elements and served as its Chair in 2003–2005 and a member of the IEEE MTT-S Technical Committee 12 on Microwave and Millimeter-Wave Packaging and Manufacturing. He served on the IEEE MTT-S AdCom Membership Services Committee (2003–2005) and was elected to the MTT-S AdCom in 2010. He received the Best Paper Award of the ISHM’97 30th International Symposium on Microelectronics.

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John Papapolymerou (SM’08) received the B.S.E.E. degree from the National Technical University of Athens, Athens, Greece, in 1993, and the M.S.E.E. and Ph.D. degrees from the University of Michigan, Ann Arbor, in 1994 and 1999, respectively. From 1999 to 2001, he was an Assistant Professor at the Department of Electrical and Computer Engineering, University of Arizona, Tucson, and during the summers of 2000 and 2003 he was a Visiting Professor at The University of Limoges, France. From 2001–2005 and 2005–2009, he was an Assistant Professor and Associate Professor, respectively, at the School of Electrical and Computer Engineering, Georgia Institute of Technology, where he is currently a Professor. He has authored or coauthored over 270 publications in peer-reviewed journals and conferences. His research interests include the implementation of micromachining techniques and MEMS devices in microwave, millimeter-wave and THz circuits and the development of both passive and active planar circuits on semiconductor (Si/SiGe, GaAs) and organic substrates (liquid crystal polymer-LCP, LTCC) for System-on-a-Chip (SOC)/ System-on-a-Package (SOP) RF front ends. Dr. Papapolymerou is the Chair of Commission D of the US National Committee of URSI. He was Associate Editor for IEEE Microwave and Wireless Component Letters (2004–2007) and the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION (2004–2010). He currently serves as Associate Editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. During 2004, he was the Chair of the IEEE MTT/AP Atlanta Chapter. He was the recipient of the 2010 IEEE Antennas and Propagation Society (AP-S) John Kraus Antenna Award, the 2009 IEEE Microwave Theory and Techniques-Society (MTT-S) Outstanding Young Engineer Award, the 2009 School of ECE Outstanding Junior Faculty Award, the 2004 Army Research Office (ARO) Young Investigator Award, the 2002 National Science Foundation (NSF) CAREER award, the best paper award at the 3rd IEEE International Conference on Microwave and Millimeter-Wave Technology (ICMMT2002), Beijing, China and the 1997 Outstanding Graduate Student Instructional Assistant Award presented by the American Society for Engineering Education (ASEE), The University of Michigan Chapter. His students have also been recipients of several awards including the Best Student Paper Award presented at the 2004 IEEE Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems, the 2007 IEEE MTT-S Graduate Fellowship, and the 2007/2008 and 2008/2009 IEEE MTT-S Undergraduate Scholarship/Fellowship.

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3D Polarized Channel Modeling and Performance Comparison of MIMO Antenna Configurations With Different Polarizations Manh-Tuan Dao, Viet-Anh Nguyen, Yun-Taek Im, Seong-Ook Park, Member, IEEE, and Giwan Yoon, Member, IEEE

Abstract—We propose a three-dimensional (3D) polarized MIMO channel model, which takes into account 3D power angular spectrum and comprehensive propagation characteristics of electromagnetic waves excited by polarized antennas. Based on the model, we derive a close form expression of the spatial correlation as a function of the physical parameters representing both characteristics of arbitrary antennas and propagation environment in 3D space. The spatial correlation expression allows to use the Von Mises Fisher (VMF) distribution, resulting in a more accurate and general channel model. Through simulation, we evaluate and compare performance, in terms of the spatial correlation and capacity, of 2 2 MIMO configurations with different polarizations, i.e., V/V, V/H, and slanted 45 polarizations, as a function of critical input parameters including elevation angle, antenna orientation, antenna spacing, cross-polarization discrimination (XPD), and signal-to-noise ratio (SNR). The effect of the parameters on the performance is analyzed, and verified in certain cases through the literature. Index Terms—Capacity, correlation, MIMO, polarization, polarized channels, 3D, XPD.

I. INTRODUCTION UTURE wireless communication systems are expected to provide high data rates and a better quality of wireless signals satisfying demanding multimedia services such as video and teleconferencing. The so-called MIMO (multi-input multi-output) systems, in which multiple antennas are used at both transmitter and receiver, have been proposed to achieve these rates due to an improvement in spectrum efficiency [1], [2]. The major issue of MIMO systems is that capacity gain is highly dependent on the spatial correlation, which is a function of the antenna array and channel characteristics (i.e., spacing,

F

Manuscript received January 12, 2010; revised November 01, 2010; accepted November 16, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported by the Ministry of Education, Science Technology (MEST) and Korea Institute for Advancement of Technology (KIAT) through the Human Resource Training Project for Regional Innovation under Contract 20080702123415 and the by Intelligent Radio Engineering Center (IREC). M. T. Dao, V. A. Nguyen, Y. T. Im, and S. O. Park are with the Microwave and Antenna Laboratory, Electrical Engineering Department, Korean Advanced Institute of Science and Technology (KAIST), Daejeon 305-732, Korea (e-mail: [email protected]; [email protected]; [email protected]; sopark@ee. kaist.ac.kr). G. Yoon is with the Communication Electronics Laboratory, Electrical Engineering Department, Korean Advanced Institute of Science and Technology (KAIST), Daejeon 305-732, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2152319

pattern, orientation, polarization, angular spread, angle of arrival, power angular profile) [3], [4]. It has been shown in [4] that, in order to achieve reasonably low correlation, antenna spacing has to be at least half a wavelength, resulting in increasing equipment size. The use of cross-polarized antenna has an advantage to reduce the spacing required on terminal. Thus, this approach is receiving considerable attention in the design of MIMO antennas. Implementation and evaluation of MIMO systems need a comprehensive understanding of MIMO channels. Despite a number of channel models, very few models are available for the use of cross-polarized antenna [5]–[8]. The limitation of these works is that the propagating waves are assumed to arrive only from the azimuth plane therefore not include the elevation spectrum, thus not fully considering the characteristics of antenna and environmental factors. There are several polarized models taking into account both the azimuth and elevation spectrum. The models in [9], [10] compute the locations of individual scatterers based a particular scatterer distribution and generate channel realizations based on the interaction of scatterers and planar wavefronts. The limitation of this approach is the complexity in parameterizing and generating the scatterer distributions for a variety of propagation environments, and simulation time. A novel 3D polarized channel model is presented in [11]. Using this model to study the impact of the elevation spectrum is reported in [12]. However, the assumption that the elevation and azimuth distributions are independent is in general not accurate. In addition, the mathematical formulation of the spatial correlation is simplified by assuming the cross-polarization discrimination (XPD) of unity and ideal dipole antennas. Although such assumptions lead to analytical expressions of the spatial correlation, they are in general not likely to accurately represent the propagation characteristics of electromagnetic waves excited by polarized antennas. Evaluation of MIMO systems can also be carried out by experiments. Several experiments have been made to study the spatial correlation and capacity in indoor office environment [13]–[15], the effect of pattern and polarization [16]–[19]. Carrying out experiments, however, is expensive, time-consuming, and applicable to particular environments and array configurations. In addition, it is difficult to investigate the effect of particular parameters representing propagation characteristics, such as XPD and angular of arrival (AoA), on MIMO performance. In this paper, we propose a 3D polarized geometry model that can accurately represent important aspects of polarized MIMO channels in 3D space, while maintaining a low complexity of simulations. In particular, we derive the close-form expression

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Fig. 1. Illustration of 3D channel model.

of the spatial correlation as a function of the physical parameters representing both characteristics of arbitrary antennas and propagation environment in 3D space. The close-form expression allows to integrate the Von Mises Fisher (VMF) distribution, which has been proved to be appropriate for modeling spherical data [20], [21], to the model, resulting in a more accurate and general channel model. Using this model, we investigate the effect of elevation angle, antenna orientation, antenna spacing, XPD, and SNR on performance of a 2 2 MIMO system using half a wavelength dipole antennas for three typical types of transmission, i.e., vertical, vertical and horizontal, and slanted . We make analysis of performance and comparison among the polarizations. The paper is arranged as follows. Section II presents the geometry MIMO channel model and derives the spatial correlation matrix. Section III describes the antenna configurations with the different polarizations and presents the calculation of the spatial correlation and capacity. Section IV presents discussion and analysis of the simulation results of MIMO performance with the different configurations. Finally, conclusions are drawn in Section V.

distribution. Similarly, the scatterers are assumed to lie on and the th receive scatterer, a spherical surface of radius , is specified by the solid denoted as angle which follows a given angular distribution. and denote the azimuth angle of departure The symbols (AAoD) and elevation angle of departure (EAoD), respectively. and denote the azimuth angle of Similarly, the symbols arrival (AAoA) and elevation angle of arrival (EAoA), respectively. The distance between the centers of the transmitter and receiver is . For the local scattering assumption, the radii and are much smaller than the distance . The antenna configuration at both transmitter and receiver is assumed to be uniform linear array (ULA). The spacing between antenna elements at the transmitter and receiver is denoted by and , respectively. The orientation of antenna array at the transmitter is specified by the angles and . Similarly, the orientation of antenna array at the receiver is specified by and . It is assumed that the system is under the angles non-line-of-sight (NLOS) and flat fading environment. The input-output relationship can be expressed as (1)

II. 3D CHANNEL MODEL Consider a MIMO system with transmit antennas and receive antennas under a 3D geometry model, as shown in Fig. 1. The model is a two-sphere 3D geometrical model, which can be considered as an extension of the two-ring model [22], [23]. It is assumed that energy contribution of remote scatterers is negligible, only local scatterers at both ends of radio link are considered. The transmitter is fixed and the receiver is in motion with the velocity vector . There are and scatterers at the transmitter and receiver, respectively. The scatterers are assumed to and the th transmit lie on a spherical surface of radius , is specified by scatterer, denoted as which follows a given angular the solid angle

, and denote transmitted vector, received where is a comvector, and noise vector, respectively. component plex channel coefficient matrix. The of , denoted as , can be written as (2)

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cient malized as

is unnormalized. If the channel coefficient is nor-

(2)

(4)

is the power transferred through the subchannel and are the gain and phase shift between V (H) component of the transmit antenna and V (H) component of the receive antenna, respectively, caused by the interaction and is the distance from of the local scatterers to the th transmit antenna is the the scatterer to the th receive antenna distance from the scatterer is the distance from the scatterer to the scatterer is the complex field pattern of the th transmit for V polarization; is the complex antenna for H polarization; field pattern of the th transmit antenna is the complex field pattern of the th receive for V polarization; is the complex antenna field pattern of the th receive antenna for H polarization; and are the inverse XPD for VV/HV and HH/VH is the inverse of the co-polar transmission, respectively; is the wavenumber, where is ratio (CPR); the wavelength; denotes the wave vector pointing in the and . It propagation direction from the scatterer is assumed that the gains are independent, finite, and positive random variables that are independent of the phase shifts, which . are i.i.d and uniformly distributed over the interval Furthermore, it is assumed that

the normalized channel coefficient has the standard . Consequently, the channel matrix normal distribution, is properly modeled as

where

.. .

..

.

.. .

(5)

Let us define the matrix (6) where is the matrix and denotes transposition. As a result, we can determine the spatial correlaas follows: tion matrix of the (7) constructed from the model is special complex Since the are comGaussian, the second-order statistics of the pletely specified by the [24]. The spatial correlaand , denoted as , is tion between written as

(8) (3) is the expectation operator. where By the central limit theorem, with the given statistical propand , the numbers of scatterers at erties of the channel, as apthe transmitter and receiver, approach infinity, the proaches a circularly symmetric Gaussian random process with . Therefore, the channel mean zero and variance becomes a purely Rayleigh-fading process. The channel coeffi-

All the expectations in (8) are in the general form of . Owing to the assumption that the gains are independent variables that are independent of the phase can be transformed shifts, the expectation into (9), shown at the bottom of the page. Evaluation of the second expectation in (9) results in that of the four expectaand . tions, The phaseshifts, and , are uniform over .

(9)

DAO et al.: 3D POLARIZED CHANNEL MODELING AND PERFORMANCE COMPARISON OF MIMO ANTENNA CONFIGURATIONS

The difference between them, denoted as distribution as follows:

, has the triangular

Then, the expectation

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can be analytically calculated as

(10) . With the distribution, the expectation is equal to zero. Consequently, the second expectation in (9) simplifies to

(16) As

, the numerator of (8) can be expressed as (17),

(11) The cross-polarization discrimination (XPD), which is a measure of depolarization in propagation environment, is defined as the ratio of the average power received in the co-polarized channel to the average power received in the cross-polarized channel:

(12)

(17)

where is the component in the xy channel. The propagation characteristics of vertically polarized waves are significantly different from those of horizontally polarized waves. The received power in the vertical-to-vertical transmission is normally higher than in the horizontal-to-horizontal transmission, . This fact is due to the Brewster or angle phenomenon for horizontally polarized transmission [25]. Therefore, we can define a unique co-polar ratio (CPR) as

where and are the scatterer distributions at the transmitter and the receiver. The expectations in (17) are is the distance from the computed by (16). The term th transmit antenna to the point on the transmit-scatfrom the array center of tering sphere at the solid angle the transmitter. The other terms are defined in a similar way. These distances are calculated using the law cosines as follows:

(13) It is observed that, the XPD and CPR, when expressed in decibel (dB), have the normal distribution, . Depending on the environment, the mean of XPD varies from 0 to 18 dB, with the standard deviation in order of 3–8 dB [11]. The mean of CPR as the XPD varies between 0 and 6 dB [26]. By denoting (CPR) in dB and as the XPD (CPR) in linear, the expectations in (11) can be computed as follow: (14)

(18) is the position vector of the th transmit antenna is the position vector of the point on the transmitfrom the array center scattering sphere at the solid angle is the angle between the two vectors; and of the transmitter; the other terms are defined in a similar fashion where

, is easily The probability distribution of , denoted as computed by transformation of the normal distribution of [27], yielding

(15)

(19)

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It is assumed that

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, (19) can be simplified to

(20) By using the approximation fied to

, (20) is simpli-

(21) The term in (21) is easily computed by taking scalar , denoted as product of the unit vector along the direction of , and the unit vector along the direction of , denoted as , as follows:

(22) where is the unit vector of antenna array at the transmitter. As a result, (21) can be evaluated as

(25)

(23) where

Similarly,

(26) (24) Eventually, the correlation in (8) can be rewritten as (25),

Several distributions, e.g., uniform, Gaussian, wrapped Gaussian, and Laplacian are used in prior work to characterize the AAoD and AAoA for 2-dimensional (2D) models [5], [6], [28]. These distributions are also used for 3D channel models [10]–[12], [29] in which the azimuth and elevation are assumed to be independent. Although this assumption can lead to closed form expressions, it is in general not likely to accurately represent the characteristics of the propagation environment. In this paper, we use the Von Mises Fisher (VMF) distribution to describe directions in space where each direction is uniquely characterized not only by its azimuth but also elevation. The VMF distribution used to model direction of scatterers in is given by [20], [21] the Euclidean (27) where and are the direction and the mean direction of scatterers, respectively, determined by their azimuth and elevation with limits, angles

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is the concentration parameter, a measure of the spread of is the normalizing conscatterers around the mean; and when , or the Euclidean . stant, The VMF above-mentioned distribution is only valid for a single cluster. A cluster here is defined as a group of scatterers located within an isolated interval in solid angle. It has been widely observed from measurements that energy in the radio channel is clustered into multiple of isolated intervals in solid angle, depending on the physical layout of the propagation environment. In this case, we use a mixture of VMF distributions as proposed in [21] (28) is the number of clusters; is defined as the prior where and are the probability that the th cluster was generated; mean direction and concentration of the th cluster, respectively. Estimation of the parameters of a VMF mixture can be found in [30]. Consequently, the scatterer distributions at the transmitter and receiver can be expressed as

Fig. 2. Configurations of antennas for different polarizations. (a) Vertical polarization. (b) V/H polarization. (c) Slanted 45 polarization.

6

(29) Fig. 3. Half-wavelength dipole antenna.

where the terms are identically defined as in (28). From the spatial correlation matrix, , whose entries are computed by (8), we can generate samples of channel matrix as follows:

The radiation field patterns of the half-wavelength dipole for vertical and horizontal polarizations and are given by [31]

(30) is matrix with complex Gaussian elements. where Because the samples of the channel maintain the spatial correlation of , we can say that its spatial characteristics are the same as those of the real channel.

III. PERFORMANCE EVALUATION AND ANTENNA

(31) where . For convenience, the field patterns of the half-wavelength dipole with the inclination angle of 0 , 45 , and 90 are shown in Fig. 4.

A. Antenna Configurations and Pattern

B. Spatial Correlation and Capacity

This paper evaluates performance of MIMO transmission in terms of capacity and correlation for different polarizations. For this purpose, the antenna configurations of a 2 2 MIMO system are shown in Fig. 2. Half a wavelength dipole antenna with its coordinates depicted in Fig. 3 is used at both transmitter and the receiver for all the configurations. Fig. 2(a) shows the configuration of vertical polarization. All dipoles are vertically polarized in this configuration. Fig. 2(b) shows the case using a cross-polarized array which composes of a vertically polarized dipole and a horizontally polarized dipole. Fig. 2(c) shows polarized dipole array which the case using a slanted dipoles. composes of slanted

The spatial correlation among channel coefficients is fully characterized by the correlation matrix deis termined by (7). It can be shown from (8) that equal to conjugate of and that is unity. Hence, the behavior of the spatial correlation matrix can be evaluated by (32) It is assumed that in the system described by (1) the transmitter has no channel-state information, and only the receiver knows the actual realizations. This implies that the signals are independent and the power is equally divided among the

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Fig. 4. Field patterns of half-wavelength dipole antenna with inclination angle of 0 , 45 and 90 .

transmit antennas. Under this assumption, the capacity of MIMO channel is given by [2] (33) where is the average signal-to-noise ratio (SNR), is the identity matrix, and is the normalized channel matrix, which is computed by (30). Dual-polarized configurations suffer from subchannel power losses, which need to be accounted for in their capacity calculation. If we use a fixed transmit power constraint, on we normalize the channel so as to achieve an average of the vertical-to-vertical (vv) link. As a result, the SNR is equal to [26] (34) Fig. 5. Contour plot of AoA profile.

As the channel capacity is a function of the channel matrix , the channel capacity is random quantity whose distribution is determined by the distribution of . Using the Monte Carlo simulation method, we generate 15000 samples of the channel and compute the statistics of MIMO capacity given by (33). IV. RESULTS OF SIMULATION AND DISCUSSION Mathematically modeled physical parameters in this model covering both propagation and antenna characteristics include the angular of arrival (AoA) and angular of departure (AoD) , mean and spread of XPD profiles , antenna orientation , antenna spacing , pattern and polarization. The effect of any parameter on the spatial correlation and capacity can be evaluated by simulation. The angular profiles vary depending on the physical layout of the propagation environment. Extraction of the VMF parameters requires measurement campaign and data processing. For purpose of simulation, we use the result of the VMF parameters for the AoA profile in [21], as shown in Fig. 5. It is noticed from Fig. 5 that the main energy of incoming waves

comes from the direction

. We set

, dB, the reference SNR on the the mean vv link dB, and the mean dB as default values. First, we study the effect of elevation angle by evaluating MIMO performance of the configurations under the 3D model and under the 2D model that is transformed from the 3D model and inteby setting the elevation angles grating the integrand in (25) over the azimuth domain only. The correlation versus receiving antenna spacing for the configurations is shown in Fig. 6. It is observed that the 2D channel model gives higher correlation for the configurations with the V/V and polarizations (Fig. 6(a) and (b)). However, it is slanted interesting to note that the 2D channel model provides the correlation of zero for the configuration with the V/H polarization (Fig. 6(c)). This is because the 2D model considers antenna pattern in azimuth plane only. As shown in Fig. 4, the vertical components of the dipole with inclination angle of 90 is only equal

DAO et al.: 3D POLARIZED CHANNEL MODELING AND PERFORMANCE COMPARISON OF MIMO ANTENNA CONFIGURATIONS

Fig. 6. Performance comparison between 3D model and 2D model for the different configurations. (a) Vertical polarization. (b) Slanted polarization.

to zero in the plane . In this case, omission of the vertical components of the 2D model makes the orthogonality of polarization between the two antennas at the transmitter and receiver, yielding zero correlation. Therefore, depending on the configurations, the 2D model would underestimate or overestimate MIMO performance. On the other hand, the influence of patterns among elements is important to the study on the effect of elevation angle. In [12], the assumption of identical dipoles for all elements only results in the underestimation of the capacity that is not true in general. Second, we investigate the effect of antenna orientation. . The correlation varies Fig. 7 shows the correlation versus and gets minimum when is approxas a function of imately equal to 60 or to 240 . This is because, at these values, the main direction of incoming waves, which is shown in Fig. 5, is perpendicular to the receive array. Intuitively, since the distance between the two sub-channels and is maximum, the correlation is minimum. It also found that the correlation of the configuration with the V/H polarization is the lowest among the configurations, and that the correlation of the configuration with the slanted polarization is lower than that of the configuration with the V/V

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polarization. (c) V/H

polarization. Similar experimental observation can be found in [19]. The behavior of the correlation results in that of the average capacity as shown Fig. 8. It is indeed found that the or at . In capacity is the highest at addition, although the correlation of the configuration with the polarV/V polarization is higher than that of the slanted ization, their capacity is almost the same, varying between 3.9 and 4.1 bps/Hz. This is because the configuration employing polarization suffers from the power loss as a the slanted consequence of the amount of depolarization. It is also found that the capacity of the configuration with the V/H polarization is almost constant at 5 bps/Hz, or 20% higher than that of the polarizations. configurations with the V/V and slanted Thus, the V/H polarization is effective from the viewpoint of the correlation to enhance the capacity in MIMO channels. On the other hand, cross-polarization diversity is a means to create uncorrelated channels across antenna elements. Third, we investigate the effect of XPD. Fig. 9 shows the correlation as a function of mean XPD . The correlation is constant when considering the V/V polarization since the receiving antennas receive vertical waves only. The behavior of pothe correlation of the configurations with the V/H and

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Fig. 9. Correlation versus XPD for the different configurations. Fig. 7. Correlation versus receiving antenna orientation for the different configurations.

Fig. 10. Capacity versus XPD for the different configurations.

Fig. 8. Capacity versus receiving antenna orientation for the different configurations.

larizations is different, getting slightly lower as decreases due to the un-correlation of the orthogonal components. Similar redB, the correlation is sults are found in [32] and [26]. At minimum, implying that if there is richness of multipath in the propagation environment where all scatterers effectively depolarize electromagnetic waves into the same power level of vertical and horizontal polarization, the correlation is reduced. The behavior of the capacity is shown in Fig. 10. The capacity of the cross-polarized configurations decreases as decreases. This is because the reduction in the SNR due to the amount of depolarization is more dominant than the reduction in the correlation as decreases. Fourth, we examine the effect of receiving antenna spacing . Figs. 11 and 12 shows the correlation and capacity versus , respectively. As can be seen, the correlation decreases as increases. The capacity of the configurations with the V/V and polarizations increases gradually as increases while that of the configuration with the V/H polarization is almost con, the correlation of the configuration with the stant. At V/V polarization is unity while that of the configuration with the

Fig. 11. Correlation versus receiving antenna spacing for the different configurations.

V/H polarization is 0.4. Consequently, the capacity of the V/V configuration is 3.44 bps/Hz while that of the V/H configuration is about 5 bps/Hz, or 47% higher than that of the V/V configuration. This means that antennas can be co-located by using orthogonality of polarization. Polarization diversity, therefore,

DAO et al.: 3D POLARIZED CHANNEL MODELING AND PERFORMANCE COMPARISON OF MIMO ANTENNA CONFIGURATIONS

Fig. 12. Capacity versus receiving antenna spacing for the different configurations.

Fig. 13. Capacity versus SNR for the different configurations.

becomes more attractive for implementing MIMO handset antennas within the limited space. Finally, we investigate the effect of SNR on the capacity for the configurations. As shown in Fig. 13, in a low SNR region, the difference in the capacity among the configurations is insignificant. When SNR is less than about 11 dB, the capacity polarization is less of the configuration with the slanted than that of the configuration with the V/V configuration. In a high SNR region, the dual polarized configurations are effective. On the other hand, when the SNR is low, the reduction in the correlation of the dual polarized configurations is not enough to compensate the power loss as a result of the depolarization. However, when the SNR is high, the benefit of the low correlation of the dual polarized configurations is very effective. V. CONCLUSION MIMO systems are considered a potential solution to the demand for high data rates of up to the order of 1 Gbps in next generation systems. An understanding of their channels is critical to the development of multi-antenna systems. This paper proposes a 3D polarized channel model for MIMO systems. In

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this model, the scatterers are distributed over the sphere centered on the antenna array. The probability distribution of scatterers is based on the VMF distribution that is suitable for directions in space. In particular, derivation of a close form expression for the spatial correlation matrix which completely characterizes the spatial properties of the channel has been presented. Comparison in terms of the correlation between the 2D and 3D channel models has been made to show the inaccurate estimation of the 2D model in the scenarios. Monte Carlo simulations are performed to evaluate and compare 2 2 MIMO configurations with different polarizations, i.e., V/V, V/H, and slanted . The results show that the dual polarized configurations outperforms the single polarized configuration in a high SNR region and a high XPD. The agreement of the behavior of the correlation in our simulation with that of the correlation in the literature [19], [26], [32] confirms the validation of our model. The proposed model can be used to evaluate the effect of arbitrary parameters on performance of MIMO systems in terms of both correlation and capacity. The proposed model is presented here for linear array, but can be extended to any kind of array type. REFERENCES [1] I. Telatar, “Capacity of multiple-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, Nov. 1999. [2] G. Foschini and M. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wirel. Pers. Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998. [3] D.-S. Shiu, G. Foschini, M. Gans, and J. Kahn, “Fading correlation and its effect on the capacity of multielement antenna systems,” IEEE Trans. Commun., vol. 48, no. 3, pp. 502–513, Mar. 2000. [4] D. Gesbert, M. Shafi, D. S. Shiu, P. Smith, and A. Naguib, “From theory to practice: An overview of MIMO space-time coded wireless systems,” IEEE J. Sel. Areas Commun., vol. 21, no. 3, pp. 281–302, Apr. 2003. [5] Spatial channel model for multiple input multiple output (MIMO) simulations 2003, TR 25.996, 6.1.0, 3 GPP. [6] H. Xu, D. Chizhik, H. Huang, and R. Valenzuela, “A generalized spacetime multiple-input multiple-output (MIMO) channel model,” IEEE Trans. Wireless Commun., vol. 3, no. 3, pp. 966–975, May 2004. [7] L. Jian, L. Thiele, and V. Jungnickel, “On the modelling of polarized MIMO channel,” presented at the Proc. Europ. Wireless, Apr. 2007. [8] C. Oestges, V. Erceg, and A. Paulraj, “Propagation modeling of MIMO multipolarized fixed wireless channels,” IEEE Trans. Veh. Technol., vol. 53, no. 3, pp. 644–654, May 2004. [9] T. Svantesson, “A double-bounce channel model for multi-polarized MIMO systems,” in IEEE 56th Vehicular Technology Conf. Proc. VTC Fall, 2002, vol. 2, pp. 691–695. [10] H. Asplund, A. Glazunov, A. Molisch, K. Pedersen, and M. Steinbauer, “The COST 259 directional channel model Part—II: Macrocells,” IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3434–3450, Dec. 2006. [11] M. Shafi, M. Zhang, A. Moustakas, P. Smith, A. Molisch, F. Tufvesson, and S. Simon, “Polarized MIMO channels in 3-D: Models, measurements and mutual information,” IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp. 514–527, Mar. 2006. [12] M. Shafi, M. Zhang, P. Smith, A. Moustakas, and A. Molisch, “The impact of elevation angle on MIMO capacity,” in Proc. ICC IEEE Int. Conf. Communications, Jun. 2006, vol. 9, pp. 4155–4160. [13] P. Kafle, A. Intarapanich, A. Sesay, J. McRory, and R. Davies, “Spatial correlation and capacity measurements for wideband MIMO channels in indoor office environment,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1560–1571, May 2008. [14] P. Kyritsi, D. Cox, R. Valenzuela, and P. Wolniansky, “Correlation analysis based on MIMO channel measurements in an indoor environment,” IEEE J. Sel. Areas Commun., vol. 21, no. 5, pp. 713–720, Jun. 2003. [15] A. Molisch, M. Steinbauer, M. Toeltsch, E. Bonek, and R. Thoma, “Capacity of MIMO systems based on measured wireless channels,” IEEE J. Sel. Areas Commun., vol. 20, no. 3, pp. 561–569, Apr. 2002.

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[16] P. Kyritsi, D. Cox, R. Valenzuela, and P. Wolniansky, “Effect of antenna polarization on the capacity of a multiple element system in an indoor environment,” IEEE J. Sel. Areas Commun., vol. 20, no. 6, pp. 1227–1239, Aug. 2002. [17] K. Sulonen, P. Suvikunnas, L. Vuokko, J. Kivinen, and P. Vainikainen, “Comparison of MIMO antenna configurations in picocell and microcell environments,” IEEE J. Sel. Areas Commun., vol. 21, no. 5, pp. 703–712, Jun. 2003. [18] P. Suvikunnas, J. Salo, L. Vuokko, J. Kivinen, K. Sulonen, and P. Vainikainen, “Comparison of MIMO antenna configurations: Methods and experimental results,” IEEE Trans. Veh. Technol., vol. 57, no. 2, pp. 1021–1031, Mar. 2008. [19] K. Nishimori, Y. Makise, M. Ida, R. Kudo, and K. Tsunekawa, “Channel capacity measurement of 8 2 MIMO transmission by antenna configurations in an actual cellular environment,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3285–3291, Nov. 2006. [20] K. Mardia and P. Jupp, Directional Statistics. New York: Wiley, 2000. [21] K. Mammasis, R. Stewart, and J. Thompson, “Spatial fading correlation model using mixtures of Von Mises Fisher distributions,” IEEE Trans. Wireless Commun., vol. 8, no. 4, pp. 2046–2055, Apr. 2009. [22] G. Byers and F. Takawira, “Spatially and temporally correlated MIMO channels: Modeling and capacity analysis,” IEEE Trans. Veh. Technol., vol. 53, no. 3, pp. 634–643, May 2004. [23] A. Molisch, “A generic model for MIMO wireless propagation channels,” in Proc. IEEE ICC’02, May 2002, vol. 1, pp. 277–282. [24] E. I. Telatar and D. Tse, “Capacity of and muatual information of broadband multipath fading channels,” in Proc. IEEE Int. Symp. Information Theory, 1998, p. 188. [25] P. Kyritsi and D. Cox, “Propagation characteristics of horizontally and vertically polarized electric fields in an indoor environment: Simple model and results,” in Proc. 54th IEEE Veh. Technol. Conf., 2001, vol. 3, pp. 1422–1426. [26] C. Oestges, B. Clerckx, M. Guillaud, and M. Debbah, “Dual-polarized wireless communications: From propagation models to system performance evaluation,” IEEE Trans. Wireless Commun., vol. 7, no. 10, pp. 4019–4031, Oct. 2008. [27] M. T. Dao, V. A. Nguyen, and S. O. Park, “Derivation and analysis of spatial correlation for 2 2 MIMO system,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 409–413, 2009. [28] L. Schumacher and B. Raghothaman, “Closed-form expressions for the correlation coefficient of directive antennas impinged by a multimodal truncated Laplacian PAS,” IEEE Trans. Wireless Commun., vol. 4, no. 4, pp. 1351–1359, Jul. 2005. [29] A. Zajic, G. Stuber, T. Pratt, and S. Nguyen, “Wideband MIMO mobile-to-mobile channels: Geometry-based statistical modeling with experimental verification,” IEEE Trans. Veh. Technol., vol. 58, no. 2, pp. 517–534, Feb. 2009. [30] A. Banerjee, I. Dhillon, J. Ghosh, and S. Sra, “Clustering on the unit hypersphere using Von Mises-Fisher distributions,” J. Machine Learning Res., vol. 6, pp. 1345–1382, 2005. [31] T. Taga, “Analysis for mean effective gain of mobile antennas in land mobile radio environments,” IEEE Trans. Veh. Technol., vol. 39, no. 2, pp. 117–131, May 1990. [32] F. Quitin, C. Oestges, F. Horlin, and P. De Doncker, “Multipolarized MIMO channel characteristics: Analytical study and experimental results,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2739–2745, Sep. 2009.

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Manh-Tuan Dao was born in Hoabinh, Vietnam, on December 2, 1978. He received the B.Sc. degree in electrical engineering from Hanoi University of Technology, Hanoi, Vietnam, in 2001 and the M.Sc. degree in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2009, where he is working toward the Ph.D. degree. From December 2001 to December 2006, he worked as an RF Engineer at the Radio Frequency Directorate (RFD), Vietnam. His current research interests include channel modeling and measurement for MIMO systems, and design of MIMO antenna arrays and MIMO radar systems.

Viet-Anh Nguyen received the B.Sc. degree in telecommunication and electrical engineering from the Post and Telecommunication Institute of Technology, Hanoi, Vietnam, in 2002 and the M.Sc. degree in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2008, where he is currently working toward Ph.D. degree. From February 2002 to February 2006, he worked as a Mobile Network Researcher at Research Institute of Post and Telecommunication, Hanoi, Vietnam. His current research interests are MIMO antenna systems, Reconfigurable beam antennas, smart antennas, and small antenna array.

Yun-Taek Im was born in ChungNam, Korea, on June 7, 1978. He received the B.S. degree in electrical engineering and computer science from Hanyang University, Korea, in 2005, the M.S. degree in electronic and electrical engineering from the Pohang University Science and Technology (POSTECH), in 2007. He is currently working toward the Ph.D. degree at the Korea Advanced Institute of Science and Technology (KAIST), Daejeon. From 2005 to 2007, he was engaged in development of RFID antennas and millimeter-wave beamforming systems. His current research interests are MIMO antenna systems and radar systems including antenna design.

Seong-Ook Park (M’05) was born in KyungPook, Korea, in December 1964. He received the B.S. degree from KyungPook National University, in 1987, the M.S. degree from Korea Advanced Institute of Science and Technology, Daejeon, in 1989, and the Ph.D. degree from Arizona State University, Tempe, in 1997, all in electrical engineering. From March 1989 to August 1993, he was a Research Engineer with Korea Telecom, Daejeon, working with microwave systems and networks. He later joined the Telecommunication Research Center, Arizona State University, until September 1997. Since October 1997, he has been with the Information and Communications University, Daejeon, first as an Associate Professor, and currently as a Professor at the Korea Advanced Institute of Science and Technology, IT convergence Campus. His research interests include mobile handset antenna, and analytical and numerical techniques in the area of electromagnetics. Dr. Park is a member of Phi Kappa Phi.

Giwan Yoon (M’94) was born in Pohang, Korea, in 1959. He received the B.S. degree from Seoul National University (SNU), Seoul, Korea, in 1983, the M.S. degree from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, in 1985, and the Ph.D. degree from the University of Texas at Austin, in 1994. From 1985 to 1990, he was with LG Group, Seoul, Korea. From 1994 to 1997, he was with Digital Equipment Corporation, MA. From 1997 to 2009, he was a Professor in the School of Engineering, Information and Communications University (ICU), Korea. Currently, he is a Professor in the Department of Electrical Engineering, KAIST. His major research areas of interest include multifunctional intelligent devices and their technologies for RF and wireless applications. Dr. Yoon is a member of KIMICS, Korea.

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Subwavelength Radio Repeater System Utilizing Miniaturized Antennas and Metamaterial Channel Isolator Kamal Sarabandi, Fellow, IEEE, and Young Jun Song, Student Member, IEEE

Abstract—Implementation of a novel high gain miniaturized radio repeater for improving wireless network connectivity in complex environment is presented in this paper. Unlike existing repeater systems, this system utilizes two closely spaced low profile miniaturized planar antennas capable of producing omnidirectional and vertical radiation pattern as well as a channel isolator layer that serves to decouple the adjacent antennas. The metamaterial based channel isolator serves as an electromagnetic shield, thus enabling it to be built in a sub-wavelength size of 0 07 20 0 014 0 , the smallest repeater ever built. A prototype of the small radio repeater is fabricated to verify the design performance through a standard free-space measurement setup. The feasibility of amplifying and re-transmitting the received signal is demonstrated through measurement which compares well with the numerical simulation results. Index Terms—Antenna array mutual coupling, electromagnetic shielding, indoor radio communication, multiaccess communication.

I. INTRODUCTION A. Background of This Study

F

OR wireless network systems, the path-loss between the transmitter and receiver is a critical factor that determines the possible range of communication between two nodes. Complex environments such as urban canyons and building interiors often contain numerous obstacles that impede the line-of-sight (LOS) communication and increase the path-loss. The existing long range ad-hoc communication network relies on multipath (multiple reflection, diffraction, and penetration through obstacles). In these environments especially at high frequencies the path-loss dramatically increases, which often requires higher transmitter power and closely spaced communication nodes. Furthermore, as transmitter power increases or as transmitting nodes become closer, the potential for mutual interference between communication cells increases which, Manuscript received January 13, 2010; revised November 04, 2010; accepted December 10, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This research was supported by the U.S. Army Research Laboratory under contract W911NF and prepared through collaborative participation in the Microelectronics Center of Micro Autonomous Systems and Technology (MAST) Collaborative Technology Alliance (CTA). The authors are with the Radiation Laboratory, Department of Electrical Engineering and Computer Science, The University of Michigan at Ann Arbor, Ann Arbor, MI 48109-2122 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.2011.2152320

if present, can cause degradation in coverage capacity. Additionally, a topology that uses closely spaced nodes will be more expensive than a similar topology sparsely populated with nodes. To overcome these situations and to help improve the ground area coverage of communication signals without increasing the transmitter power, radio repeaters have been extensively used in various application scenarios. Numerous studies regarding feasibility and operation of the radio repeater have been presented in [1]–[4]. Additionally, numerous commercial products utilizing the concept of the radio repeater have been introduced and fabricated such as in [5] and [6]. The main objective of the radio repeater in these scenarios is to achieve enhanced connectivity by amplifying a radio signal through an active device as shown in Fig. 1. For the downlink communication, from a base station to an end-node/unit, the signal originating at the base station is linked through the Receive antenna (RX) of repeater, amplified, and retransmitted through the Transmit antenna (TX), and vice versa for the uplink direction. However, the mutual coupling between a repeater’s RX and TX antennas generates a positive feedback loop as shown in Fig. 1. When the gain of the RF amplifier is greater than the isolation level of the RX and TX antennas, the overall system will start to oscillate, and the communication coverage of that micro cell cannot be established. Thus, the level of mutual coupling limits the performance of a radio repeater as well as the dimension and cost of the overall system. To circumvent this intrinsic problem, generally two approaches have been proposed. The first method is to divide the frequencies of the uplink and downlink signals. This methodology utilizes a frequency division duplex (FDD) to reduce the mutual coupling by separating signal frequencies. However, it requires complex circuitry, larger size, and a common protocol to manage frequency allocation, which all imply higher cost and much more power consumption. The second method is to adapt a time division duplex (TDD) in time domain. This also introduces additional logic circuitry, latency, and knowledge of the repeater, transmitter and receiver locations, as presented in [7] and [8]. In this paper, a new device is proposed to overcome the adverse effect of various complex environments by reducing the path-loss. To enhance radio connectivity and maintain low-power communication, a very small radio repeater with a large radar cross section (RCS) and an omnidirectional radiation pattern is proposed. The proposed radio repeater receives the LOS signal from the transmitter, amplifies it, and then retransmits the amplified signal omnidirectionally, which establishes a secondary LOS to arbitrary receivers. Thus, a

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Fig. 1. Schematic of radio link using radio repeater.

Fig. 3. Modified MMA: (a) topology of modified MMA for small radio repeater; (b) design parameters.

TABLE I DESIGNED PARAMETERS OF MODIFIED MMA

Fig. 2. Schematic of proposed small radio repeater.

II. DESIGN SPECIFICATIONS A. Miniaturized Repeater Antenna

series of small radio repeaters can enhance radio connectivity by establishing a LOS with the repeater nodes. B. Benefits of the Proposed Small Radio Repeater Fig. 2 shows the proposed Small Radio Repeater. It consists of two miniaturized planar antennas capable of supporting an omnidirectional pattern and vertical polarization. Additionally, a metamaterial based isolator structure is used, and active RF amplification circuitry as well as a battery is integrated into the radio repeater platform. Since RX/TX of the proposed repeater shows omnidirectionality over the H plane, the uplink and downlink signal paths can be established through a single circuit path. This can reduce complexity and power consumption of RF circuitry. In addition, pure vertical polarization allows for a simple antenna structure for the base station and end-node/units as well as a decreased path-loss along the channel. It is well understood that for near-earth wave propagation scenarios vertically polarized waves experience much less path-loss compared to horizontally polarized waves as presented in [9]. To achieve the compact dimension of the radio repeater, a metamaterial based channel isolator is utilized. By generating the normal magnetic field along the signal path between RX and TX, artificial magnetic walls are generated that serve to suppress the electromagnetic wave propagation from RX to TX antennas. The proposed radio repeater occupies a very small area with a very short height without its active circuitry. The passive components such as two miniaturized antennas and metamaterial based isolator are presented and verified in this paper. The prototype is fabricated using a commercially available dielectric substrate. In addition, a commercially available RF amplifier and battery are used to verify the operational feasibility of the proposed repeater. This configuration is shown to boost the power level of the received signal by 32 dB.

According to antenna theory, the intensity and polarization of the radiated field are proportional to the level and direction of the current distribution over the antenna. In many practical miniaturized antennas, the level of the excited current is limited by the impedance mismatch between the feeding network and the antenna itself. To achieve a low-profile miniaturized antenna, a quarter-wave microstrip resonator fed near the short-circuited end is used. To achieve miniaturization and impedance matching a four-arm spiral shape quarter-wave resonator structure was utilized and presented in [10]. Although a good input reflection coefficient at the frequency of operation can be obtained, this antenna requires two layers; an upper layer consisting of the open-ended spiral shape line and a lower layer consisting of the short-circuited transmission line for the matching network to radiate the power effectively. This physical structure increases complexity and cost for fabrication. And any misalignment between the two layers can shift the operation frequency. The proposed new approach is to place the matching network at the same layer of the miniaturized antenna as shown in Fig. 3. Although the main radiation is from the shorted pins, some radiation is emanated from the spiral arms. The polarization of the radiated field from the spiral arms is horizontal. Therefore, each of the spiral arms should be placed in a symmetrical manner in order to minimize horizontally polarized radiated field. Symmetry of these arms is essential to cancel such radiated fields and eventually achieve an omnidirectional vertically polarized radiation pattern. In addition, symmetry and of the shorting pins enable close spacing the excited currents through these pins to be in-phase. The short circuited currents that pass through the four vertical pins are the dominant radiating elements and responsible for the vertically polarized radiated field. The optimized miniaturized multi-elements monopole antenna (MMA) is designed using a commercial finite element method solver (Ansoft’s HFSS ver. 11.1). To be incorporated into the small radio repeater, the optimized MMA must be further modified to achieve smaller dimensions. Because of the sub-wavelength dimension of the proposed radio

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Fig. 4. Simulated responses of modified MMA: (a) S H(xy) plane.

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response; (b) radiation pattern in E(zx) plane; (c) radiation pattern in E(yz) plane; (d) radiation pattern in

repeater, a pair of the optimized MMA with four arms can produce a high level of mutual coupling within the small ground plane. Thus, the objective is to modify the MMA geometry so that mutual coupling can be reduced while maintaining the polarization purity and the desired radiation pattern. By utilizing only two arms of the MMA with a symmetric topology as shown in Fig. 3, the horizontal current cancellation and reduction of mutual coupling can be achieved at the expense of an asymmetric radiation pattern in the E plane. The modified MMA is designed and verified using Ansoft’s HFSS. The designed topology and parameters are shown in Fig. 3. The physical dimensions are optimized for operation around 2.5 GHz and summarized in Table I. And all line length and spacing (between traces) values are set to 0.4 mm, except for , and which are set to 0.6 mm. Fig. 4(a) presents the simulated input reflection coefficient of the modified MMA. As can be seen, the modified MMA shows a 18.5 dB of input reflection coefficient at the design frequency. The radiation pattern of the modified MMA is shown in Fig. 4(b), (c), and (d). The vertical polarization in the H-plane shows an omnidirectional pattern, similar to that of a monopole. In addition, the level of the horizontal polarization in the H-plane is negligible compared to the vertical polarization, which implies that the cancellation between horizontal electric current is achieved effectively. Furthermore, the transmission line based antenna can be fabricated using the

standard printed circuit technology. This serves to reduce the alignment error observed in the multi-layer design, and hence the fabrication process and the discrepancy between the simulation and measurement are reduced significantly. B. Metamaterial Based Channel Isolator In practical antenna systems, the mutual coupling between adjacent antennas restricts compact integration of multiple antennas in a small area for applications such as multiple input and multiple output (MIMO) communication systems. To suppress the mutual coupling, various approaches have been studied and presented in [11]–[13]. In general, these studies can be categorized into two approaches. The first method is to engineer the electric and magnetic properties of the material, such as the permittivity and permeability, by introducing an artificial structure. For example, a mushroom-like structure can suppress the mutual coupling by introducing a negative refractive index, as shown in [11] and [12]. Another method utilizes metamaterial insulators to block the EM energy from being transmitted across the insulation boundary, as shown in [13]. This metamaterial insulator consists of magneto-dielectric embedded circuits, which can be modeled as parallel LC resonant circuits. However, both approaches have intrinsic limitations when attempting to suppress the mutual coupling of antennas at the commercial ISM

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TABLE II DESIGNED PARAMETERS OF CHANNEL ISOLATOR

Fig. 5. Unit cell of channel isolator: (a) topology of unit cell; (b) square loop by image theory.

frequency band (around 2.5 GHz). The artificial structure requires large physical dimensions, and the metamaterial insulator causes fabrication complexity and cost. To address these limitations, a metamaterial based channel isolator is proposed and designed as shown in Fig. 5(a). The proposed isolator is designed to resonate at the desired channel frequency and decrease the mutual coupling by suppression of the surface waves in the substrate generated by the vertical pins of the MMA. The vertical pins create a transverse magnetic (TM) wave in the substrate with zero cutoff frequency. The magnetic field is parallel to the ground plane and perpendicular to the pins. To inhibit propagation of the TM surface wave, an electromagnetic band-gap (EBG) metamaterial layer can be utilized. The advantage of the band-gap material is that it creates an equivalent open circuit to the surface wave as opposed to a short-circuit that a metallic wall can produce. The band-gap metamaterial consists of an array of the parallel LC resonant circuits that are magnetically coupled with the substrate mode. This is realized using unit cells that consist of vertical wires and horizontal conducting strips, which behave like a distributed inductor and a distributed capacitor, respectively. Using image theory, when each loop is imaged a larger loop having a larger inductance and a smaller capacitance is formed as shown in Fig. 5(b). Assuming the fundamental mode propagates from the TX to the RX through the substrate, the horizontally polarized magnetic field linked by the square loop induces an electric current on the vertical wires. In addition, this induced current generates a magnetic field which is perpendicular to the loop. When the unit cells are closely spaced to each other, the inductance of the loops is increased and the periodic array acts like a solenoid. At resonance the periodic layer acts as a perfect magnetic conductor (PMC) plane. Due to the mutual coupling of the adjacent loops, the self inductance of the square loop as shown in Fig. 5(b) can be obtained from [14]

(1)

values and low Q factor will cause the suppression of the mutual coupling to deteriorate. To simultaneously reduce the deviation of these values, improve the Q factor, and lower the cost of fabrication, printed circuit technology can be utilized to implement the capacitors. As mentioned, the magnetic field induces the electric current on the vertical wires, and this current transforms to a displacement current (electric field) as it gets through the gaps between the fingers of the series interdigital capacitor. As most of the electric field between the metallic strips of this capacitor is in the gap and perpendicular to the metallic edges, its capacitance can be computed from the capacitance per unit length of two thin co-planar strips given by [15] (2)

(3) where is the metallization ratio and the complete elliptic integral of first kind defined by

is

(4) Since the individual capacitors between fingers are connected in parallel, the total capacitance per unit length of interdigital capacitor is equal to (5) is between where is the capacitance between inner strips, outer and inner strips, and is the number of fingers. Hence, the total capacitance of the proposed isolator can be calculated easily from , where is the length of the fingers. The interdigital capacitor is centered between two vertical wires. The designed parameters are summarized in Table II. The length and height of the unit cell are chosen to be 6.25 mm 1.57 mm. The corresponding inductance and capacitance of the unit cell are thus found to be 15.2132 nH and 0.1869 pF, respectively. Based on this calculation, the self resonant frequency is calculated to be 2.98 GHz. III. PARAMETRIC STUDIES AND OPTIMIZATION

where is the internal area of the loop and is the periodicity of unit cells. The quality factor (Q) of the equivalent single pole isolator affects the performance and isolation bandwidth. Thus, commercial lumped capacitors with finite deviation of capacitance

A. Optimal Configuration Without the Isolator In Section II, the principle of operation of the MMA was established. The modified MMA with two symmetric arms and vertical pins was designed on a small ground plane and was

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Fig. 6. Repeater platform without channel isolator.

shown to produce a very good impedance match and vertical polarization. This design is further optimized to be integrated into the channel isolator as shown in Fig. 6. As mentioned before, two arms of the original MMAs that would come close to the metamaterial isolator are removed. The magnetic field from these arms could have coupled to the isolator loops and established a link between the two antennas instead of isolating them. In a miniaturized antenna, the size of the ground plane can also affect the performance of the antenna. The edge currents on the ground plane affect the radiation pattern, directivity, and polarization. To maintain small physical dimensions, the design parameters for the optimized configuration include the position of the TX and RX antennas as well as the dimensions of the ground plane. The optimization is performed using HFSS to achieve impedance matching at the desired frequency, maintaining an isolation level smaller than 20 dB, and minimizing the size of the ground plane. Fig. 7(a) represents a parametric study where the simulation responses of (coupling) between RX and TX antennas are displayed. In this simulation the spacing between the antenna and edges of the ground plane are fixed and the distance between the two antennas, is varied. As can be seen, the distance between the two antennas does not play a major role in the mutual coupling between the two antennas. This implies that the amount of coupling from surface wave propagation is not affected by the separation distance within the specified range of distances shown in Fig. 7(a). However, in choosing the ground plane size the overall dimension of the small radio repeater platform and the space for the channel isolator should be taken into account to avoid any interaction between the two antennas and the channel isolator. The optimized distance between the two antennas is found to be 25 mm. In addition, the width of the ground plane affects the level of mutual coupling due to excitation of edge currents. This effect is shown in Fig. 7(b) where all other dimensions are fixed ( , ) and is changed. To account for the integration of an RF amplifier on the backside, the dimensions of the platform are finally chosen to be 40.01 mm 20.68 mm, which corresponds to . B. Isolator and Antenna Integration The geometry of the proposed miniaturized radio repeater is composed of two miniaturized low-profile antennas capable of radiating vertical polarization and a metamaterial isolator layer as shown in Fig. 8. As mentioned before, close spacing between the antennas and the isolator causes the mutual coupling, and

Fig. 7. Simulated coupling between antennas shown in Fig. 6: (a) varying the distance between RX and TX; (b) varying ground plane width.

Fig. 8. Repeater platform with channel isolator.

therefore it affects the performance of the repeater. Specifically, the antennas input impedances and the resonant frequency of the isolator both change as a result of the placement of the antennas and the isolator. The current distribution on the ground plane is used to evaluate the optimal placement of the isolator. As all of the physical parameters are related to each other through various electromagnetic interactions, optimization is achieved through adjusting the length of the isolator loop and the strip iteratively using HFSS. As an initial step in the design, the RX and TX antennas as well as the isolator are designed separately. For integration, since multiple resonant structures within sub-wavelength dimensions are used, the use of manual mesh modifications in HFSS is required to capture the details of the fields around the isolator. The designed topology is in Fig. 8. Physical parameters

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TABLE III DESIGNED PARAMETERS OF THE SMALL RADIO REPEATER

Fig. 10. Simulated radiation pattern: (a) single ground plane without isolator; (b) separated ground planes without isolator; (c) single ground plane with channel isolator. Fig. 9. Simulated S-parameters of the small radio repeater with and without the metamaterial isolator.

are optimized for the repeater to operate around 2.72 GHz and are reported in Table III. The optimized simulation response and measured data are discussed in Sections IV and V. IV. REPEATER SIMULATION RESULTS In this section full-wave analysis is carried out to examine the performance of the proposed repeater. Fig. 9 shows the simulated S-parameters of the optimized small radio repeater platform. As shown, a 20 dB of transmission coefficient can be achieved between the TX and RX antennas without the channel isolator with designed dimensions (assuming the two antennas are well matched.) Incorporating the channel isolator, the transmission coefficient drops to 30 dB. Also shown is that the antenna response is affected due to the interaction between the antennas and the isolator. In fact, per our design the antennas are well matched (over 15 dB of input reflection coefficient), and the center frequency is at the desired value in the presence of the isolator. The presence of the antenna also affects the isolator frequency response. As shown before, the resonant frequency of an isolated unit cell of the metamaterial isolator is at 2.98 GHz. With some small adjustments, in the presence of the antenna, this resonance occurs at 2.72 GHz as shown in Fig. 9. It should be noted here that if the ground planes of transmit and receive antennas are disconnected improved isolation can be achieved. However, this way no amplifier can be inserted between the transmit and receive antennas. The simulated radiation patterns in H plane are represented in Fig. 10. In order to provide enough area for two modified

antennas and a channel isolator, the ground plane should be extended along the longitudinal direction, which breaks the symmetry of the ground plane. Although the modified MMA with square ground plane shows pure vertical polarization in H plane (see Fig. 4(d)), the horizontal induced current on the rectangular ground plane generates horizontal polarization in H plane as shown in Fig. 10(a). In all cases, however, the antenna shows around 9 dBi of gain in vertical polarization. The gain of the antenna is limited due to dielectric and metallic losses. Increasing the dielectric thickness and increasing antenna dimensions increase the gain. Also it should be noted that the distance between two antennas is about 25 mm. For such small separations, the coupling mainly comes from the antennas’ near field. Fig. 11 shows the magnetic field distribution over the ground plane for the repeater with and without the isolator. As expected, the horizontal magnetic field generated from the lower antenna in Fig. 11(a) propagates through the substrate and produces the mutual coupling to the top antenna. When incorporating the channel isolator, Fig. 11(b) indicates that the horizontal magnetic field is maximized within the channel isolator. This implies that the surface currents (perpendicular to the direction of magnetic field in the channel isolator) are interrupted by the isolator, as shown in Fig. 12. V. EXPERIMENTAL RESULTS The prototype of the proposed small radio repeater system is fabricated using a 1.57 mm-thick Rogers RO-5880 substrate , as shown in Fig. 13. Fig. 14 shows the measured S-parameters of the small radio repeater and indicates that the resonant frequency is located at 2.72 GHz. Since the fabrication process includes physical limitations such as under cutting

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Fig. 13. Fabricated small radio repeater w/wo channel isolator.

Fig. 11. Simulated H field distribution: (a) small radio repeater without channel isolator; (b) small radio repeater with channel isolator.

Fig. 14. Measured S-parameters of proposed small radio repeater with and without channel isolator.

Fig. 15. Verification for operation of small radio repeater.

Fig. 12. Simulated current distribution: (a) small radio repeater without channel isolator; (b) small radio repeater with channel isolator.

of the copper in the etching process and errors in alignment, the frequency shift between the computer based design and actual fabrication is unavoidable. However, this discrepancy can be corrected after a few trials. In addition, post tuning and optimization can be used to obtain the designed performance. Fig. 15 shows the system setup for the measuring of S-parameters. To feed each of antennas, two microstrip transmission lines are used where one is connected to an amplifier with variable gain and its output is connected to the other antenna through a directional coupler. The directional coupler is inserted between RF amplifier and transmit antenna to monitor the occurrence of oscillation when it happens as the gain is increased. As can be seen in Fig. 14, the repeater without the channel isolator shows a 18 dB of transmission coefficient, and the proposed repeater shows a 42 dB of transmission coefficient, which indicates 24 dB of suppression improvement. With the channel

isolator, 28 dB of peak level of is observed, however, in spite of 1 dB of insertion loss from the directional coupler, a maximum gain of 32 dB for a wideband RF amplifier can be utilized with this repeater. Since the antennas are slightly mismatched near to the desired frequency (2.72 GHz), the max. Therefore, it imum gain can be higher than the peak level of is verified that a commercial wideband RF amplifier with a gain of 32 dB can be integrated into the proposed repeater without oscillation. VI. CONCLUSIONS In this paper, a new concept for implementation of miniaturized radio repeater is presented. To construct the radio repeater, two miniaturized low-profile antennas radiating vertical polarization and a very thin metamaterial isolator layer are integrated into a compact configuration. The antennas are designed to have an omnidirectional radiation pattern to make the repeater insensitive to the positions of the transmitter and receiver. In addition, the proposed

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isolator is shown to suppress the mutual coupling, improving the transmission coefficient from 18 dB to 42 dB. The dimensions of the TX/RX antennas and a unit cell of the isolator are and , respectively. The overall dimensions of the proposed radio repeater are , which corresponds . The proposed radio repeater to system has been simulated and verified experimentally. The prototype of the design has been fabricated using printed circuit technology, which serves to reduce fabrication complexity and allows for easy commercial production at a large scale. Such a radio repeater system can mitigate the adverse effects of obstacles in radio connectivity for ad-hoc networks in complex environments. REFERENCES [1] S. K. Park, P.-J. Song, and G. S. Bae, “Joint optimization of radio repeater location and linking in WLL systems with 2.3 GHz frequency band,” in Proc. IEEE Int. Conf. on Communications, 1999, vol. 3, pp. 1617–1621. [2] P. Slobodzian, “Estimation of the repeater gain required for a wireless link,” in Proc. 15th Int. Conf. on Microwaves, Radar and Wireless Communications, May 2004, vol. 2, pp. 656–659. [3] J. Borkowski, J. Niemela, T. Isotalo, P. Lahdekorpi, and J. Lempiainen, “Utilization of an indoor DAS for repeater deployment in WCDMA,” in Proc. IEEE 63rd Vehicular Technology Conf., May 2006, vol. 3, pp. 1112–1116. [4] A. H. Naemat, A. Tee, A. S. M. Marzuki, B. Mohmd, K. Khalil, and A. R. A. Rahim, “Achieving optimum in-building coverage of 3 G network in Malaysia,” in Proc. IEEE Int. RF and Microwave Conf., Sep. 2006, pp. 343–346. [5] B. W. Lovinggood, M. D. Judd, and W. P. Kuiper, “Integrated Repeater,” U.S. Patent 6,934,511 B1, 2005. [6] D. L. Runyon, S. B. Thompson, J. W. Maxwell, and D. L. M. SR., “Wireless Repeater With Feedback Suppression Features,” U.S. Patent 2006/0 205 343 A1, 2006. [7] T. W. Ban, B. Y. Cho, W. Choi, and H.-S. Cho, “On the capacity of a DS/CDMA system with automatic on-off switching repeaters,” in Proc. IEEE Int. Conf. on Communications, 2001, vol. 3, pp. 780–784. [8] M. Lee, B. Keum, Y. Son, J.-W. Kim, and H. S. Lee, “A new low-complex interference cancellation scheme for WCDMA indoor repeaters,” in Proc. IEEE Region 8 Int. Conf. on Computational Technologies in Electrical and Electronics Engineering, Jul. 2008, pp. 457–462. [9] D. Liao and K. Sarabandi, “Terminal-to-terminal hybrid full-wave simulation of low-profile, electrically-small, near-ground antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 806–814, Mar. 2008. [10] W. Hong and K. Sarabandi, “Low-profile, multi-element, miniaturized monopole antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 72–80, Jan. 2009. [11] D. Sievenpiper, L. Zhang, R. Broas, N. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Microw. Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [12] F. Yang and Y. Rahmat-Samii, “Microstrip antennas integrated with electromagnetic band-gap (EBG) structures: A low mutual coupling design for array applications,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2936–2946, Oct. 2003. [13] K. Buell, H. Mosallaei, and K. Sarabandi, “Metamaterial insulator enabled superdirective array,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1074–1085, April 2007. [14] H. Mosallaei and K. Sarabandi, “Design and modeling of patch antenna printed on magneto-dielectric embedded-circuit metasubstrate,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 45–52, Jan. 2007. [15] R. Igreja and C. J. Dias, “Analytical evaluation of the interdigital electrodes capacitance for a multi-layered structure,” Sensors Actuat. A: Phys., vol. 112, no. 2–3, pp. 291–301, 2004.

Kamal Sarabandi (S’87–M’90–SM’92–F’00) received the B.S. degree in electrical engineering from the Sharif University of Technology, Tehran, Iran, in 1980, the M.S. degree in electrical engineering in 1986, and the M.S. degree in mathematics and the Ph.D. degree in electrical engineering from The University of Michigan at Ann Arbor, in 1989. He is currently the Director of the Radiation Laboratory and the Rufus S. Teesdale Professor of Engineering in the Department of Electrical Engineering and Computer Science, The University of Michigan at Ann Arbor. His research areas of interest include microwave and millimeterwave radar remote sensing, metamaterials, electromagnetic wave propagation, and antenna miniaturization. He possesses 25 years of experience with wave propagation in random media, communication channel modeling, microwave sensors, and radar systems and leads a large research group including two research scientists, 14 Ph.D. students. He has graduated 36 Ph.D. and supervised numerous post-doctoral students. He has served as the Principal Investigator on many projects sponsored by the National Aeronautics and Space Administration (NASA), Jet Propulsion Laboratory (JPL), Army Research Office (ARO), Office of Naval Research (ONR), Army Research Laboratory (ARL), National Science Foundation (NSF), Defense Advanced Research Projects Agency (DARPA), and a large number of industries. Currently he is leading the Center for Microelectronics and Sensors funded by the Army Research Laboratory under the Micro-Autonomous Systems and Technology (MAST) Collaborative Technology Alliance (CTA) program. He has published many book chapters and more than 185 papers in refereed journals on miniaturized and on-chip antennas, metamaterials, electromagnetic scattering, wireless channel modeling, random media modeling, microwave measurement techniques, radar calibration, inverse scattering problems, and microwave sensors. He has also had more than 442 papers and invited presentations in many national and international conferences and symposia on similar subjects. Dr. Sarabandi is a member of Commissions F and D of URSI. He was the recipient of the Henry Russel Award from the Regent of The University of Michigan. In 1999, he received a GAAC Distinguished Lecturer Award from the German Federal Ministry for Education, Science, and Technology. He was also a recipient of the 1996 EECS Department Teaching Excellence Award and a 2004 College of Engineering Research Excellence Award. In 2005 he received the IEEE GRSS Distinguished Achievement Award and the University of Michigan Faculty Recognition Award. He also received the best paper Award at the 2006 Army Science Conference. In 2008 he was awarded a Humboldt Research Award from The Alexander von Humboldt Foundation of Germany and received the best paper award at the IEEE Geoscience and Remote Sensing Symposium. He was also awarded the 2010 Distinguished Faculty Achievement Award from the University of Michigan. The IEEE Board of Directors announced him as the recipient of the 2011 IEEE Judith A. Resnik medal. He served as a member of NASA Advisory Council appointed by the NASA Administrator for two consecutive terms from 2006–2010. He is serving as a Vice President of the IEEE Geoscience and Remote Sensing Society (GRSS) and is a member of the Editorial Board of the PROCEEDINGS OF THE IEEE. He was an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and the IEEE SENSORS JOURNAL. In the past several years, joint papers presented by his students at a number of international symposia (IEEE APS’95,’97,’00,’01,’03,’05,’06,’07; IEEE IGARSS’99,’02,’07; IEEE IMS’01, USNC URSI’04,’05,’06,’10,’11 AMTA ’06, URSI GA 2008) have received best paper awards.

Young Jun Song (S’07) received the B.S. degree in electrical engineering (summa cum laude) from the Seoul National University, Seoul, Korea, in 2006, and the M.S. degree in electrical engineering from The University of Michigan at Ann Arbor, in 2009, where he is currently working toward the Ph.D. degree.

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Improved Two-Antenna Direction Finding Inspired by Human Ears Rongguo Zhou, Member, IEEE, Hualiang Zhang, Member, IEEE, and Hao Xin, Senior Member, IEEE

Abstract—This paper presents an improved two-antenna microwave passive direction finding system inspired by the human auditory system. By incorporating a head-like scatterer between two monopole antennas, both phase and magnitude information can be utilized to estimate the direction of arrival (DOA) of a microwave signal, thus eliminating ambiguities associated with phase wrapping at high frequency, just like human ears. In addition, better DOA sensitivity is demonstrated with the incorporation of the head-like scatterer. Both numerical modeling and experimental results of a simple X-band two-monopole configuration with symmetric and asymmetric scatterers are presented. Index Terms—Biological inspired, direction of arrival (DOA) estimation, monopole.

I. INTRODUCTION ICROWAVE passive direction finding is a very important technology that has many military and commercial applications including electronic warfare [1], mobile communications [2], etc. A typical microwave direction finding system may use an array consists of a large number of antenna elements and sophisticated algorithms to achieve high degree of accuracy [1]. However, the size, weight, speed and cost associated with the large number of hardware components and the complicated signal processing can be impractical, especially for portable and commercial applications. For example, a hand-held direction finding gadget that is capable of identifying and locating the source of incoming microwave signals is highly desirable for a soldier in battlefield. In addition, accurate and efficient direction finding will be very useful in next generation wireless systems for location based services and applications. A very interesting biological system that is capable of direction finding for acoustic waves is the human auditory system which includes a pair of pinnae located at each side of the human head to collect acoustic signals, ear canals, eardrums and cochlea to guide and detect incoming acoustic signals and auditory nerves and neurons in the brain to process the detected signals. In fact, human ears have many amazing and intriguing abilities related to direction finding, among them, estimating

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Manuscript received August 11, 2010; revised November 03, 2010; accepted December 10, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported in part by the Army Research Office (ARO). R. Zhou and H. Xin are with the Department of Electrical and Computer Engineering and the Department of Physics, University of Arizona, Tucson, AZ 85721 USA (e-mail: [email protected]; [email protected]). H. Zhang is with the Department of Electrical Engineering, University of North Texas, Denton, TX 76207 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.2011.2152348

arrival angle with accuracy up to 1 without ambiguity under binaural (utilizing two ears) condition, the famous “cocktail party phenomenon,” the ability of “learning the surrounding environment within seconds,” sound source localization with a single ear (mono-aural), to name a few [3]. In this paper, we propose a novel DOA estimation technique using only two antennas, which is inspired by the human auditory system. The idea is to utilize a lossy scatterer between two antennas, which emulates the low-pass filtering function of the human head at high frequency [3], [4], to achieve more accurate DOA estimation. A simple 2-monopole example at X-band frequency (8 to 12 GHz) is studied and the multiple signal classification (MUSIC) algorithm [5] is applied to calculate the DOA. Our theoretical and experimental studies have shown that by incorporating a head-like scatterer between two antennas, not only high frequency ambiguity associated with phase wrapping is eliminated, but also the DOA sensitivity can be improved significantly. This paper is organized as the following. In Section II, the analogy between the human auditory system sound source localization and microwave passive direction finding is discussed. In Section III, the effectiveness of the proposed technique is demonstrated by numerical examples. Measurement results of various two-antenna direction finding configurations are reported in Section IV. A brief conclusion is given in Section V. II. ANALOGY BETWEEN HUMAN SOUND LOCALIZATION AND MICROWAVE DIRECTION FINDING Passive direction finding (or sometimes referred to as direction of arrival, DOA, estimation) for microwave signal is very analogous to the direction finding of acoustic wave by human ears. Even the wavelengths of audible sound waves are comparable to that of microwave frequency, for example, 3 KHz sound has a wavelength of 113 mm which is about the wavelength of a 2.65 GHz microwave signal. Fig. 1 illustrates the analogy between a microwave direction finding system and the human auditory system. The microwave antennas are similar to the pinnae which are natural directional antennas for acoustic waves; the band-pass filters, amplifiers, mixers and detectors provide similar functions as the guiding and detecting parts of the human auditory system; the signal processing component can be thought as the brain. The remarkable localization (mainly in the azimuth plane) capabilities of human ears for both continuous waves (CW) and transient signals have long been recognized and studied quite extensively [3], [6], [7]. Many intriguing facts and phenomena were experimentally observed and underlying mechanisms were proposed and proved. As early as 1936, Stevens and Newman

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Fig. 1. Comparison of a passive microwave direction finding system (left) and the human auditory system (right).

reported free space experimental data on localization of sound sources by human ears which revealed the two main mechanisms of binaural sound localization, one operating best at high frequency and the other at low frequency [8]. Later on, more studies in anechoic chambers confirmed the earlier results [9], [10]. For most of the audible frequency range (20 Hz–20 KHz), human ears are able to estimate sound sources with up to 1 of accuracy (this is rather impressive considering there are only two “antennas”—pinnae). , the phase For low frequency sound difference between the acoustic signals received by the two ears (referred to as the binaural case) serves as the most important cue. The typical distance between two human ears is about 23 cm which is equal to the wavelength of a 1.5 KHz sound wave (speed of sound in air is 340 m/s). The physical reason of this low frequency limit is similar to the antenna element spacing limitation for microwave direction finding or phased array an, the tennas: to avoid the phase ambiguity of multiples of antenna elements should be spaced less than half a wavelength, . The front-back ambiguity is eliminated by the directivity of human ears (analogous to antenna radiation pattern) [3]. For , the human hearing higher frequency sound system incorporates a simple but elegant solution: using the head as a low-pass filter! For most incident angles, one ear receives without the influence of the head while the other receives after the incident signal goes through (or around) the low-pass filter—human head, whose response function is incident angle dependent and can have an attenuation as much as 20 dB [7]. This effect is often referred to as the head-related-transfer function (HRTF) which provides important cue for sound source localization for high frequencies. Fig. 2 illustrates the HRTF efis the incident sound signal; and fect, where are the head-related responses at the left ear and right ear, reand are spectively; the received sound signals at the left ear and right ear, respecand leads to both a tively. The difference between phase and a magnitude difference between the received signals at two ears. The combination of the phase (or time for transient signals) and amplitude information enables the human auditory system to have great localization capabilities for both low and high frequency ranges. Both of the binaural mechanisms mentioned above have analogies or may be directly applied to microwave systems.

Fig. 2. Utilizing the HRTF, human auditory system can achieve unambiguous direction finding for high frequency signals.

The low-frequency phase difference method is widely used in microwave direction finding [1]. However, the high-frequency scheme utilizing an effective low-pass filter (the shadowing effect of human head) has not been reported in the literature, at least to our knowledge. Furthermore, because of the spacing between the adjacent antenna elements can now be much larger , the mutual coupling issue that are common to antenna than array systems can be greatly reduced.

III. NUMERICAL SIMULATIONS To evaluate the feasibility of applying some of the human sound localization mechanisms in microwave direction finding, a simple two-antenna (omni-directional monopoles are used here for simplicity) configuration is considered. In this study, by introducing a lossy scatterer in between the two adjacent antennas (emulating the head effect of the human auditory system), accurate direction finding without phase ambiguity for high frequency signals is achieved. In this paper, we confine our discussions in the azimuth plane for simplicity without losing generality. We also assume a single signal case with the understanding that this technique may be generalized to multiple signals case with more than two antenna elements. Take an electromagnetic (EM) signal coming from an azimuth direction and impinging on the two monopole antennas separated by a distance as shown in Fig. 3, assuming without the scatterer first, the phase difference between received , where signals at these two antennas is, is the wavelength of the signal. It can be derived that for both EM and acoustic DOA that if is greater than half wavelength, there may be ambiguities in the estimated DOA. To avoid this kind of ambiguity at high frequency, it is proposed here that a lossy scatterer is placed between the two antennas, providing the similar low-pass filtering function as the human head between of sigtwo ears. Without the scatterer, the phase difference nals measured at the two antennas is the only key information to estimate the DOA. With the scatterer, the magnitude differ, and are the received ence signal magnitudes of the two antennas) provides an additional important cue in the DOA estimation, eliminating the phase ambiguity issue.

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Fig. 3. A finite-element model illustrating the geometry of the two-monopole and scatterer configuration with an incoming signal from an azimuth angle.

The simple configuration in Fig. 3 (with/out a lossy scatterer) working at X-band (monopole length is 7 mm with a center freat 10 GHz) is modquency near 10 GHz; eled by the full-wave finite-element EM solver High Frequency Structure Simulator (HFSS) [11]. Our results have shown that the exact dielectric constant and loss tangent do not impact the direction finding performance qualitatively. As an explicit example, the scatterer used here is an absorbing material with a and assigned to have geometry of and . The steering vectors (based on phase and ) magnitude differences, defined as as a function of frequency and incident angle with all the EM environments included are obtained by two different methods: far-field illumination by a horn antenna; direct plane-wave exand ) citation. The phase and magnitude differences ( received by two monopoles without a scatterer in between using these two methods are consistent [4]. The plane-wave excitation method is then adopted to sweep the entire azimuth dimento 360 since it is much less computationally sion from intensive. and at 10 and 12 GHz, Fig. 4 plots the simulated with and without the lossy scatterer. Several interesting and important points are worth noting. First, because of the mutual coupling between the two antennas, the magnitude differences for the case without the lossy scatterer are not the ideal 0 dB for all the incident angles, instead, it can be as high as 2.7 dB for certain incident angles. However, compared to the lossy scatterer case, the magnitude difference for the case without the lossy less than 2.7 dB at 10 GHz and scatterer is much smaller ( less than 0.22 dB at 12 GHz), as expected. Second, the phase difference versus incident angle curve with the scatterer is significantly steeper than that without the scatterer, which should lead to higher sensitivity in the DOA estimation. For example, with the scatterer, a same error in the measured phase difference will cause a smaller error in the calculated DOA compared to the case without the scatterer. Third, although the phase ambiguity issue seems to be more severe for the case with the scatterer, for example, the same phase difference may lead to four different incident angles, the large magnitude differences should be sufficient to overcome the ambiguities. In addition, both the phase and magnitude differences with and without the scatterer are the same for supplementary incident angles (or, it can be thought as the front/back ambiguity), which is due to the symmetry respect to the y-axis (see Fig. 3). A simple solution for this issue is to

Fig. 4. Simulated phase (a) and magnitude (b) differences at 10 GHz and phase (c) and magnitude (d) differences at 12 GHz of two monopoles with and without a lossy scatterer in between.

shift the monopoles positions so that the system becomes asymmetric (this can be achieved by using directional antennas as

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Fig. 5. Simulated MUSIC output of the three scenarios with a 12 GHz signal incident from 80 direction: without a scatterer (dotted dashed line), with a symmetrically positioned scatterer (dashed line) and with an asymmetrically positioned scatterer (solid line).

well just as the human ears are). In the following examples, the symmetry is broken by shifting the two monopoles toward one side of the scatterer block by 7.5 mm. Several scatterer materials with various dielectric constants and loss tangents are studied. Although different phase and magand ) are obtained for each case, nitude differences ( all of the important features are the same. A cylindrical shaped scatterer made of the same material is also studied. The simulation results show very similar improvements of direction finding ability in terms of ambiguity elimination and sensitivity enhancement, indicating this biological inspired technique is not very sensitive to the exact form factor of the scatterer. In the following examples, a lossy foam absorber material, ARC-LSand 10211 (made by ARC Technologies Inc.), with and a geometry of is used. This material is adequate to provide sufficient magnitude difference information even though it has much lower loss than the scatterer used in the previous example. To evaluate the DOA performance of the two-monopole/scatterer system, the phase and magnitude differences between the two monopoles as a function of incident angle from 0 to 360 are first simulated and saved as the calibration steering vectors. Then, for an arbitrary incident angle , assuming a phase error , the multiple signal classification (MUSIC) algorithm is used to estimate the DOA, corresponding to the steering vectors that are orthogonal to the noise subspace of the autocorrelation matrix of the received signal. Therefore, the DOA can be obtained as the peaks of the MUSIC spectra by searching the steering vectors from 0 to 360 . For the cases of two-monopole with a lossy scatterer in between, because of the additional magnitude difference information in the steering vectors, false peaks due to phase ambiguity in the MUSIC spectra are eliminated. In Fig. 5, the MUSIC output for the simulated case with a 12 GHz signal coming from 80 direction are plotted for the three scenarios mentioned above (without scatterer, symmetric scatterer and asymmetric scatterer). It is clear that the addition of the attenuating scatterer (similar to the low-pass-filter function of human head) in between the two antennas eliminates the phase wrapping. Moreover, it can ambiguity caused by the be observed from Fig. 5 that the asymmetrically placed scatterer breaks the front/back symmetry and eliminates the corre-

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Fig. 6. Simulated averaged DOA estimation errors assuming 0:25 dB magnitude difference and 2 phase difference errors: (a) versus frequency (averaged over all incident angles from 0 to 360 with 2 step) and (b) versus incident angle (averaged over all frequencies from 8 to 12 GHz with 0.5 GHz step).

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Fig. 7. A photograph of the X-band two-monopole and symmetric scatterer prototype.

sponding ambiguity as well. For the no scatterer case, there are four peaks near and that can be observed, for which the 80 and 222 ambiguity is due to the greater than spacing of the monopoles. With the scatterer, this ambiguity is clearly eliminated by the added magnitude information, which can be seen from both the symmetric and asymmetric cases in Fig. 5. The 80 and 100 ambiguity manifested in the no scatterer and symmetric scatterer cases is due to the front/back sym-

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Fig. 8. Comparison of measured (solid lines) and simulated (dashed lines) magnitude (left) and phase (right) differences at 10 GHz for the three cases. (a) No scatterer, (b) symmetric scatterer, (c) asymmetric scatterer.

metry of the two-antenna system which is absent in the asymmetric configuration. To verify the predicted accuracy enhancement by the addition of a head like scatterer, average DOA estimation errors as functions of frequency and incident angle are evaluated, assuming the ambiguities can be resolved. For example, assumed errors in phase and in magnitude) are first introduced ( to the simulated steering vectors; then the modified steering vectors are fed back to the MUSIC algorithm to find the average incident angles which are compared to the original correct incident angles. Numerical studies for signals from 8 to 12 GHz and coming from 0 to 360 are performed. Fig. 6 plots the average DOA estimation errors as functions of frequency (a) and incident angle (b).

As shown in Fig. 6, with scatterers (both symmetric and asymmetric cases), the DOA estimation errors are smaller compared to those without the scatterer, indicating accuracy improvement as expected. In addition, the asymmetric scatterer case has slightly smaller errors than those of the symmetric scatterer case. IV. EXPERIMENTAL RESULTS An X-band prototype of the same configuration as illustrated in Fig. 3 is built and tested to experimentally verify the proposed concept. Fig. 7 shows a photograph of the two-monopole antenna with an absorber material (ARC-LS-10211) placed symmetrically in between the two antennas. The ground plane size . As described previously, the monopoles is

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Fig. 9. MUSIC output of a 12 GHz signal incident from 90 for the two-antenna configurations without scatterer (dotted dashed line), with the symmetric scatterer (dashed line) and with the asymmetric scatterer (solid line).

have a length of 7 mm and are separated by a distance of 15 and , mm. The absorber properties are: with a geometry of . The asymmetric configuration is again realized by shifting the absorber along the x-direction by 7.5 mm. The X-band two-antenna systems without and with symmetric and asymmetric scatterers are tested using a vector network analyzer in an anechoic chamber. The steering vectors (spanning 8 to 12 GHz with 0.5 GHz step, 0 to 360 with 1 step) were measured with a transmitting horn antenna, one receiving monopole while the other one is terminated by a 50- load and vice versa. Fig. 8 compares the measured and at 10 GHz, without the lossy and simulated scatterer (Fig. 8(a)), with the symmetric (Fig. 8(b)) and with the asymmetric lossy scatterer (Fig. 8(c)). The measured phase and , or steering vectors) and magnitude differences ( agree very well with simulation results, verifying the accuracy of our simulation procedure. Moreover, it is confirmed that with the incorporation of the lossy scatterer, the magnitude difference becomes larger and provides useful information in is greater estimating DOA and the phase slope which leads to better DOA sensitivity. The DOA performance of the experimental prototypes is then evaluated by transmitting a signal using a horn antenna in the far field region from 0 to 360 with 15 step. MUSIC algorithm is used to estimate the DOA based on the previously measured steering vectors. As predicted by simulations, it is observed that ambiguities at all frequencies and incident angles are completely eliminated with the incorporation of the asymmetric scatterer. Fig. 9 plots the MUSIC output for an incident 12 GHz signal coming from 90 , for all three antenna/scatterer configurations. As predicted by the simulation results, there are ambiguities in the estimated DOA for the two-antenna system without a scatterer in between because the antenna distance is greater than half wavelength. The asymmetric scatterer configuration performs the best, also as expected. The overall DOA estimation accuracy for all three cases versus frequency and incident angles are shown in Fig. 10. Compared to the simulated DOA estimation accuracies in Fig. 6, quite good agreement is observed. It can be seen that for the entire X-band, the average DOA estimation

Fig. 10. Measured averaged estimation errors for all three two-antenna configurations (a) versus frequency (averaged over all incident angles from 0 to 360 with 15 step) and (b) versus incident angle (averaged over all frequencies from 8 to 12 GHz with 0.5 GHz step).

error decreased from more than 3 to less than 1 with the incorporation of the scatterer in between the two antennas. The average errors plotted as a function of incident angle also agree well with simulation results and show significant improvement with the incorporation of the scatterer. In summary, the asymmetric configuration has slightly better accuracy than the symmetric case, in addition to its ability of eliminating all ambiguities including the front/back ambiguity. V. CONCLUSIONS A human ears inspired two-antenna direction finding configuration is proposed. Theoretical and experimental investigations of a two-antenna direction finding system at X-band incorporating a human head-like scatterer are performed. Our results have shown that the incorporation of the head-like scatterer not only eliminates the phase ambiguity issue at higher frequencies, but also improves the general sensitivity of the two-antenna direction finding system. This kind of biological inspired microwave technique may lead to future direction finding systems that are low-cost, compact and light weight. ACKNOWLEDGMENT The authors wish to thank Dr. Dev Palmer at ARO for useful discussions.

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REFERENCES [1] S. E. Lipsky, Microwave Passive Direction Finding. New York: Wiley, 1987. [2] L. C. Godara, “Application of antenna arrays to mobile communications. II. Beam-forming and direction-of-arrival considerations,” Proc. IEEE, vol. 85, no. 8, pp. 1195–1245, 1997. [3] B. C. J. Moore, Introduction to the Psychology of Hearing. Baltimore, MD: University Park Press, 1977. [4] H. Xin and J. Ding, “An improved two-antenna direction of arrival (DOA) technique inspired by human ears,” presented at the IEEE AP-S Int. Symp, Jul. 2008. [5] R. O. Schmid, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag., vol. 34, pp. 276–280, 1986. [6] E. Villchur, Acoustics for Audiologists. San Diego, CA: Singular Publishing Group, 2000. [7] W. A. Yost, Fundamentals of Hearing: An Introduction, 4th ed. New York: Academic Press, 2000. [8] S. S. Stevens and E. B. Newman, “The localization of actual sources of sound,” Am. J. Psychol., vol. 48, pp. 297–306, 1936. [9] T. T. Sandel, D. C. Teas, W. E. Feddersen, and L. A. Jeffress, “Localization of sound from single and paired sources,” J. Acous. Soc. Am., vol. 27, pp. 842–852, 1955. [10] L. A. Jeffress, “Detection and lateralization of binaural signals,” Audiology, vol. 10, pp. 77–84, 1971. [11] High Frequency Structure Simulator11th ed. Ansoft Corporation, 2008.

Rongguo Zhou (M’10) received the B.S. degree in physics from University of Science and Technology of China, in 2004 and the Ph.D. degree in physics from University of Arizona, Tucson, in 2010. Currently, he is with the Department of Physics, University of Arizona, Tucson. His research interests are RF/microwave component designs, including amplifier design, antenna design and measurement, metamaterial design for antenna applications, etc. He also works on the algorithm and implementation of microwave signal direction of arrival estimation.

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Hualiang Zhang (M’08) was born in Wuhan, China. He received the Bachelor degree in electrical engineering from the University of Science and Technology of China (USTC), in September 2003 and the Ph.D. degree in electronic and computer engineering from the Hong Kong University of Science and Technology (HKUST), in January 2007. At HKUST, his research interests include design and synthesis of microwave filters, MEMS technologies especially their applications to the RF passive components and optimization techniques. From 2007 to 2009, he was with the University of Arizona, Tucson, as a Postdoctoral Research Associate, conducting research related to RF/microwave circuits, antenna design and meta-materials based circuits. Since August 2009, he has been with the Department of Electrical Engineering, University of North Texas, Denton, as an Assistant Professor.

Hao Xin (SM’06) received the Ph.D. degree in physics from the Massachusetts Institute of Technology (MIT), Cambridge, in 2001. He performed research studies for five years at MIT’s Physics Department and at Lincoln Laboratory, where he investigated power dependence of the surface impedance of high-Tc superconducting films and Josephson junction properties at microwave frequencies. From November 2000 to November 2003, he was a Research Scientist with the Rockwell Scientific Company, where he conducted research as Principal Manager/Principal Investigator in the area of electromagnetic band-gap surfaces, quasi-optical amplifiers, electronically scanned antenna arrays, MMIC designs using various III-V semiconductor compound devices and random power harvesting. From 2003 to 2005, he was a Sr. Principle Multidisciplinary Engineer at Raytheon Missile Systems, Tucson, AZ. He is now an Associate Professor of the Electrical and Computer Engineering Department and the Physics Department at the University of Arizona, Tucson. He has published over 80 refereed technical papers in the areas of solid-state physics, photonic crystals and the applications thereof in microwave and millimeter wave technologies. He has 12 patents issued and one pending in the areas of photonic crystal technologies, random power harvesting based on magnetic nano-particles and microwave nano-devices. His current research focus is in the area of microwave, millimeter wave and THz technologies, including solid state devices and circuits, antennas, passive circuits and applications of new materials such as metamaterials and carbon nanotubes.

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Foliage Attenuation Over Mixed Terrains in Rural Areas for Broadband Wireless Access at 3.5 GHz Kin Lien Chee, Member, IEEE, Saúl A. Torrico, Senior Member, IEEE, and Thomas Kürner, Senior Member, IEEE

Abstract—This paper reports the modeling of foliage attenuation for a broadband wireless access system deployed over mixed terrains in rural areas at 3.5 GHz. The foliage is composed of leaves and leafstalks, and are the dominant scatterers along the transmission paths. The foliage attenuation is determined using the Torrico-Lang model combined with digital topography information. In this model, leaves are modeled as thin lossy circular dielectric discs whereas leafstalks (petioles) are modeled as thin lossy dielectric cylinders. Three measurement campaigns were performed using the mobile WiMAX system (IEEE 802.16 e) deployed in Hetzwege/Abbendorf during winter, spring and mid-summer. Using the winter data as a baseline, the foliage loss due to different degrees of foliation in spring and in winter is studied. The derived foliage loss is then verified and compared with an empirical exponential decay model. Index Terms—Broadband wireless access, foliage attenuation, IEEE 802.16 e, multiple scattering theory, radio propagation, scattering parameters, skin depth, vegetation.

I. INTRODUCTION

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HE advancements in broadband wireless access systems have been marked as important solutions to shorten the communication links between isolated villages located in rural areas. However, wave propagation in such areas, where vegetation is dominant, always poses challenges for radio network planning. The link budget of the deployed system which is optimized at one time may fade towards the limit at the cell edge during seasonal changes. This is especially obvious for a system deployed in mixed terrains covered with dense vegetation (mainly deciduous trees) where the scattering effects due to leaves and leafstalks (including petioles and stipules) of the canopy is strongest in mid-summer. In the subsequent paragraphs of this paper, a cluster of leaves, petioles or stipules will be collectively termed as foliage. Manuscript received June 15, 2010; revised November 10, 2010; accepted December 16, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported in part by the Stiftung Zukunfts- und Innovationsfonds Niedersachsen under the project “WiMAX in Niedersachsen”. K. L. Chee is with the Institut für Nachrichtentechnik, Technische Universität Braunschweig, 38106 Braunschweig, Germany (e-mail: [email protected]). S. A. Torrico is with the Department of Electrical and Computer Engineering, The George Washington University, Washington, DC 20052 USA and also with Comsearch, an Andrew Company, Ashburn, VA 20147-2405 USA (e-mail: [email protected]). T. Kürner is with the Institut für Nachrichtentechnik, Technische Universität Braunschweig, 38106 Braunschweig, Germany (e-mail: [email protected]. de). 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.2011.2152340

In order to better understand the attenuation effects produced by the tree canopy, it is important to know the biophysical parameters as well as the electrical and geometrical characteristics of branches and leaves. However, due to the lack of universal parameters which can effectively describe the tree canopy, the use of many theoretical models with high accuracy is mostly limited to a specific tree or a group of trees in a region. In the ITU-R P.833 model [1], the specific attenuation (dB/m) due to vegetation is calculated within the frequency range of 30 MHz and 30 GHz, however, the model does not specify the type of trees or group of trees that have been used for the analysis, rather the electrical and geometrical characteristics of the branches and the leaves. Also in this model, to avoid overestimating the attenuation due to a long propagation path, the attenuation is bound by a maximum total additional attenuation which unfortunately depends closely on the vegetation type and depth. In practice, the vegetation loss derived from this model is claimed to vary widely due to the irregular nature of the canopy medium and the wide range of species. Hence, in order to correctly model the vegetation loss at a specific area, the databases of vegetation parameters are yet to be established and justified. It is observed that the multiple scattering theory has been widely adopted to calculate the specific attenuation of a tree canopy [3], however, in vegetated residential areas the combination of the multiple scattering theory and the multiple forward diffraction theory has been used to account for tree attenuation [3]. Investigation from [2], [3] showed that tree attenuation in mixed terrain scenarios can be adequately modeled using either the uniform theory of diffraction (UTD) or the multiple knife-edge model. This is especially true in the scenario where the mixed terrain tree canopies extend above the average building rooftop height and block the LOS propagation path by inducing excessive diffraction loss. However, in the mixed terrain scenarios with random building and tree heights, the vegetation loss is mainly attributed to the multiple scattering of fields produced by the tree canopy, thus multiple scattering theories such as the wave theory or the radiative transport theory are preferred to quantify foliage attenuation. Several publications [4]–[6] investigated wave propagation along mixed paths in forested environments using scattering theories; where the vegetation medium was modeled as a dissipative dielectric slab at frequencies below 200 MHz. However, as the frequency increases the vegetation can no longer be modeled as a homogeneous medium, instead the vegetation is modeled using a four-layered medium for frequencies up to 900 MHz [7], [8]. The four-layered medium model determines the scattered fields at each of the four layers, namely air layer, canopy layer, trunk layer and ground layer. Recent contributions include the extension of the four-layered model up to 3 GHz and

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CHEE et al.: FOLIAGE ATTENUATION OVER MIXED TERRAINS IN RURAL AREAS FOR BROADBAND WIRELESS ACCESS AT 3.5 GHz

subsequently the research group has summarized this model to a more general case, simply by considering the fields with two anisotropic layers, namely the canopy layer and the trunk layer [9]–[12]. Reckoning the importance of including the scattered fields produced by the tree canopy, the multiple scattering theory that was well-developed in the middle of last century was clearly the alternative option to compute foliage attenuation. For this, a tree scattering model developed from the Foldy-Lax multiple scattering theory was proposed and has been reported in [13]–[16]. This paper adopts an analytical model proposed by Torrico-Lang [17], which was originally developed from the Foldy-Lax multiple scattering theory, to study foliage attenuation along the propagation path of mixed terrains in rural areas at 3.5 GHz. Following the Torrico-Lang model [17], we predict the propagation constant, the skin depth, and the specific attenuation of a tree canopy for different transmitted and received polarizations. The analytical model is constructed by replacing the canopy of a tree with a discrete random medium whose statistical characteristics are related to the physical quantities of the tree. The tree canopy is represented as an ensemble of leaves and branches all having prescribed location and orientation statistics. Leaves are modeled as flat circular lossy dielectric discs of finite size, and branches (including leafstalks) are modeled as finitely long circular lossy dielectric cylinders. The mean field in the canopy is calculated using the Foldy-Lax multiple scattering theory [13]–[15], [17]. It is found that the propagation constant of the mean field propagating through the tree canopy has both real and imaginary components. The imaginary component of the propagation constant over the tree volume leads to expressions for the specific attenuation and the skin depth of a tree canopy. The skin depth serves as a measure of the decay of the mean field as it propagates along the tree canopy. The range of validity of the model occurs when the depth of the tree canopy is less than one or two skin depths, since only the mean field is considered. This paper is organized as follows. Section II describes the tree scattering model which is used to study the foliage attenuation in this paper. Section III provides background information about the WiMAX field trial project in Lower Saxony which explains the motivation of this investigation. Also, a general description of the environment as well as the measurement campaigns carried out in the region are briefly discussed in this section. Section IV presents the numerical results and discussions, with the main results covered in Subsection A. Subsection B briefly compares the results with the empirical model. In Section V conclusions are drawn.

II. MODELING OF FOLIAGE ATTENUATION The canopy of a deciduous tree is modeled by a random ensemble of leaves, primary branches (tree trunks) and secondary branches (leafstalks, petioles or stipules) all having prescribed location and orientation statistics as shown in Fig. 1(a). Consider a plane wave of unit amplitude and polarization is incident upon the tree canopy, as shown in Fig. 1(a). The total field in the medium is composed of the mean field and the incoherent field. The incoherent field is produced by the statistical superposition

(a)

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(b)

Fig. 1. (a): Incident plane wave on canopy constructed with thin discs (to represent leaves) and thin cylinders (to represent branches). (b): Leaf orientation with respect to incident wave direction i.

of the scattered waves due to the random distribution of the scatterers in the tree canopy. The mean field in the tree canopy is calculated using the discrete scattering theory of Foldy-Lax [13], [14], and [17]. By solving the mean wave equation and using the Foldy-Lax approximation, it is found that the mean field in the tree canopy has a propagation constant that has real and imaginary components. We note that the mean field propagates in the direction of the transmission path and decays exponentially as it propagates along the canopy. A measure of the attenuation as the wave propagates through the tree canopy is known as skin depth, which is given as [17]: (1) is the complex propagation constant of the medium. where The range of validity of the model is when the physical depth of the tree canopy is one or two skin depths. Under this assumption, the mean field is the main contributor to the total field supposing the incoherent fields are small compared with the mean field. Therefore it is assumed negligible in this investigation. A. Flat Leaves as Scatterers As opposed to its coniferous counterpart which has typical needle shape leave to sustain in cool climate, the leaflets of deciduous trees are rather wide and quasi-circular in shape. Each leaf in the canopy is modeled as a lossy dielectric circular thin disc as shown in Fig. 1(b). The orientation of the thin disc is described by an inclination angle from the z-axis and an azimuth angle from the x-axis. The specific attenuation of a random ensemble of leaves is obtained by knowing the scattered field of a single leave. The scattered field is derived by determining the induced field in the disc under the assumption that the disc radius is much larger than the thickness of the disc, hence, the phase variations in the induced fields normal to the disc can be neglected. The specific attenuation for an ensemble of leaves in decibels per meter for the vertical polarization component can be represented as [17]

(2)

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Fig. 2. Secondary branches (leafstalks or petioles) orientation with respect to incident wave direction i. Fig. 3. 3D DEM of Hetzwege, in the county of Rotenburg (Wümme), Germany.

where is the complex component of the relative susceptibility of the leaves, is the propagation constant in free space, is the density of the leaves [m ], is the thickness of the discs [m], is the radius of the discs [m], is the incident angle of the fields [radian], and are the polar inclinations of the leaves [radian]

[m], is the incident angle of the fields [radian], and the polar inclinations of the secondary branches [radian]

are

(6) (3)

(7)

(4)

B. Thin Leafstalks/Petioles as Scatterers Depending on the densities, cross sectional radii and lengths, branches can be categorized into groups. Following the tree scattering model reported from [16], branches of a single tree can be classified into 5 groups. Bearing in mind that the scattering model presented in this paper has an aim to predict the foliage loss for various transmission paths within the villages, detailed classification of branches for every single tree in the region seems unrealistic, hence branches are classified into two main groups, namely the primary trunks and secondary branches. Primary trunks are arches which remain in the propagation paths when defoliated due to season changes. Secondary branches such as leafstalks and petioles connect leaflets to the primary trunks. In the scattering model presented in this paper they are modeled as lossy thin dielectric cylinders as shown in Fig. 2. The orientations of the thin cylinders are described by an inclination angle from the z-axis and by an azimuth angle from the x-axis. The specific attenuation of a random ensemble of branches is obtained by knowing the scattered field of a single cylinder. The scattered field of a cylinder is derived from the induced electric fields within the cylinder using the quasi-static approximation. The specific attenuation for an ensemble of leafstalks/petioles in decibels per meter for the vertical polarization component can be represented as [17]

(5) is the complex component of the relative susceptiwhere bility of the secondary branches, is the propagation constant in free space, is the density of the leafstalks/petioles [m ], is the length of the branches [m], is the radius of the branches

It is assumed that the statistics of the inclination angle and azimuth angle are independent and that the probability densities of the azimuth angles for the branches and the leaves are uniformly distributed from 0 to 360 . Owing to the azimuthal symmetry of the scatterers, the mean fields of the vertical and horizontal polarizations do not couple, so that no depolarization effects occur to the level of the mean field. Furthermore, the total scattered fields in the direction of propagation can be deemed as the individual summation of the scattering components from leaves and leafstalks/petioles with similar incident polarizations. III. WIMAX FIELD TRIAL IN LOWER SAXONY Under the framework of the project “WiMAX Field Trial in Lower Saxony,” a WiMAX system based on IEEE 802.16e has been deployed in rural areas of Hetzwege/Abbendorf, Germany. As part of the research activities of field strength prediction, wave propagation through vegetation in such rural areas, which are largely covered by dense vegetation has become a core research field in this project. A self-tailored 3D digital elevation model (DEM) with clutter information of the rural areas with 2 meters resolution has been developed for the purpose of wave propagation studies [18]. Typical clutter groups identified in the region include houses, trees, green fields and streets. To study the effect of terrain irregularities and clutter distribution on the wave propagation, the terrain profile of each transmission path has been generated using double Bresenham algorithm [19]. From each of the terrain profiles, trees along the transmission path have been identified. This information forms an important background for the foliage attenuation investigated in this paper. A. Measured Environment Fig. 3 shows the self-tailored 3D DEM of Hetzwege, in the county of Rotenburg (Wümme), Germany. The area is covered by dense vegetation where pixels in light green represent green

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(a)

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(b)

Fig. 4. (a) Base station (BS) antenna mounted on 25 m tower. (b) Receiving antenna (Rx) and GPS receiver mounted on top of vehicle.

fields and pixels in dark green represent groups of trees identified in the region. Additionally, houses (blocks in amber) and streets (yellow) are other clutter types that are typically identified in the region. A WiMAX base station is erected at the north-west area of the region (marked with “x”). B. Measurement Campaigns The measurement campaigns were carried out at Hetzwege. The base station which was erected on a 25 m tower (Fig. 4(a)), was meant to provide internet access via WiMAX (IEEE 802. 16e) to residents within an area of 2 km by 2 km. A four column-array antenna from Andrew, APW43512014-0N, was assembled with 120 azimuth and 2 mechanical tilting. The sector antenna operated with a gain of 23 dBi at the boresight at 3.5 GHz, with a horizontal beamwidth of 25 and a vertical beamwidth of 50 . The antenna transmitted vertically polarized signals with an input power of 3.2 W from the base station. A Rohde & Schwarz TSMW WiMAX Scanner with an omniantenna mounted to the roof of a vehicle (1.5 m height from ground) was used to detect the WiMAX signal as shown in Fig. 4(b). The vehicle drove through all possible tracks in the village with a constant speed of 40 km/h. External GPS was used to correlate the GPS coordinates with the measured signal strength and afterwards, both sets of data were streamed to a laptop operating the dedicated software, ROMES v4.40, delivered with the WiMAX Scanner. Three measurement campaigns in July 2009, early April and mid-May 2010 were conducted in Hetzwege. The measurement campaign in early April 2010 measured the signal strength in winter, when residual snow from the previous winter had completely melted and before the deciduous trees were due to start growing back their foliage (both leaves and petioles/stipules). The measurement campaign in mid-May 2010 determined the signal strength in spring, when the major part of the deciduous tree canopy had recovered but was still not in full blooming foliage. The measurement campaign in July 2009 measured the signal strength in mid-summer when all deciduous trees were in full foliage. IV. NUMERICAL RESULTS AND DISCUSSION A. Main Results Starting from the self-tailored 3D DEM with clutter information reported in [18], the terrain profiles between the base station and each of the receiving points are generated [19]. From

Fig. 5. Terrain profile generated from double Bresenham algorithm with identified trees (blue) and houses (red).

each of the terrain profiles, trees are identified, allowing the incident angle of the plane wave from the base station to be derived accordingly. Fig. 5 shows an example of the terrain profile generated with identified trees. The received signal determined from the three measurement campaigns, namely winter, spring and summer, are correlated, thus allowing the different signal strengths to be derived. It has been reported that the wind speeds above and within a row crop canopy are generally higher at noon time than at night time due to the higher air temperature [20], while it is rather constant during the day time. The three measurements were performed during the day time to avoid the unnecessary dynamic effect which may have resulted in fading of the received signal, as reported in [21]. The time varying phase changes of the scattered radio wave from the swaying tree components at this frequency are assumed to be negligible, thus allowing the measurement results from the three campaigns to be directly comparable based on the GPS coordinates. By keeping the same measurement setups, the difference in path loss derived from the measurement campaigns can be attributed to the foliage loss which has been observed in spring and summer. Under the assumption that the primary trunks of the deciduous trees remain in the propagation channel throughout different measurement campaigns, and the scattered fields induced from the primary trunks do not correlate with the scattered fields induced from the leaves and secondary branches in quasi-static approximations, the foliage loss derived from the measurement campaign is fully attributed to the scattering induced from leaves and secondary branches. As discussed in Section II, the foliage attenuation is modeled with two components; the scattered fields due to leaves and secondary branches. The radii of the leaves are assumed to be constant in spring and summer, i.e., about 5 cm [22]. The thicknesses of the leaves are also assumed to be constant in spring and summer, about 0.2 mm [17]. The leaf density in summer is assumed to be 300 m whereas it is assumed to be about 200 m in spring. Adopting the semi-empirical formula for calculating the leaf permittivity for GHz with salinity of 1% as proposed in [23] (8)

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where is the moisture content assumed to be 80%, is the complex permittivity of saline water, which is assumed to be 71-28i at 10 C in spring and 70-24i at 20 C in summer at 3 GHz [24], [25]. The complex permittivities of leaves are found i and i. The suscepto be of leaves in summer and spring can then be derived tibility accordingly as (9) The polar orientations of the leaves are assumed to be uniformly distributed between 0 and 180 , which means that there is no preferred inclination. The attenuation per meter due to the scattered fields of leaves in canopy is derived from the equations presented in Section II. Similarly, the radii of the secondary branches are assumed to be 0.2 cm in both spring and summer. The lengths of the secondary branches are assumed to be 10 cm in spring and 14 cm [17] in summer, the densities of the secondary branches are about 11.4 m in spring and 14 m [17] in summer. The susceptibilities of these branches are about 28-7i [16], [17], which are assumed to be constant in spring and summer. Also, the polar orientations of the secondary branches are assumed to be uniformly distributed between 0 and 180 , which corresponds to no preferred inclination in both spring and summer. The effective foliage attenuation per meter, also known as effective foliage attenuation coefficient for each receiving position, can be derived using equations presented in Section II. To determine the total foliage attenuation ( [dB]) for each transmission path, the number of trees present at each transmission path has to be derived from the terrain profiles, so that

Fig. 6. Number of trees

N , calibrated from spring measurement data.

(10) (11) where is the effective foliage attenuation [dB/m], is the lateral distance [m] where the scattered fields propagate before reaching the receiver, is the number of trees identified from each of the terrain profiles, and [m] is the resolution of each pixel of the terrain profiles which is 2 m. 1) Calibration of N From Spring Measurement Data: Trees in the self-tailored 3D DEM are modeled as pixel groups characterized by perimeters identified from the aerial picture [18], i.e., each pixel that falls within a perimeter is deemed as single tree with 2 m resolution. Hence, simply taking the number of trees derived directly from each of the terrain profiles generated from the 3D DEM will result in overestimating the foliage loss. To counteract the overestimation and the accuracy limitation presented in the 3D model, the number of trees for each transmission path is calibrated from the measurement data in spring, as shown in Fig. 6, and the derived number of trees for each transmission path is used to derive the foliage loss modeling in summer. Fig. 7(a) and (b) show the comparisons between the results of foliage loss modeling and measurement data in spring and summer, respectively. A direct comparison between the measurement data shows that the foliage loss is higher in summer when trees are in full foliage. The presence of leaves and branches as scatterers in full foliage increases the fluctuation of propagation fields in the

Fig. 7. (a) Comparison of foliage loss modeling with measurement in spring. (b) Comparison of foliage loss modeling with measurement in summer.

canopy, which can be observed from the measurement data in summer. The effective foliage coefficient in summer is about 0.7 to 0.75 dB/m while it is about 0.55 to 0.67 dB/m in spring. The mean tree attenuations reported from previous work done are in the order of 1.1 dB/m at 869 MHz [26] and 1.1 dB/m–1.3 dB/m at 1.6 GHz [27]. A tree scattering model proposed using

CHEE et al.: FOLIAGE ATTENUATION OVER MIXED TERRAINS IN RURAL AREAS FOR BROADBAND WIRELESS ACCESS AT 3.5 GHz

the multiple scattering theory of Foldy-Lax and reported in [16] revealed the total attenuation to be in the order of 1.1 dB/m for trees in-leaf and 0.8 dB/m for defoliated trees at 2 GHz. When the frequency increases to 3.5 GHz, the mean total tree attenuation measured in [28] is found to be in the order of 1.7 dB/m. It is to be emphasized that the foliage loss presented in this paper does not include the scattering effects due to primary branches, which are assumed to remain in the propagation channel throughout various measurement campaigns. The specific attenuation due to primary branches for incident angles between 40 and 90 with vertical polarization at both antenna ends was found in previous work [17] to be in the order of 0.5 to 0.6 dB/m at 1.9 GHz. Summing up the effective foliage loss determined in this paper with the specific attenuation due to primary branches gives an overall foliage attenuation of 1.52 dB/m. This value is slightly lower than the mean foliage coefficient at 3.5 GHz reported from [28], which was in the order of 1.7 dB/m. The foliage attenuation presented here assumes that the tree canopies are less than one or two skin depths. In the case where the transmission path consists of dense trees which are close together and overlapping, this assumption may no longer be held hence the transmission path specific effective foliage attenuation per meter may appear slightly smaller than the mean foliage attenuation per meter derived from a single tree. 2) Without Calibration: In the case where calibration from measurement is not possible, the prediction of foliage loss can be achieved by fitting the scattering parameters individually over many small local groups of trees in a region, where each local group of trees is characterized by a separate set of scattering parameters. Fig. 8(b) and (c) presents the foliage loss modeling along two routes in the region with scattering parameters in summer described in Table II. The effective foliage attenuation coefficient derived here is about 2.41 dB/m for route 1 and 0.86 dB/m for route 2. The local modeling of each transmission path using a separate group of parameters allows the foliage attenuation to be correctly modeled without calibrating from measurement. To further investigate how significantly the foliage loss modeling changes with the biophysical scattering parameters, the scattering parameters from Table I are again used to model the foliage loss at these two routes, with the outcomes plotted in Fig. 8(b) and (c). The outcomes show that the foliage loss modeling at routes 1 and 2 derived from Table I parameters are far below the measured foliage loss. This also implies that the biophysical parameters of the vegetation play a significant role in the study of foliage loss at this frequency. Also, it is to be emphasized that the specific attenuations (dB/m) in Fig. 7(a),(b) and Fig. 8(b),(c) are modeled on a per route basis, which means that the vegetation loss for all receiving points along a particular route is modeled with the same set of biophysical scattering parameters which are collectively quantified using the effective scattering parameters for all trees along the corresponding route. To further improve the modeling of the vegetation loss along a route, it is necessary that the scattering parameters of each individual group of trees along the route be established. However, this may be unrealistic for a broadband wireless access system deployed in dense forested rural areas due to a massive amount of data to be handled and

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Fig. 8. (Top, a): Routes map, (middle, b) and (bottom, c): foliage loss modeling with individual set of scattering parameters used to characterize trees along specific streets in the region.

maintained. It is importantly emphasized that defoliation is a natural environmental process. B. Comparison With the Empirical Model Recalling the well-known empirical model for tree attenuation from the Weissberger exponential decay model [29], the tree attenuation can be derived with the following: (12)

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TABLE I SUMMARY OF SCATTERING PARAMETERS

TABLE II SCATTERING PARAMETERS ALONG ROUTE 1 AND ROUTE 2 IN SUMMER

where is the tree loss [dB], is the frequency [GHz] and is the depth of trees in the direction of propagation [m]. Generally, the model is valid from 230 MHz to 95 GHz. Since the model is limited to a foliage depth of up to 400 m in the direction of propagation, it will be much more interesting to redefine the model in a general form (13) where is the tree loss [dB], is the frequency [MHz] and is the depth of trees in the direction of propagation [m]. and are coefficients which characterize the trees and the environment. By performing least-square fitting with the foliage loss derived from the modeling results presented in Fig. 7, the coefficients can be determined and the model in summer can be rewritten in the following form as: (14) The exponential decay model from Weissberger indicates that the path loss differences for trees in-leaf and out-of-leaf are in the order of 3 to 5 dB [29]. The foliage loss derived here is about 3.15 dB in spring and 4.68 dB in summer, which is comparable with previous works. It is to be emphasized that the foliage loss derived using empirical models in Fig. 9(a) (b) is fitted only with the tree scattering model derived from a single set of parameters, as presented in Table I for all transmission paths in the region with a distance of 600 m to 1000 m from the transmitter. The foliage loss modeled using the empirical model gives only an implicative value which helps to characterize the foliage loss in the village with dense vegetation. For comparison purposes, the foliage loss in summer, determined previously from the multiple scattering theory, is plotted again in Fig. 9(b). The foliage loss determined from the multiple scattering theory gives a standard deviation of 1.1 dB compared with the measurement, whereas the empirical model gives

Fig. 9. (a) Foliage loss modeling using exponential decay model fitted with foliage attenuation in spring, as derived from Fig. 7(a). (b) Foliage loss modeling using exponential decay model fitted with foliage attenuation in summer, as derived from Fig. 7(b).

CHEE et al.: FOLIAGE ATTENUATION OVER MIXED TERRAINS IN RURAL AREAS FOR BROADBAND WIRELESS ACCESS AT 3.5 GHz

a standard deviation of 2.53 dB compared with the measurement. The 50% reduction in standard deviation indicates a significant improvement in vegetation loss modeling using the multiple scattering theory with the aid of terrain information. The discussed model compensates the limitation of the conventional empirical model and allows the fluctuation of vegetation loss for each transmission path in the region to be correctly modeled. Further improvements can be expected if the scattering parameters of individual tree groups within each terrain profile are varied according to the species, densities and other biophysical parameters, if such information is available. Furthermore, the discussed model overcomes the random nature of the canopy medium and allows the vegetation loss to be effectively studied in a collective manner. V. CONCLUSION The foliage attenuation over mixed terrains in rural areas at 3.5 GHz is discussed in this paper. The tree scattering model discussed in this paper was derived using the Foldy-Lax discrete multiple scattering theory, where the tree canopy is modeled as dielectric medium constructed from leaves and branches which can be described as lossy dielectric thin circular discs and lossy dielectric thin cylinders, respectively. Data from three measurement campaigns scheduled in winter, spring and mid-summer is used to differentiate varying degrees of defoliation which distinguish the scattering mechanisms within the canopy of trees along the propagation paths. Using the signal strength in winter as a baseline, the signal levels in spring and mid-summer along the same propagation paths in rural areas are compared. The path loss differences between winter and spring, as well as between winter and mid-summer, are fully attributed to the scattering effects due to moderately in-leaf and completely in-leaf, respectively. This effective foliage attenuation due to foliation is found to be about 0.7 dB/m under the assumption that the primary trunks of trees along the propagation paths remain in the channel throughout various defoliation processes, hence the scattering effects due to these primary trunks are excluded. A brief comparison with the empirical model from Weissberger shows good agreement with previous works, where the mean foliage loss is concerned. In addition to the scattering parameters which can be fitted according to the environment as commonly practiced in the Weissberger empirical model, the approach discussed in this paper allows the lateral distance which the wave propagates before reaching the receiving end to be determined from the terrain profile generated from a 3D DEM. An improvement of 50% in the standard deviation is achieved by comparing both the discussed model and the empirical model to the measurement data in summer, when foliage loss is the highest. The improvement achieved indicates the significance of the role of the terrain information in the modeling of vegetation loss, especially when terrain data with sufficient resolution is widely available for research purposes. REFERENCES [1] Recommendation ITU-R P. 833-2, Attenuation in Vegetation 1999, ITU.

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[2] W. Zhang, “Formulation of multiple diffraction by trees and buildings for radio propagation predictions for local multipoint distribution service,” J. Res. Natl. Inst. Stand. Technol., vol. 104, no. 6, pp. 579–585, Nov.–Dec. 1999. [3] S. A. Torrico, H. L. Bertoni, and R. H. Lang, “Modeling tree effects on path loss in a residential environment,” IEEE Trans. Antennas Propag., vol. 46, no. 6, Jun. 1998. [4] T. Tamir, “On radiowave propagation in forest environments,” IEEE Trans. Antennas Propag., vol. 15, no. 6, pp. 806–817, 1967. [5] D. Dence and T. Tamir, “Radio loss of lateral waves in forest environments,” Radio Sci., vol. 4, no. 4, pp. 308–318, 1969. [6] T. Tamir, “Radio waves propagation along mixed paths in forest environments,” IEEE Trans. Antennas Propag., vol. 25, no. 4, pp. 471–477, 1977. [7] G. P. S. Cavalcante, D. A. Rogers, and A. J. Giardola, “Radio loss in forests using a model with four layered media,” Radio Sci., vol. 18, no. 5, pp. 691–695, 1983. [8] G. P. S. Cavalcante, M. A. R. Sanches, and R. A. N. Oliveira, “Mobile radio propagation along mixed paths in forest environment,” J. Microw. Optoelectron., vol. 1, no. 4, Sep. 1999. [9] L. W. Li, T. S. Yeo, P. S. Kooi, and M. S. Leong, “Radio wave propagation along mixed paths through a four-layered model of rain forest: An analytic approach,” IEEE Trans. Antennas Propag., vol. 46, no. 7, pp. 198–1111, 1998. [10] J. H. Koh, L. W. Li, P. S. Kooi, T. S. Yeo, and M. S. Leong, “Dominant lateral waves in canopy layer of a four-layered forest,” Radio Sci., vol. 34, no. 3, pp. 681–691, 1999. [11] L. W. Li, T. S. Yeo, P. S. Kooi, M. S. Leong, and J. H. Koh, “Analysis of electromagnetic wave propagation in forest environment along multiple paths,” J. Electromagn. Waves Applicat., vol. 13, no. 8, pp. 1057–1059, 1999. [12] L. W. Li, C. K. Lee, T. S. Yeo, and M. S. Leong, “Wave mode and path characteristics in an inhomogeneous anisotropic forest environment,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2445–2455, 2004. [13] L. L. Foldy, “The multiple scattering of waves,” Phys. Rev., vol. 67, no. 3, pp. 107–119, 1945. [14] M. X. Lax, “Multiple scattering of waves,” Rev. Mod. Phys., vol. 23, no. 4, pp. 287–310, Oct. 1951. [15] V. Twersky, “Multiple scattering of electromagnetic waves by arbitrary configurations,” J. Math. Phys., vol. 8, no. 3, pp. 589–610, Mar. 1967. [16] Y. L. C. De Jong and M. H. A. J. Herben, “A tree scattering model for improved propagation prediction in Urban microcells,” IEEE Trans. Veh. Technol., vol. 53, no. 2, pp. 503–512, Mar. 2004. [17] S. A. Torrico and R. H. Lang, “A simplified analytical model to predict the specific attenuation of a tree canopy,” IEEE Trans. Veh. Technol., vol. 56, no. 2, pp. 696–703, Mar. 2007. [18] K. L. Chee and T. Kürner, “Towards a realistic propagation prediction model—A self-tailored 3D-digital elevation model with clutter information,” presented at the Wave Propagation and Scattering in Communications, Microwave Remote Sensing and Navigation, WFMN 2009 ITG-Conf. Chemnitz, Germany, 2009. [19] K. L. Chee and T. Kürner, “Effect of terrain irregularities and clutter distribution on wave propagation at 3.5 GHz in suburban area,” presented at the 4th Eur. Conf. on Antennas and Propagation, Spain. [20] A. F. G. Jacobs, J. H. van Boxel, and R. M. M. EI-Kilani, “Vertical and horizontal distribution of wind speed and air temperature in a dense vegetation canopy,” J. Hydrol. 166, pp. 313–326, 1995. [21] M. Cheffena and T. Ekman, “Modeling the dynamic effects of vegetation on radiowave propagation,” in Proc. IEEE Int. Conf. on Communications, pp. 4466–4471. [22] S. A. Torrico and R. H. Lang, “Wave propagation in a vegetated residential area using the distorted born approximation and the FresnelKirchhoff approximation,” presented at the 3rd Eur. Conf. on Antennas and Propagation, Berlin, 2009. [23] C. Mätzler, “Microwave (1–100 GHz) dielectric model of leaves,” IEEE Trans. Geosci. Remote Sensing, vol. 32, pp. 947–949, Sept. 1994. [24] A. Stogryn, “Equations for calculating the dielectric constant of saline water,” IEEE Trans. Microw. Theory Tech., Aug. 1971. [25] W. Ellison, A. Balana, G. Delbos, K. Lamkaouchi, L. Eymard, C. Guillou, and C. Prigent, “New permittivity measurements of seawater,” Radio Sci., vol. 33, no. 3, pp. 639–648, May–Jun. 1998.

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[26] W. J. Vogel and J. Goldhirsh, “Tree attenuation at 869 MHz derived from remotely piloted aircraft measurements,” IEEE Trans. Antennas Propag., vol. AP-34, no. 12, pp. 1460–1464, Dec. 1986. [27] W. J. Vogel and J. Goldhirsh, “Earth-satellite tree attenuation at 20 GHz: foliage effects,” Electron. Lett., vol. 29, no. 18, pp. 1640–1641, 1993. [28] B. Benzair, H. Smith, and J. R. Norbury, “Tree attenuation measurements at 1–4 GHz for mobile radio systems,” in Proc. 6th Int. Conf. on Mobile Radio and Personal Communications, 1991, pp. 16–20. [29] M. A. Weissberger, “An initial critical summary of models for predicting the attenuation of radio waves by foliage,” presented at the Electromagnetic Compatibility Analysis Center, Annapolis, MD, 1981, ECAC-TR-81-101. Kin Lien Chee (M’10) was born in Pahang, Malaysia. He received the B.Eng. degree (Hons.) in electrical and electronics engineering from the Nanyang Technological University, Singapore, in 2003 and the M.Sc. degree in electronics engineering from the University of Applied Sciences Bremen, Germany, in 2008. From 2003 to 2006, he worked at Smith Detection Asia Pacific in Singapore and Australia, dealing with baggage inspection system using X-ray technology in aviation, ports and borders. From 2008 to present, he is a Researcher at the Institut für Nachrichtentechnik, Technische Universitaet Braunschweig, Germany, working on radio wave propagation for broadband wireless access system in rural areas.

Saúl A. Torrico (M’93–SM’99) was born in Cochabamba, Bolivia. He received the B.S. and M.S. degrees in electrical engineering, in 1983 and 1992, respectively, and the Ph.D. degree in electrophysics in 1998, all from The George Washington University, Washington DC. He joined Comsearch, an Andrew Company, Ashburn, VA, in 1985, where he is currently Principal Scientist. He has been responsible for directing Comsearch’s efforts in research and development in the areas of radiowave propagation and system design pertinent to mobile communications systems, terrestrial microwave communications systems, and mobile satellite systems. He has contributed to the inclusion of the wireless location method U-TDOA (Up Link Time difference of Arrival) and the Stand Alone SMLC (Serving Mobile Location Center) SAS-Centric solution as part of the 3GPP global UMTS standards. He acts as a consultant to different PCS and cellular carriers for planning, designing, implementing, and optimizing their cellular/PCS networks. As a lead researcher, he has developed a workstation-based PCS/cellular software package for AMPS, TDMA, and CDMA technologies, and a microwave frequency management software package for a U.K. personal communication network carrier as well as for a German cellular carrier. These software packages have been considered in Germany as the standard and are used in several European and Latin American countries to design personal communication systems and microwave systems. Between 1983 and 1985, he was involved in designing and implementing AM, FM, TV, and cellular systems. He teaches radiowave propagation and radio network planning for microwave and mobile systems for the telecommunications industry. Since 2000, he has been a Professorial Lecturer in applied electromagnetics and antenna and propagation for wireless systems at The George Washington University, Washington, DC. He has published articles on the topic of outdoor radiowave propagation. He is the coauthor of a chapter in the Handbook on Antennas in Wireless Communications (CRC Press, 2002). His current research topics include wave propagation in natural and urban terrain, radiation and scattering of electromagnetic waves in complex media, wave propagation in random media, and spectrum management techniques. Dr. Torrico is a senior member of the IEEE Antenna and Propagation Society, the Vehicular Technology Society, and the Geoscience and Remote Sensing Society. He is an elected Member of the International Scientific Radio Union (URSI/USNC) Commission F (Radio Wave Propagation and Remote Sensing) and the National Spectrum Managers Association (NSMA). At the NSMA he serves as the Co-Chairman of the Radiowave Propagation group. He is a contributor/participant of the European Cooperation in the Field of Scientific and Technical Research (COST) group. He was part of the organizing committee for several IEEE-APS/URSI International Symposiums and professional meetings. He is a member of the New York Academy of Science.

Thomas Kürner (S’91–M’94–SM’01) received the Dipl.-Ing. degree in electrical engineering and the Dr.-Ing. degree from Universitaet Karlsruhe, Germany, in 1990 and 1993, respectively. From 1990 to 1994, he was with the Institut fuer Hoechstfrequenztechnik und Elektronik (IHE), Universitaet Karlsruhe, working on wave propagation modeling, radio channel characterization and radio network planning. From 1994 to 2003, he was with the Radio Network Planning Department at the headquarters of the GSM 1800 and UMTS operator E-Plus Mobilfunk GmbH & Co KG, Duesseldorf, where he was Team Manager responsible for radio network planning tools, algorithms, processes and parameters. Since 2003, he has been a Full Professor of mobile radio systems at the Institut fuer Nachrichtentechnik (IfN), Technische Universitaet Braunschweig. His working areas are propagation, traffic and mobility models for automatic planning of mobile radio networks, planning of hybrid networks, car-to-car communications as well as indoor channel characterization for high-speed short-range systems including future terahertz communication systems. He has been engaged in several international bodies such as ITU-R SG 3, UMTS Forum Spectrum Aspects Group and COST 231/273/259/2100. H Prof. Dr.-Ing. Kürner has been a participant in the European projects ISTMOMENTUM and ICT-SOCRATES. Currently, he is chairing IEEE802.15 IG THz. He has served as Vice-Chair Propagation at the European Conference on Antennas and Propagation (EuCAP), in 2007 and 2009, and has been an Associate Editor of the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY since 2008. He is a member of VDE/ITG, VDI, and an elected member of the International Scientific Radio Union (URSI/USNC) Commission F (Radio Wave Propagation and Remote Sensing).

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Communications Modification of Radiation Patterns of First Harmonic Mode of Slot Dipole for Dual-Frequency Operation You-Chieh Chen, Shih-Yuan Chen, and Powen Hsu

Abstract—The effect of parasitic slots on the radiation patterns of the first harmonic mode of a coplanar waveguide-fed slot dipole antenna is presented. By adding four parasitic slots, two on each of the two arms of the slot dipole, the radiation patterns of the first harmonic mode of the slot dipole can be modified to be similar to those of its fundamental mode, while the performance of the fundamental mode is retained. Therefore, the antenna would operate as a dual-band operator with similar radiation patterns. Details of the design and measured and simulated results are presented and discussed. Index Terms—Coplanar waveguides, multi-frequency antennas, slot dipole antennas.

I. INTRODUCTION Due to the spurious radiations, the harmonic modes of an antenna are useless or need to be suppressed in some special applications such as active integrated antennas (AIAs) [1]–[3] and rectifying antennas (Rectennas) [4]. In other dual-frequency applications, such as GSM900/1800, the frequency ratio of the resonant frequencies in the two operating bands is two. Therefore, a slot dipole can easily cover the two bands by operating at its fundamental and first harmonic modes. However, the first harmonic mode of a slot dipole is not considered to meet the requirements because its radiation patterns have a null in the broadside direction, whereas those of the fundamental mode are broadside and bi-directional. In order to achieve desirable multi-band operation, the radiation patterns in all operating bands must be similar. This is the reason why all other previously proposed dual or tri-band slot dipole antennas [5]–[8] have their resonant frequencies designed at their corresponding fundamental modes of different path lengths to have similar radiation patterns in all operating bands. If one can modify the radiation patterns of the first harmonic mode of a slot dipole to be broadside and bi-directional as that of the fundamental mode, it would then be a better candidate to apply to such application due to its uniplanar and relatively simple structure and less work on impedance tuning. Nevertheless, there are few works making efforts to modify the radiation patterns of the first harmonic mode of a slot dipole.

Manuscript received September 09, 2010; revised October 24, 2010; accepted December 16, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported by the National Science Council, Taiwan, R.O.C., under Contracts NSC 97-2221-E-002-061-MY3 and NSC 98-2221-E-002-049. The authors are with the Department of Electrical Engineering and the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C. (e-mail: [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.2011.2152351

Fig. 1. Configuration of the proposed antenna.

The present work focuses on the modification of the radiation patterns of the first harmonic mode of a slot dipole. By adding four coupling slots, the first harmonic radiation patterns of the coplanar waveguide (CPW)-fed slot dipole are modified to be similar to those of its fundamental mode, while the performance of the fundamental mode in terms of bandwidth, gain, and patterns changes only slightly. Therefore, this antenna can be used for dual-frequency operation and is especially suitable for frequency ratio of two, such as GSM900/1800 systems. The simple technique for pattern modification can also be applied to other antenna designs including wide-band antennas and multi-band antennas with large frequency ratio to overcome the problems of distorted radiation patterns due to harmonics so that the pattern stability can be obtained. Besides, the bandwidth of the lower band of this antenna is wider than those of the previously proposed dual- or tri-band slot dipole antennas [6]–[8]. II. ANTENNA STRUCTURE AND DESIGN The configuration of the proposed CPW-fed slot dipole antenna with four parasitic slots is shown in Fig. 1. It consists of a CPW-fed slot dipole with length L and width W and four parasitic slots with length Ls and width Ws . The length of slot dipole L is corresponding to a half guided-wavelength of its fundamental mode while the width W is used for impedance tuning. The length of the parasitic slot Ls is equal to half of that of the slot dipole L. The width of the parasitic slot Ws is used for impedance tuning and pattern modification at the first harmonic frequency of the slot dipole. In order to maximize the coupling between slot dipole and parasitic slots, the spacing d between them is chosen as small as possible. When operating at the fundamental mode, most of the magnetic currents are flowing on the slot dipole and the length of slot dipole L corresponds to a half guided-wavelength while few are flowing on the parasitic slots as can be seen from Fig. 2(a). Therefore, the performance of the fundamental mode of a CPW-fed slot dipole with four parasitic slots is not much degraded with respect to the case without parasitic slots. When operating at the first harmonic mode, the magnetic currents both flow on the slot dipole and parasitic slots. The length of slot dipole L equals to one guided-wavelength of the first harmonic mode and the length of parasitic slots Ls corresponds to a half guided-wavelength. Each one of the slot dipole arms can be divided into two equal sections with opposite phases which will cause out-of-phase couplings on the parasitic slots. The couplings will cancel out each

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Fig. 4. Measured and simulated input reflection coefficients of CPW-fed slot dipole with and without parasitic slots.

Fig. 2. Simulated instantaneous magnetic current distributions of (a) fundamental mode and (b) first harmonic mode.

Fig. 5. Simulated input reflection coefficients of the proposed antenna with various d.

Fig. 3. Photograph of the prototype antenna.

other but the section sandwiched between parasitic slots has larger coupling than the other section does due to the closer distance. Therefore, the phases of the parasitic slots and the corresponding slot section sandwiched in between them are out-of-phase as shown in Fig. 2(b). Consequently, the radiations from these two regions cancel each other out. Only the other section of the slot dipole contributes to the radiations so that the broadside radiation patterns can be obtained. On the contrary, without parasitic slots, the radiation patterns of the first harmonic mode of the slot dipole will have a null in the broadside direction. It should be noted that the width of parasitic slots s should be large enough to have acceptable modified radiation patterns. The above discussion also applies to placing parasitic slots above and below the other sections of the slot dipole near the CPW.

W

III. SIMULATION AND MEASUREMENT RESULTS A hardware-proof-of-concept of the proposed CPW-fed slot dipole antenna with four parasitic slots was fabricated on an FR-4 substrate with dielectric constant r = 4 3, loss tangent tan = 0 02, and thickness = 1 6 mm. The ground plane dimension is 140 2 60 mm2 .

h

:



:



:

Fig. 6. Simulated input reflection coefficients of the proposed antenna with various W .

S LWL

The widths of the strip and gap of the 50- CPW feed line, and , are determined to be 3.0 and 0.3 mm, respectively. The dimensions of the slot dipole, parasitic slots, and their spacing , , s , are chosen to be 42, 5, 21, 7, and 0.3 mm, respectively, s , and such that its fundamental and first harmonic resonant frequencies are at about 2.3 and 4.6 GHz, respectively. For comparison, a CPW-fed slot dipole without parasitic slots was also fabricated with the same

G W

d

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Fig. 7. Measured radiation patterns at fundamental frequencies with parasitic slots: (a) E-plane (y -z plane) and (b) H-plane (x-z plane), and without parasitic slots: (c) E-plane (y -z plane) and (d) H-plane (x-z plane). Solid line: co-polarization, dashed line: cross-polarization.

parameters. All the simulations were carried out by using Ansoft Ensemble 8.0. Fig. 3 shows the photograph of a prototype of the proposed CPW-fed slot dipole antenna with parasitic slots built in the laboratory. Fig. 4 shows the measured and simulated input reflection coefficients of the antenna with and without parasitic slots. We can observe from this figure that the measured resonant frequencies of the fundamental modes with and without parasitic slots are 2.22 and 2.30 GHz, respectively, while those of the first harmonic modes are 4.64 and 4.88 GHz, respectively. The difference between first harmonic frequencies with and without parasitic slots may be interpreted as that the mutual coupling between parasitic slots and slot dipole becomes more significant 010 dB) at higher frequency. The measured bandwidths (jS11 j of the fundamental modes of the antenna with and without parasitic slots are 17.4% (2.00–2.38 GHz) and 20.8% (2.03–2.50 GHz), respectively, while those of the first harmonic modes are 8.8% (4.47–4.88 GHz) and 8.6% (4.68–5.10 GHz), respectively. Parametric studies are also conducted and shown in Figs. 5 and 6. Fig. 5 shows the effects of varying d while keeping other parameters unchanged. From this figure, we can observe that d does not significantly affect the frequency response. However, our simulations show that a smaller value of d can result in better patterns modification. The effects of the width of parasitic slot are investigated in Fig. 6 by varying Ws while keeping other parameters unchanged. As the width of parasitic slot Ws increases, the first harmonic frequency decreases. The radiation patterns of the fundamental mode of the slot dipole with and without parasitic slots measured at 2.22 and 2.30 GHz, respectively, are broadside and bi-directional and are nearly the same as can be seen from Fig. 7. The radiation patterns of the first harmonic mode of the slot dipole with and without parasitic slots measured at 4.64 and 4.88 GHz, respectively, are shown in Fig. 8. Comparing this figure with Fig. 7, we can clearly observe that the radiation patterns of the first harmonic mode of the antenna without parasitic slots are quite different from those of the fundamental modes. However, as the parasitic slots are added, the broadside and bi-directional radiation patterns are obtained both at the two operating bands. The measured peak

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Fig. 8. Measured radiation patterns at first harmonic frequencies with parasitic slots: (a) E-plane (y -z plane) and (b) H-plane (x-z plane), and without parasitic slots: (c) E-plane (y -z plane) and (d) H-plane (x-z plane). Solid line: co-polarization, dashed line: cross-polarization.

antenna gains of the slot dipole with and without parasitic slots at the fundamental frequencies are 5.9 and 6.0 dBi, respectively, and those at the first harmonic frequencies are 5.1 and 6.7 dBi, respectively. The antenna efficiencies obtained by using simulator Ansoft HFSS 10.0 of the proposed antenna at 2.22 and 4.64 GHz are 91% and 81%, respectively, and those obtained by measurement are 93% and 84%, respectively. IV. CONCLUSION Characteristics of a CPW-fed slot dipole antenna with four parasitic slots have been investigated. The four parasitic slots can modify the radiation patterns of the first harmonic mode of the slot dipole to be similar to those of its fundamental mode with only minor gain loss, while the performance of the fundamental mode of the slot dipole with and without parasitic slots is retained. Therefore, the proposed antenna can be used for dual-frequency operation and is especially suitable for frequency ratio of two, such as GSM900/1800 systems. Moreover, the simple technique for pattern modification can also be applied to other broad-band or multi-band antenna designs to overcome the problems due to distorted radiation patterns of harmonic resonances.

REFERENCES [1] H. Kim and Y. J. Yoon, “Microstrip-fed slot antennas with suppressed harmonics,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2809–2817, Sep. 2005. [2] A. Guraliuc, G. Manara, P. Nepo, G. Pelosi, and S. Selleri, “Harmonic tuning for Ku-band dielectric resonator antennas,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 568–571, 2007. [3] N.-A. Nguyen, R. Ahmad, Y.-T. Im, Y.-S. Shin, and S.-O. Park, “A T-shaped wide-slot harmonic suppression antenna,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 647–650, 2007. [4] Y.-J. Ren, M. F. Farooqui, and K. Chang, “A compact dual-frequency rectifying antenna with high-orders harmonic-rejection,” IEEE Trans. Antennas Propag., vol. 55, no. 7, pp. 2110–2113, Jul. 2007. [5] X.-C. Lin and C.-C. Yu, “A dual-band CPW-fed inductive slot-monopole hybrid antenna,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 282–285, Jan. 2008.

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[6] N. Behdad and K. Sarabandi, “A wide-band slot antenna design employing a fictitious short circuit concept,” IEEE Trans. Antennas Propag, vol. 53, no. 1, pp. 475–482, Jan. 2005. [7] S.-Y. Chen and P. Hsu, “Broadband radial slot antenna fed by coplanar waveguide for dual-frequency operation,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3448–3452, Nov. 2005. [8] S.-Y. Chen, Y.-C. Chen, and P. Hsu, “CPW-fed aperture-coupled slot dipole antenna for tri-band operation,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 535–537, 2008.

Low-Profile Composite Helical-Spiral Antenna for a Circularly-Polarized Tilted Beam H. Nakano, N. Aso, N. Mizobe, and J. Yamauchi

Abstract—A wire composite helical-spiral (CHES) antenna fed from the central point of its bottom is investigated, where the number of helical turns is selected to be extremely small (one turn). The CHES is designed such that it radiates a circularly-polarized tilted beam under the condition that the diameter of the CHES is smaller than that of a corresponding tilted-beam spiral antenna. It is found that the direction of the tilted beam in the elevation plane is not sensitive to a change in frequency. The input impedance and the gain are relatively constant within a design frequency range of approximately 9%.

Fig. 1. Composite helical-spiral (CHES) antenna. (a) Perspective view. (b) Top view. (c) Side view.

Index Terms—Composite helical-spiral antenna, low-profile antenna, tilted beam.

I. INTRODUCTION An antenna that radiates a tilted beam is often required, for example, for mobile communications systems, satellite communications systems, or wireless LAN systems. In response to this requirement, several tilted beam antennas have been developed; the antennas in [1], [2] are for linearly-polarized tilted radiation and the antenna in [3] is for circularly-polarized (CP) tilted radiation. The latter antenna [3] consists of a single spiral arm (not the conventional two spiral arms) and a CP tilted beam is generated by superimposing the radiation from a ring active region of two-wavelength circumference in the spiral plane onto the radiation from a ring active region of one-wavelength circumference [4]. Inevitably, the diameter of the spiral (the horizontal dimension or HD) is larger than 0.64 wavelength. This communication presents a novel CP tilted-beam antenna, designated as the composite helical-spiral (CHES) antenna. This antenna is reduced to 77% of the conventional spiral [3] in the HD but still achieves high radiation efficiency. As Kraus stated [5], a helical antenna above a ground plane (the x-y plane) radiates a CP axial beam, when the following two conditions are met: (1) the circumference of the helical arm, CHX , is between 3/4 wavelength and 4/3 wavelengths, and (2) the length of the helical arm is long enough to support a traveling wave current from the feed point to the arm end. As the circumference CHX is increased above Manuscript received January 05, 2010; revised November 08, 2010; accepted January 05, 2011. Date of publication May 10, 2011; date of current version July 07, 2011. The authors are with the College of Engineering, Hosei University, Koganei, Tokyo 184-8584, Japan (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.2011.2152318

4/3 wavelengths, the radiation beam becomes an off-axial beam. This phenomenon is used for designing the CHES in this communication. To make the height of the helical antenna small, the helical arm in this communication is truncated with a small number of turns. Such a truncated helical antenna does not radiate a CP wave due to the large reflected current from the helical arm end. One solution to this issue is to add a resistive material near the helical arm end to suppress the current reflected from the arm end [6]. However, addition of the resistive material has the disadvantage that it greatly reduces the radiation efficiency. Therefore, in this communication, a different technique is proposed; instead of using a resistive material, a spiral arm is connected to the truncated helical arm. The current on the spiral arm gradually attenuates, radiating into free space, and thereby the reflected current from the end of the spiral arm is reduced. This technique leads to a high radiation efficiency. II. CONFIGURATION A CHES is designed for Ku-band operation (from 11.7 GHz to 12.75 GHz), where frequency f = 12:225 GHz ( f12:225 ; wavelength 24:54mm  12:225 ) is the design center frequency. Fig. 1 shows the configuration of the CHES, whose arm is made of a single wire of radius  and fed from the coordinate origin. The helical section is specified by the pitch angle HX , radius rHX , and number of helical turns n. The spiral section is defined by an equiangular spiral function of rSP = rHX e0a( 0 ) , where rSP is the distance from the central point of the spiral, o’, to a point on the spiral arm; “a” is the growth rate; and 0 is the winding angle of the spiral, ranging from st to end . To design the CHES as a low-profile antenna, the number of helical turns, n, is chosen to be extremely small: n = 1 with HX = 2 rad (leading to a starting angle of st = 2 rad). The spiral growth rate and wire radius are fixed to be a = 0:105 mm=rad and  = 0:2 mm, respectively. The notation used in this communication is summarized in Table I.

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TABLE I CONFIGURATION PARAMETERS

= 6 8 mm at a frequency of 12.225 GHz, = 8 . (a)  = 2 rad. = 6 rad

Fig. 2. Radiation pattern for r : where the helix pitch angle is held at    . (c)  . (b) 

= 4 rad

Major parameters, including the helix radius rHX , pitch angle HX , and end angle of the spiral end , are varied subject to the objectives of the analysis. Note that the ground plane (GP) is assumed to be of infinite extent in the theoretical (numerical) analysis. III. PARAMETER STUDY The current distribution along the antenna arm is obtained by solving an integral equation [7] for an antenna of an arbitrarily curved wire, with the help of the method of the moment [8]. The antenna characteristics are calculated on the basis of the obtained current distribution. A. Effects of Helix Radius rHX and End Angle end The current distributed from the feed point F to the arm end

Send at f12:225 is investigated; for this, the end angle is varied  to  , with the helix radius rHX from end as a parameter (rHX : 12:225 =  rHX0 : , rHX : 12:225 =  rHX0 : , and rHX rHX+ : ), holding the pitch angle at 12:225 =   HX0 , which is designated as the initial pitch angle. It HX is found that, as the end angle end is increased, the decay in the amplitude of current I becomes smoother. This means that the reflected

2

= 2 radians 6 radians = 15 2   5 9 mm = 1 75 2   6 8 mm = 2   7 8 mm =8  jj

current traveling toward the feed point F decreases with an increase in end . As a result, the radiated wave is circularly polarized [see an example for rHX0 , shown in Fig. 2, where ER and EL are a right-hand CP wave component and a left-hand CP wave component, respectively. Note that the direction of maximum radiation (; ) = (max ; max ) expressed in the spherical coordinate system is illustrated by two axes (z1 and x1 )]. B. Effect of Pitch Angle HX and Decomposition of the Radiation Pattern On the basis of the discussion in Section III-A, we choose a CHES that has a helix radius of rHX = 6:8 mm (already designated as rHX0 , or the initial radius) and an end angle of end = 6 radians ( end0 , designated as the initial end angle). This antenna is expressed as CHES(rHX0 ; end0 ). Analysis reveals that, as the helix pitch angle HX for the CHES(rHX0 ; end0 ) is increased, the beam direction angle max , measured from the z axis, increases (from max = 24 for HX = 4 to max = 39 for HX = 12 ), as shown in Fig. 3. In other words, the beam direction in the elevation plane can be controlled by the pitch angle HX . It is worth investigating how each of the currents distributed along the helical and spiral arms contributes to the formation of

CHES(r

Fig. 3. Radiation pattern for : , with the pitch angle 

12 225 GHz

)

; at a frequency of f as a parameter.

=

the co-polarization component ER of the tilted CP beam. For this investigation, a CHES(rHX0 ; end0 ) with the initial pitch angle HX0 is used. Hereafter, the antenna having the configuration parameters

(rHX ; end ; HX ) = (rHX0 ; end0 ; HX0 ) = (6:8 mm; 6 rad; 8) is referred to as CHES(rHX0 ; end0 ; HX0 ). The first row of Fig. 4 presents the radiation field intensity jER j and phase R of the

right-hand CP wave component; these are calculated using the entire current distributed along CHES(rHX0 ; end0 ; HX0 ). The radiation field in the first row has a maximum intensity in the  = 32 ( max;12:225 ) direction and a minimum intensity in the  = 030 (null;12:225 ) direction. The second row presents the radiation intensity and phase of the right-hand CP component from the helical arm only, calculated on the basis of the current distributed along the helical arm (partial current of the entire current along the antenna arm). Similarly, the results calculated on the basis of the current distributed along the spiral arm only (partial current of the entire current along the antenna arm) are shown in the third row. As jER j shows, the helical arm radiates a tilted beam and the spiral arm radiates an axial beam (broadside beam). It is also found that, in the max;12:225

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Fig. 6. Frequency response of the axial ratio and gain.

Fig. 4. Decomposition of the radiation field from CHES (r

;

;

).

Fig. 7. Frequency response of the input impedance (Z ; ; ). CHES (r

Fig. 5. Frequency response of the beam direction ( (r ; ; ).

;

) for CHES

direction, the phase relevant to the radiation field from the helical arm   (R-Helix = 032 ) and that from the spiral arm (R-Spiral = 070 ) are constructive (not exactly in-phase but roughly in-phase) for forming a beam. However, in the null;12:225 direction, the phases with respect to R are approximately out-of-phase;R-Helix = 85 and R-Spiral = 096 . Due to these phase relationships, the overall radiation forms a tilted beam, as shown by the radiation pattern in the first row. IV. FREQUENCY RESPONSE OF THE BEAM DIRECTION, AXIAL RATIO, INPUT IMPEDANCE, AND GAIN The frequency response of the radiation characteristics of a representative CHES(rHX0 ; end0 ; HX0 ) is analyzed in this section. The diameter for this antenna is reduced to 77% of that for the conventional tilted-beam spiral antenna [3]. Within a design frequency range of 11.7 GHz and 12.75 GHz, the radiation efficiency is found to be 100% for perfectly conducting arm and ground plane. It is also found that the beam angle max in the elevation plane (x1 -z plane) remains relatively unchanged, as seen from Fig. 5; max is between 28 and 34 . On the other hand, the beam angle max varies from 69 to 121 within the design frequency range. This variation in max is attributed to the phase relationship between the two fields generated from the currents on the helical and spiral arms, as mentioned in Section III-B.

=R

+ jX ) for

The solid line in the bottom of Fig. 6 shows the frequency response for the theoretical axial ratio (AR), where the observation is performed always in the (; ) = (32 ; 95 ) direction, which is the beam direction at the design center frequency 12.225 GHz. It is confirmed that the tilted beam is circularly polarized within the design frequency range: less than 2.1 dB (a detailed calculation shows that the frequency bandwidth for an AR = 3 dB criterion is 16.9%, ranging from 11.275 GHz to 13.35 GHz). For checking the validity of the theoretical axial ratio, experimental work is performed, where a finite circular ground plane of radius 200 mm (7.8 wavelengths at the lowest design frequency of 11.7 GHz) is used to approximate a ground plane of infinite extent (used for the theoretical investigation). Good agreement between the theoretical and experimental data is found. Note that a similar good AR result is obtained in the beam direction (; ) = (max ; max ) varying with frequency (see Fig. 5). Note that the gain illustrated in Fig. 6 is commented in the last paragraph of this Section IV. Fig. 7 shows the frequency response of the input impedance (Zin = Rin + jXin ) for the CHES(rHX0 ; end0 ; HX0 ). It can be seen that the theoretical and experimental data are in good agreement. The input impedance is almost constant within the design frequency range, where the resistive and reactive components show, respectively, approximately Rin = 100 ohms and Xin = 0 ohm. Final comments are made for the gain. As shown in Fig. 3, the radiation pattern for ER in the max 0  plane [i.e., the azimuth plane around the z -axis at  = max or (; ) = (max , variable) plane] is very wide. This wide beam holds within the design frequency range. Therefore, the frequency response for the gain in the fixed direction   (; ) = (32 ; 95 ), which is the same direction used for the axial ratio observation, remains almost unchanged; in fact, the gain is between 9.1 dBi and 8.7 dBi within the design frequency range, as shown in Fig. 6. It can be said that the beam rotation (variation in max with frequency) does not cause a serious damage to the gain in the fixed direction.

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V. CONCLUSIONS A proposed CHES antenna is small in the horizontal dimension (diameter), compared with a corresponding spiral antenna that radiates a tilted CP beam based on the first and second radiation modes; the diameter for the CHES is reduced to 77% of that for the spiral. Analysis of the frequency response for the CHES reveals that the angle of the maximum radiation in the elevation plane, max , remains relatively unchanged; max is between 28 and 34 in a design frequency range of 11.7 GHz to 12.75 GHz. Within this frequency range, the axial ratio in the beam direction is less than 3 dB, the input impedance is almost constant, and the gain reduction in the fixed direction (always observed in the beam direction fixed at the center design frequency) due to the beam rotation is very small.

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A High-Isolation Dual-Polarization Microstrip Patch Antenna With Quasi-Cross-Shaped Coupling Slot Jie Lu, Zhenqi Kuai, Xiaowei Zhu, and Nianzu Zhang

Abstract—A novel dual-polarization microstrip patch antenna fed by quasi-cross-shaped slot with high isolation (greater than 50 dB) over a wide bandwidth from 3 to 4 GHz is investigated. Four identical slots—referred to “quasi-cross slot”—are arranged symmetrically in an unconnected cross configuration in the center. The isolation between two orthogonal polarizations of the proposed antenna can be greatly improved over the conventional cross slot. A prototype was fabricated and measured. The maximum achievable gain of the proposed antenna is 9.6 dBi with a gain variation of 0.6 dBi over the bandwidth of 200 MHz (3.4–3.6 GHz). The design is suitable for array application in MIMO system. Details of the proposed design and experimental results are presented and discussed. Index Terms—Aperture coupling, dual polarization, high isolation.

ACKNOWLEDGMENT The authors thank V. Shkawrytko, S. Kirita, and H. Mimaki for their assistance in the preparation of the manuscript.

REFERENCES [1] H. Nakano, Y. Ogino, and J. Yamauchi, “Array antenna composed of bent four-leaf elements,” in Proc. 3rd Eur. Conf. on Antennas and Propagation (Eucap2009), Berlin, Germany, Mar. 2009, pp. 1187–1190. [2] H. Nakano, Y. Ogino, and J. Yamauchi, “Bent two-leaf antenna radiating a tilted, linearly polarized, wide beam,” IEEE Trans. Antennas Propag., vol. 58, no. 11, pp. 3721–3725, Nov. 2010. [3] H. Nakano, Y. Shinma, and J. Yamauchi, “A monofilar spiral antenna and its array above a ground plane—Formation of a circularly polarized tilted fan beam,” IEEE Trans. Antennas Propag., vol. 45, no. 10, pp. 1506–1511, Oct. 1997. [4] J. A. Kaiser, “The Archimedean two-wire spiral antenna,” IRE Trans. Antennas Propag., vol. AP-8, no. 3, pp. 312–323, May 1960. [5] J. D. Kraus and R. J. Marhefka, Antennas, 3rd ed. New York: McGraw Hill, 2003, ch. 8. [6] H. Nakano, J. Yamauchi, and S. Iio, “Tapered backfire helical antenna with loaded termination,” Electron. Lett., vol. 18, no. 4, pp. 158–159, Feb. 1982. [7] H. Nakano, “Antenna analysis using integral equations,” in Analysis Methods for Electromagnetic Wave Problems, E. Yamashita, Ed. Norwood, MA: Artech House, 1996, vol. 2. [8] R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968.

I. INTRODUCTION Microstrip patch antennas have been applied to compact and multifunctional MIMO systems for years, as they are compact, have a low profile, and are lightweight. The well-known aperture-coupled patch antennas first presented by Pozar [1] have been developed into a variety of dual-polarization slot-coupled patch antennas reported in [2]–[11]. Their designs, available designs in the open literature, have two orthogonal offset slots [2], [3], cross-shaped slots [4]–[7] or H-shaped slots [8]–[10]. Moreover, a combined edge/aperture feeding and feedforward circuit provides good isolation [11], [12]. In general, better isolation results when the field excited is more symmetrical. The geometry of common orthogonal offset slots [2] can only provide the isolation of 18 dB over the impedance bandwidth. The isolation, though, is better than 35 dB, which can be achieved by the symmetrical arrangement of slots [3]. To decrease the coupling between two orthogonal polarizations, H-shaped slots are also a good choice. The isolation between two feeding ports with two orthogonal H-shaped slots is more than 36 dB over the operating band [8]. Tzung-Wern chiou and Kin-Lu Wong propose a hybrid feeding combining in-phase aperture-coupled feeding (H-shaped coupling slot) and out-of-phase gapcoupled probe feeding to obtain a high isolation of more than 40 dB [9]. A slot with a simple configuration, a cross-shaped coupling slot, is used to excite the two orthogonal modes and thereby generate dual linear polarizations. A power combiner joins the two feed lines located symmetrically with respect to the center of the cross, and an air bridge provides a crossover between the two orthogonal feed circuits; this was introduced by C. H. Tsao et al. to construct an aperture-coupled antenna [4]. Using the geometry of a cross slot that had a narrow central part, Mariano Barba showed that the isolation between two polarizations is typically 36 dB over a 24% impedance bandwidth (return losses > 24 dB) [5]. Bjorn Lindmark adopted a single 50 feed line exciting Manuscript received May 17, 2010; revised October 31, 2010; accepted January 15, 2011. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported in part by the National 973 project under Grant 2010CB327400, in part by the National High-Tech Project under Grant 2009AA011801, and in part by NSFC under Grant 60921063. The authors are with the State Key Laboratory of Millimeter Waves, School of Information Science and Engineering, Southeast University, Nanjing 210096, 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.2011.2152333

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Fig. 2. Geometries of antenna A and B.

feed lines is employed to decouple feed lines and patch edges—this is discussed in Section II, C . The isolation between two polarizations is enhanced to more than 50 dB over a broadband. Furthermore, the gain of the antenna is 9.6 dBi at operating frequency, higher than that of the conventional cross-shaped slot which has a gain of about 7.5 dBi [7]. A prototype of the proposed antenna has been implemented; the simulated and measured results are also presented. II. ANTENNA ARCHITECTURE AND DESIGNS The architecture of the proposed antenna is shown in Fig. 1. It consists of quadrate radiating patch, air gap, slots, and feed lines. These are discussed in greater detail in Sections III and IV. A. Patch and Air Gap It is known that patches on high dielectric constant substrates have a lower bandwidth than the ones on low dielectric substrates. As a result of this, it could be that the best substrate is air (in fact, a patch printed on low dielectric substrate with an air space). The air space between the upper layer and patch is h = 8 mm, which is about 0 =10 (see Fig. 1(a)). B. Slots

Fig. 1. Geometry of dual-polarized quasi-cross slot-coupled microstrip patch antenna. (a) Side view of the antenna, the height and relative permittivity of the and " : , respectively; the height and patch substrate are h relative permittivity of the feed substrates are h : and " : , . respectively; the air space between the upper layer and patch is h (b) Top view of the antenna. (c) Coupling slots on the ground plane; single slot L : ,W ; the ; the space between two slots, g side length of the patch, : .

= 1 mm

= 19 2 mm

= 2 mm a = 27 8 mm

= 2 65

= 0 508 mm

=22 = 8 mm = 4 mm

one leg of the cross-shaped aperture and two 100 lines exciting the other leg symmetrically to achieve an isolation greater than 35 dB [6]. The results reported in the above-mentioned papers shows that high isolation (e.g., > 45 dB) between two polarizations over the entire impedance bandwidth is difficult to reach when a conventional crossshaped coupling slot is used for dual-polarization, because (1) The purity of two linear polarizations generated by a cross slot is not high enough to keep them strictly orthogonal; and (2) Strong coupling will occur between two orthogonal polarizations in the center of the cross slot if the field around the slot is not strictly symmetrical. These disadvantages are overcome in the model proposed in this communication by using a quasi-cross-shaped coupling slot—discussed in Section II, B . Further, the structure of U-shaped folded

The quasi-cross-shaped slot consists of four slots, which are of the same size and arranged symmetrically in a cross configuration (see Fig. 1(c)). The space between two slots, which are parallel, is g. The value of g should be greater than the width of one slot (2 mm), as much as possible, because when g is less than or slightly greater than 2 mm, the orthogonal slots will either overlap or be too close, and strong coupling will occur between the two polarizations. Further, in order to avoid strong coupling between the feed lines in the upper layer and the edges of the patch, the value should be less than a0L (assume that the slots are fed centrally), as low as possible. For the proposed antenna, within the available range of 2–8.6 mm, when the value of g is chosen to be 4 mm (or twice the width of one slot), it is found in the simulation that good isolation and radiation performance can be obtained. As mentioned in Section I, the impurity of two linear polarizations and the coupling between two orthogonal polarizations in the center of the cross slot can be overcome by using a quasi-cross-shaped slot. First, as there is no overlap in the center of the quasi-cross, the coupling in the center of the cross slot can be easily eliminated. Second, In order to demonstrate that the purity of two linear polarizations and the antenna gain can be improved with the geometry of Fig. 1(c) compared to that of the conventional cross slot, two antennas fed by different slots have been simulated in HFSS [13], as shown as Fig. 2: Antenna A—a patch antenna with a conventional single coupling slot; and Antenna B—a patch antenna with the proposed dual coupling slots. It is well known that the distribution of surface currents on the patch can adequately reflect the purity of the linear polarization generated by the coupling slot, so the surface currents between the two antennas

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Fig. 3. The distribution of surface currents on the patch (a) Antenna A and (b) Antenna B.

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Fig. 5. The feed networks of the antenna. (a) The general design. (b) The modified design.

Fig. 4. The simulated radiation patterns of antennas A and B (a) E-plane radiation patterns and (b) H-plane radiation patterns.

were compared. Fig. 3(a) depicts the surface current distribution on the patch of antenna A, on which the surface current distribution is whirly. The purity of linear polarization is poor because the direction of surface currents varies greatly. Moreover, the surface currents distributed in the middle and edges of the patch are opposite and their far-field radiation cancels out in the z -axis direction, which leads to a drop in the antenna gain. In Fig. 3(b), however, the current distribution on the patch is in one direction, not whirly, so the purity of the linear polarization of antenna B is much higher than in the case of antenna A. Besides, since the surface current intensity distributed in the middle and edges of the patch is in the same direction, the far-field radiation increases. As a result, the gain of antenna B is higher than that of antenna A. Moreover, the cross polarization level of antenna B is lower than that of antenna A, as shown in Fig. 4. Therefore, the purity of linear polarization and antenna gain is higher in the dual-polarized microstrip antenna with

Fig. 6. Photograph of the proposed antenna.

quasi-cross-shaped slot, than that in the dual-polarized microstrip antenna with conventional cross slot. C. Feed Lines The feed networks of the proposed antenna consist of the upper layer (f2) and the lower layer (f1), which are separated by the ground plane (see Fig. 1(a)). The general design of the feed networks is shown in Fig. 5(a); the stubs of microstrip lines are straight. However, there was a spur lying on S21 as shown in Fig. 7(a), which deteriorated the isolation over the bandwidth (3.0–4.0 GHz). This spur appears on S21 mainly because of the strong electric field coupling between the feed lines, which feed the horizontally polarized slots, and two edges of patch

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Fig. 8. Measured radiation patterns of the proposed antenna at (a) 3.4 GHz, (b) 3.5 GHz, (c) 3.6 GHz.

TABLE I SIMULATED AND MEASURED GAIN OF THE PROPOSED ANTENNA

S parameters against frequency; ground-plane size = 120 mm 2 120 mm, a = 27:8 mm, L = 19:2 mm, W = 2 mm, g = 4 mm. (a) Simulated and measured S for the design in Fig. 5(a). (b) The Fig. 7. Measured

simulated and measured S for the design in Fig. 5(b). (c) The measured S and S for the design in Fig. 5(b).

that are part of the radiation area for vertical polarization. The regions of the coupling between the two orthogonal polarizations are shown in Fig. 5(a). To keep the slots fed centrally and the feed lines as far away from the edges of patch as possible, the design of feed lines were modified to allow them to be folded U-shaped structures, as shown in Fig. 5(b). With this modification, the spurs disappear from S21 and the curves are more flat, as can be seen in Fig. 7(b). In the lower layer (f1) of the modified design, the dual slots are fed uniformly in amplitude and phase. The upper layer (see Fig. 5(b)) has a similar configuration as the lower one. III. EXPERIMENTAL RESULTS The antenna was fabricated, as shown in Fig. 6. The center frequency was designed to be at 3.5 GHz. In the lower layer, f 1 t = 3:2 mm, f 1 u1 = 1:57 mm, and f 1 u2 = 0:8 mm. In the upper layer (f2), f 2 t = 3:2 mm, f 2 u1 = 1:57 mm, f 2 u2 = 1 mm. The

other dimensions of the antenna are given in the caption of Fig. 7. The measured impedance bandwidth (return loss > 10 dB) was 570 MHz (3.23–3.8 GHz) and 610 MHz (3.3–3.91 GHz) for port 1 and port 2, respectively, which met the bandwidth requirement (3.4–3.6 GHz) of antennas in MIMO system, as shown in Fig. 7(c). The simulated and measured isolation between port 1 and port 2 is shown in Fig. 7(b). Good agreement between simulated and measured results is obtained. It can be seen that the isolation against frequency across the entire measured bandwidth (3.0–4.0 GHz) was more than 50 dB. The E-plane and H-plane radiation patterns of ports 1 and 2 within the operating bandwidth (3.4–3.6 GHz) were measured and are presented in Fig. 8. It is shown that the cross-polarization components are all 038 dB down from the copolarization components on boresight in E and H planes, respectively. In all three cases (Fig. 8(a)–(c)), the cross-polarization level is less than 017 dB within 3-dB beamwidth of 45 and 46 for E plane and H plane, respectively. The front-to-back ratio of the proposed antenna is less than 10 dB, which is worse than a conventional cross-shaped slot coupled antenna. That is mainly because the length of the slot in Fig. 3(b) is longer than that of the slot in Fig. 3(a) and hence the backward radiation of a quasi-cross-shaped slot

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is stronger than that of a conventional cross slot. Table I lists the simulated and measured antenna gain. The maximum gain of the proposed antenna within the operating bandwidth was 9.6 dBi.

Omnidirectional Circularly Polarized Antenna Utilizing Zeroth-Order Resonance of Epsilon Negative Transmission Line

IV. CONCLUSIONS

Byung-Chul Park and Jeong-Hae Lee

The design of a dual-polarized aperture-coupled patch antenna with a quasi-cross-shaped slot was presented. The symmetrical feeding and the quasi-cross-shaped slot of the proposed antenna provide purer linear polarization, a much higher isolation (> 50 dB) and higher gain (9.6 dBi) than the conventional cross slot antenna. Further, the modified design of folding feed lines, which is introduced to suppress the coupling between the edge of the patch and feed lines, improves the isolation performance.

REFERENCES [1] D. M. Pozar, “A reciprocity method of analysis for printed slot and slot coupled microstrip antennas,” IEEE Trans. Antennas Propag., vol. 34, no. 12, Dec. 1986. [2] A. Adrian and D. H. Schaubert, “Dual aperture-coupled microstrip antenna for dual or circular polarization,” Electron. Lett., vol. 23, no. 23, Nov. 1987. [3] P. Brachat and J. M. Baracco, “Printed radiating element with two highly decoupled input ports,” Electron. Lett., vol. 31, no. 4, Feb. 1995. [4] C. H. Tsao, Y. M. Hwang, F. Kilburg, and F. Dietrich, “Aperture-coupled patch antennas with wide-bandwidth and dual-polarization capabilities,” in Proc. Antennas and Propagation Society Int. Symp., 1988, vol. 3. [5] M. Barba, “A high-Isolation, wideband and dual-linear polarization,” IEEE Trans. Antennas Propag., vol. 56, no. 5, May 2008. [6] B. Lindmark, “A novel dual polarized aperture coupled patch element with a single layer feed network and high isolation,” in Proc. Antennas and Propagation Society Int. Symp., Jul. 1997, vol. 4. [7] J. R. Sanford and A. Tengs, “A two substrate dual polarized aperture coupled patch,” in Proc. Antennas and Propagation Society Int. Symp., Jul. 1996, vol. 3. [8] S.-C. Gao, L.-W. Li, M.-S. Leong, and T.-S. Yeo, “Dual-polarized slotcoupled planar antenna with wide bandwidth,” IEEE Trans. Antennas Propag., vol. 51, no. 3, Mar. 2003. [9] T.-W. Chiou and K.-L. Wong, “Broad-band dual-polarized single microstrip patch antenna with high isolation and low cross polarization,” IEEE Trans. Antennas Propag., vol. 50, no. 3, Mar. 2002. [10] 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, Feb. 2002. [11] I. Ac’imovic, D. A. McNamara, and A. Petosa, “Dual-polarized microstrip patch planar array antennas with improved port-to-port isolation,” IEEE Trans. Antennas Propag., vol. 56, no. 11, Nov. 2008. [12] S. L. Karode and V. F. Fusco, “Dual polarised microstrip patch antenna using feedforward isolation enhancement for simultaneous transmit/ receive applications,” presented at the National Conf. on Antennas and Propagation, Apr. 1, 1999. [13] Ansoft HFSS Version 11.

Abstract—The omnidirectional circularly polarized (CP) antenna using a circular mushroom structure with curved branches is proposed. The antenna is based on the zeroth-order resonance (ZOR) mode of epsilon negative (ENG) transmission line (TL) to obtain a vertical polarization and an omnidirectional radiation pattern. Also, the horizontal polarization is obtained by the curved branches. The 90 phase difference between two orthogonal polarizations is inherently provided by the zeroth-order resonator. Therefore, the antenna has an omnidirectional CP radiation pattern in the azimuthal plane. In addition, this antenna is planar type and simply designed without a dual feeding structure and 90 phase shifter. The measured average axial ratio and left-hand (LH) CP gain are 2.03 dB and 0 40 dBic, respectively, in the azimuthal plane. Index Terms—Epsilon negative transmission line (ENG TL), omnidirectional circularly polarized antenna, zeroth-order resonance (ZOR) antenna.

I. INTRODUCTION The omnidirectional circularly polarized (CP) antenna is very attractive in wireless communications due to its omnidirectional radiation pattern and CP property [1], [2]. Because the omnidirectionality and the CP property of an antenna are not only capable of covering a large service area but also support a free alignment between the receiving and transmitting antennas, an omnidirectional CP antenna is very useful for wireless communication systems such as Global Position Systems (GPSs) and personal mobile systems [3], [4]. Recently, the linearly polarized (LP) metamaterial antennas with omnidirectionality have been reported using the zeroth-order resonance (ZOR) [5], [6]. These ZOR antennas having mushroom patches generate a uniform vertical electric field against ground plane, so that the antennas can be omnidirectional radiator. Some metamaterial CP antennas composed of dual feed and 90 phase shifter have been published [7], [8]. However, these metamaterial antennas have directional radiation patterns. The omnidirectional circularly polarized antenna using a metamaterial transmission line was first presented in 2009 [9]. The antenna in [9] utilized the ZOR mode of double negative (DNG) transmission line (TL). However, the measured performance of an antenna did not show a good omnidirectional axial ratio since the antenna should have an asymmetrical feed due to its structure. In this communication, the omnidirectional CP metamaterial antenna is proposed using the ZOR mode of epsilon negative (ENG) transmission line (TL) to improve the axial ratio. The CP antenna using the ENG TL is good for the symmetrical feed and simple because it has no gaps between the cells. The antenna utilizes the ZOR mode of epsilon negative (ENG) transmission line (TL) to obtain an omnidirectional radiation pattern and a vertical polarization. The

Manuscript received June 13, 2010; revised November 06, 2010; accepted December 16, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported by the Mid-career Researcher Program through an NRF grant funded by the MEST (No. 2010-0013273). The authors are with the Department of Electronic Information and Communication Engineering, Hongik University, Seoul 121-791, 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.2011.2152337 0018-926X/$26.00 © 2011 IEEE

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Fig. 1. Structure of antenna.

horizontal polarization is obtained by the curved branches. Also, the 90 phase difference between the vertical and the horizontal polarization is inherently supported by the zeroth-order resonator. The antenna, therefore, has an omnidirectional CP radiation pattern in the azimuthal plane. The CP antenna is planar type and simply designed without 90 phase shifter and dual feed line. In addition, the electrical size of the proposed antenna is reduced by 44%, compared with that of the previously presented omnidirectional CP antenna in [1]. To confirm the characteristics of the CP antenna, the proposed antenna was fabricated and measured. II. DESIGN OF ANTENNA Fig. 1 shows the structure of the proposed antenna. As shown in Fig. 1, the CP antenna is based on the circular mushroom structure which consists of four unit cells (2 by 2). It is noted that a unit cell corresponds to a quarter of a circular patch including one via. Since the CP antenna uses the ZOR of ENG TL, it has no gaps between the cells as stated above. For the symmetrical axial ratio and radiation pattern, the coaxial feed is placed on the center of the circular mushroom structure as shown in Fig. 1. The ZOR mode of a circular mushroom structure gives the omnidirectional radiation pattern and vertical polarization. To obtain the horizontal polarization, the curved branch is attached to each unit cell. Also, the 90 phase difference between the vertical and the horizontal polarization is inherently supported by the zeroth-order resonator. The antenna, therefore, has an omnidirectional CP radiation pattern in the azimuthal plane. The dispersion curves in Fig. 2 are calculated using a unit cell of a rectangle patch since the ZOR mode is used for the CP antenna. The dispersion curves are obtained from full wave simulation (Ansoft HFSS) by assuming the infinite periodic structure. Thus, a unit cell of a quarter circular patch in Fig. 1 is converted to a unit cell of a rectangle patch since the quarter circular patch cannot be applied to the periodic boundary condition for simulation. Instead, the area of a unit cell is set to be equal to that of a quarter circular patch in Fig. 1. It is also calculated that the ZOR frequency decreases as the length of the curved branch increases because the longer length of L provides larger shunt capacitance (CR ). Note that the ZOR frequency is proportional to (LL CR )00:5 where LL and CR are shunt inductance (left-handed component) and shunt capacitance (right-handed component), respectively. Fig. 3 shows the electric field (on the left) and the surface current (on the right) distribution of the CP antenna at t = 0; T=4; T=2, and 3T=4. Since this antenna utilizes the ZOR mode to radiate, it behaves as a capacitor. Thus, the curved branch attached to each unit cell is charged

Fig. 2. Dispersion curves.

and discharged repeatedly in a period of time (T). At t = 0, the curved branches are fully charged and the positive charges are on the top of the branches. At this time, no surface current flows and the antenna radiates the vertically polarized wave as shown in Fig. 3(a). At t = T=4, the positive charges on the top of the branches are fully discharged, resulting in the surface current flow as shown in Fig. 3(b). At this time, the antenna radiates the horizontal polarized wave which is generated by the horizontal surface current on the branches. At t = T=2, the curved branches are fully charged and the negative charges are on the top of the branches. Thus, no surface current flows and the antenna radiates the vertically polarized wave which is reversely polarized with respect to the vertically polarized wave in Fig. 3(a). At t = 3T=4, the negative charges on the top of the branches are fully discharged, resulting in the current flow as shown in Fig. 3(d). Therefore, the curved branches of the zeroth-order resonator inherently provide the 90 phase difference between the vertically and horizontally polarized wave because there is always the 90 phase difference, corresponding to time difference of T/4, between the charged and discharged state. Table I shows the phase difference, magnitude ratio, axial ratio, and center frequency when the length of the curved branch (L) changes from 25 to 31 mm and the other values (D = 56 mm, d = 40 mm, w = 1:5 mm, s = 9 mm) are fixed. As shown in Table I, it is found that the magnitude ratio (E =E ) between two orthogonal components is controlled by the length of the curved branch (L). On the other hand, the phase difference between two orthogonal components is simulated to be independent of the length of the curved branch (L), indicating that the curved branches of the zeroth-order resonator inherently provide the 90 phase difference. It is also observed that the center frequency decreases as the length of L increases because the longer L provides a larger shunt capacitance, as discussed before. For a good CP radiation, the magnitude ratio and the phase difference between two orthogonal components are close to 1 and 90 , respectively, and thus, the good average axial of 1.04 dB is obtained in the simulation at the length L of 31 mm as shown in the Table I. However, due to the employment of balun to prevent the cable radiation in the measurement, the good average axial ratio of 2.03 dB is shown at the length L of 29 mm rather than at the length L of 31 mm in the azimuthal x-y plane ( = 90 ). Note that four curved branches on the substrate are directed to the clockwise direction, resulting in the left-hand (LH) CP characteristic of the antenna. Conversely, if the direction of arrangement of four curved branches is in the counterclockwise direction, the right-hand (RH) CP characteristic will be obtained.

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Fig. 4. Return loss.

Fig. 3. Electric field and surface current distribution of the proposed antenna (a) t = 0, (b) t = T=4, (c) t = T=2 (d) t = 3T=4.

TABLE I THE PHASE DIFFERENCE, MAGNITUDE RATIO, AXIAL RATIO, AND CENTER FREQUENCY VERSUS THE LENGTH OF THE CURVED BRANCH (L)

III. EXPERIMENTAL RESULTS The CP antenna is fabricated on an RT/Duroid 5880 substrate ("r = confirm the omnidirectional radiation pattern and CP property. The dimensions of design values are D = 56 mm (0:2770 ); d =

2:2) to

Fig. 5. Simulated and measured axial ratio at  = 90 .

mm (0:1980 ); w = 1:5 mm (0:0070 ); s = 9 mm (0:0450 ), and L = 29 mm (0:1440 ), respectively. A Bazooka balun is used in the measurement to suppress the current through the outside conductor of coaxial cable. The simulated and measured return loss are shown in Fig. 4. The measured return loss shows a good agreement with simulated data. The simulated and measured 10 dB bandwidth are 8 MHz (1.475 to 1.483 GHz and 1.481 to 1.489 GHz), corresponding to fractional bandwidth of 0.541% and 0.539%, respectively. Fig. 5 shows the simulated and measured axial ratio in the azimuthal plane ( = 90 ). The simulated and measured average axial ratio at each ZOR frequency (1.479 GHz and 1.485 GHz) are 1.83 dB (1.65 to 2.04 dB) and 2.03 dB (1.64 to 2.48 dB), respectively. The results show that the antenna has a good CP property. The measured axial ratio is still less than 3 dB at  = 86 ; 88 , and 90 within a 10 dB bandwidth as shown in Fig. 6. The simulated and measured gain are presented in Fig. 7. The simulated and measured average LHCP gain are 0.56 dBic (0.51 to 0.60 dBic) and 00:40 dBic (00:73 to 00:14 dBic), respectively, in the azimuthal x-y plane ( = 90 ). On the other hand, the simulated and measured average RHCP gain are 019:02 dBic (019:89 to 018:10 dBic) and 019:10 dBic (021:02 to 017:66 dBic), respectively, in the azimuthal x-y plane ( = 90 ). Thus, these results prove 40

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Fig. 6. Measured average axial ratio ( frequency.

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= 84 ; 86 ; 88 ; 90

, and 92 ) versus

( = 90 ) versus frequency.

Fig. 8. Measured LHCP gain 

IV. CONCLUSION The omnidirectional CP antenna has been introduced using a circular mushroom structure with the curved branches. The antenna uses the ZOR mode of ENG TL to obtain the omnidirectional radiation pattern and the vertical polarization. Also, the horizontal polarization is achieved by the curved branches. The 90 phase difference between two orthogonal polarizations is inherently provided by the zeroth-order resonator. The CP antenna has an omnidirectional radiation pattern with the average LHCP gain of 00:40 to 00:95 dBic with ripple of less than 1 dB. The measured average axial ratio and LHCP gain in the azimuthal x-y plane ( = 90 ) are 2.03 dB and 00:40 dBic, respectively, which show a good agreement with the simulated results.

REFERENCES

Fig. 7. Simulated and measured gain (a) x-y plane and (b) x-z plane.

that the proposed antenna has a LHCP characteristic. The average LHCP gain of 00:40 to 00:95 dBic with ripple of less than 1 dB was measured within a 10 dB bandwidth, as shown in Fig. 8. The proposed antenna efficiency of 56% was measured by the Wheeler cap method [10], [11] and the simulated antenna efficiency is 76%. The difference of antenna efficiency between simulation and measurement is thought to be due to the use of Bazooka balun in the measurement.

[1] F. R. Hsiao and K. L. Wong, “Low-profile omnidirectional circularly polarized antenna for WLAN access point,” Microw. Opt. Technol. Lett., vol. 46, no. 3, pp. 227–231, 2005. [2] Y. Xu and C. Ruan, “A novel design of circularly polarized omni-directional antenna for Ka band,” in Proc. Millimeter Waves Global Symp., 2008, pp. 378–379. [3] K. W. Leung and H. K. Ng, “The slot-coupled hemispherical dielectric resonator antenna with a parasitic patch: Applications to the circularly polarized antenna and wideband antenna,” IEEE Trans. Antennas Propag., vol. 53, no. 5, pp. 1762–1769, 2005. [4] R. R. Ramirez, F. D. Flaviis, and N. G. Alexopoules, “Single-feed circularly polarized microstrip ring antenna arrays,” IEEE Trans. Antennas Propag., vol. 48, no. 7, pp. 1040–1047, 2000. [5] J. G. Lee and J. H. Lee, “Zeroth order resonance loop antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 994–997, 2007. [6] 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, 2007. [7] C. J. Lee, K. M. K. H. 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. Y. F. Yang and A. Z. Elsherbeni, “A dual band circularly polarized ring antenna based on composite right and left handed metamaterials,” Progr. Electromagn. Res., vol. 78, pp. 73–81, 2008. [9] B. C. Park and J. H. Lee, “Omnidirectional circularly polarized antenna base on meta material transmission line,” presented at the IEEE AP-S, Jun. 2009. [10] D. M. Pozar and B. Kaufaman, “Comparison of three methods for the measurement of printed antenna efficiency,” IEEE Trans. Antennas Propag., vol. 36, pp. 136–139, 1988. [11] 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, 2005.

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A Compact Sequential-Phase Feed Using Uniform Transmission Lines for Circularly Polarized Sequential-Rotation Arrays Shih-Kai Lin and Yi-Cheng Lin

Abstract—A compact sequential-phase (SP) feed for circularly polarized sequential-rotation (SR) arrays is presented. Unlike traditional SP feed using multiple segments for impedance transformations, the presented SP feed employs only a single-stage transition where the transmission line width is uniform, making the whole SP feed very compact and neat in layout. The radial design of SP-feed lines to antenna elements may further facilitate the optimization work in adjusting the element spacing. Two 4 and 3 8 are given and miniature square SP feeds of the sizes compared. Experimental prototypes were built and verified with measured 2 results. The proposed compact SP feed can be extended to 2 feeding networks, and is very suitable for large-scale printed CP arrays. Index Terms—Antenna array feeds, antenna impedance, broadband antennas, circular polarization (CP), sequential rotation (SR).

I. INTRODUCTION Sequential rotation (SR) techniques are of interest in the design and implementation of broadband circularly polarized (CP) antenna arrays. Through proper sequential rotations of the array elements with the excitation of sequential phases, the CP antenna arrays may perform remarkable CP bandwidths, even though the element is linearly polarized [1]–[3]. Theoretical and experimental studies have shown that SR techniques may improve the overall bandwidth performances of CP arrays, in terms of polarization purity, impedance matching, and pattern symmetry [4], [5]. In recent years, several broadband designs of CP antenna elements have been proposed [6]–[8]. All suggest that SR techniques may further improve CP bandwidth when the antenna is extended to 2 2 2 arrays/sub-arrays. For instance, a traveling wave CP antenna [8] can increase the 3 dB-AR bandwidth from 20% to 50% with the SR techniques. The sequential phase (SP) feed in conjunction with SR elements plays an important role in practical CP-array designs. Various SP-feed designs have been reported, including corporate, series, and hybrid types [9]–[11]. These SP feeds commonly employ multiple segments of quarter wavelength transformers for impedance matching. Therefore, the entire feeding networks have become complicated and might occupy a large area. A large SP feed may cause the antenna element spacing to be increased, particularly when the SP feed and SR elements are placed on the same layer of PCB. The increased element spacing would, in turn, raise the side lobe level in the array radiation patterns. A desirable SP feed requires the following features: (1) providing antenna elements with balanced signals of equal magnitude and an incremental 90 phase, (2) transforming the antenna impedance to the feed I/O port impedance for arbitrary values, and (3) occupying a small area as to be accommodated in the array layouts with flexibility to adjust Manuscript received July 20, 2010; revised October 24, 2010; accepted November 08, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported in part by the National Science Council of Taiwan under Grant NSC 99-2219-E-002-006 and in part by Excellent Research Projects of National Taiwan University under Grant 99R80302. S.-K. Lin is with the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan. Y.-C. Lin is with the Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan (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.2011.2152346

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the element spacing. In this communication, we present an SP-feed design incorporating the above features. The following sections describe the design scheme’s details, the performance analysis, and alternative layouts. II. CONFIGURATION AND DESIGN SCHEME A. Evaluation of SP-Feed Size for Optimal Performance

Fig. 1 shows the configuration of a sequential rotation 2 2 2 CP antenna array associated with the SP feed. The CP antenna element is represented by a singly-fed notched circular patch that is sequentially rotated and fed by the corresponding excitation through the SP feed. In practice, the element spacing d is the key parameter as it may affect the broadside gain, the side lobe level, and the cross polarization discrimination (XPD) caused by the coupling effects. From the experimental results [6], the suggested optimal element spacing d is about 0.7 0 , which is equivalent to a guided wavelength g for a transmission line in general PCBs of a low dielectric constant ("r = 3 0 5). Insofar as the general resonant type CP antenna has a size of about a half-wavelength 0.5 g and a margin of 0.1 g for the distance between SP feed and the antenna element, the desirable size of a square SP feed Lf should be evaluated at 0.3–0.4 g . Fig. 1(a) shows the overall configuration of 2 2 2 CP arrays with an emphasis on the evaluated size and spacing of array elements and the SP feed. Fig. 1(b) shows the topology of the proposed radial SP feed for the example of Lf = g =4. B. Compact SP Feed of g =4 2 g =4 In Fig. 1(b), the I/O port of the entire SP feed, denoted by Port 1, has impedance Zin determined from the testing connector or RF circuitry. Port 2 to Port 5 are antenna-feed ports designed with equal amplitude and an incremental 90 phase delay, connected to the corresponding antenna elements, with impedance denoted by Za . Note that the unit length here is g =16, used as the basis length in the topology of SP-feed layout, where g is the guided wavelength of the transmission line in the PCB at the operating frequency. In Fig. 1(b), the two meander lines have total lengths of one-quarter wavelengths and three-quarter wavelengths, respectively, performing the quarter-wavelength impedance transformation. The width of these meander lines is determined by the antenna-port impedance and the I/O-port impedance, which is explored at greater length in a later section. Note that the presented SP feed in Fig. 1(b) contains only a single transition of impedance transformation, leading to uniformity in the width of the transmission line in SP feed. Furthermore, the entire layout is simple and compact, occupying an area of only g =4 2 g =4, which is the minimal limit for an SP feed using straight delay line segments for Port 2–3 and Port 4–5. C. Equivalent Circuit Model Fig. 2 shows the design schematic diagram through the equivalent transmission line model for the proposed SP feed. The transmission line segment is expressed in terms of the characteristic impedance and the propagation phase (electric length) in the figure, corresponding to the line width and length of microstrip transmission lines in the physical layout. The single-transition impedance transformers are realized by the central segments of characteristic impedance Zt , which are implemented by the meander line sections for miniaturization purposes, as shown in Fig. 1(b). Note that the three-quarter-wavelength segment performs the quarter-wave impedance transformation and, at the same time, provides an extra 180 phase delay with respect to the quarter-wavelength segment. Thus, the impedance of transformer segments is designed to convert the impedance of Za =2 at the joint of two adjacent antenna ports to the impedance of 2Zin at the joint of the SP feed and the I/O port. The relation of Zin ; Za , and Zt is expressed by

0018-926X/$26.00 © 2011 IEEE

1

1

2 Zt = (2Zin ) (Za =2) = Zin Za :

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Fig. 1. (a) Conceptual configuration of the sequential-phase (SP) feed and 2 mensions of the proposed SP feed using uniform transmission lines.

2 2 sequential-rotation CP arrays (RHCP example). (b) Topology and nominal di-

Fig. 3. Implementation and prototype of the presented SP feed: (a) optimal dimensions; (b) photo.

Fig. 2. Schematic diagram of the proposed SP feed.

If the antenna impedance is designed to match the input port impedance, then we have Zin = Za = Zt . Under this condition, the entire layout of the SP feed and connecting transmission lines become very neat in that all segments have uniform width, like the layout shown in Fig. 1(b). In practice, when the width of a microstrip line is enormous owing to a thick PCB substrate or process limitations (particularly at millimeter wave frequencies), the meander line sections may become bulky and dense so that coupling effects become manifest. In this case, some optimal tuning may be needed to compensate for the coupling effects in the proposed SP feed. III. EXPERIMENTAL RESULTS AND DISCUSSION The presented SP feed design is illustrated with an FR4 substrate with a dielectric constant of 4.4 and a thickness of 0.6 mm. Fig. 3 shows the optimal dimensions of the proposed SP feed of area g =4 2 g =4 designed at 2.5 GHz and the prototype photo. Note that a minor tuning for the meander segment’s dimensions is rendered to compensate for the coupling effects, in comparison to the nominal dimensions in Fig. 1(b). Fig. 4 shows the measured and simulated S-parameter performances of the developed SP feed. It can be observed that the measured 10-dB return loss bandwidth covers a wide band from 2.08 GHz to 2.95 GHz,

which is about 36% in the fractional bandwidth. Regarding the transmission performances, Fig. 4 shows that the input power is uniformly divided into the SP-feed ports, namely Port 2 to Port 5. In practice, the bandwidth of magnitude balance can be defined as the maximum difference among the output ports within a specific level L (1jSj1 j L dB), depending on the applications. Usually, the balance level and the balance bandwidth constitute a trade-off between the two factors. In this study, we optimize the SP-feed prototype for the balance bandwidth at the expense of the balance level, which is specified at L = 1:5 dB. Fig. 4 shows that the measured balance bandwidth for the 1.5 dB balance level ranges from 2.1 GHz to 2.9 GHz, about 32%. The maximum imbalance appears at 2.5 GHz, the controlled center frequency. Fig. 5 shows the measured and simulated transmission phase balance of the developed SP feed. The incremental phase difference between the adjacent ports is plotted against frequencies. One can define the bandwidth of phase balance by the off-set phase 5 or 10 with respect to the nominal 90 . It is observed that the bandwidth of the 10 phase balance is measured from 2.22 GHz to 2.76 GHz, about 22% in the fractional bandwidth. As mentioned earlier herein, the meander line sections in the proposed SP feed may endure coupling effects when the segment width becomes enormous. In this case, one may consider slightly extending the SP-feed size to mitigate the coupling effects. To do so, one may increase the constant phase  in Fig. 2. An example of the proposed SP feed with the area expanded to 3g =8 2 3g =8 is given in Fig. 6, where  = 22:5 is employed. Compared to the SP feed of g =4, the SP feed

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Fig. 4. Measured and simulated S-parameter in magnitude of the presented SP feed. (The points represent the measured data and the lines represent the simulated data).

Fig. 7. Configuration of the 2

2 2 SR arrays using the presented SP feed.

Fig. 5. Measured phase difference between the adjacent ports of the presented SP feed. (The points represent the measured data and the lines represent the simulated data).

Fig. 8. Impedance matching of the antenna element and array shown in Fig. 7.

Fig. 6. Layout of the proposed SP feed in an area of 3

=8 2 3 =8.

of 3g =8 yields performances of greater reliability because coupling effects are reduced.

terms of impedance matching, axial ratio, and gain. Fig. 7 shows the optimized dimensions of the patches and SP feed in 2 2 2 arrays. Fig. 8 shows the improved impedance bandwidth of the array, compared to the single patch antenna. Fig. 9 shows the axial-ratio and gain-level performances, where both appeared to undergo improvements. These performance improvements were also observed in [12]. In this illustration, the impedance is designed at 100 ohms for the antenna element and 50 ohms for the input port, where the 1.6 mm thick PCB of FR4 substrate was used. Additionally, the presented SP feed can be easily extended to the large CP planar array for high-gain applications. Using the compact and uniform features, the presented 2 2 2 SP feed and array elements can be repeated and iteratively extended to the larger scale 2N 2 2N arrays. For example, Fig. 10 shows the configuration of 4 2 4 arrays using the group-and-repeat process based on the module unit of the presented 2 2 2 SP feed of 3g =8.

IV. APPLICATION OF SP FEED IN ANTENNA ARRAYS

To illustrate the presented SP feed with applications to 2 2 2 arrays, we employed a CP patch for an antenna element with a 3g =8 2 3g =8 SP feed. The array performances were optimized for the bandwidth in

V. CONCLUSION This communication presents a compact and simple sequential-phase (SP) feed, including a design scheme, implementation, and

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2

[5] U. R. Kraft, “An experimental study on 2 2 sequential-rotation arrays with circularly polarized microstrip radiators,” IEEE Trans. Antennas Propag., vol. 45, no. 10, pp. 1459–1466, Oct. 1997. [6] A. R. Weily and Y. J. Guo, “Circularly polarized ellipse-loaded circular slot array for millimeter-wave WPAN applications,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 2862–2870, Oct. 2009. [7] 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, 2010. [8] K.-F. Hung and Y.-C. Lin, “Novel broadband circularly polarized cavity-backed aperture antenna with traveling wave excitation,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 35–42, Jan. 2010. [9] S. S. Yang, R. Chair, A. A. Kishk, K. F. Lee, and K. M. Luk, “Study on sequential feeding networks for subarrays of circularly polarized elliptical dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 321–333, Feb. 2007. [10] H. Evans, P. Gale, B. Aljibouri, E. G. Lim, E. Korolkeiwicz, and A. Sambell, “Application of simulated annealing to design of serial feed sequentially rotated 2 2 antenna array,” Electron. Lett., vol. 36, no. 24, pp. 1987–1988, Nov. 2000. [11] K. H. Lu and T.-N. Chang, “Circularly polarized array antenna with corporate-feed network and series-feed elements,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3288–3292, Oct. 2005. [12] A. A. Kishk, “Performance of planar four elements array of single-fed circularly polarized dielectric resonator antenna,” Microw. Opt. Technol. Lett., vol. 38, no. 5, pp. 381–384, 2003.

2

+

Fig. 9. Simulated AR and CP gain at zenith ( z-axis) of the antenna element and array shown in Fig. 7.

A New Formula for the Pattern Bandwidth of Fabry-Pérot Cavity Antennas Covered by Thin Frequency Selective Surfaces S. A. Hosseini, F. Capolino, and F. De Flaviis

Fig. 10. Configuration of the 4

2 4 SR arrays using the presented SP feeds.

measurement. Using only a single transition of impedance transformation, the core layout of the proposed SP feed is uniform and neat. Two topologies of the presented SP feed are provided for the sizes of a g =4 and a 3g =8. The SP feed of g =4 can be employed for the case of critical demands on the small element spacing, while the design of 3g =8 can be used for general planar array applications. An example of 3g =8 SP feed with a patch as an array element is discussed in relation to imperfect performance. The presented SP feed can be extended to a 2N 2 2N feeding network, and is highly suitable for large-scale PCB arrays with less feed loss due to the shorter transmission line length in the total feeding network.

Abstract—A new closed form expression is introduced to estimate the 3 dB pattern bandwidth of a Fabry-Pérot cavity antenna covered by a thin frequency selective surface (FSS) radiating at the broadside direction. The new formula has been obtained by using reciprocity, transmission line theory, and the susceptance model of the FSS. This formula estimates the 3 dB pattern bandwidth more accurately than previous expressions. Index Terms—Fabry-Pérot cavity (FPC) antenna, frequency selective surface (FSS), 3 dB pattern bandwidth.

I. INTRODUCTION A Fabry Pérot cavity (FPC) covered by a partially reflective surface was devised by Von Trentini [1] as a directive antenna. The FPC was later conceived as a grounded dielectric, covered by a denser layer as described in [2]. In [3], two layers of periodic rods made of alumina were placed above a patch antenna to increase its directivity. However, for fabrication reasons it is advantageous to cover the FPC by a frequency selective surface (FSS) made by an array

REFERENCES [1] T. Teshirogi, M. Tanaka, and W. Chujo, “Wideband circularly polarized array antenna with sequential rotations and phase shifts of elements,” in Proc. Int. Symp. Antennas Propagat. ISAP, Tokyo, Japan, Aug. 1985, pp. 117–120. [2] J. Huang, “A technique for an array to generate circular polarization using linearly polarized elements,” IEEE Trans. Antennas Propag., vol. 34, no. 9, pp. 1113–1124, Sep. 1986. [3] P. S. Hall, “Application of sequential feeding to wide bandwidth, circularly polarized microstrip patch arrays,” in Proc. Inst. Elect. Eng., May 1989, vol. 136, pp. 390–398, pt. H. [4] K. D. Palmer, J. H. Cloete, and J. J. van Tonder, “Bandwidth improvement of circularly polarized arrays using sequential rotation,” in Proc. IEEE Antennas and Propagation Symp., Jul. 1992, vol. 1, pp. 135–138.

Manuscript received June 28, 2010; revised October 29, 2010; accepted January 15, 2011. Date of publication May 10, 2011; date of current version July 07, 2011. S. A. Hosseini is with the Department of Electrical Engineering and Computer Sciences, Henry Samueli School of Engineering, University of California, Irvine, CA 92697 USA (e-mail: [email protected]). F. Capolino and F. De Flaviis are with the Department of Electrical Engineering and Computer Sciences, University of California, Irvine, CA 92697 USA. 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.2011.2152343

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Fig. 1. Side view of the FPC antenna fed by either (a) an electric current in the middle of the cavity, or by (b) a magnetic current, modeling a slot, on the ground plane. The cavity is homogeneous in the x-y plane.

of either metallic patches or slots, as was used for the leaky wave (LW) antennas in [4]–[7]. The thin FSS made of periodic metallic patches or slots has been modeled in [5], [6] as a pure imaginary shunt admittance Y = j^bY0 , where Y0 is free space admittance and ^b is the normalized FSS susceptance, in a transmission line (TL). Based on the mentioned model, general formulas describing the pattern bandwidth (BW), half-power beamwidth, and the maximum radiated electric and magnetic fields as a function of ^b have been derived in [7], for this class of planar LW antennas. In [8], fundamental radiation properties of this class of LW antennas have been investigated in depth. Among other results, in [7], [8], a closed form formula was introduced to approximate the 3 dB pattern bandwidth of a FPC antenna covered by a thin FSS. The expression yields accurate results only for very large ^b values. The TL model of the FPC antenna was used in [9], to derive a general expression to calculate the theoretical gain of a FPC antenna fed by either electric or magnetic dipoles. Using the expressions in [9, Appendix A], one can determine that normalized susceptance values of j^bj = 1; 5, and 10 yield gains of 8.5 dB, 19.5 dB, and 25.5 dB for an FPC antenna filled with air, respectively. Therefore, large values of ^b, e.g., j^bj > 10, would imply gains larger than 25.5 dB. Since FPC antennas constitute a useful design also for planar, low-profile antennas with moderate gains (between 10 and 20 dB for instance), it is believed that a more accurate BW formula is needed for this class of FPC antennas. Therefore, this work provides a better closed form approximation of the 3 dB pattern BW that is also applicable to moderate gain antennas using more precise approximations on the radiated power density. For large gain antennas (i.e., for large ^b values) this proposed BW expression tends to coincide with that in [7], [8]. Finally, analytical and numerical comparisons are made between the results calculated from the new formula and the numerically computed 3 dB pattern BW. II. IMPROVED FORMULA FOR THE PATTERN BANDWIDTH Since the radiation properties of the FPC antenna (directivity, gain, gain BW, pattern BW) strongly depend on the cavity dimensions and the type of FSS used, the FPC antenna fed by an elementary dipole point-source is analyzed here. Ideally, a FPC antenna can be fed by either an electric elementary dipole inside the cavity at distance hs from the ground plane, Fig. 1(a), or by an elementary magnetic dipole on its ground plane as shown in Fig. 1(b) which models a slot on the same plane. The arrows shown in Fig. 1 illustrate the polarization of the elementary electric (a) or magnetic (b) dipoles. The two above-mentioned designs are similar in their radiation characteristics. Although all calculations presented here are based on having the electric current (dipole) positioned in the middle of the cavity, the results achieved are also valid for a FPC antenna fed by a magnetic current. Utilizing the reciprocity theorem [10], [11], as an alternative to calculate directly the radiated far-field of antenna fed by an electric dipole, the far-field radiation pattern is found by determining the induced electric field at the feeding point in receiving mode, when the antenna is illuminated by a plane-wave. The received field is calculated by using a TL model, as already done in [12]. Therefore, it is assumed without loss

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Fig. 2. Transmission Line (TL) model for a FPC antenna in the receiving mode illuminated by a plane-wave, fed by an electric dipole inside the cavity.

of generality that the feeding electric dipole in the middle of the cavity, at z = hs in Fig. 1(a), is oriented along the x direction. According to reciprocity property detailed in [12, p. 1330], the far-zone x-polarized electric field Ex radiated by an antenna at broadside ( = 0 ) is related to the electric field induced at the feeding point z = hs of the antenna in the receiving mode Expw due to an incident x-polarized plane wave Exinc (z ) = E0 ejk z propagating from the z direction (Fig. 2), as Ex = Expw . The magnitude of the plane-wave is E0 = 0j!0 =(4) as calculated using the reciprocity theorem as disp cussed in [12, p. 1330], k0 = ! 0 "0 is the free space wavenumber and ! the radian frequency. The 3 dB pattern BW of a planar FPC antenna, placed in xy -plane, is evaluated only for the broadside radiation (or reception) direction, i.e., where the antenna radiates (receives) its maximum power at its operational frequency. That is why calculations, here, are restricted to the receiving mode with plane-wave illumination from the z-direction (Fig. 2). Using the TL model of the receiving antenna [12, p. 1331], the received electric field Expw at the point z = hs is modeled by a voltage as Expw = V (z = hs ), produced by an incident traveling wave V + ejk z with V + = E0 , as in Fig. 2. Therefore, the far-field radiation by the antenna at broadside is found as

Ex

= V (z = hs ):

(1)

In the TL model Z = Z0 r ="r is the characteristic impedance of the material filling the cavity, and Z0 = Y001 = 0 ="0 is the free space impedance. A very thin FSS is modeled as a lumped imaginary admittance Y as shown in Fig. 2, [5]–[8], whose normalized FSS-susceptance is ^b = Y=(jY0 ). Capacitive and inductive FSSs result in positive and negative ^b values respectively. The TL voltage induced at the feeding point z = hs of a FPC antenna is calculated as

2V + sin(khs ) = sin( kh) 1 + jZ0 Btot

(2)

Btot = Y0 [^b 0 r cot(kh)]

(3)

V

where k = !=v is the wavenumber inside the cavity, with p v = 1= r 0 "r "0 , and Btot is the total imaginary admittance of the FPC antenna, looking leftward at z = h + 0+ (Fig. 2)

where r = "r =r . The broadside radiation power density of a FPC antenna is therefore determined as (detailed in [8])

P

2 Y0 V + 2 sin2 (khs ) = 2VZ0 = 2sin 2 (kh) 2 1 + Z02 Btot j

j

j

j

:

(4)

The total imaginary admittance Btot is zero at the operational frequency !op of the FPC antenna (which is defined as the resonance frequency of the FPC), which implies that the resonance height h is given by

h=

1 tan01 r = op 1 + r ^b 2 ^b

kop



(5)

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where the most-right hand side is an approximation for large values of ^b. Here, it is assumed that the dipole feed is placed in the center of the cavity (hs = h=2, where the cavity electric field is maximum) which, for moderate or high-gain FPC antennas implies that sin(khs )  = 1. Therefore, using (4), the maximum power density at broadside is approximated as



j + j2 =sin2 (!op h=v):

P (!op ) = 2Y0 V

(6)

The relative 3 dB pattern BW is defined as

BW3 dB =

!3+dB

0 !30dB

(7)

!op

where !36dB are the frequencies that half the power radiated at broadside

6

P (!3 dB ) =

1 P (!op ): 2

(8)

Equation (8) can be solved numerically to find the 3 dB pattern BW of the antenna, but without providing for the capability of further analytical investigations. Since for a FPC antenna covered by a thin FSS several radiation parameters can be defined as a function of ^b [5]–[8], a closed-form formula is useful in determining the 3 dB pattern BW of the antenna, which can also be employed for further studies to increase the 3 dB pattern BW, as in [13]. Furthermore, general trends can be inferred by inspecting the closed form formula. In order to find the !36dB values analytically a few considerations are necessary. Note that the resonance condition Btot (!opt ) = 0 for large values of ^b also implies that cos(!op h=v )  = 01 whereas sin(!op h=v ) is approximated as

sin !op

 0r : =

h v

h v

 0r 0 = ^b

h  r cos !36dB = 01 + ^ v b

h v

(10-1)

16 :

(10-2)

For thin FSSs modeled as inductive, ^b = 0Z0 =(!op L), or capacitive, ^b = Z0 !op C , shunt loads [5]–[7], we can assume that changes of ^b are negligible within the 3 dB BW of the antenna, which is usually narrow. Thus, using (9) and (10), (8) is approximated as A12 + B 1 + C = 0 where

A=

h v

b B = 2r ^

2

^b4 + ^b2 2r2 + 1 + r4 h v

;

C=

0r2 :

(11)

Solving for 1, the upper and lower limits of the 3 dB pattern BW are found as

16 = r

v h

01 6

BW3 dB = r

op h

^b4 + ^b2 (2r2 + 2) + r4 ^b4 + ^b2 (2r2 + 1) + r4

:

(13)

Note that the cavity height h is determined by (5) and its value also depends on the inductive or capacitive choice of the FSS; in other words, besides the magnitude of ^b, also its sign affects the resonance value of h. In (13), the ratio v=h has been equivalently rewritten as b, which v=h = !op op =(2h). Note that for very large values of ^ imply high gain FPC antennas, using (5), h = op =2, and retaining only the ^b4 terms, (13) can be further simplified as

^b4 + ^b2 (2r2 + 2) + r4

^b4 + ^b2 (2r2 + 1) + r4

BW3 dB =

2

^ b2

r

(14)

which is the approximated BW formula for high gain antennas covered by a thin FSS obtained in [7], [8]. III. ILLUSTRATIVE EXAMPLES

16

h v

From (7), one has BW3 dB = (1+ 0 10 )=!op , which leads to

(9)

^b

Also, defining 1 = ! 0 !op and assuming that 1=!op is a small number, i.e., much less than unity, and using the Taylor approximations around the operational frequency of the antenna (i.e., for 1  = 0) one has

sin !36dB

Fig. 3. Comparison between “exact” values of the 3 dB pattern BW of a FPC antenna covered by inductive or capacitive FSS and the results calculated from the approximated formulas: our result (13), and previous result (14).

:

(12)

Comparisons are made between the 3 dB pattern BW values calculated analytically using (13) and (14) and the numerical results carried out by full-wave simulations for the same FPC antennas to demonstrate that (13) can be used as design BW estimation with high accuracy. These comparisons also verify the better accuracy of (13) over previously found (14). The comparison is shown in Fig. 3 for negative and positive FSS susceptance ^b and for "r = 1 and r = 1, against an “exact” result obtained by solving (8) numerically. As expected, both (13) and (14) are accurate for high values of ^b, however for lower susceptance values ^b, i.e., in case of low-gain FPC antennas, (13) yields more accurate results with respect to (14). Furthermore, one can notice that (13) provides more accurate results also in distinguishing between capacitive and inductive FSSs, while this is not possible with (14). In Table I, the relative 3 dB pattern BW values calculated using (13) and (14) are compared with the numerical results calculated for the same structures simulated by a full-wave (FW) simulation (Ansys HFSS). The comparison is carried out for FPCs with inductive and capacitive low values of the FSS susceptance ^b, designed at 10 GHz. The simulated FPC antenna is fed by an ideal magnetic dipole (slot) on its ground plane, and covered by an FSS (made of Copper with thickness of 10 m) with infinite extent along x and y . The inductive and capacitive FSSs, with a square FSS-unit-cell of 12 2 12 mm2 , are

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TABLE I RELATIVE 3 DB PATTERN BW OF A FPC ANTENNA FED BY IDEAL MAGNETIC DIPOLE (SLOT) COVERED WITH INDUCTIVE/CAPACITIVE FSSS, COMPARISON BETWEEN FW RESULTS AND APPROXIMATED EXPRESSIONS (13) AND (14)

respectively made of periodic rectangular slots and strips with width of 5 mm and length L. Different values of FSS susceptance ^b are obtained by varying L (see [14]), i.e., for the inductive FSSs (slots), L = 10:93 mm, 10.40 mm, and 9.48 mm resulting in ^b = 01:5; 02, and 03, respectively. While for the capacitive FSSs (strips), L = 9:58 mm, 10:25 mm, and 10:95 mm resulting in ^b = 1:5; 2, and 3 respectively. Then, the antenna broadside radiation power, gain, and pattern and gain BWs are calculated using the formulas described in [9, Appendix A]. As shown in Table I, the pattern BW values calculated from (13) and those obtained from the full-wave simulation results are in better agreement than the previous BW formula (14) as expected. The accuracy of (13) is also investigated by considering the example in [15, Table I], in which a FPC antenna fed by a patch (6 2 6 mm2 ) and covered by a finite-size (7 2 28 elements) FSS made of periodic patches (0.95 2 0.1 cm2 ), and FSS-unit-cell size of 1.15 2 0.3 cm2 , was designed at 12.4 GHz for a cavity partially filled with air (12.792 mm) and partially with a dielectric substrate ("r = 3:38 and thickness of 0.508 mm (20 mil)). As reported in [15], the FPC cavity fed by a patch antenna provided the maximum gain of approximately 18 dB at 12.4 GHz, whereas, the patch antenna (without FPC) provided a gain of 5–6.5 dB in the entire operational frequency band of 10 to 14 GHz. The reported 3 dB BW of the directivity of the FPC antenna, as shown in [15, Fig. 7], was approximately 875 MHz (7.05%). The procedure discussed in [14], when applied to the FSS in [15], yields ^b = 2:35. For simplicity, in our calculation the FPC antenna is filled with air (which is what the FPC in [15] is mostly filled with), and the FPC resonates at 12.4 GHz for a FPC height, using (5), h = 13:65 mm (the total height in [15] was 13.3 mm). Using (8), the 3 dB pattern BW of the FPC antenna, with FSS susceptance ^b = 2:35 (and thus we assume it constant in the narrow 3 dB BW frequency range), h = 13:65 mm, and fed by an ideal magnetic dipole, is 7.92%. Instead, using the analytic formulas (13) and (14), the estimated 3 dB pattern BW is calculated as 8.6% and 11.5%, respectively. Alternatively, by using a more accurate frequency-dependent model of the FSS based on the method discussed in [14] and the formulas in [9, Appendix A], the theoretical gain and broadside power density of the FPC fed by an ideal magnetic dipole on the ground plane are calculated, with a maximum gain of 12 dB (which can be interpreted as the difference between the gain of the FPC antenna and that of the patch at 12.4 GHz), and its 3 dB pattern BW is numerically calculated to be equal to 7.13%. In summary, the estimated 3 dB pattern BW value of 8.6%, by (13), is very close to the pattern BW obtained from (8), (7.92%), and the pattern BW of the FPC fed by an ideal magnetic dipole calculated numerically based on the values calculated using [9], (7.13%), and to the directivity BW of the FPC fed by the patch (7.05%, obtained from [15, Fig. 7]), demonstrating a better accuracy with respect to the pattern BW calculated by (14), (11.5%). This result also shows that the pattern BW of a FPC antenna (usually narrow) is mainly determined by the resonant cavity and not by the feed.

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IV. CONCLUSION A new closed form expression (13) was proposed for the estimation of the 3 dB pattern bandwidth of a FPC antenna covered by a thin FSS that, for very high-gain antennas, converges to a previous formula (14) proposed in [7], [8]. Both analytical and numerical comparisons prove the new formula (13) to be a significant improvement over the previous one (14) for medium-gain antennas. The proposed formula is well suited for FPCs covered by either capacitive (patches) or inductive (slots) FSSs, and it is an effective tool for engineers for providing an estimation of the resulting pattern bandwidth before starting a computationally expensive simulation campaign for designing the FPC and FSS.

ACKNOWLEDGMENT The authors would like to thank Ansys (HFSS) for providing them their simulation tool that was instrumental in this work.

REFERENCES [1] G. V. Trentini, “Partially reflecting sheet arrays,” IRE Trans. Antennas Propag., vol. AP-4, pp. 666–671, 1956. [2] D. R. Jackson and N. G. Alexopoulos, “Gain enhancement methods for printed circuit antennas,” IEEE Trans. Antennas Propag., vol. 33, no. 9, pp. 976–987, Sep. 1985. [3] M. Thevenot, M. S. Denis, A. Reineix, and B. Jecko, “Design of a new photonic cover to increase antenna directivity,” Microw. Opt. Tech. Lett., vol. 22, no. 2, pp. 136–139, July 1999. [4] A. P. Feresidis and J. C. Vardaxoglou, “High gain planar antenna using optimised partially reflective surfaces,” in Proc. Inst. Elect. Eng. Microw. Antennas Propag., Dec. 2001, vol. 148, no. 6, pp. 345–350. [5] T. Zhao, D. R. Jackson, J. T. Williams, H. D. Yang, and A. A. Oliner, “2-D periodic leaky-wave antennas—Part I: Metal patch design,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3505–3514, Nov. 2005. [6] 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. [7] 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. 3525–3533, Nov. 2005. [8] G. Lovat, P. Burghignoli, and D. R. Jackson, “Fundamental properties and optimization of broadside radiation from uniform leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1442–1452, May 2006. [9] 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. [10] N. G. Alexopoulos, P. B. Katehi, and D. B. Rutledge, “Substrate optimization for integrated circuit antennas,” IEEE Trans. Microw. Theory Tech., vol. 31, no. 7, pp. 550–557, Jul. 1983. [11] C. A. Balanis, Antenna Theory: Analysis and Design. New York: Wiley, 1982, pp. 127–132. [12] P. Burghignoli, G. Lovat, F. Capolino, D. R. Jackson, and D. R. Wilton, “Directive leaky-wave radiation from a dipole source in a wire-medium slab,” IEEE Trans. Antennas Propag., vol. 56, no. 5, pp. 1329–1339, May 2008. [13] G. Lovat, P. Burghignoli, F. Capolino, and D. R. Jackson, “Bandwidth analysis of highly-directive planar radiators based on partially-reflecting surfaces,” presented at the EuCAP , Nice, France, Nov. 2006. [14] S. A. Hosseini, F. Capolino, and F. De Flaviis, “Design of a single-feed 60 GHz planar metallic Fabry-Perot cavity antenna with 20 dB gain,” presented at the iWAT2009, Santa Monica, CA, Mar. 2009. [15] Y. J. Lee, J. Yeo, R. Mittra, and W. S. Park, “Design of a high-directivity electromagnetic bandgap (EBG) resonator antenna using a frequency-selective surface (FSS) superstrate,” Microw. Opt. Technol. Lett., vol. 43, no. 6, pp. 462–467, Dec. 2004.

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Switchable Frequency Selective Slot Arrays B. Sanz-Izquierdo, E. A. Parker, and J. C. Batchelor

Abstract—A switchable frequency selective surface (FSS) made of square loop slots and PIN diodes connected to a novel separate biasing circuit is presented. The structure uses a very thin, flexible substrate sandwiched between two physically independent metallic layers to create the active filter. An application is the modification of the EM architecture of buildings, where propagation could be controlled using active FSS. The relatively small number of elements employed creates a compact FSS structure which could fit in an aperture within a wall of a building. The Fabry-Perot approach is used to design a cascaded version for improved filter selectivity. Index Terms—Electromagnetic wave propagation, EM architecture, frequency selective surfaces (FSS), PIN diodes, switching.

I. INTRODUCTION The electromagnetic architectural configuration of a building determines the electromagnetic wave propagation conditions inside the structure as well as reflection or transmission to the exterior. Most existing buildings have been designed and built with no account of the variety of wireless technologies that nowadays co-exist, and their impact on network efficiency and security. As the structure of buildings cannot generally be easily modified, the installation of frequency selective surfaces (FSS) into buildings could be the answer to some of these issues [1]–[4]. FSS can transmit or reflect electromagnetic waves striking the surface at predetermined operational frequency bands. The surfaces could be applied to a partition wall or sections of buildings, allowing efficient frequency re-use or increasing security. Several projects funded in the U.K. by Ofcom have studied FSS in the built environment, some reported in [1], [2]. Active frequency selective surfaces can add a higher level of control over the electromagnetic wave propagation in buildings. They permit the time dependent modification of the characteristics of the surface by applying an external magnetic [5]–[7] or electric [7]–[20] control. Two related procedures might be considered when applying active devices to frequency selective surfaces: switching [7]–[17] and tuning [5]–[7], [18]–[20]. Switching between transmitting and reflecting states using PIN diodes on metallic dipole patches is well known [7]–[11], and has already been tested in the built environment [12]. The development of switched slot form FSS has been reported more recently, for example [13]–[17]. In that work, band pass active FSS consisted of slots etched on one side of an FR4 substrate, with the biasing circuit on the other side, and metallic vias connected both sides. An equivalent circuit for that configuration was derived in [16], [17]. In this communication, we discuss active square loop slot arrays fabricated on a double sided structure, sandwiching a very thin, flexible Manuscript received September 05, 2009; revised May 11, 2010; accepted November 09, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported in part by the U.K. Engineering and Physical Sciences Research Council and by the National Policing Improvement Agency. The authors are with the Department of Electrical Engineering, Department of Electronics, University of Kent, Canterbury CT2 7NT, U.K. (e-mail: [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.2011.2152312

dielectric substrate. The thin substrate employed allows placing the biasing circuit with the diodes on the rear side without physical connection to the front surface which contains the slots. These in turn could lay on a second, thick dielectric layer if necessary for support, or for additional bandpass shaping. The studies presented focus on the application of this novel biasing technique to dual-polarized square loop slots. The Fabry-Perot approach [21] is also explored to design a cascaded version for improved filter selectivity. II. SWITCHABLE SQUARE LOOP SLOT ARRAYS A. Design and Fabrication The square loop is a relatively compact geometry commonly employed as FSS elements. The transmission response of a closely packed array of this element is typically characterized by wide bandwidth and relatively good resonance stability with angle of incidence [22], [23]. A double-sided switchable structure using square loop slots on one side and the biasing circuit on the other side is described in this section. The front and the rear views of the unit cell of the structure are presented in Fig. 1(a) and (b). A side view showing the slots on the front, the biasing circuit with the PIN diodes at the rear and the thin dielectric layer in the middle can be seen in Fig. 1(c). There is no physical connection between the two sides and the PIN diodes capacitively couple to the area surrounding the slots. A flexible polyester substrate 0.05 mm thick, with "r = 3:0 and loss tangent  = 0:04 was used for all the designs presented here. The FSS consisted of a 5 2 5 array of square loop slots in a square lattice of periodicity p = 37 mm. The sides of the squares (L) were 29 mm in length, and the slot width w was 0.3 mm. The dimensions of the biasing circuit lines at the rear were: A = 27:6 mm, B = 3:6 mm, C = 30:3 mm, D = 32:8 mm, E = 32:4 mm, F = 1:5 mm, G = 16:8 mm, H = 5:0 mm. The circuit contained 4 diodes per square loop slot, connected in series within a cell and simultaneously connected to the 5 cells in a column, making 20 diodes in series per column. The five columns had different feed lines that could be controlled independently if desired. Photographs of the rear side of the 5 2 5 slot array and a single cell of the array are shown in Fig. 2(a) and (b). BAR64-02 silicon PIN diodes with forward resistance Rs = 2:1 ohms and capacitance at 0 volts of Cs = 0:17 pF were employed for the switching. CST Microwave Studio was used for simulations of the design with diodes added as lumped capacitor/resistor for the OFF/ON states. The very thin dielectric substrate increased significantly the meshing process and computational requirements. All the FSS structures here were simulated as infinite arrays. B. Measurements Measurements were carried out using two log periodic antennas at 0.6 m from a 3 2 3 m absorbing panel containing the FSS. The system was calibrated relative to an open aperture of approximately 190 2 190 mm in the centre of the panel. Fig. 3 shows the measured transmission response of the switchable FSS with the diodes OFF, at normal wave incidence angle, and at 45 in TM and TE. At normal incidence, the bandpass was centered at around 2.5 GHz with insertion loss of 0.6 dB and a 010 dB fractional bandwidth of 75%. At TE45 , the insertion loss increased to 1.2 dB and the 010 dB bandwidth decreased to 43%. At TM 45 , the insertion loss was 0.6 dB and the 010 dB bandwidth increased to 84%. In all three measurements, the insertion loss for the 2.4 GHz to 2.5 GHz Bluetooth band was kept below 1.7 dB. The response when the diodes were in the ON state is shown in Fig. 4. Transmission levels were well below 018 dB for normal incidence, TE45 and TM45. Simulations compared well with the measurements,

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Fig. 1. Unit cell of the active FSS: (a) Front view with the square loop slot, (b) rear view with the biasing circuit, (c) magnified section view of the structured, (d) equivalent circuit model for the FSS with diodes switched OFF, (e) simplified equivalent circuit model for FSS with diodes switched ON.

Fig. 2. (a) Rear side of the 5

2 5 array (b) a magnified view of a unit cell. Fig. 4. Simulated and measured transmission response of the active FSS with diodes switched ON.

C. Analysis of the Active FSS

Fig. 3. Simulated and measured transmission response of the active FSS with diodes switched OFF.

showing good angular stability and polarization performance. In the simulations, diodes were found to be the largest contributors to the insertion losses of the FSS, with the diodes in the OFF state adding over 0.3 dB. In addition, as infinite arrays were simulated while finite FSS were measured, much of the discrepancy between the simulations and measurements in Figs. 3 and 4 is likely to be due to edge effects.

Simulations were run to analyze the effect of the biasing circuit on the performance of the FSS. The main findings are summarized as follows. — The FSS was stable to angle of incidence for both vertical and horizontal polarization. The difference in resonance frequency between the two polarizations in the OFF state was less than 1%. — The lossy dielectric substrate attenuated any possible resonance due to the vertical and horizontal lines of the biasing circuit, particularly in the ON state. — Fig. 5 shows transmission response of the FSS with the diodes in the OFF/ON state, and also the FSS without the biasing circuit but with the diode’s characteristic capacitance/resistance connected directly to the slot FSS. The bias lines were found to behave mainly as parallel capacitors (Cb in Fig. 1(d)), decreasing slightly the resonant frequency in the OFF state and by 8% in the ON state, reducing the ON/OFF transmitted power ratio by 2 dB. — A reduction in the length of A, C, D, E and G (Fig. 1(b)) to the minimum length needed to connect the diodes (A0 = 15 mm, C 0 = D0 = 17:5 mm, E 0 = 19:5 mm, G0 = 15 mm) de-

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Fig. 6. Two cascaded layers of the switchable FSS separated by polystyrene foam, one layer rotated by 90 degrees. Fig. 5. Simulated and measured transmission response of the active FSS configuration with diodes switched ON.

graded the ON state performance of the FSS by 2 dB (Fig. 5 as A’C’D’E’G, ON). It also reduced capacitance Cb , increasing slightly the resonant frequency. Equivalent circuit models for switchable square loop slots were described in [16], [17]. Using that model and the findings described in this section, a simplified equivalent circuit model of our active FSS design with the diodes in OFF states is presented in Fig. 1(d), where: L1 is the inductance created by the metallic grid at each side of the slot (L1 =2 corresponds to the two sides in parallel), Cs the capacitance of the slot, Ch the capacitance between the biasing circuit and the FSS and Cb the parallel capacitance added by the biasing lines. The diode adds Cd , rd and Ld to the circuit, behaving mainly as Cd in the OFF state [9]. Ch can be omitted as it has very low impedance at 2.5 GHz. In addition, the inductance of the centre patch Lp could be disregarded in the OFF state due to its low impedance value compared to Cs and Cd [16], [17]. The resonant frequency of the active FSS with diodes in the OFF state would be:

w0 = p

2

L1 CT

(1)

where

Fig. 7. Measured transmission response of the two cascaded layers of the active FSS with diodes switched OFF.

(0.16 ) as illustrated in Fig. 6. In addition, one of the layers was rotated 90 degrees, improving the control capabilities of the surface and allowing individual switching of columns in one layer and rows in the second one. Fig. 7 shows the transmission response of the cascaded FSS with the diodes switched OFF. There is a clear improvement in the roll-off rate and width of the band-pass peak compared with the single layer structure described in Section I (Fig. 3). The bandwidth of the 10 dB passband has decreased to 40%, 27% and 40% at normal incidence, TE45 and TM45 respectively. At normal incidence, the insertion loss is 1 dB at 2.5 GHz and below 2 dB between 2.35 GHz and 2.7 GHz. At TE45 , the insertion loss is now 3 dB at 2.5 GHz and under 3.4 dB from 2.4 GHz to 2.7 GHz. AT TM45, the insertion loss is less than 2.7 dB between 2.35 GHz and 2.5 GHz. Simulations and measurements showed that diodes contributed to over 0.5 dB of the insertion losses at normal incidence. The losses might be reduced by using higher quality diodes such as those employed in [26]. The transmission response of the cascaded active FSS with the diodes in ON state is shown in Fig. 8. At 2.5 GHz the transmission levels dropped to 25 dB, 23 dB and 17 dB at normal incidence, TE45 and TM45 respectively. Across this range of illumination angle, the difference in transmission coefficient between the diode OFF and ON states was over 14 dB for the 2.4 GHz to 2.5 GHz Bluetooth frequency band. It is worth considering that in mobile communications and wireless local area networks in the built environment, comparatively small reductions in signal interference can give very significant reductions in the system outage probability. In [25], for example, a 15 dB increase in the carrier-to-interference ratio has been demonstrated to reduce the outage probability by a factor of almost 30.

0

CT

= Cs + Cb + Cd :

(2)

Fig. 1(e) illustrates the slot created when the diodes are switched ON and its simplified equivalent circuit model. The square slot is split in four sections and the capacitance of Cs and Cb reduced to Cs2 and Cb2 . The inductance of the patch Lp2 should now be taken into account as it is in series with L1 =4. The main effect of the diodes and the biasing circuit was a reduction in the electrical length of the slot from the typical 0.5  to 0.45 . III. CASCADING ACTIVE SQUARE LOOP SLOT ARRAYS A. Design and Measurements Two cascaded FSS arrays can be regarded as a basic Fabry-Perot interferometer (FPI) [21], [24]. FPI behave as high Q filters, with very narrow passbands at frequencies where the sum of the path length phase between the layers, and the phase of the surface reflection coefficient satisfies a specific condition. In order to improve the filter selectivity, two layers of the active FSS were separated by a polystyrene foam layer, by a distance S of 20 mm

0

0

0

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Fig. 8. Measured transmission response of the two cascaded layers of the active FSS with diodes switched ON.

IV. CONCLUSIONS AND DISCUSSION Active frequency selective surfaces with diodes capacitively coupled to slots arrays via a very thin, flexible substrate have been presented. A switching structure using square loop slots has been demonstrated and analyzed. In addition, two cascaded layers of the active FSS improved the roll-off rate and widened the passband peaks, while maintaining a similar ON/OFF power ratio. Simulations as infinite arrays compared well with measurements of the 5 2 5 array for the two configurations presented here. The main aim of this research work is to control electromagnetic propagation in buildings by modifying their Electromagnetic Architecture. The relatively small number of elements employed makes the structure attractive for applications to small apertures in buildings, reducing the cost of installation and maintenance. The methodology described has been filed as Tunable Surface filing no. GB 0902380.6 ACKNOWLEDGMENT The authors would like to thank S. Jakes for assisting in the experiments.

REFERENCES [1] M. Philippakis, C. Martel, D. Kemp, R. Allan, M. Clift, S. Massey, S. Appleton, W. Damerell, C. Burton, and E. A. Parker, “Application of FSS structures to selectively control the propagation of signals into and out of buildings,” ERA Rep. 2004-0072, 2004 [Online]. Available: http://stakeholders.ofcom.org.uk/binaries/research/spectrum-research/exec_summary.pdf [2] M. Hook and K. Ward, “A project to demonstrate the ability of frequency selective surfaces and structures to enhance the spectral efficiency of radio systems when used within buildings,” Ofcom ref. AY4462A, 2004. [3] E. A. Parker, J. B. Robertson, B. Sanz-Izquierdo, and J. C. Batchelor, “Minimal size FSS for long wavelength operation,” Electron. Lett., vol. 44, no. 6, pp. 394–395, Mar. 2008. [4] B. Sanz-Izquierdo, I. T. Ekpo, J.-B. Robertson, E. A. Parker, and J. C. Batchelor, “Wideband EM architecture of buildings: Six-to-One dualpassband filter for indoor wireless environments,” Electron. Lett., vol. 44, no. 21, pp. 1268–1269, 2008. [5] E. A. Parker and S. B. Savia, “Active frequency selective surfaces with ferroelectric substrates,” IEE Proc. Microw., Antennas Propag., vol. 148, no. 2, pp. 103–108, 2001. [6] Y. C. Chan, G. Y. Li, T. S. Mok, and J. C. Vardaxoglou, “Analysis of a tunable frequency-selective surface on an in-plane biased ferrite substrate,” Microw. Opt. Technol. Lett., vol. 13, no. 2, pp. 59–63, Oct. 1996. [7] T. K. Chang, R. J. Langley, and E. A. Parker, “Active frequency selective surfaces,” IEE Proc., vol. 143, pt. Part H, pp. 62–66, 1996.

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[8] E. A. Parker and S. Massey, “Application of FSS structures to selective control the propagation of signals into and out of buildings,” Ofcom ref AY4464A, Annex 5: “Survey of Active FSS”, Aug. 2010 [Online]. Available: http://stakeholders.ofcom.org.uk/binaries/research/spectrum-research/survey.pdf [9] T. K. Chang, R. J. Langley, and E. A. Parker, “An active square loop frequency selective surface,” IEEE Microw. Guided Wave Lett., vol. 3, no. 10, pp. 387–388, Oct. 1993. [10] B. Philips, E. A. Parker, and R. J. Langley, “Active FSS in an experimental horn antenna switchable between two beamwidths,” Electron. Lett., vol. 31, no. 1, 5, pp. 1–2, Jan. 1995. [11] A. Tenant and B. Chambers, “Experimental dual polarized phase-switched screen,” Electron. Lett., vol. 39, no. 1, pp. 119–121, Jan. 2003. [12] B. M. Cahill and E. A. Parker, “Field switching in an enclosure with active FSS screen,” Electron. Lett., vol. 37, no. 4, 15, pp. 244–245, Feb. 2001. [13] G. I. Kiani, K. P. Esselle, A. R. Weily, and K. L. Ford, “Active frequency selective surface using pin diodes,” in Proc. IEEE Antennas and Propagat. Int. Symp., Jun. 2007, vol. 9–15, pp. 4525–4528. [14] G. I. Kiani, K. L. Ford, K. P. Esselle, and A. R. Weily, “Oblique incidence performance of an active square loop frequency selective surface,” in Proc. Eur. Conf. on Antennas and Propagation, Nov. 2007, pp. 1–4. [15] G. I. Kiani, K. L. Ford, K. P. Esselle, A. R. Weily, C. Panagamuwa, and J. C. Batchelor, “Single-Layer bandpass active frequency selective surface,” Microw. Opt. Technol. Lett., vol. 50, no. 8, pp. 2149–2151, Aug. 2008. [16] K. Chang, S. I. Kwak, and Y. J. Yoon, “Equivalent circuit modeling of active frequency selective surfaces,” in Proc. IEEE Radio and Wireless Symp., Jan. 2008, vol. 22–24, pp. 663–666. [17] K. Chang, S. I. Kwak, and Y. J. Yoon, “Active frequency selective surfaces using incorporated PIN diodes,” IECEI Trans. Electron., vol. E91, no. 12, Dec. 2008. [18] A. Tennant and B. Chambers, “A single-layer tuneable microwave absorber using an active FSS,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 1, pp. 46–47, Jan. 2004. [19] C. Mias, “Waveguide and free-space demonstration of tunable frequency selective surface,” Electron. Lett., vol. 39, no. 14, pp. 1060–1062, Jul. 2003. [20] C. Mias and C. Tsakonas, “Waveguide demonstration of varactor-diode-tunable band-pass frequency selective surfaces,” Microw. Opt. Technol. Lett., vol. 45, no. 1, pp. 62–66, Feb. 2005. [21] A. C. de. C. Lima and E. A. Parker, “Fabry-Perot approach to the design of double layer FSS,” IEE Proc., Microw., Antennas Propag., vol. 143, pp. 157–162, Apr. 1996. [22] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [23] R. J. Langley and E. A. Parker, “Equivalent circuit model for arrays of square loops,” Electron. Lett., pp. 294–296, 1982. [24] C. Antonopoulos and E. A. Parker, “A design procedure for FSS with wide transmission band and rapid rolloff,” IEE Microw., Antennas Propag., vol. 145, pp. 508–510, Dec. 1998. [25] A. H. Wong, M. J. Neve, and K. W. Sowerby, “Performance analysis for indoor wireless systems employing directional antennas in the presence of external interference,” in Proc. IEEE AP-S Int. Symp., Washington, D.C., 2005, vol. 1A, pp. 799–802. [26] F. Costa, A. Monorchio, S. Talarico, and F. M. Valeri, “An active high impedance surface for low profile tunable and steerable antennas,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 676–680, 2008.

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Full-Wave Scattering From a Grooved Cylinder-Tipped Conducting Wedge Anastasis C. Polycarpou and Marios A. Christou

Abstract—The problem of full-wave electromagnetic scattering from a cylinder-tipped conducting wedge with either a sectoral or annular groove is formulated using the mode-matching technique. The mode expansion of the sector involves ratios of Bessel functions with large order as the order is inversely proportional to the inner angle of the sector. An asymptotic expansion of Bessel functions with large orders is introduced in order to overcome the numerical difficulty involved using a regular series expansion. Numerical results indicate good agreement with data obtained using the finite element method (FEM). Index Terms—Conducting grooved wedge, electromagnetic scattering, mode expansion, mode-matching technique.

I. INTRODUCTION The rigorous formulation of full-wave electromagnetic scattering from canonical structures has always been a problem of great interest. Not only it provides the scientist or engineer with more insight and understanding of the problem in hand, but it allows for quick and accurate simulations and parameterization of the geometry under investigation. However, it is often the case that a rigorous formulation of such problems presents numerical challenges that are difficult to overcome and, as a result, they need special treatment. A conducting wedge with cylindrical grooves or blunted tips is considered a canonical geometry as it conforms to the cylindrical coordinate system. Scattering from conducting wedges with cylindrical grooves, slots, and cavities were considered using a number of methods including a hybrid FEM/uniform theory of diffraction (UTD) [1]–[3], a method of moments (MM) approach through the use of an appropriate Green’s function [4], or a MM hybridized with the geometrical theory of diffraction (GTD) [5]. A mode-matching technique in conjunction with transform theory [6], [7] was recently used to solve for the scattering of a slotted conducting wedge. The mode-matching technique was also used by Cugiani et al. to obtain the bistatic radar cross section of dihedral corner reflectors [8]. The problem of scattering from a cylinder-tipped wedge was formulated in the mid 1980s by Hallidy [9] using uniform asymptotic expressions that were derived from the governing Green’s function. In [9], Hallidy has shown results that compared well with the eigenmode expansion for cylindrical tips with diameters larger than 1.5 . In this communication, the mode-matching technique is used to formulate the problem of scattering from a cylinder-tipped conducting wedge where the cylindrical tip has either a sectoral or annular groove, as shown in Fig. 1. The analysis of this structure was inspired by an on-going research project where multi-groove cylindrical tips are placed at sharp edges in order to reduce undesired diffraction in the shadow region. Manuscript received July 19, 2010; revised September 28, 2010; accepted November 08, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. A. C. Polycarpou is with the Department of Electrical and Computer Engineering, University of Nicosia, Nicosia, Cyprus (e-mail: [email protected]. cy). M. A. Christou is with the Department of Mathematics, University of Nicosia, Nicosia, Cyprus (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2152345

Fig. 1. Cylinder-tipped conducting wedge with annular groove.

II. ANALYSIS A. Problem Formulation The geometry of a cylinder-tipped conducting wedge with annular groove is shown in Fig. 1 ( = 0 for a sectoral groove). The wedge is infinite in the z direction and, therefore, the problem can be treated in two dimensions. The structure is excited with either an electric (for TMz ) or a magnetic (for TEz ) line source placed at a distance i from the center of the tip and an angle i measured from the positive x axis. The radius of the cylindrical tip is defined as , whereas the start and the end of the cylindrical sector is specified by 1 and 2 , respectively. A time-harmonic dependence exp (j!t) is assumed throughout the communication. The modal fields for Ez and Hz , which are solutions of the Helmholtz’s equation with the proper boundary conditions, can be written for the region external to the groove (Region I), as well as for the region internal to the groove (Region II). In particular, the modal field expansions in these two regions, for TMz and TEz polarizations, are given below. 1) Region I: The longitudinal components of the total electric and magnetic fields, for TMz and TEz polarizations, respectively, are given by

Ez (; ) = Hz (; ) =

1

Ao H(2) ( i ) J ( ) + BpT M H(2) ( ) 1 sin (i ) 1 sin ();     i ; 0    o p=1

1

Ao H(2) ( i ) J ( ) + BpT E H(2) ( ) (1) 1 cos (i ) 1 cos ();     i ; 0    o p=0

where  is the radial distance to the observation and  is the observation angle measured from the positive x axis. The order of Bessel and Hankel functions is specified by the exterior wedge angle, denoted as o , and is given by  = p=o . The constant Ao can be evaluated by (i) imposing the continuity of the magnetic field at  = i , (ii) expressing the magnetic current density at the line source in terms of a cosine Fourier series, and (iii) using the Wronskian [10] to simplify the expressions. Following this procedure, it can be shown that Ao is given by

Ao = 0

 Ie o

for TMz ; Ao =

0

 Im o

for TEz

(2)

where is the phase constant, Ie;m is the impressed electric/magnetic current at the line source,  is the intrinsic impedance of the free space,  = 2 for  = 0, and  = 1 for  6= 0.

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2) Region II: Due to the presence of metallic walls at angles 1 and 2 , the longitudinal components of the electric and magnetic fields in

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where qp is the Kronecker delta function, and TM Sqp

this region can be written, for the sectoral groove, as

1

Ez (; ) =

k=1

1

Hz (; ) =

k=0

CkT M J ( ) 1 sin  ( 0 1 ); 1    2 CkT E J ( ) 1 cos  ( 0 1 ); 1    2 (3)

1

TM Sqp

k=1

1

Hz (; ) =

k=0

CkT M [J ( ) Y ( ) 0 J ( ) Y ( )] sin  ( 0 1 ) ; 1    2

J ( )Y ( ) k ; 1 = 2 0 1 : ; = 1 J ( )Y ( )

(4)

0

0

(5)

It is also important to note that 0 = @=@ ( ). 3) T Mz Polarization: Let us first consider the cylinder-tipped wedge with a sectoral groove. Imposing the continuity of the tangential electric fields (Ez ) at  = , multiplying with sin(), where  = q=o , and integrating on [0; o ], yields the following expression:

o

Ao H(2) ( i ) J ( ) + BqT M H(2) ( )

2

o A H (2) ( i )J ( ) + BqT E H(2) ( ) ~1 o  0

1

= k=1

TM CkT M J ( ) 1 Iqk

k=0

cos () cos [ ( 0 1 )] d;  ~1 =

TE Cm

=

 TM Iqk

sin () sin [ ( 0 1 )] d:

=

1

p=0

Imposing now the continuity of the tangential magnetic fields (H ) at = , multiplying with sin( ( 0 1 )), where  = m=1, and integrating on [1 ; 2 ], one can obtain



1

2

Ao H(2) ( i ) J ( )+ BpT M H(2) ( )

TM sin (i) 1 Imp

(8) TM is given by (7) with q , k replaced by p, m. Substituting (8) where Imp into (6) yields the following system of equations with unknowns being the coefficients BpT M 1

p=1

BpT M sin (i ) H(2) ( ) qp 0 1

= p=1

p = q 6= 0 (14) p = q = 0.

Ao H(2) ( i ) J ( ) (2) +BpT E H ( )

TE cos (i ) 1 Imp

(15)

where

=

m ;~2 = 1

2;

1;

k = m 6= 0 k=m=0

(16)

TE is given by (14) with q , k replaced by p, m. Substituting (15) and Imp into (13) yields

1J0 ( ) 0

p=1

2;

1;

~2 1J ( )

1

(7)



1

(13)

Furthermore, imposing the continuity of Hz at  = , multiplying with cos( ( 0 1 )), and integrating on [1 ; 2 ], we obtain

where

TM Cm =

TE CkT E J ( ) 1 Iqk

where TE Iqk =

(6)

cos i

0

=



sin i

(12)

4) T Ez Polarization: The cylinder-tipped wedge with sectoral groove is considered first. Using the governing field expressions in the two regions, given by (1)–(5), and using the Maxwell’s equations, the -component of the electric field can be evaluated. By imposing the continuity of E at  = , multiplying with cos(), and integrating on [0; o ], the following expression can be obtained:



1

(11)

0

R ( ) = J ( ) Y ( ) 0 J ( ) Y ( ) :

where is the inner radius of the annular sector, and

Dk =

R ( ) T M T M I I R ( ) qk kp k=1  1

=

where

1

CkT E [J ( ) Y ( ) 0 Dk J ( ) Y ( )] 1 cos  ( 0 1 ) ; 1    2

(10)

0

Now, in the case of the cylinder-tipped wedge with an annular groove, the final system of equations will be identical to that of the TM previous case, shown in (9), except for the definition of Sqp which is now defined as

and, for the annular groove, as

Ez (; ) =

J ( ) T M T M I I : J ( ) qk kp k=1  1

=

Ao H(2) ( i ) sin (i )

(2) TM 4H ( ) Sqp

o 1

0

p=0

BpT E cos (i )

TE H(2) ( ) Sqp o 1 TE J ( ) Sqp o 1 = Ao H(2) ( i ) cos (i ) 1  0J ( ) p=0  ~

2

H(2) ( )

qp ~1

0

1

(17)

0

o 1

TM 4J ( ) Sqp 0

1

which corresponds to a matrix system with the unknown vector being TE is defined as the coefficients of BpT E . The function Sqp

J ( ) qp (9)

TE Sqp

J ( ) T E T E I I ~ : J ( ) qk kp 2 k=0  1

=

0

(18)

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Now, for the cylinder-tipped wedge with annular groove, the final set T E which of equations is identical to (17) except for the definition of Sqp is given by TE Sqp

=

1

Q ( ) T E T E I I ~ Q ( ) qk kp 2 k=0  0

(19)

where

Q ( ) = J ( ) Y ( ) 0 Dk J ( ) Y ( )

(20)

and Dk is defined in (5). B. Asymptotic Expansions of High-Order Bessel Functions Evaluating the governing equations in order to fill-in the matrix system of either TMz or TEz polarization, a problem appeared during T M or S T E . The order of the Bessel the computation of the term Sqp qp function is inversely proportional to the inner angle of the sector; thus, for a small inner angle and a moderately large modal index, the order of the Bessel function becomes large. On the other hand, the argument of the Bessel function is fixed and independent of the order and the inner angle of the groove. In such a case, we were faced with major computational problems of the form 0/0 and 1=1. To overcome this obstacle, we adopted specific asymptotic expansions of the Bessel function of large order and fixed argument. According to Watson [11], the Bessel function of first kind can be approximated as

J (z )  exp  +  ln

z

2 0

+

1 Ck 2 ln( ) 1 k=0  k 1

(21)

where only the zero-order constant is given in [11]; i.e., Co = 1= 2=. In order to obtain a higher-order approximation, the Stirling’s formula for the Gamma function [12], i.e., 0( + 1)  p 2 (=e) , must be substituted into the regular series expansion of the Bessel function and set equal to the Watson’s approximation in (21). Doing so, one may obtain the remaining coefficients given by z 2 z 4 z 6 2 ; C2 = p2 ; C3 = 0 p2 ; etc: C1 = 0 2 p 2 2 2 2 6 2

(22)

For the case of the cylinder-tipped wedge with sectoral groove, the T M and S T E involve the computation of J terms Sqp  +1 =J which is qp judiciously evaluated using the higher-order approximation of (21) Now, for the cylinder-tipped wedge with annular sector, the term T M involves both kinds of Bessel function. Assuming first-order apSqp proximation, these can be written as [12]   p 1 2ze ; Y (z)  0 2 2ze : (23) 2 Utilizing the above formulas, R ( )=R ( ) in (11) can be approx-

J (z ) 

0

0

imated, for large orders of  , as

R ( ) R ( )

 0  

Q ( ) Q ( )

  0 

0





0









+





:

(24) Similarly, for the TEz polarization, Q0 ( )=Q ( ) in (19) can be approximated, for large orders of  , as 0





0





Fig. 2. Asymptotic expansion of a ratio of Bessel functions for large orders and fixed argument ( = 5; 1 = 10 ).





+





: (25)

Fig. 3. TM scattering from a cylinder-tipped wedge with sectoral groove ( = 270 ;  = 180 ;  = 180 ;  = 270 ;  = =2;  = ).

III. RESULTS The accuracy of the aforementioned asymptotic expansions of the Bessel function for large orders and fixed argument was tested before running any particular cases of scattering. Fig. 2 illustrates a case where  = 5 and 1 = 10 . J+1 =J is evaluated using both the regular series expansion and the asymptotic expansion. The ratio is plotted as a function of =( ). It is observed that the regular series expansion fails to compute the ratio for values of =( ) larger than 10. The reason is because the numerator and denominator become extremely small and the division cannot be performed. The asymptotic expansion is used in this case to overcome the numerical difficulty. The mode-matching formulation presented in Section II was verified for the case of TMz polarization by comparing with an in-house developed nodal FEM code using first-order Absorbing Boundary Conditions (ABCs). Fig. 3 shows a comparison between the two methods when the source is in the far-field and the observation is one wavelength away from the center of the cylindrical tip. The line source is positioned at an angle i = 180 . The sectoral groove is set by 1 = 180 and 2 = 270 . The wedge angle is o = 270 , whereas the radius of the cylindrical tip is  = =2. As seen from Fig. 3, there is an excellent agreement between the two methods. A comparison with the nodal FEM is shown in Fig. 4 for a cylindrical tip radius  =  and an annular groove with inner radius = =2. The groove is set between 180 and 270 . The wedge angle is 270 , the source is placed in the far field with i = 180 , and the observation distance from the center of the tip is 1.5 . As illustrated, there is a very good agreement between the two methods for both polarizations.

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Zeroth-Order Complete Discretizations of Integral-Equation Formulations Involving Conducting or Dielectric Objects at Very Low Frequencies E. Ubeda, J. M. Tamayo, J. M. Rius, and A. Heldring

Fig. 4. Scattering from a cylinder-tipped wedge with annular groove ( = 270 ;  = 180 ;  = 180 ;  = 270 ;  = ; = =2;  = 1:5).

Abstract—We present the Self-Loop basis functions, a divergence-conforming set with zero charge density. These basis functions allow a rearrangement of the Linear-linear basis functions set to overcome the low-frequency breakdown. We show the accuracy and stability at very low frequencies of the Linear-linear discretizations of (i) electric-field integral equation for perfectly conducting objects and (ii) the Poggio-Miller-Chang-Harrington-Wu-Tsai formulation, for dielectric bodies. Index Terms—Electromagnetic scattering, integral equations, method of moments (MoM), numerical analysis, radar cross section (RCS).

A small discrepancy is observed in the backscattered direction, which is mainly attributed to the use of first-order ABCs in the FEM. IV. CONCLUSION The mode-matching technique was formulated to solve the problem of scattering by a cylinder-tipped wedge with a sectoral or annular groove. Numerical difficulties appeared in the evaluation of Bessel functions when the inner angle of the groove was small. Asymptotic expansions of Bessel functions were derived and implemented in the formulation in order to overcome the problem. Numerical results and comparison with the FEM illustrate the accuracy and validity of the proposed formulation.

REFERENCES [1] G. Pelosi, R. Coccioli, G. Manara, and A. Monorchio, “Scattering from a wedge with cavity backed apertures in its faces and related configurations: TE case,” IEE Proc. Microw. Antennas Propag., vol. 142, no. 2, pp. 183–188, Apr. 1995. [2] A. Borgioli, R. Coccioli, G. Pelosi, and J. L. Volakis, “Electromagnetic scattering from a corrugated wedge,” IEEE Trans. Antennas Propag., vol. 45, no. 8, pp. 1265–1269, Aug. 1997. [3] S. Alfonzetti, G. Borzi, and N. Salerno, “An iterative solution to the scattering from cavity-backed apertures in a perfectly conducting wedge,” IEEE Trans. Magn., vol. 34, no. 5, pp. 2704–2707, Sep. 1998. [4] J. W. Silvestro and C. M. Butler, “TE scattering from a perfectly conducting wedge loaded by a slot,” IEE Proc. H Microw. Antennas Propag., vol. 139, no. 1, pp. 105–109, Feb. 1992. [5] J. W. Silvestro, “Scattering from slot near conducting wedge using hybrid method of moments/geometrical theory of diffraction: TE case,” Electron. Lett., vol. 28, no. 11, pp. 1055–1057, May 1992. [6] J. J. Kim, H. J. Eom, and K. C. Hwang, “Electromagnetic scattering from a slotted conducting wedge,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 222–226, Jan. 2010. [7] K. C. Hwang, “Scattering from a grooved wedge,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2498–2500, Aug. 2009. [8] C. Cugiani, R. Orta, P. Savi, and R. Tascone, “Full-wave investigation of a corner reflector proposed as reference line target for automotive applications,” IEEE Trans. Antennas Propag., vol. 45, no. 12, pp. 1823–1829, Dec. 1997. [9] W. Hallidy, “On uniform asymptotic Green’s functions for the perfectly conducting cylinder tipped wedge,” IEEE Trans. Antennas Propag., vol. 33, no. 9, pp. 1020–1025, Sep. 1985. [10] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [11] G. N. Watson, A treatise on the Theory of Bessel Functions. Cambridge, U.K.: Cambridge at the University press, 1922. [12] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Washington DC: National Bureau of Standards, 1964.

I. INTRODUCTION The divergence-conforming basis functions ensure normal continuity of the current across the edges arising from the discretization. The polynomial order of completeness of their divergence [1] establishes the order of the discretization. The divergence-conforming basis functions are well-suited for the method of moments (MoM) discretization [2] of the electric-field integral equation (EFIE) involving perfectly conducting objects. The Rao, Wilton and Glisson (RWG) [3] set exhibits constant normal/linear tangential (CN/LT) variation along the edge. The variation of the linear-linear (LL) set [4]–[6] is linear normal/linear tangential (LN/LT), which doubles the number of unknowns. Even though the LL set carries out a higher-order expansion of the current, both RWG and LL sets produce a piecewise constant expansion of the charge density, whereby they are zero-order examples of divergence-conforming sets. A MoM-discretization of the EFIE for perfectly conducting objects (PeC-EFIE) with the traditional divergence-conforming basis functions suffers from the so-called low-frequency breakdown. At very low frequency regime, the contribution of the vector potential to the impedance matrix becomes negligible, according to the finite machine precision, compared with the scalar potential. This makes the discretization of the PeC-EFIE ill-conditioned and the solution inaccurate. For double machine precision and direct solution of the resulting matrix, the low-frequency breakdown for a RWG discretization of the PeC-EFIE appears with sizes of the mesh cells below 1008  [7]. The discretization of the magnetic-field integral equation in the analysis of PeC-objects (PeC-MFIE) produces impedance matrices with stable condition number. However, EFIE discretizations of PeC-objects are in general preferred because they offer better accuracy and because they can be applied to open surfaces. Manuscript received December 02, 2009; revised October 22, 2010; accepted January 21, 2011. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported in part by the Spanish Interministerial Commission on Science and Technology (CICYT) under Projects TEC201020841-C04-02, TEC2009-13897-C03-01, TEC2007-66698-C04-01/TCM, and CONSOLIDER CSD2008-00068 and in part by the “Ministerio de Educación y Ciencia” through the FPU fellowship program. The authors are with the Signal Theory and Communications Department, Universitat Politécnica de Catalunya, 08034 Barcelona, Spain (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.2011.2152316

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For the numerical analysis of penetrable objects, the Poggio-MillerChang-Harrington-Wu-Tsai (PMCHWT) formulation [8]–[10], which comes from the imposition of the continuity of the surface-tangential component of the electric and magnetic fields across the dielectric regions, also suffers from the low-frequency breakdown. The PMCHWT discretization results from the combination of PeC-EFIE and PeC-MFIE discretizations, particularized for the inner and outer regions of the dielectric bodies. The testing and subtraction of the innerregion and outer-region PeC-MFIE surface integrals discards the spectrally stable Gram-matrix contributions. Therefore, a discretization of the PMCHWT in Method of Moments inherits spectral properties of the PeC-EFIE discretization, such as the low-frequency breakdown. The rearrangement of the divergence-conforming basis functions as the combination of solenoidal and nonsolenoidal, quasi-irrotational, sets of functions leads to MoM-discretizations of the PeC-EFIE and of the PMCHWT also valid at very low frequencies. The solenoidal subspace captures the null space of the scalar potential and thus allows a stable and accurate discretization at very low frequencies. The Loop-Tree [11] or the Loop-Star [11], [12] basis functions develop a rearrangement of the RWG current space. The Loop basis functions, expanding the RWG-solenoidal space, are defined around vertices and have zero divergence. They gather all the closed paths arising from the RWG discretization of closed or open and simply connected surfaces. The Tree or the Star basis functions capture the remaining nonsolenoidal space, whereby they have non-zero divergence. Loopstar Galerkin-discretizations of the electric current for the PeC-EFIE [11] and of the electric and magnetic currents for the PMCHWT [13] are readily available. Furthermore, there exist generalized Loop-Tree decomposition schemes for higher-order MoM-discretizations of the EFIE for perfectly conducting objects [14]. Such schemes, though, are not required for the RWG and LL basis functions, which are complete to zero divergence-order. In this communication, we present a new scheme for a stable MoMdiscretization of the EFIE, involving conducting objects, and of the PMCHWT, involving penetrable bodies, with the Linear-linear basis functions in the very low frequency regime. Our strategy is based upon combining a novel set of basis functions, the Self-Loop basis functions [15], with the well-known Loop-Star basis functions. In this communication, we prefer to use the unnormalized forms [1] of the RWG and Linear-linear basis functions sets. The definition of the loop, self-loop and star basis functions as linear combinations of the unnormalized RWG and LL basis functions become then simpler.

Fig. 1. Simplex-coordinates of the vertex j defined, respectively, on the pair of triangles T and T .

Fig. 2. Self-Loop basis function.

Moreover, the unnormalized RWG basis function f~ results from the summation of the two unnormalized Linear-linear basis functions associated to that edge: f~(1) ; f~(2) ~ f

=

(1) + f~(2)

~ f

(3)

which are also called, respectively, terms of first- and second-kind [17]. The definition of the unnormalized Linear-linear basis functions of first- and second-kind becomes then

r j 2 ^+ 2 + 0 2 0 i r j 2^ 2 + (2) ( ) = 0 j r i 2 ^ + 0 2 0 0 j r i 2^ (1) (~r) =

~ f

~ f

~ r

i



n

~ r

T





n

~ r

T





n

~ r

T





n

~ r

T

(4) (5)

:

We define the Self-Loop basis functions f~SELF as the subtraction of the unnormalized first-kind and second-kind Linear-linear basis functions

SELF = f~(1) 0 f~(2)

~ f

II. SELF-LOOP BASIS FUNCTIONS

(6)

or, equivalently, The unnormalized form [1] of the RWG basis functions [3], associated to the edges arising from the discretization, can be defined in terms of the curl-conforming “Whitney forms” [16] k ; k of the opposed vertices k+ and k0 as



~ f

k =

k 2 ^0

2 n^+ n

+ ~ r 2 T ~ r

2 0

(1)

T

^+ ; n^0 stand for the surface unit normal vectors on the triangles where n + 0 T ;T (see Fig. 1). The “Whitney forms” are defined in terms of i ; j and i ; j , the simplex coordinates of, respectively, the vertices i and j on each triangle T + ; T 0 , shown in Fig. 1, as

k = ( i r j 0 



j

ri ) 

:

(2)

SELF (~r) =

~ f

(i rj + (i rj + 



j





j

r i ) 2 ^+ r i ) 2 ^0 

n

~ r



n

~ r

2 + 2 0 T T

:

(7)

The Linear-linear basis functions can then be expressed in terms of the unnormalized RWG and the Self-Loop sets as

1( + 2 (2) = 1 ( 0 2 (1) =

SELF )

(8)

SELF ):

(9)

~ f

~ f

~ f

~ f

~ f

~ f

Since the divergences of the first- and second-kind Linear-linear basis functions is the same, the Self-Loop basis functions have zero divergence. The Self-Loop basis functions are associated to the edges arising from the discretization and ensure null charge by compensating the sign of the continuous normal component of the current across the edge (see Fig. 2).

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III. SOLENOIDAL SPACES A MoM-discretization of the PeC-EFIE stable at very low-frequencies requires the complete expansion of the null-space of the scalar potential operator. This space, so-called solenoidal, requires the divergence of the current to be zero. The remaining non-solenoidal subspace of current must therefore have non-zero divergence. The solenoidal subspace in the RWG-discretized space over closed or open and simply connected surfaces is expanded by the Loop basis functions. The Loop basis function associated to a vertex arising from the discretization can (^ n ) [18]–[20], where  denotes the simplex-cobe defined as ordinate associated to that vertex over the triangles around the vertex. This scalar function is called “solenoidal potential” [18] and shows a piecewise linear variation between neighboring vertices arising from the discretization. Similarly, the definition in (7) of the Self-Loop basis functions can be simplified as

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PeC-MFIE discretizations in the outer region (Eo or Ho) and in the inner region (Ei or Hi). The discretizations of the PMCHWT formulation adopt the same basis functions sets for the expansion of the electric and magnetic currents. Therefore, both are decomposed in their solenoidal (s) and nonsolenoidal (ns) parts with the same basis functions sets. The resulting matrix system under this notation becomes

Esi i Ens Hsi i Hns

r2

f~SELF (~r) =

r 2 ( r 2 (

i i

j n^ + ) ~r 2 T + j n^ 0 ) ~r 2 T 0

r2

IV. CONSTRUCTION OF THE IMPEDANCE MATRIX The impedance elements in the matrices associated to the [Loop;Star] or [Loop;Self-loop;Star] MoM-discretizations of the PeC-EFIE result from the combination of the impedance elements in the matrices resulting from the unnormalized RWG or LL discretizations of the PeC-EFIE. After the rearrangement of the discretized current space in the solenoidal and nonsolenoidal subspaces, the impedance matrices are still unstable at very low frequencies

Esi i Ens

=

ZsEout 0s (!) Eout (! ) Zns 0s

ZsEout 0ns (!) Eout (1=!) Zns 0ns

Js Jns

(11)

i denote the incident electric field tested, respectively, where Esi and Ens with the solenoidal and the nonsolenoidal sets. Esi can also be com-

puted through the inner-product of the “solenoidal potential” with the surface-normal component of the incident magnetic field ([20], (37)), which makes visible the ! -dependence and results in better accuracy in the computation of the real part of Esi for electrically very small loops. Moreover, the nonsolenoidal contribution of the current must become zero at zero frequency because the divergence of the static current must be zero [21]. It is therefore reasonable to define the following fre0 = Jns =! and Esi = Esi =! , which quency-normalized magnitudes Jns make the final matrix system stable at very low frequencies [20]

Esi i Ens

=

Z 0 Eo ZsEo s0s (1) 0ns (!) Eo Zns0s (!) Z 0 Eo ns0ns (1)

Js : Jns

(12)

In the scattering analysis of penetrable objects, the impedance elements resulting from the MoM-discretizations of the PMCHWT formulation with the [Loop;Star] or the [Loop;Self-Loop;Star] sets accordingly result from the combination of the impedance elements of the unnormalized RWG or LL discretizations of the PMCHWT. The integral formulations for dielectrics combine the computation of the PeC-EFIE or

:

(13)

Ei P Eo ZEs 0Js = Zs0s + Zs0s P Eo Ei ZEs 0Jns = Zs0ns + Zs0ns P Eo Ei ZEns0Js = Zns0s + Zns 0s P Eo Ei ZEns 0Jns = Zns0ns + Zns0ns

r2

0 r2

P P P ZEs 0Jns ZEs 0Mns 0Ms ZEs P P P ZEns Z Z 0Jns Ens0Ms Ens0Mns P P P ZHs 0Jns ZHs 0Ms ZHs 0Mns P P P ZHns Z Z 0Jns Hns0Ms Hns0Mns

These submatrices can be defined in terms of the solenoidal and nonsolenoidal discretizations of the PeC-EFIE as

(10)

6 which is derived by replacing the first term in (7) as i (j n ^ )= 6 6 (i j n ^ ) j (i n ^ ). In view of (10), now, for the SelfLoop basis functions, the “solenoidal potential” becomes quadratic. The solenoidal subspace in the LL-discretized space can be readily obtained from (8) and (9) through the combination of the Self-Loop basis functions and the Loop basis functions. In consequence, doubling the number of unknowns in the LL-discretized space involves increasing only the dimension of the solenoidal space of the original RWG-discretized space.

=

P ZEs 0Js P ZEns 0Js P ZHs 0Js P ZHns 0Js Js Jns 2 Ms Mns

(14)

and 2 Ei 2 P Eo ZHs 0Ms = Zs0s =o + Zs0s =i P Eo 2 Ei ZHs0Mns = Zs0ns =o + Zs0ns =i2 P Eo 2 Ei 2 ZHns 0Ms = Zns0s =o + Zns0s =i P Eo 2 Ei ZHns0Mns = Zns0ns =o + Zns0ns =i2

(15)

where i and o stand for the inner-region and outer-region impedances. Furthermore, the submatrices in (13) derived from the PeC-MFIE formulation are P Ho Hi ZHs 0Js = Zs0s + Zs0s P Ho Hi ZHs 0Jns = Zs0ns + Zs0ns P Ho Hi ZHns0Js = Zns0s + Zns 0s P Ho Hi ZHns 0Jns = Zns0ns + Zns0ns

(16)

which are related with the remaining submatrices as P P ZEs 0Ms = 0ZHs0Js P P ZEs0Mns = 0ZHs 0Jns P P ZEns 0Ms = 0ZHns0Js P P ZEns 0Mns = 0ZHns0Jns :

(17)

The PMCHWT formulation requires the Galerkin-testing of the surface-tangential components of the fields. The testing in these submatrices is thus done differently as in the conventional PeC-MFIE MoMdiscretizations, which carry out a Galerkin testing of the cross product of the normal vector with the field. Moreover, the summation in (16) enhances the Cauchy principal values of the integrals and cancels the limiting value of the integrals of the singular terms of the Kernel. The very low frequency performance of the PMCHWT discretization yields [13] P P ZEs ZEs 0Js (!) 0Jns (!) P P ZEns0Js (!) ZEns0Jns (1=!) P P ZHs 0Js (!2 ) ZHs 0Jns (1) P P ZHns0Js (1) ZHns0Jns (1)

P P ZEs 0Ms (!2 ) ZEs 0Mns (1) P P ZEns0Ms (1) ZEns0Mns (1) P P ZHs 0Ms (!) ZHs 0Mns (!) P P ZHns0Ms (!) ZHns0Mns (1=!)

(18) which is still spectrally unstable. The matrix becomes stable after the solenoidal testing of the electric- and magnetic-fields and the

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 7, JULY 2011

Fig. 3. Condition number at low and very low frequencies of the matrix resulting from the discretization with the basis functions sets RWG, LL, [Loop;Star] and [Loop;Self-Loop;Star] of the EFIE for a perfectly conducting regular tetrahedron with side 0.1 m discretized with 100 triangular facets.

Fig. 4. Condition number at low and very low frequencies of the matrix resulting from the discretization with the RWG, LL, [Loop;Star] and [Loop;Selfand raLoop;Star] sets of the PMCHWT for a Penetrable sphere with " dius 0.1 m discretized with 128 triangular facets.  stands for the free space wavelength.

=3

nonsolenoidal components of the currents are frequency-normalized as

Z PEs P ZEns Z PHs P ZHns 0

0

Js (1) 0Js (! ) 0Js (! ) 0Js (1) 0

P ZEs Jns (!) P ZEns Jns (1) P ZHs Jns (1) P ZHns Jns (!) 0

0

0

0

Z PEs P ZEns P ZHs P ZHns 0

Ms (!) 0Ms (1) 0Ms (1) 0Ms (! ) 0

P ZEs Mns (1) P ZEns Mns (!) : P ZHs Mns (!) P ZHns Mns (1) 0

0

0

0

(19)

Fig. 5. Bistatic yz-plane for a perfectly conducting square pyramid with same m. edge lengths of 0.1 m meshed with 200 triangular facets and 

=1

Fig. 6. Bistatic xz-plane for a perfectly conducting prism with dimensions: m. 0.1 m 0.05 m 0.05 m meshed with 180 triangles and 

2

2

= 10

the current space spanned by the Linear-linear set, the discretization of the EFIE due to both sets offers the same RCS patterns. Moreover, we show in Figs. 6 and 7 some RCS results at very low frequencies for both formulations. In Fig. 6, we show the RCS for a perfectly conducting prism with sides 0.1 m 2 0.05 m 2 0.05 m meshed with 180 triangles and  = 1011 m. In Fig. 7, we show the RCS for a penetrable sphere with "r = 2 meshed with 120 triangles and a wavelength in the free space of 0 = 1012 m. There is in all cases an impinging x-polarized plane wave with +z-propagation. In view of Figs. 6 and 7, the RCS performance at such very low frequencies is very similar for both discretizations [Loop;Star] and [Loop;Self-Loop;Star] when applied to both formulations PeC-EFIE and PMCHWT.

V. RESULTS

VI. RADIATION AT VERY LOW FREQUENCIES

The condition number of the resulting impedance matrices with the frequency-normalized discretizations in solenoidal and nonsolenoidal subspaces must be bounded as the electrical dimensions of the object decrease. We show in Figs. 3 and 4 the condition number of the MoM-matrices resulting from the EFIE and the PMCHWT discretizations, respectively, of a perfectly conducting regular tetrahedron with side 0.1 m and of a penetrable sphere with radius 0.1 m and relative permittivity 3. It is clear that both [Loop;Star] and [Loop;Self-Loop;Star] discretizations are stable at very low frequencies, while the other discretizations RWG and LL blow up beyond the machine precision. Note how the resulting matrices in the dielectric case have higher condition numbers than in the perfectly conducting case. This is a consequence of the inherent less stable spectral behavior of the PMCHWT at very low frequencies. In Fig. 5, we show the bistatic RCS cut over the yz-plane of a perfectly conducting square pyramid with equal edge lengths of 0.1 m and  = 1 m. Since the [Loop;Self-Loop;Star] basis functions rearrange

The expression of the radiation vector N in the Rayleigh frequency region, where the object dimensions are much smaller than the wavelength, can be approximated with the two most important terms in the Taylor’s expansion of the Kernel for small values of the wavenumber k as

J ejkr^ ~r ds '

N=

1

J ds

0

S

0

S

+

jk J (^r 1 ~r )ds 0

0

(20)

S

where S denotes the surface embracing the discretized object and J denotes the discrete expansion of the current. The first term in the previous expression can be expressed for each cartesian component in a different manner by taking into consideration that 0

Ji ds

0

S

r 1 (i J ) 0 i r 1 J ds i = x; y; z 0

=

S

0

0

S

0

0

(21)

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 7, JULY 2011

Fig. 7. Bistatic yz-plane for a dielectric sphere and " m.

0:1 m meshed with 128 triangles and  = 10

= 2 with radius =

The first term in (21) becomes null because of the normal-continuity across the edges arising in the discretization of the expansion of the current with the divergence-conforming basis functions. Therefore, (20) becomes equivalently

N ' 0 ~r r 1 J ds + jk J (^r 1 ~r )ds : 0

0

0

0

S

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Fig. 8. Relative error of the backward bistatic RCS against the number of unknowns for a perfectly conducting regular tetrahedron meshed with uniform triangulations with side 1 m under an impinging x-polarized +z-propagating planewave and  m. The MoM-EFIE current for a nonuniform [Loop;Star] discretization leading to 24057 edges with very fine degree of meshing near pointed vertices and sharp edges is adopted as reference.

=1

(22)

S

Furthermore, the current can be decomposed in terms of its solenoidal and nonsolenoidal contributions, respectively, s and ns . Their expansion in Taylor’s series around = 0 results in

J J k J ns = (jk)J (1) ns + 1 1 1 (0) J s = J s + (jk)J (1) s + 111

(23)

k

which provides for the null nonsolenoidal contribution when ! 0. The first term of the Taylor’s expansion in (23) is sufficient for the expression of at very low frequencies so that

N

N ' 0(jk)

~r r 1 J (1) Js(0) (^r 1 ~r )ds ns ds 0 0

S



0

0

0

0

(24)

S

Fig. 9. Relative error of the backward bistatic RCS against the number of unknowns for a perfectly conducting regular tetrahedron meshed with uniform triangulations with side 1 m under an impinging x-polarized +z-propagating m. The MoM-EFIE current for a nonuniform planewave and  [Loop;Star] discretization leading to 24057 edges with very fine degree of meshing near pointed vertices and sharp edges is adopted as reference.

= 10

04

which justifies the -scaling of the RCS at very low frequencies. Note how at very low frequency regime, well inside the Rayleigh frequency region, the RCS performance depends in essence on the static solenoidal and nonsolenoidal EM-contributions of the current. The rearrangement at very low frequency regime with the [Loop;Self-loop;Star] basis functions allows to extend the validity of the Linear-linear MoM-codes over a broad band of frequencies, embracing the low-frequency breakdown bound. The adoption of the Linear-linear basis functions is advantageous for the accurate RCS computation for some objects with moderately small electrical dimensions, outside the RCS Rayleigh region. For example, the 1 2 1 perfectly conducting plate with normal incidence in [4] or a regular perfectly conducting tetrahedron with side 1 (see Fig. 8). In view of our tests at very low frequency ( = 109 m), the adoption of the [Loop;Star] set in front of the [Loop;Self-Loop;Star] set becomes more favorable for smooth objects like the sphere (see Fig. 10) than for a sharp-edged object like the tetrahedron (see Fig. 9). Moreover, for the regular tetrahedron, contrary relative performance is observed for these decompositions at different frequency regime (see Figs. 8 and 9), outside and inside the RCS Rayleigh frequency region. Indeed, the leading decompositions for a sharp-edged object with electrically small or moderate dimensions become, respectively, the [Loop;Star] and the [Loop;Self-Loop;Star] decompositions. In consequence, the broad-band RCS-computation for a sharp-edged object









with the same decomposition, [Loop;Star] or [Loop;Self-Loop;Star], does not become most accurate against the number of unknowns over the whole frequency band. All these comparisons assume mesh refinement to improve the accuracy on the computed RCS. However, it is quite common to perform a broad-band multi-frequency analysis by keeping the adopted mesh fixed. The meshing of the object at the highest frequency, which represents the most stringent case in terms of required facets, is reused at lower frequencies. Quite often the EM-solvers do not work together with advanced mesh generators, required to capture the intricacy of complex structures, whereby the mesh may have been generated somewhere else. Moreover, the generation of meshes at lower frequencies from this provided mesh may become complicated, especially if the original mesh is unstructured and the body under analysis is curved. Sometimes, for simplicity, the object is not meshed at lower frequencies if there is a single valid mesh available at the highest frequency. In this scenario, we cannot gain accuracy through mesh refinement because the same meshing must be adopted at each frequency. However, we can improve the accuracy by increasing the order of the discretization [1], [14]. Both [Loop;Star] and [Loop;Self-Loop;Star] basis functions represent zeroth order divergence-conforming sets because the divergence

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Fig. 10. Relative error of the backward bistatic RCS against the number of unknowns for a perfectly conducting sphere with radius 1 m under an impinging m. The Mie series solution x-polarized -propagating planewave and  is adopted as reference.

+z

= 10

Fig. 11. Relative error of the backward bistatic RCS for a perfectly conducting regular tetrahedron with side 1 m meshed with a uniform triangulation of 676 facets under an impinging x-polarized -propagating planewave and the folm, m, m. The MoM-EFIE curlowing set of wavelengths: 1 m, rent for a nonuniform [Loop;Star] discretization leading to 24057 edges with very fine degree of meshing near pointed vertices and sharp edges is adopted as reference.

10

+z 10

10

is piecewise constant. The [Loop;Self-Loop;Star] decomposition, though, contains the Self-Loop contribution, derived from a quadratic, higher-order, “solenoidal potential”, which increases the dimension of the solenoidal subspace. We expect then to obtain better current expansion for order-demanding bodies, like sharp-edged objects. In Figs. 11 and 12, we show for two different sharp-edged objects and a fixed mesh the broad-band multi-frequency backscattered RCS computed with the MoM-discretization of the EFIE with the [Loop;Star] and [Loop;Self-Loop;Star] decompositions. Each frequency step sets the electrical dimensions of these objects to ; 1003 ; 1006  and 1009 . We now observe how the [Loop;Self-Loop;Star] set offers better accuracy over the whole frequency regime, from moderate to very small electrical dimensions. VII. CONCLUSION We present the Self-Loop basis functions. They stand for a divergence-free divergence-conforming set of basis functions. Together with the Loop and the Star basis functions, they provide stable impedance matrices in the very low frequency regime for the Linear-linear discretization of the EFIE and of the PMCHWT. These formulations tackle the scattering analysis, respectively, of perfectly conducting and dielectric bodies. The contribution of the Self-loop set in the RCS computation at very low frequencies becomes less relevant than the contribution from the

Fig. 12. Relative error of the backward bistatic RCS for a perfectly conducting square pyramid with lifted basis meshed with a triangulation of 800 facets under an impinging x-polarized -propagating planewave and the following set of wavelengths: 1 m, m, m, m. The MoM-EFIE current for a nonuniform [Loop;Star] discretization leading to 23925 edges with very fine degree of meshing near pointed vertices and sharp edges is adopted as reference.

10

+z 10

10

Loop set, especially for smooth objects. In any case, this stable basis function rearrangement at very low frequencies provides a broad-band range of validity to the Linear-linear discretization of the EFIE and of the PMCHWT, which excel as appropriate choices for the accurate RCS computation for some perfectly conducting [4] or dielectric [6] bodies of moderate electrical dimensions. In the broad-band multi-frequency numerical analysis of perfectly conducting sharp-edged objects with a fixed predetermined meshing, the decomposition with Loop, Self-Loop and Star basis functions provide, for the sharp-edged objects tested, better RCS-accuracy than the conventional RWG decomposition with Loop and Star basis functions. The Self-Loop basis functions contribute to the expansion of the solenoidal part of the current with a “solenoidal potential” of higher, quadratic, order than the Loop basis functions.

REFERENCES [1] 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. [2] R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968. [3] 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. [4] L. C. Trintinalia and H. Ling, “First order triangular patch basis functions for electromagnetic scattering analysis,” J. Electromagn. Waves Appl., vol. 15, pp. 1521–1537, 2001. [5] Ö. Ergül and L. Gürel, “Linear-linear basis functions for MLFMA solutions of magnetic-field and combined-field integral equations,” IEEE Trans. Antennas Propag., vol. 55, no. 4, Apr. 2007. [6] P. Yla-Ölijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering methods with iterative methods,” Radio Sci., vol. 40, 2005, RS6002, doi: 10.1029/ 2004RS003169. [7] Z. G. Qian and W. C. Chew, “A quantitative study on the low frequency breakdown of the EFIE,” Microw. Opt. Technol. Lett., vol. 50, no. 5, May 2008. [8] A. J. Poggio and E. K. Miller, “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, Ed. Oxford, U.K.: Pergamon Press, 1973, ch. 4. [9] T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci., vol. 12, pp. 709–718, Sept. –Oct. 1977. [10] Y. Chang and R. F. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag., vol. AP-25, pp. 789–795, Nov. 1977.

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[11] W. Wu, A. W. Glisson, and D. Kajfez, “A study of two numerical procedures for the electric field integral equation at low frequency,” Appl. Computat. Electromagn. Soc. J., vol. 10, no. 3, Nov. 1995. [12] J. Lee, R. Lee, and R. J. Burkholder, “Loop star basis functions and a robust preconditioner for EFIE scattering problems,” IEEE Trans. Antennas Propag., vol. 51, no. 8, Aug. 2003. [13] S. Y. Chen, W. Cho Chew, J. M. Song, and J. Zhao, “Analysis of low frequency scattering from penetrable scatterers,” IEEE Trans. Geosci. Remote Sensing, vol. 39, no. 4, Apr. 2001. [14] R. A. Wildman and D. S. Weile, “An accurate broad-band method of moments using higher basis functions and tree-loop decomposition,” IEEE Trans. Antennas Propag., vol. 52, no. 11, Nov. 2004. [15] J. M. Tamayo, E. Ubeda, and J. M. Rius, “Novel self-loop basis functions for the stability of the linear-linear discretization of the electric field integral equation at very low frequencies,” presented at the IEEE AP-S Int. Symp. and URSI Radio Science Meeting, Charleston, SC, Jun. 2009. [16] A. Bossavit, “Whitney forms: A class of finite elements for three-dimensional computations in electromagnetics,” Inst. Elect. Eng. Proc., vol. 135, no. 8, pt. A, pp. 493–500, 1988. [17] Ö. Ergül and L. Gürel, “Improving the accuracy of the magnetic field integral equation with the linear-linear basis functions,” Radio Sci., vol. 41, 2006, RS4004, doi:10.1029/2005RS003307. [18] G. Vecchi, “Loop-star decomposition of basis functions in the discretization of the EFIE,” IEEE Trans. Antennas Propag., vol. 47, no. 2, Feb. 1999. [19] E. Arvas, R. F. Harrington, and J. R. Mautz, “Radiation and scattering from electrically small conducting bodies of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 34, no. 1, pp. 66–77, Jan. 1986. [20] J. Zhao and W. C. Chew, “Integral equation solution of Maxwell’s equations from 0 frequency to microwave frequencies,” IEEE Trans. Antennas Propag., vol. 48, no. 10, pp. 1635–1645, Oct. 2000. [21] Y. Zhang, T. J. Cui, W. C. Chew, and J. Zhao, “Magnetic field integral equation at very low frequencies,” IEEE Trans. Antennas Propag., vol. 51, no. 8, Aug. 2003.

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Fast Convergence of Fast Multipole Acceleration Using Dual Basis Function in the Method of Moments for Composite Structures Mei Song Tong and Weng Cho Chew

Abstract—The dual basis function proposed by Chen and Wilton in 1990 is used to represent the magnetic current for solving electromagnetic (EM) surface integral equations (SIEs) with penetrable materials and the solution process is accelerated with multilevel fast multipole algorithm (MLFMA) for large problems. The MLFMA is a robust accelerator for matrix equation solvers by iterative method, but its convergence rate strongly relies on the conditioning of system matrix. If the MLFMA is based on the method of moments (MoM) matrix in which the electric current is represented with the Rao-Wilton-Glisson (RWG) basis function, then how one represents the magnetic current in electric field integral equation (EFIE) and magnetic field integral equation (MFIE) really matters for the conditioning of system matrix. Though complicated in implementation, the dual basis function is ideal to represent the magnetic current because it is similar to the RWG basis function in properties but approximately orthogonal to it in space. With a simple testing scheme, the resultant system matrix is well-conditioned and the MLFMA acceleration can be rapidly convergent. Numerical examples for EM scattering by large composite objects are presented to demonstrate the robustness of the scheme. Index Terms—Composite structure, dual basis function, fast multipole algorithm, method of moments.

I. INTRODUCTION The surface integral equations (SIEs) for describing electromagnetic (EM) problems with penetrable structures include both equivalent electric current and magnetic current as unknowns on the material interfaces [1]. In the method of moments (MoM) solution for the SIEs [2], one has to represent both currents with suitable basis functions. If the Rao-Wilton-Glisson (RWG) basis function [3] is used to represent the primary electric current, how one represents the magnetic current really matters in solving electric field integral equation (EFIE) and magnetic field integral equation (MFIE). Using the RWG basis function ^ 2 RWG basis function, where n ^ is a unit normal vector on again or n the interfaces, could partially resolve the problem, but one cannot feel fully comfortable due to the involved drawbacks. The use of RWG basis function again will result in an ill-conditioned system matrix because part of operators in the SIEs are not well tested. The use of n ^ 2 RWG will produce fictitious charges on the boundaries of triangular meshes because it is not normally continuous across the boundaries [4]. Chen and Wilton proposed the dual basis function in 1990 for representing the magnetic current [5]. Although Buffa and Christiansen proposed a similar basis function named Buffa-Christiansen (BC) basis function recently [6], [7], it was mainly designed for preconditioning. Note that the BC basis function represents a subset of the dual basis function [7]. The dual basis function is defined on a polygon pair formed by connecting the centroids to midpoints of edges in all triangles associated with the common edge of a RWG triangle pair. The dual basis function is approximately orthogonal to the RWG Manuscript received June 06, 2010; revised October 22, 2010; accepted December 04, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. M. Tong is with Tongji University, School of Electronics and Information Engineering, Shanghai 201804, China (e-mail: [email protected]). W. C. Chew is with the Department of ECE, University of Illinois, Urbana, IL 61801 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2152336

0018-926X/$26.00 © 2011 IEEE

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basis function in space but it is similar to it in properties, namely, the charge density within each polygon is constant and the total charge within a polygon pair vanishes. Due to the unique way of defining the dual basis function, there is no current component normal to each polygon boundary and thus no fictitious line charges exist across the boundaries. Also, it is associated with each RWG basis function so the number of two basis functions is the same when representing the magnetic current and electric current, respectively. Although it is complicated in implementation, the dual basis function looks ideal as a counterpart of RWG basis function. Fast multipole algorithm (FMA) is a robust accelerator for solving dense matrix equations and has been recognized as one of top ten algorithms in the 20th century [8]. The FMA has been deeply studied in EM community and was enhanced to a more advanced multilevel scheme or MLFMA [9]. Through its acceleration, the extremely large problems with over 370 millions of unknowns have been solved on parallel workstations recently [10]. The distinctive feature of the MLFMA is the rapid evaluation of the matrix-vector multiply in the iterative solution of matrix equations. Also, the matrix elements are not explicitly stored except for those near-interaction entries, leading to a matrix-free scheme with dramatic reduction on memory requirement. The MLFMA performs the matrix-vector multiply through radiation pattern, translator and receiving pattern based on the addition theorem for the Green’s function. With the use of the multilevel tree structure, the diagonalization of the translation operator and the interpolation or anterpolation for wave expansions at different levels, the MLFMA can reduce the original ( 2 ) complexity of iterative solution, both in CPU time and in memory usage, to ( log ) complexity, where is the number of unknowns. This is a significant reduction in computational is very large. costs when However, the convergence of MLFMA is strongly related to the conditioning of system matrix because it uses iterative methods in the process. If the matrix is ill-conditioned, the iteration could be very slow or even divergent. To guarantee the conditioning of matrix, one usually uses preconditioners to improve the conditioning of system matrix but this will increase much complexity [11]–[14]. In this work, we employ the dual basis function to expand the magnetic current and this choice not only resolves the accurate representation for the magnetic current but also the system matrix can be well-conditioned with a simple testing scheme in a natural way. This is because the approximate orthogonality between the two basis functions can guarantee the good testing of each operator in the integral equations and the system matrix is diagonally dominant. This is very desirable in fast algorithms with iterative methods since the rapid convergence is achievable. Numerical examples for scattering by large penetrable or composite objects are presented to illustrate the effectiveness of the scheme.

ON

N

ON N

N

II. SURFACE INTEGRAL EQUATIONS FOR COMPOSITE OBJECTS

S

n^ 2 n^ 2

; M) = 0 n^ 2 ; M) = 0

1

n

where ^ is the unit normal vector on the surface and E is the electric field. The subscript “inc” represents an incident field and the number “1” or “2” denotes the region index of the corresponding fields and media. The superscript “ + ” denotes the interface where observation points are located and “+” and “ ” on indicate the interior side and exterior side of the interface, respectively. The above SIEs are known as the electric field integral equations (EFIEs). The involved electric field relates to the source current J and M through and operators, i.e.,

S

S Einc (1)

0

S

L

E(J; 0) =

0 i!

E(0; M) =

 (r; r ) 1 J(r )dS G 0

S

M(r ) 2 rg(r; r )dS 0

S

0

0

0

=

0

=

K

0ikL(J)

K(M)

(2)

where

rr g(r; r ) (3) k2 is the dyadic Green’s function. In the above, ", ,  and k denote the permittivity, permeability, wave impedance and wave number, respectively, which are related to the involved medium. Also, ! is the angular frequency, I is an identity dyad and g (r; r ) = eikR =(4R) is the 3D scalar Green’s function in which R = jr 0 r j is the distance between  (r; r ) =  G I+ 0

0

0

0

a field point r and a source point r . The above derivation for SIEs can be easily extended to the multiple-layer case. For example, if the object is a penetrable core fully coated with two-layer distinct penetrable media as shown in Fig. 1(b), we have three material interfaces where equivalent currents (J1 M1 ), (J2 M2 ) and (J3 M3 ) reside, respectively. The corresponding EFIE can be written as S 0

;

;

When the penetrable objects are assumed to be internally homogeneous, the EM scattering by them can be governed by SIEs which can be derived through the equivalent electric and magnetic current method [5]. Consider the scattering by a three-dimensional (3D) homogeneous dielectric or magnetic object as shown in Fig. 1(a). There exist equivalent electric and magnetic surface currents J and M on the object surface to produce the original fields inside and outside the object. The boundary condition, which is the continuity of tangential components of fields at the boundary, leads to the following SIEs

ES2 (J ES (J

Fig. 1. Scattering by 3D composite objects. (a) A homogeneous penetrable object. (b) A penetrable core fully coated with two-layer distinct penetrable media. (c) A PEC object partially connected with a penetrable object.

S

n^ 2 [E2 n^ 2 [E2S n^ 2 [E3S n^ 2 [E3S

n^ 2 E1 (J1 ; M1) = 0 S (J1 ; M1 ) 0 E2 (J2 ; M2 )] = 0 S (J1 ; M1 ) 0 E2 (J2 ; M2 )] = 0 S (J2 ; M2 ) 0 E3 (J3 ; M3 )] = 0 S (J2 ; M2 ) 0 E3 (J3 ; M3 )] = 0 S n^ 2 E4S (J3 ; M3) = 0 n^ 2 Einc :

;

(4)

Furthermore, if a PEC object is partially coated with a penetrable material or a PEC object and a penetrable object are partially connected

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together as shown in Fig. 1(c), then the related EFIE can be expressed as follows

S

S

S n^ 2 [E2 (J1 ; 0) 0 E2 (J2 ; M2 )] = 0 n^ 2 Einc S S S n ^ 2 [E2 (J1 ; 0) 0 E2 (J2 ; M2 )] = 0 n ^ 2 Einc S n ^ 2 E1 (J2 ; M2 ) = 0:

(5)

In the numerical solution for (5), the contact region should be meshed consistently for the two sides and both the electric current on the PEC surface and the electric current on the surface of penetrable material are represented with the RWG basis function, so that the electric current is continuous on the contact region. The corresponding MFIE for the above equations can be obtained immediately by replacing E with H. The combination of the EFIE and MFIE with different strategies can form the combined-field integral equation (CFIE), Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) equation or Müller formulation [1]. These equations are more stable than the individual EFIE or MFIE in numerical solutions but require higher costs [15]. The EFIE and MFIE are easier to solve but how one expands the magnetic current with a suitable basis function could be a problem. We use the dual basis function to resolve the problem and the solutions are accelerated with the MLFMA. III. DUAL BASIS FUNCTION The dual basis function was proposed by Chen and Wilton in 1990 and there is a detailed introduction for it in Chen’s dissertation [5]. We also present clarifications for its definition especially the determination of weighting coefficients in our previous work [16]. The dual basis function is defined on a polygon pair formed by respectively enclosing the two endpoints of the common edge of an RWG triangle pair. It is actually a collection of RWG basis functions defined over small triangles formed by subdividing each polygon with a weighting coefficient, i.e.,

~ n (r) = 3

L

01 6 6 C 3 (r); nl nl

l=0 0;

r 2 n6 , P

0 n(l+1) =

0 l

(6)

Otherwise

1

0 l0 + q0 A0 ] [Cnl n nl nl

n(l+1) 0 l = 0; 1; 2; . . . ; Ln 0 2

for the polygon Pn0 , where qn0 = 0(ln00 + ln+0 )=AP and density in Pn0 with a total area of A

t1

i!2

S

G 2 (r r0 ) 1 J(r0 ) 0 + 21 t 1 [^ 2 M(r0 )] ;

+ t 1 0S

t1

i!1

S

dS

n

r 2 (r r0 ) 2 M(r0 ) 0 = t 1 Einc r 2 g

;

dS

;

S

G 1 (r r0 ) 1 J(r0 ) 0 0 21 t 1 [^ 2 M(r0 )] ;

+ t 1 0S

dS

n

r 1 (r r0 ) 2 M(r0 ) 0 = 0 r 2 g

;

dS

;

S

(9)

where the integrals in the K operator are in the Cauchy principal value (CPV) sense. When the testing function t is chosen as the RWG basis function as usually done, the matrix equation for the above EFIE will be very ill-conditioned if both the electric and magnetic currents are expanded with the RWG basis function. This is because the residue ^ 2 operation on the magnetic current term in the K operator has a n and this dominant term will vanish after being tested, implying that the K operator is not well tested and one cannot obtain a correct solution in general [17]. If the magnetic current M is represented by the dual basis function, then the residue term in the K operator will be diagonally dominant because the dual basis function is approximately orthogonal to the RWG basis function in space in the self-interaction patches. Therefore, both the L and K operators are well tested in such a scheme and the resultant impedance matrix is well-conditioned. This is extremely desirable in fast algorithms like MLFMA since the convergence of iterations strongly relies on the conditioning of impedance matrix.

The solutions for the above matrix equation can be accelerated with the MLFMA for electrically large problems. In the MLFMA, the matrix-vector products for far interactions are calculated through aggregation, translation and disaggregation stages and the scalar Green’s function is expanded in terms of the addition theorem [9]

ikr

e

ji

r

2 ^ ik1(r

=

d ke

ik

(8) is the charge

L

4 l=0

(7) is the charge

0r

)



mm (k; rmm )

(10)

where

mm (k; rmm ) =

C

for the polygon Pn+ , where qn+ = (ln00 + ln+0 )=AP density in Pn+ with a total area of AP .

The SIEs for penetrable objects can be solved with the MoM in which the electric current is represented by the RWG basis function while the magnetic current is expanded by the dual basis function. Consider the scattering by a single dielectric object as sketched in Fig. 1(a), the explicit EFIE consists of L and K operators and can be written as



P

1 + + + + + n1 = + [0Cn0 ln0 + qn An0 ] ln1 1 + + + + + Cn(l+1) = [Cnl lnl + qn Anl ] + l n(l+1) + l = 1; 2; 3; . . . ; Ln 0 2

IV. CONDITIONING OF IMPEDANCE MATRIX

V. FAST MULTIPOLE ACCELERATION

6 is the total number of small triangles within the polygon where Ln 6 6 is the weighting coefficient on the common edge of the lth Pn , C nl 6 (r) is the RWG basis function defined over small triangle pairs and 3nl the lth small triangle pair. The weighting coefficient is used to enforce the divergence of the basis function or the charge density to be constant within each polygon and the total charge to be zero within each polygon pair. It can be determined with the following recursive formula C

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l (2l +1)h(1) (kr ^) (11) rmm 1 k mm )Pl (^ l

i

(1)

is the translator in which hl is the spherical Hankel function of the first kind of order l, Pl is the Legendre polynomial of the first kind of order l and L is the number of series terms after truncation. Also, rm and rm represent the centers of the mth and m0 th groups in which the field point rj and source point ri reside, respectively, rjm denotes the distance vector from the field point to its group center, rim denotes the distance vector from the source point to its group center and rmm denotes the distance vector from the field group center to source group center. Substituting (10) into the L operator of the SIEs, we have the following matrix element for the L operator

L = ji

A

2^

d k

L ( ^) 1 Vfmj k

L ^) mm (k; rmm )Vsm i (k



(12)

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where

L (k^) = 1 Vfmj 4 L (k^) = Vsm i

1S

eik1r

I 0 k^k^) 1 ji(rim ) dS

e0ik1r

1S

I 0 k^k^) 1 tj (rjm) dS

( (

0

(13)

t

are the receiving pattern with the testing function j and radiation pattern with the basis function i , respectively. For the K operator of the SIEs, the unknown current is expanded with the dual basis function which is a summation of RWG basis functions defined over small triangle pairs within a polygon pair, so the matrix element can be written as

j

K Aji

=

K (k^) 1 0 ( ; 0 ) K (k^) d2 k^ fmj mm mm sm i

kr

V

V

(14) Fig. 2. Geometry of various scatterers (a) A pencil-like object with a PEC cylinder and dielectric cone (b) A dielectric or magnetic sphere with one-layer full dielectric coating (c) A PEC sphere with one-layer full dielectric coating (d) A dielectric sphere with two-layer full dielectric coating.

where

K (k^) = 0 ik k^ 2 Vfmj 4 K (k^) = Vsm i

L

1

1S

0

l=0

6 Cnl

1S

eik1r e0ik1r

tj (rjm) dS ji (rim ) dS

0

(15)

are the corresponding receiving pattern and radiation pattern, respectively. In the above, we have used the subscripts b and s on the triangles 6 0 1S or 1S to denote a big RWG triangle pair Tn and a small RWG triangle pair within polygons, respectively. By feeding the above radiation and receiving patterns to an MLFMA tree structure which is created in advance, we can then accelerate the solution process. VI. NUMERICAL EXAMPLES The proposed MLFMA is demonstrated through solving for EM scattering by large penetrable or composite scatterers. Since the MFIE and EFIE are fully dual to each other and their implementations are very similar, we only solve the EFIE here. To see the good conditioning of system matrices when using the dual basis function to expand the magnetic current, we first consider a small problem without using the MLFMA acceleration for scattering by a composite scatterer as shown in Fig. 2(a). The scatterer is a pencil-like object with a PEC cylinder and a dielectric cone which are partially connected together. The geometry is defined by a = 0:1, h1 = 0:2 and h2 = 0:1, where  is the wavelength of incident plane wave. The material of cone is characterized by a relative permittivity "r = 3:0 and the relative permeability for all materials is unity except stated otherwise. It is assumed that the incident plane wave has a frequency f = 300 MHz and is illuminating the objects from the 0z direction in free space in all examples. We calculate the bistatic radar cross section (RCS) observed along the principal cut ( = 0 and  = 0 to 180 ) for the scatterers in both vertical-vertical (VV) polarization and horizontal-horizontal (HH) polarization. Fig. 3 shows the bistatic RCS solutions (normalized by a2 ) for the pencil-like object when its cylinder part and cone part are discretized into 1842 and 766 triangular patches, respectively (note that 332 patches on the contact region are shared by the two parts). We can see that the solutions are in good agreement with the ones obtained from the Nyström method (NM) [18] which is a quite different method from the MoM. It is found that the system matrix is well-conditioned even for fine geometrical discretization if using the dual basis function to represent the magnetic current. In the contrast, it is extremely ill-conditioned if using the RWG basis function to expand the magnetic current. The conditioning numbers (CN ) with different number of total triangular patches P for both cases are presented in Table I.

Fig. 3. Bistatic RCS solutions (normalized by a ) for a pencil-like object with a PEC cylinder and dielectric cone. The geometry is defined with a : , h :  and h : . The cone has a relative permittivity " : .

=02

= 01 =30

=01

TABLE I CONDITION NUMBERS CN OF SYSTEM MATRICES FOR A PENCIL-LIKE SCATTERER WITH DIFFERENT MESH SIZES

(

)

With the demonstrated good conditioning of system matrices for the dual basis function representation of magnetic current, we now solve the large scattering problems with the MLFMA acceleration. The scatterers are chosen as radially stratified spheres as sketched in Figs. 2(b)–2(d) so that the available analytic solutions or Mie series solutions [19] can be used for validation. Because the solutions include much oscillation due to the large dimensions of objects, we only show the solutions within one half of principal cut in order to see the comparison clearly. The first example of the large problems is the scattering by a dielectric sphere with one-layer full dielectric coating as shown in Fig. 2(b). The inner radius is a1 = 10 and the outer radius is a2 = 12. The dielectric core and coating materials are characterized by a relative permittivity "r1 = 3:0 and "r2 = 5:0, respectively. We discretize the inner and outer interfaces

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Fig. 4. Bistatic RCS solutions for a dielectric sphere with one-layer full dielec and a , retric coating in VV. The radii of two interfaces are a spectively. The dielectric core and coating have a relative permittivity " : and " : , respectively.

= 12 =30

Fig. 6. Bistatic RCS solutions for a dielectric sphere with two-layer full dielectric coating in VV. The radii of three interfaces are a , a  and , respectively. The relative permittivities of three layers are " : , a : and " : , respectively. "

Fig. 5. Bistatic RCS solutions for a PEC sphere with one-layer full dielectric : and coating in HH. The dielectric coating has a relative permittivity " the radii of two interfaces are a  and a , respectively.

Fig. 7. Bistatic RCS solutions for a dielectric sphere with two-layer full di, a  and electric coating in HH. The radii of three interfaces are a , respectively. The relative permittivities of three layers are " : , a : and " : , respectively. "

= 10

=50

= 10

=30

= 12

into 168,342 and 215,680 triangular patches, respectively, resulting = 1 152 066 unknowns in total. The bistatic RCS solution in in VV is shown in Fig. 4 and we can see that it is close to the analytic counterpart. In the second example of the large problems, we consider the scattering by a PEC sphere with one-layer full dielectric coating as sketched in Fig. 2(c). The inner and outer radii are the same as in the first example and the coating material has a relative permittivity r = 3 0. We use the same meshes as in the first example, resulting in = 899 553 unknowns in total. Fig. 5 plots the bistatic RCS solution in HH and it is well consistent with the corresponding analytic solution. The third example of the large problems illustrates the scattering by a dielectric sphere with two-layer full dielectric coating as shown in Fig. 2(d). The radii of three interfaces are 1 = 4 0 , 2 = 6 0 and 3 = 8 0 , respectively and the materials of three layers are characterized by the relative permittivities r1 = 4 0, r2 = 3 0 and r3 = 2 0, respectively. We discretize the first, second and third interfaces into 41,726, = 87,324 and 139,692 triangular patches, respectively, resulting in 806 226 unknowns in total. The bistatic RCS solutions in VV and HH are sketched in Figs. 6 and 7, respectively and again they are in good agreement with the analytic versions. We summarize the CPU time and memory usage for these examples (the first case is taken in the first example) and they are shown in

N

;

;

" N

a

"

;

:a

: "

:

: "

:

a

=8 =30

=8 =30

=4

=20

=4

=20

=6 =40

=6 =40

TABLE II SUMMARY OF CPU TIME (T) AND MEMORY USAGE (M) IN THE THREE EXAMPLES FOR LARGE PROBLEMS

;

:

:

N

Table II. To see the fast convergence of the scheme, we also include in the table when the relative root-meanthe number of iterations square (RMS) error reaches 1004 in the generalized minimal residual (GMRES) iterative method. All the calculations are performed on a Dell Precision 690 machine with two dual-core 3.0 GHz CPUs and 16 GB RAM, but only one core is used.

K

VII. CONCLUSION In this work, we choose the dual basis function proposed by Chen and Wilton in 1990 to represent the magnetic current in the MoM for solving the SIEs with penetrable materials. The dual basis function is defined over a polygon pair formed through enclosing the two ends of

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common edge of an RWG triangle pair. The benefits of the basis function include the approximate orthogonality to the RWG basis function, the association with the RWG basis function, the constant charge density within each polygon, the vanishing charges within a polygon pair and no fictitious line charges across polygon boundaries. When used with the RWG basis function together to represent the magnetic current and electric current, respectively, the resultant system matrix is naturally well-conditioned with a simple testing scheme. This is because each operator in the SIEs is well tested and the system matrix is diagonally dominant. The good conditioning of the system matrix is very desirable in fast algorithms because they use iterative methods to solve the matrix equations and the convergence strongly relies on the conditioning. We implement the above dual basis function with the partnership of RWG basis function in the MoM and the process is accelerated with the MLFMA for solving the SIEs with penetrable media. Numerical examples for scattering by large composite objects are presented to demonstrate the scheme and fast convergence of solutions can be observed.

[16] M. S. Tong, W. C. Chew, B. J. Rubin, J. D. Morsey, and L. Jiang, “On the dual basis for solving electromagnetic surface integral equations,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3136–3146, Oct. 2009. [17] X. Q. Sheng, J. M. Jin, J. M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag., vol. 46, no. 11, pp. 1718–1726, Nov. 1998. [18] M. S. Tong and W. C. Chew, “A higher-order Nyström scheme for electromagnetic scattering by arbitrarily shaped surfaces,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 277–280, 2005. [19] G. T. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum, Radar Cross Section Handbook. New York: Plenum Press, 1970.

Validation of Rain Spatial Classification for High Altitude Platform Systems Stanislav Zvanovec and Pavel Pechac

REFERENCES [1] W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves. San Rafael, CA: Morgan & Claypool, 2008. [2] E. K. Miller, L. Medgyesi-Mitschang, and E. H. Newman, Computational Electromagnetics: Frequency-Domain Method of Moments. New York: IEEE Press, 1992. [3] 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, Mar. 1982. [4] S. M. Rao and D. R. Wilton, “E-field, H-field and combined field solution for arbitrarily shaped three-dimensional dielectric bodies,” Electromagn., vol. 10, no. 4, pp. 407–421, Oct. 1990. [5] Q. L. Chen, “Electromagnetic modeling of three-dimensional piecewise homogeneous material bodies of arbitrary composition and geometry,” Ph.D. dissertation, Univ. Houston, TX, 1990. [6] A. Buffa and S. H. Christiansen, “A dual finite element complex on the barycentric refinement,” Math. Comput., vol. 76, no. 260, pp. 1743–1769, Oct. 2007. [7] F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative Calderón preconditioner for the electric field integral equation,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2398–2412, Aug. 2008. [8] J. Board and K. Schulten, “The fast multipole algorithm,” IEEE Comput. Sci. Eng, vol. 2, no. 1, pp. 76–79, Jan./Feb. 2000. [9] W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics. Boston: Artech House, 2001. [10] Ö. Ergül and L. Gürel, “Rigorous solutions of electromagnetics problems involving hundreds of millions of unknowns,” IEEE Antennas Propag. Mag., Jun. 2010. [11] K. Sertel and J. L. Volakis, “Incomplete LU preconditioner for FMM implementation,” Microw. Opt. Tech. Lett, vol. 26, no. 4, pp. 265–267, Aug. 2000. [12] J. Lee, J. Zhang, and C.-C. Lu, “Sparse inverse preconditioning of multilevel fast multipole algorithm for hybrid integral equations in electromagnetics,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2277–2287, Sep. 2004. [13] B. Carpentieri, I. S. Duff, L. Giraud, and G. Sylvand, “Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations,” SIAM J. Sci. Comput., vol. 27, no. 3, pp. 774–792, Nov. 2005. [14] T. Malas and L. Gürel, “Incomplete LU preconditioning with the multilevel fast multipole algorithm for electromagnetic scattering,” SIAM J. Sci. Comput., vol. 29, no. 4, pp. 1476–1494, Jun. 2007. [15] B. M. Kolundzija, “Electromagnetic modeling of composite metallic and dielectric structures,” IEEE Trans. Microw. Theory Tech, vol. 47, no. 7, pp. 1021–1032, Jul. 1999.

Abstract—Previous work on rain spatial classification for terrestrial PMP radio systems has been extended in this communication to stratospheric systems—high altitude platform systems. It was found that the rain spatial parameter concept can also be effectively applied to HAP systems, although specific adjustments have had to be made to the method. The differences between terrestrial and stratospheric systems are discussed with regard to the new derived dependences. Index Terms—Diversity methods, millimeter wave propagation, rain.

I. INTRODUCTION High altitude platforms (HAP), also known as stratospheric repeaters/systems [1], could offer the possibility of very fast additional coverage or could be utilized as an alternative to satellite or terrestrial systems. Broadband reliable transmissions in the millimeter waveband are also envisaged [2], therefore the influence of rain on propagated electromagnetic waves is of a great importance for system planning. Among other factors, rain spatial properties and the corresponding fade mitigation techniques, e.g., route diversity schemes, should be considered. The route diversity given by the recommendation ITU-R P. 618 [3] for satellite links cannot be accurately applied, since the link geometry is different in the case of HAP systems. Rather than analyzing only two HAP link joint statistics (e.g., elaborately studied in [4]), many simultaneous diversity links representing overall system performance were tested under specific conditions, both in time and space, within a defined area. The methodology and a rain spatial classification method relevant to rainfall influence on terrestrial PMP radio systems were proposed in [5]. This communication extends this work (briefly summarized in Section II) to the specific case of stratospheric systems. Based on extensive simulations, the method is Manuscript received February 24, 2010; revised November 02, 2010; accepted December 10, 2010. Date of publication May 10, 2011; date of current version July 07, 2011. This work was supported by Czech Science Foundation Grant 102/08/P346 and research project no. OC09075 of the Czech Ministry of Education, Youth and Sports. The authors are members of the Department of Electromagnetic Field, Czech Technical University in Prague, Czech Republic (e-mail: [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.2011.2152325

0018-926X/$26.00 © 2011 IEEE

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validated for a HAP network and particular adjustments of the method are proposed. II. APPROACH USED FOR TERRESTRIAL SYSTEMS To validate HAP systems, the approach to rain spatial classification for terrestrial systems [5] was utilized. It is based on an enumeration of the outage improvement probability P defined as a percentage of terminal stations with a successfully established diversity link (i.e., with received power PP div higher than the threshold PP treshold ) throughout the network from nout links having a received power level PP main below the threshold due to rain attenuation

= nout = P

(PP div > PP treshold ) 100 nout (PP main < PP treshold ):

n 1

(1)

1

(2)

In the case of terrestrial systems, the outage improvement was derived as a function of angular separation # (rad) and the ratio of the main and diversity link lengths dmain =ddiv [5]

P

= aconst

1

1

0

2

#0  0 bconst

1

0

d d 1

dmain ddiv

c

:

(3)

An empirical parameter aconst is dependent on the maximum rain rate RMAX taken from the whole rainfall radar scan and on the rain fade margin PMARG set in the system as [5]

aconst = a1 + a2 1 PMARG 0 RMAX

(4)

where a1 = 110:64 and a2 = 2:14 were derived for terrestrial systems from the radar and simulated data. The other coefficients, bconst and cconst , are then highly dependent on the rate spatial distribution described by the rain spatial parameter S in km2 as [5]

cconst = S c 1 (1 0 exp(0c1 1 S )) bconst = =4 0 exp(b1 1 cconst )

(5) (6)

with c1 = 0:12, and b1 = 1:38 obtained by a curve fitting algorithm. The rainfall spatial parameter is able to characterize the spatial properties of the rain event in terms of rain cell sizes and rain rate slopes, relating to their impact on the point-to-multipoint terrestrial system within a given area [5]. It is calculated by normalizing a particular rain rate spatial distribution by its maximum rain rate and then by dividing it into 9 contour levels; then, by averaging areas of the rain cell for each contour threshold and subsequently, by taking the mean value of the average rain cell areas for all contour levels [5].

Fig. 1. Scenario of HAP diversity.

considered, a circular polarization was set up in the case of HAP links [2]. The main focus was to derive corresponding relations among rain event parameters and the availability of HAP systems independently of the particular deployment of HAP stations and spatial distribution of users. Therefore for particular rain distribution, the user was moved step-by-step in a 1 km raster and at each position the likelihood that the connection to a HAP station (situated in an altitude of 20 km; land distances from 2 to 20 km and all azimuths) would be significantly affected by rain was investigated. In the case the rain loss exceeded the rain fade margin, an additional diversity link with a second HAP station (having lengths of up to 20 km) was tested for all angular separations. See the diversity scenario considered for HAP systems in Fig. 1. In the next step, the dependences of the parameters aconst ; bconst and cconst were derived from the obtained set of probabilities of outage reductions within HAP systems by use of genetic algorithms. Even though simulations of HAP systems again resulted in a linear dependence of aconst on RMAX and PMARG as in [5], the relations denoting the influence of the maximum rain rate—especially the slope of this dependence—have to be described differently with respect to the rain fade margins. It was observed that the value of the rain fade margin parameterizes aconst dependence on the maximum rain rate. A new derived relationship valid for HAP systems can be expressed by

aconst = a1 +

1 PMARGRMAX + a2 RMAX PMARG :

a1

1

0

The constants a1 = 50:16 and a2 = 00:454 were obtained by regressive fitting of the simulated data. Examples demonstrating results from simulations and a fitting by the new relation for rain fade margins of 10 dB and 20 dB can be seen in Fig. 2. The influence of the rain spatial distribution on the outage improvement probability was analyzed for HAP systems with a route diversity scheme. From the simulations performed, the resulting dependence of cconst on rain spatial parameter was determined in the form of

III. STATISTICS AND MODELING FOR HAP SYSTEMS An analysis of a hypothetical HAP system was performed based on meteoradar scans for the Czech Republic for the period from 2002–2005 [6]. During simulations, the influence of the outage improvement probability on the rain fade margin constant within the whole system (set from 5 to 30 dB) was tested at a frequency of 48 GHz (chosen in accordance with [1]). Contrary to terrestrial system simulations [5], where a linear polarized electromagnetic wave was

(7)

cconst = (c2 1 S )c

0

c02 1 :

(8)

The parameters c1 and c2 in (8) were optimized using genetic algorithms to the values of c1 = 0:309 and c2 = 0:520, respectively. The dependence of the last empirical parameter, bconst , on the parameter cconst obtained from the simulation of HAP systems is depicted in Fig. 3.

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Fig. 2. The dependence of the parameter a on the maximum rain rate within the particular area and on rain fade margins of 10 dB and 20 dB.

Fig. 3. The dependence of the parameter b systems.

on parameter c

for HAP

The derived dependence for HAP systems has a linear increase and can be expressed as

= b1 cconst + b2 (9) where b1 = 0:283 and b2 = =16. It should be mentioned that these bconst

1

statistics were generated for the region of the Czech Republic. The rainfall spatial parameter can be applied in general to any climatic regions but its definition would have to be based on appropriate experimental data using the same methodology. IV. COMPARISON OF HAP AND TERRESTRIAL SYSTEM PERFORMANCES In this point, it is useful to demonstrate the similarities and differences between HAP and terrestrial systems based on the simulation results. Knowledge of two basic types of rainfall is essential for a detailed understanding of the simulation results. The first type, stratiform rain, is usually spread over large areas and can be characterized by the slow decay of the low rain rate from its maximum (which is usually less than 10 mm/h). The second type, convective rain, represents smaller heavy

Fig. 4. A comparison of dependences of the parameter c parameter derived for HAP and terrestrial systems.

on the rain spatial

rain cells with high rain intensity. From a physical point of view, wide spread stratiform rainfalls result in a lower number of millimeter link drops, unlike convective rains which cause high numbers of system outages. Nevertheless, since convective rains are spread over a smaller area, they can be over-traced by HAP links. Note that, according to [7], the mean height of the rain is equal to 3.36 km above mean sea level (figures valid for central Europe). In the case of terrestrial systems, a higher percentage of links can cross the rainy area even in the cases when the diversity base station is not directly affected by rain and therefore different outage improvement probabilities can be observed. A remarkable difference in the utilization of route diversity within both systems can be observed when changing the main to diversity link length ratio. A constant length of the main link and an increasing length of the diversity link lead to a decrease in the outage improvement probability in the case of terrestrial systems, unlike HAP systems, where the outage improvement probability increases under specific conditions. The parameter cconst , determining a value of the power of the main and diversity link lengths ratio in (3) is a vital measure, particularly in this case. A comparison of dependences of the parameter cconst on the rain spatial parameter for HAP systems and for terrestrial point-to-multipoint systems is illustrated in Fig. 4. In the case of both systems, the parameter cconst increases with the increasing rain spatial parameter. Note that for HAP systems, the cconst parameter reaches negative values for the low rain spatial parameter. Therefore, in the case of the rain spatial parameter below 15.8 km2 , with an increasing length of the diversity HAP link an increase of the outage improvement probability, caused by the negative value of the parameter cconst , can be observed. This distinction can be explained by a difference between link configurations, when majority parts of terrestrial links are affected by the rain event and the lengthening of diversity links does not significantly improve rain losses. HAP links are, on the other side, only influenced below rain heights, which leads to better system performance in the case of a low rain spatial parameter. The mentioned features of both systems are demonstrated in Figs. 5 and 6. Two rain distributions with the maximum rain rate of 30 mm/h were chosen as an example to show differences between terrestrial and HAP systems. First a rain distribution with the rain spatial parameter less than the mentioned threshold of 15.8 km2 (S = 5 km2 )—see the outage improvement probabilities in Fig. 5—and the second rain distribution with S = 40 km2 —demonstrated in Fig. 6. Rain fade margins of 10 dB were considered during the simulations.

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Fig. 5. A dependence of the outage improvement probability for rain distribution with S km ; R mm/h for (a) a terrestrial system, (b) a HAP dB. system, both with P

=5

= 30 = 10

The rain spatial parameter has also a different influence through the parameter cconst on the angular separation dependence of the outage improvement probabilities in the case of HAP systems than was observed in the case of terrestrial systems [5]. From a physical point of view this can be explained by the section of link crossing a rainy area. For angular separations between links of up to 57 degrees, rain distributions along HAP link sections of the main and diversity links are highly correlated (note, the angular limit was empirically derived for correlation coefficients exceeding 0.7), having close maximum rain rates. This means, that utilization of route diversity is not as efficient for the same link angular separations as in the case of terrestrial systems, where such a high correlation between rain distributions along paths was not determined. Therefore essentially higher changes in the outage improvement probability can be observed for these systems. An overall decrease in the dependence of the outage improvement probability on the maximum rain rate was observed in HAP systems for users employing the route diversity when compared to terrestrial systems [5]. For instance for users with similar lengths of main and diversity links and a particular rain fade margins set, the utilization of route diversity in a HAP network reaches approximately one-third of the values determined in the case of terrestrial systems [5]. This can be explained by the fact that the path affected by rain is generally shorter in the case of HAP links at high elevation angles.

2749

Fig. 6. The dependence of the outage improvement probability for rain distribution with S km ; R mm/h for (a) a terrestrial system and (b) a HAP system, both with P dB.

= 40

= 30 = 10

From the analyses performed, it can nevertheless be concluded that the rain spatial parameter concept can be very well adapted for use on HAP systems. The resulting equation for the outage improvement probability in the case of a HAP system with route diversity has to be expressed in the new form P

= aconst 10 1



0 +4:516c 1

#

0

2

10 dd

1

dmain ddiv

c

:

(10)

V. CONCLUSION Based on the results of the analyses presented in the communication, it can be concluded that the method of rain spatial classification for the evaluation of rain influences on terrestrial systems can also be adapted for HAP systems. It was derived that both approaches are quite distinctively dependent on the rain spatial parameter. This can be explained by different link geometries and system deployments. Furthermore, different dependences of HAP system performances on the variability of maximum rain rates within the areas served and

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on the rain fade margins set in systems compared to terrestrial systems were observed and discussed in the communication.

REFERENCES [1] “Preferred Characteristics of Systems in the Fixed Service Using HighAltitude Platform Stations Operating in the Bands 47.2–47.5 GHz an 47.9–48.2 GHz,” 2000, ITU-R Recommendation F.1500. [2] C. Spillard, E. Falletti, J. D. Penin, J. L. Ruíz-Cuevas, and M. Mondin, “Mobile Link Propagation Aspects, Channel Model and Impairment Mitigation Techniques,” FP6 IST-2003-506745 CAPANINA, 2004 [Online]. Available: http://www.capanina.org/deliverables.php, Deliverable D14

[3] “Propagation Data and Prediction Methods Required for the Design of Earth-Space Telecommunication Systems,” 2009, International Telecommunications Union, ITU-R Recommendation P.618-10. [4] A. D. Panagopoulos, E. M. Georgiadou, and J. D. Kanellopoulos, “Selection combining site diversity performance in high altitude platform networks,” IEEE Commun. Lett., vol. 11, no. 10, pp. 787–789, 2007. [5] S. Zvanovec and P. Pechac, “Rain spatial classification for availability studies of point-to-multipoint systems,” IEEE Trans. Antennas Propag., vol. 54, no. 12, pp. 3789–3796, 2006. [6] Z. Sokol, “The use of radar and gauge measurements to estimate areal precipitation for several Czech river basins,” Stud. Geophys. Geod., no. 47, pp. 587–604, 2003. [7] “Rain Height Model for Prediction Methods,” 2001, International Telecommunications Union, ITU-R Recommendation P.839-3.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 7, JULY 2011

2751

Comments Comments on “3-D Numerical Mode-Matching (NMM) Method for Resistivity Well-Logging Tools” Yueqin Dun, Yu Kong, Li Zhang, and Jiansheng Yuan

Abstract—In this comment, we give a detailed derivation procedure to obtain the local reflection and transmission matrix describing the converfor a half-space problem, which sion of modes from layer m to layer is based on analyzing Fan et al. “3-D numerical mode-matching (NMM) method for resistivity well-logging tools,” IEEE Trans. Antennas Propag., vol. 48, pp. 1544–1552, Oct. 2000. Our conclusions show that the local transmission matrix, (13) in the abovementioned paper, has an error.

Fig. 1. Two semi-infinitely thick layers.

m+1

Index Terms—Half-space problem, local reflection matrices, local transmission matrices, numerical mode-matching method.

In [1], we found the authors presented a very elegant method of 3-D NMM for resistivity well-logging tools. A comment that we want to make is that the derivation is not detailed enough regarding [1, p. 1546, Eq. (13)]. The derivation method has been given in [1], that is, using the continuity conditions for electric potentials and the normal electric currents at z = zm , and applying the orthogonality relation of the eigenfunctions, but the local transmission matrix in [1, Eq. (13)] has an error. The detailed derivation is given as follows. For a half-space problem with an interface at z = zm , assuming a source at z = z 0 in region m as shown in Fig. 1. The fields in the two regions are given by (10) and (11) in [1, p. 1546]. According to the continuity conditions at z = zm , that is, (12) in [1, p. 1546], we have

8tm () e K z z + R m;m+1 e K (z z ) Sm ( ) (1) = 8tm+1 () T m;m+1 e K (z z ) Sm ( ) t  m 8m () Kmz   m;m+1e K (z z ) Sm ( ) e K z z +R = m+1 8tm+1 () ( K m+1;z ) T m;m+1 e K (z z ) Sm ( ): (2) 0

1

j

0

1

0

1

1

1

j

0

0

0

0

1

0

j

0

j

0

0

0

0

0

1

1

1

1

0

1

0

1

Simplifying (1) and (2), we can rewrite (1) and (2) as

8tm () I + R m;m+1 = 8tm+1 () T m;m+1  mz I + R m;m+1 m 8tm () K = m+1 8tm+1 () K m+1;z T m;m+1 : 1

1

1

1

0

1

1

1

8m () m on both sides of (3), and then in1

8m () m 8tm () I + R m;m+1 dS = 8m () m 8tm+1 () T m;m+1 dS: 1

1

1

1

S

1

1

(5)

Considering the orthogonality relation of the eigenfunctions, we can derive (6) from (5)

I + R m;m+1 = Q m;m+1 T m;m+1

(6)

1

where  m;m+1 is defined as [1, p. 1546, Eq. (15)]. First multiplying by m+1 () on both sides of (4), and then integrating, we have

Q

S

8

8m+1 () m 8tm() K mz [ I + R m;m+1 ] dS = 8m+1 () m+1 8tm+1 () K m+1;z T m;m+1 dS: 1

0

1

1

1

S

1 0

1

1

1

(7)

Considering the orthogonality relation of the eigenfunctions, we can derive (8) from (7)

t +1 K mz (R m;m+1 I) = K m+1;z T m;m+1 : Q m;m  m;m+1 form (6). We can get R 1

1

0

0

1

R m;m+1 = Q m;m+1 T m;m+1 I: 1

(3)

(8)

(9)

0

Taking (9) into (8), we can rewrite (8) as

0

1

S

0

1

1

0

First multiplying by tegrating, we have

(4)

t +1 K  mz (Q m;m+1 T m;m+1 2I) Q m;m = K m+1;z T m;m+1 :  m;m+1 from (10) So we can obtain T 1

Manuscript received October 27, 2010; accepted December 28, 2010. Date of publication May 12, 2011; date of current version July 07, 2011. This work was supported by the Program for the Natural Science Foundation of Shandong Province under Grant ZR2010EQ012. Y. Dun and L. Zhang are with the School of Electronic Information and Control Engineering, Shandong Institute of Light Industry, Jinan 250353, China (e-mail: [email protected]). Y. Kong is with Information Center of Shandong Medical College, Jinan 250002, China (e-mail: [email protected]). J. Yuan is with Department of Electrical Engineering, Tsinghua University, Beijing 100084, China, (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2152352

1

1

0

1

0

t +1 K  mz Q m;m+1 ) 1 T m;m+1 = 2(K m+1;z + Q m;m t +1 K mz : Q m;m 1

(10)

0

1

1

1

(11)

Therefore, it has been proven that [1, p. 1546, Eq. (13)] has an error, that is, [1, Eq. (13)] does not have the coefficient 2 in (11).

0018-926X/$26.00 © 2011 IEEE

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REFERENCES [1] G. X. Fan, Q. H. Liu, and S. P. Blanchard, “3-D numerical mode-matching (NMM) method for resistivity well-logging tools,” IEEE Trans. Antennas Propag., vol. 48, pp. 1544–1552, Oct. 2000.

Comments on “Topology Optimization of Sub-Wavelength Antennas” Herbert L. Thal, Jr. A method for optimizing the performance of an electrically-small antenna formed by a conductor pattern on the surface of an empty sphere is described in [1]. The authors cite several references that define the lower bound on the Q of an antenna based only on the energy in the fields outside of the spherical surface. Their results Manuscript received January 15, 2011; accepted February 26, 2011. Date of current version July 07, 2011. The author is at Wayne, PA 19087-5337 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2152356

average about 1.7 times this bound. Unfortunately, they do not cite [2] which applies exactly to their configuration and provides a much tighter bound by including the effect of energy within the sphere. Specifically, they report Q values of 7.74 and 7.58 for k0 a of 0.588 and 0.596. The Q bounds obtained from the method of [2] are 6.75 and 6.53. The post-processed results based on the topology optimized pattern are 1.15 and 1.16 times the tighter bounds. Furthermore, if a hemi-spherical antenna actually achieved the tighter bound, it would have a unique surface current distribution and pattern, both known a priori. The gain on axis (TE) would be 4 dB lower than the gain along the ground plane (TM) [2, Table IV]. The results in [1] for each design are approaching both the characteristic pattern as well as the Q bound from [2].

REFERENCES [1] A. Erentok and O. Sigmund, “Topology optimization of sub-wavelength antennas,” IEEE Trans. Antennas Propag., vol. 59, no. 1, pp. 58–69, Jan. 2011. [2] H. L. Thal, “New radiation Q limits for spherical wire antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2757–2763, Oct. 2006.

0018-926X/$26.00 © 2011 IEEE

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 7, JULY 2011

2753

Corrections Corrections to “Traveling Waves on Three-Dimensional Periodic Arrays of Two Different Alternating Magnetodielectric Spheres” Yang Li and Robert A. Shore

6

In [1], a number of errors need to be remedied as follows. n=0 1. Equation (2.4d) = should be replaced by = . . should be replaced by 2. In (2.4e) m;l= 3. 4. 5. 6.

1 01 1 In (2.5e) b02;n should be replaced by b02;0 . In (2.5g) b01;0 should be replaced by b01;n . In (2.5j) 62n (2kh; d=2h=0) should be replaced by 62n (2kh; d=2h; 0). In [1 , p. 3079], the first line of footnote 3 f1 (2m; 2l; m; kh; d=h; 0) should be replaced by f1 (2m; 2l; n; kh; d=h; 0). In the fourth line of the same footnote, “. . . over all m and n . . .” should be replaced by “. . . over all m and l . . .”

10. In (2.23) 63s (kh; d=h; 0) should be replaced by 63s (kh; 0), and K0 ((2l 1)) (2m)2 (kh)2 ) should be replaced by K0 ((l 1=2) (2m)2 (kh)2 ). 11. In (2.25) ikd

0

should be replaced by

8khei2kd cos d sin kd : 1 0 2 cos 2 dei2kd + ei4kd 12. In (2.26)

p e0(2n01)(d=h) (2) (m +l )0(kh)

1

8

ein d [62n(2kh; d=2h=0) + . . .]

0(2n01)(d=2h)p(2) (l +m )0(2kh) e 2 (2)2(l2 + m2 ) 0 (2kh)2 14. In (2.28)

should be replaced by

1

1 in d ein d [. . .] + 18 e [62n(2kh; d=2h; 0) + . . .]:

9. In (2.17)

(2)2 (m2 + l2 ) 0 (kh)2

p e0(2n01)(d=h) (2) (m +l )0(2kh)

should be replaced by

p e0(2n01)(d=2h) (2) (m +l )0(2kh) 0

1

should be replaced by

e

:

:

0

15. In [1, p. 3084], the fifth line of Section III, “kd d (2.14)” should be replaced by “kd d equation (2.14)”. 16. In [1, Table I] row 9, the values of (kd)l1 and (kd)l2 should be interchanged. 17. Equation (A.10) should be replaced by

p e0nd=h (2) (m +l )0(kh) (2)2 (m2 + l2 ) 0 (kh)2 02nd=hp(2) (m +l )0(kh)

ein d 63n (2kh; d=2h; 0)

ei2kd cos d sin kd = 1160kh 2 cos 2 d ei2kd + ei4kd 1 0 4 cos(2n 0 1) d n=1 1 2 (01)m+l [(2m)2 0 (2kh)2 ]

1 1 ein d [62n (2kh; d=2h; 0) + . . .]: 8

1

:

13. Equation (2.27) should be replaced by

should be replaced by

ein d [. . .] + 1

p e0n(d=h) (2) (m +l )0(kh)

should be replaced by

1 1 ein d [62n(2kh; d=2h=0) + . . .] 8

1

0

8kh e cos d sin kd 1 0 2 cos 2 dei2kd + ei4kd

7. In (2.10b)

8. In (2.10d)

0

0

:

Manuscript received November 13, 2010; accepted March 04, 2011. Date of current version July 07, 2011. Y. Li is with the Department of Electrical and Computer Engineering, and Center for Nondestructive Evaluation, Iowa State University, Ames, IA 50011 USA (e-mail: [email protected]). R. A. Shore is with the Air Force Research Laboratory/RYHA, Hanscom AFB, MA 01731 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2152355

ein d 63n (2kh; d=2h; 0) = 04ikh

1 sin(2kd) 0 4kh cos(2 d cos(2n d) 0 4  ) 0 cos(2kd) n=1 1 2 (01)m+l [(2m)2 0 (2kh)2 ] 0n(d=h)p(2) (l +m )0(2kh) e 2 (2)2 (l2 + m2 ) 0 (2kh)2 :

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

2754

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 7, JULY 2011

18. Equation (A.12) should be replaced by

1

e

should be replaced by

0 n0

n d 6 (kh; d=h; 0) 4n

i

e

sin(2 d) = 2kh cos(2 d) cos(2kd)

1

2

n=1

sin(2n d)

m+l (01)

2 e0 nd=h 2

p

) (m

(2

l

+

0 kh

)

(

63s (kh; 0) = 2i (kh)

2

1

08

0

m

H0(1) ((2l 0 1)kh=2) 2

2

8kh eikd cos( d) sin(kd) 1 2 cos(2 d) ei2kd + ei4kd

0

should be replaced by 8kh ei2kd cos( d) sin(kd) : 1 2 cos(2 d) ei2kd + ei4kd

0

22. The LHS of (A.18) should be replaced by

1 n0 = d e jn0 = jkd : sgn(n 0 1=2)e n 01 p  m l 0 kh In (A.19) e0npd=h should be replaced by  m l 0 kh . e0 n0 d=h i(

1 2)

i

1 2

=

23.

(

(2

1)(

(2

)

(2

)

) (

) (

+

)

+

)

(

)

(

)

24. Equation (A.20) should be replaced by

1 =

n d 6 (2kh; d=2h; 0) 3n

i

e

16khei2kd cos d sin kd 1 2 cos 2 dei2kd + ei4kd

0

0 4 2 2

1

n=1

cos(2n

1

(2

d=2h)

1)(

p

) (l

m

(2

+

0 n0 (2

p

d=h)

1)(

2

2

) (m

(2

0

)

kh)

(2

0 (2kh)

(2 ) (l + m ) 2

25. In (A.21) e

m+l [(2m)2 0 (2kh)2 ]

( 1)

0 n0

e

0 1) d

0

2

l

+

kh) :

(2

(B.1e)

0

)

kh)

(2

i

2

2

3

:

i

2

i

3

i

(B.1f) (B.1g)

Efficient algorithms are available for calculating polylogarithm functions; see, for example, [3]. Polylogarithm functions can also be calculated with Mathematica. 27. The summations l=1 J0 (kx) and l=1 Y0 (kx) preceding (B.2) should be

1

20. In the left hand side (LHS) of (A.16) 2 i should be replaced by 2 ikh. 21. In (A.17)

2

0

)

00:901542 . . . :

1

0 (kh) ]

( 1) [(2m)

l

+

0 2i [Li (e x) 0 Li (e0 x )] 1 x 0 x )]: Cl (x) = [Li (e ) + Li (e

m;l=1 2 K0 ((l 0 1=2) (2m)2 0 (kh)2 ):

2 ikh

= 1:20205 . . . ; Cl3 ( ) =

3

l=1

) (m

(2

Cl2 (0) = Cl2 ( ) = Cl2 (2 ) = 0; Cl3 (0) = Cl3 (2 )

Cl2 (x) =

19. Equation (A.14) should be replaced by

1

p

The Clausen functions are also given exactly in terms of the dilogarithm function, Li2 (x), and the trilogarithm function, Li3 (x), [2] by

:

)

d=2h)

1)(

26. In [1, Appendix B] after (B.1d) add the following:

1

0 4kh

0

(2

k=1

J0 (kx) and

1

1

k=1

Y0 (kx):

28. In (B.3) ln(kh=(4)) should be replaced by ln(x=(4)). 29. The correct title of the paper in [5] is “Traveling electromagnetic waves on linear periodic arrays of lossless penetrable spheres.” 30. The correct title of the paper in [15] is “Dispersion diagram characteristics of periodic array of dielectric and magnetic materials based spheres.” 31. Reference [17] should be R. A. Shore and A. D. Yaghjian, “Complex waves on 1D, 2D, and 3D periodic arrays of lossy and lossless magnetodielectric spheres,” Air Force Res. Lab. In-House Rep., AFRL-RY-HS-TR-2010-0019, 2010. (PDF file available on request.) 32. The author of [18] is A. Ishimaru. A PDF file of the corrected paper is available on request.

REFERENCES [1] R. A. Shore and A. D. Yaghjian, “Traveling waves on three-dimensional periodic arrays of two different alternating magnetodielectric spheres,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3077–3091, Oct. 2009. [2] L. Lewin, Polylogarithms and Associated Functions. New York: Elsevier North Holland, 1981. [3] R. E. Crandall, “Note on fast polylogarithm computation,” vol. 77, no. 4, pp. 045104-1–045104-11, Jan. 2008 [Online]. Available: http:// people.reed.edu/~crandall/papers/Polylog.pdf, pdf file, 2006

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2755

Corrections to “Stability Analysis and Improvement of the Conformal ADI-FDTD Methods” Jian Dai

g

In [1, p. 2257], errors occurred in the equation. The correct equation now follows. j

g g

j

j

g g

j

j

g g

j

j

g g

j

j

g

j

j

41

28

j

58

Sxy ji;j;k

j

Sxy ji;j;k ylx ji+1=2;j;k g60 Sxy ji;j;k ylz ji;j +1;k+1=2 Syz ji;j;k ylz ji;j;k+1=2; g50 Syz ji;j;k zly ji;j +1=2;k+1 Syz ji;j;k zly ji;j +1=2;k g52 Syz ji;j;k zlx ji+1=2;j;k+1 Szx ji;j;k zlx ji+1=2;j;k g56 Szx ji;j;k xlz ji+1;j;k+1=2 Szx ji;j;k xlz ji;j;k+1=2 g54 Szx ji;j;k

= g7 = g19 = g49 = 1 = g8 = g20 = g47 =

= g21 = g43 = g51 = 1 = g11 = g22 = g44 =

= g13 = g25 = g55 = 1 = g14 = g26 = g35 =

= g23 = g31 = g53 = 1

=1

g

j

g

j

=1

=1

j

0

Sxy ji;j 01;k xly ji;j 01=2;k Sxy ji;j 01;k

g

j

g

j

g

j

g

j

j

g

j

g

j

g g

j

g

j

= g4 = 1

Manuscript received June 02, 2011. Date of current version July 07, 2011. J. Dai is with the Department of Electronics, China Astronautics Standards Institute, Beijing 100071, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2160495

12

0

Syz ji;j;k01 ylz ji;j;k01=2 Syz ji;j;k01 zly ji;j +1=2;k Syz ji;j;k01 zlx ji01=2;j;k+1 Szx ji01;j;k zlx ji01=2;j;k Szx ji01;j;k xlz ji;j;k+1=2 Szx ji01;j;k xlz ji+1;j;k01=2 Szx ji;j;k01 xlz ji;j;k01=2 Szx ji;j;k01 zlx ji+1=2;j;k Szx ji;j;k01 ylx ji01=2;j +1;k Sxy ji01;j;k ylx ji01=2;j;k Sxy ji01;j;k xly ji;j +1=2;k Sxy ji01;j;k zly ji;j 01=2;k+1 Syz ji;j 01;k zly ji;j 01=2;k Syz ji;j 01;k ylz ji;j;k+1=2 : Syz ji;j;k01

= g15 = 1 =g = 1 16

= g18 = 1 =g = 1 = g34 = 1 =g = 1 36

= g39 = 1 =g = 1 40

= g42 = 1 =g = 1 45

j

0

= g10 = 1 =g = 1

33

j

g

=1

= g17 = g24 = g32 = = 1 = g = 1xly i+1;j 1=2;k

j

g

j

j

9

j

g

= 1ylx i+1=2;j+1;k

= g5 = g30 = g38 =

3

j

g

2

= g29 = g37 = g59

j

g

j

j

j

g

g

;j +1=2;k = g1 = g27 = g57 = 1xlySxyi+1i;j;k = g = g = g = g = 1xly i;j+1=2;k j

g

= g6 = 1Sylxyx ii;j+1=12;k;j;k = g = 1ylz i;j+1;k 1=2 j

g

= g46 = 1 =g = 1 48

REFERENCES [1] J. Dai, Z. Chen, D. Su, and X. Zhao, “Stability Analysis and Improvement of the Conformal ADI-FDTD Methods,” IEEE Trans. Antennas Propag., vol. 59, no. 6, pt. II, pp. 2248–2258, Jun. 2011.

0018-926X/$26.00 © 2011 IEEE

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION Editorial Board Sara K. S. Hanks, Editorial Assistant Michael A. Jensen, Editor-in-Chief e-mail: [email protected] Department of Electrical and Computer Engineering (801) 422-3903 (voice) 459 Clyde Building Brigham Young University Provo, UT 84602 e-mail: [email protected] (801) 422-5736 (voice) Senior Associate Editor Karl F. Warnick Dimitris Anagnostou Hiroyuki Arai Ozlem Aydin Civi Zhi Ning Chen Jorge R. Costa Nicolai Czink George Eleftheriades Lal C. Godara

Associate Editors Yang Hao Derek McNamara Stuart G. Hay Andrea Neto Sean V. Hum George W. Pan Ramakrishna Janaswamy Athanasios Panagopoulos Buon Kiong Lau Patrik Persson Jin-Fa Lee K.V. S. Rao Kwok Wa Leung Shanker Balasubramaniam Duixian Liu Satish Sharma

Jamesina Simpson Mei Song Tong Jon W. Wallace Fan Yang Ali Yilmaz Zhengqing Yun Zhijun Zhang Yue Ping Zhang

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Digital Object Identifier 10.1109/TAP.2011.2160831

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Digital Object Identifier 10.1109/TAP.2011.2160832