IEEE Transactions on Antennas and Propagation [volume 58 number 4]

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APRIL 2010

VOLUME 58

NUMBER 4

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas Vertically Multilayer-Stacked Yagi Antenna With Single and Dual Polarizations . . . . . . . . . . . . . . . . . . . . O. Kramer, T. Djerafi, and K. Wu A Compact Tri-Band Monopole Antenna With Single-Cell Metamaterial Loading . . . . J. Zhu, M. A. Antoniades, and G. V. Eleftheriades Investigation Into the Polarization of Asymmetrical- Feed Triangular Microstrip Antennas and its Application to Reconfigurable Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Sung Ultrawideband Dielectric Resonator Antenna With Broadside Patterns Mounted on a Vertical Ground Plane Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. S. Ryu and A. A. Kishk Transparent Dielectric Resonator Antennas for Optical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. H. Lim and K. W. Leung A Low-Profile Linearly Polarized 3D PIFA for Handheld GPS Terminals . . . . . . . . . . . A. A. Serra, P. Nepa, G. Manara, and R. Massini Analysis of Strong Coupling in Coupled Oscillator Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Seetharam and L. W. Pearson Arrays Compact Elongated Mushroom (EM)-EBG Structure for Enhancement of Patch Antenna Array Performances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Coulombe, S. Farzaneh Koodiani, and C. Caloz Low-Profile PIFA Array Antennas for UHF Band RFID Tags Mountable on Metallic Objects . . . . . . . . . . . . . H.-D. Chen and Y.-H. Tsao CMOS Phased Array Transceiver Technology for 60 GHz Wireless Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Fakharzadeh, M.-R. Nezhad-Ahmadi, B. Biglarbegian, J. Ahmadi-Shokouh, and S. Safavi-Naeini Direction of Arrival Estimation in Time Modulated Linear Arrays With Unidirectional Phase Center Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Li, S. Yang, and Z. Nie Reactive Energies, Impedance, and Q Factor of Radiating Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. A. E. Vandenbosch Electromagnetics Electromagnetic Boundary Conditions Defined in Terms of Normal Field Components . . . . . . . . . . . . . . . . . I. V. Lindell and A. H. Sihvola A Cloaking Metamaterial Based on an Inhomogeneous Linear Field Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Maci The Design of Broadband, Volumetric NRI Media Using Multiconductor Transmission-Line Analysis . . . S. M. Rudolph and A. Grbic Uniform Asymptotic Evaluation of Surface Integrals With Polygonal Integration Domains in Terms of UTD Transition Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Carluccio, M. Albani, and P. H. Pathak Non-Orthogonal Domain Parabolic Equation and Its Tilted Gaussian Beam Solutions . . . . . . . . . . . . . . . . . . . . . . . Y. Hadad and T. Melamed Emissivity Calculation for a Finite Circular Array of Pyramidal Absorbers Based on Kirchhoff’s Law of Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Wang, Y. Yang, J. Miao, and Y. Chen Near-Field Electromagnetic Holography in Conductive Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. G. Williams and N. P. Valdivia Thin Microwave Quasi-Transparent Phase-Shifting Surface (PSS) . . . . . . . . . . . . . . . . . . . . . . . . N. Gagnon, A. Petosa, and D. A. McNamara Polarization Rotating Frequency Selective Surface Based on Substrate Integrated Waveguide Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. A. Winkler, W. Hong, M. Bozzi, and K. Wu Design and Analysis of a Tunable Miniaturized-Element Frequency-Selective Surface Without Bias Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Bayatpur and K. Sarabandi A Novel Band-Reject Frequency Selective Surface With Pseudo-Elliptic Response . . . . . . . . . . . . . . . . . . . . . . . . . . . A. K. Rashid and Z. Shen Broadening of Operating Frequency Band of Magnetic-Type Radio Absorbers by FSS Incorporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. N. Kazantsev, A. V. Lopatin, N. E. Kazantseva, A. D. Shatrov, V. P. Mal’tsev, J. Vilˇcáková, and P. Sáha

1022 1031 1039 1047 1054 1060 1067

1076 1087 1093 1105 1112 1128 1136 1144 1155 1164 1173 1181 1193 1202 1214 1220 1227

(Contents Continued on p. 1021)

(Contents Continued from Front Cover) Numerical Methods Embedding Calderón Multiplicative Preconditioners in Multilevel Fast Multipole Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Peeters, K. Cools, I. Bogaert, F. Olyslager, and D. De Zutter 3D Isotropic Dispersion (ID)-FDTD Algorithm: Update Equation and Characteristics Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W.-T. Kim, I.-S. Koh, and J.-G. Yook Aperture Antenna Modeling by a Finite Number of Elemental Dipoles From Spherical Field Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Serhir, J. M. Geffrin, A. Litman, and P. Besnier Invasive Weed Optimization and its Features in Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Karimkashi and A. A. Kishk Propagation Radio-Wave Propagation Into Large Building Structures—Part 1: CW Signal Attenuation and Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. F. Young, C. L. Holloway, G. Koepke, D. Camell, Y. Becquet, and K. A. Remley Radio-Wave Propagation Into Large Building Structures—Part 2: Characterization of Multipath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. A. Remley, G. Koepke, C. L. Holloway, C. A. Grosvenor, D. Camell, J. Ladbury, R. T. Johnk, and W. F. Young Numerical Investigations of and Path Loss Predictions for Surface Wave Propagation Over Sea Paths Including Hilly Island Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Apaydin and L. Sevgi On the Effective Low-Grazing Reflection Coefficient of Random Terrain Roughness for Modeling Near-Earth Radiowave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Liao and K. Sarabandi Truncated Gamma Drop Size Distribution Models for Rain Attenuation in Singapore . . . . . . . . . . . L. S. Kumar, Y. H. Lee, and J. T. Ong

1236 1251 1260 1269

1279 1290 1302 1315 1325

Scattering Comparison of TE and TM Inversions in the Framework of the Gauss-Newton Method . . . . . . . . . . . . . . . . . . . . . . . P. Mojabi and J. LoVetri Experimental Study of the Invariants of the Time-Reversal Operator for a Dielectric Cylinder Using Separate Transmit and Receive Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Davy, J.-G. Minonzio, J. de Rosny, C. Prada, and M. Fink

1336

Wireless Wireless Communication for Firefighters Using Dual-Polarized Textile Antennas Integrated in Their Garment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Vallozzi, P. Van Torre, C. Hertleer, H. Rogier, M. Moeneclaey, and J. Verhaevert A Compact Six-Port Dielectric Resonator Antenna Array: MIMO Channel Measurements and Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Tian, V. Plicanic, B. K. Lau, and Z. Ying

1357

1349

1369

COMMUNICATIONS

Dual-Band Multiple Beam Antenna System Using Hybrid-Cell Reuse Scheme for Non-Uniform Satellite Communications Traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Wang, S. K. Rao, M. Tang, and C.-C. Hsu A Low Cross-Polarization Smooth-Walled Horn With Improved Bandwidth . . . L. Zeng, C. L. Bennett, D. T. Chuss, and E. J. Wollack Design of Compact Differential Dual-Frequency Antenna With Stacked Patches . . . . L. Han, W. Zhang, X. Chen, G. Han, and R. Ma A Broadband Impedance Matching Method for Proximity-Coupled Microstrip Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Sun and L. You Electromagnetically Coupled Band-Notched Elliptical Monopole Antenna for UWB Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Eshtiaghi, J. Nourinia, and C. Ghobadi Bandwidth Enhancement of Printed E-Shaped Slot Antennas Fed by CPW and Microstrip Line . . . . . . . . . . . . . A. Dastranj and H. Abiri Handling Sideband Radiations in Time-Modulated Arrays Through Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Poli, P. Rocca, L. Manica, and A. Massa Experimental Validation of a Linear Array Consisting of CPW Fed, UWB, Printed, Loop Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. M. Tanyer-Tigrek, I. E. Lager, and L. P. Ligthart Limits on the Amplitude of the Antenna Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Foltz and J. McLean AIM Analysis of 3D PEC Problems Using Higher Order Hierarchical Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Lai, X. An, H.-B. Yuan, N. Wang, and C.-H. Liang Generalized Stability Criterion of 3-D FDTD Schemes for Doubly Lossy Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Y. Heh and E. L. Tan Low Observable Targets Detection by Joint Fractal Properties of Sea Clutter: An Experimental Study of IPIX OHGR Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X.-K. Xu

1380 1383 1387 1392 1397 1402 1408 1411 1414 1417 1421 1425

CALL FOR PAPERS

Joint Special Issue on Multiple-Input Multiple-Output (MIMO) Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joint Special Issue on Ultrawideband (UWB) Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1430 1431

IEEE ANTENNAS AND PROPAGATION SOCIETY All members of the IEEE are eligible for membership in the Antennas and Propagation Society and will receive on-line access to this TRANSACTIONS through IEEE Xplore upon payment of the annual Society membership fee of $24.00. Print subscriptions to this TRANSACTIONS are available to Society members for an additional fee of $36.00. For information on joining, write to the IEEE at the address below. Member copies of Transactions/Journals are for personal use only. ADMINISTRATIVE COMMITTEE R. D. NEVELS, President Elect 2011 2012 A. AKYURTLU *J. T. BERNHARD W. A. DAVIS H. LING M. OKONIEWSKI

M. ANDO, President 2010 P. DE MAAGT G. ELEFTHERIADES G. MANARA P. PATHAK *A. F. PETERSON

M. W. SHIELDS, Secretary-Treasurer 2013

Honorary Life Members: R. C. HANSEN, W. R. STONE *Past President Committee Chairs and Representatives Antenna Measurements (AMTA): S. SCHNEIDER Antennas & Wireless Propagation Letters Editor-in-Chief: G. LAZZI Applied Computational EM Society (ACES): A. F. PETERSON Awards: A. F. PETERSON Awards and Fellows: C. A. BALANIS Chapter Activities: L. C. KEMPEL CCIR: P. MCKENNA Committee on Man and Radiation: G. LAZZI Constitution and Bylaws: O. KILIC Digital Archive Editor-in-Chief: A. Q. MARTIN Distinguished Lecturers: J. C. VARDAXOGLOU Education: D. F. KELLY EAB Continuing Education: S. R. RENGARAJAN Electronic Design Automation Council: M. VOUVAKIS Electronic Publications Editor-in-Chief: S. R. BEST European Representatives: B. ARBESSER-RASTBURG Fellows Nominations Committee: J. L. VOLAKIS AP Transactions website: http://www.ict.csiro.au/aps Albuquerque: L. H. BOWEN Atlanta: D. M. KOKOTOFF Australian Capital Territory: E. S. LENSSON Beijing: D. ZU Beijing, Nanjing: W. X. ZHANG Benelux: D. VANHOENACKER Bombay: M. B. PATIL Buffalo: M. R. GILLETTE Bulgaria: K. K. ASPARUHOVA Calcutta: P. K. SAHA Central New England: B. T. PERRY Chicago: F. ARYANFAR Coastal Los Angeles: F. J. VILLEGAS Columbus: M. A. CARR Connecticut: C. ALVAREZ Croatia: Z. SIPUS Czechoslovakia: M. POLIVKA Dayton: A. J. TERZUOLI

Finance: M. W. SHIELDS Gold Representative: R. ADAMS Historian: K. D. STEPHAN IEEE Press Liaison: R. J. MAILLOUX IEEE Magazine Committee: W. R. STONE IEEE Public Relations Representative: W. R. STONE IEEE Social Implications of Technology: R. L. HAUPT Institutional Listings: T. S. BIRD Joint Committee on High-Power Electromagnetics: C. E. BAUM Long-Range Planning: C. RHOADS Magazine Editor-in-Chief: W. R. STONE Meetings Coordination: S. A. LONG Meetings Joint AP-S/URSI: M. A. JENSEN Membership: S. BALASUBRAMANIAM Nano Technology Council: G. W. HANSON New Technology Directions: S. C. HAGNESS Nominations: J. T. BERNHARD

PACE: J. M. JOHNSON Publications: R. J. MARHEFKA RAB/TAB Transnational Committee Liaison: D. R. JACKSON Region 10 Representative: H. NAKANO Sensor Council: A. I. ZAGHOUL, T. S. BIRD, M. W. SHIELDS Standards Committee—Antennas: M. H. FRANCIS Standards Committee—Propagation: D. V. THIEL TABARC Correspondent: C. A. BALANIS TAB Magazines Committee: W. R. STONE TAB New Technology Directions Committee: A. I. ZAGHLOUL TAB Public Relations Committee: W. R. STONE TAB Transactions Committee: T. S. BIRD Transactions Editor-in-Chief: T. S. BIRD Transnational Committee: D. R. JACKSON USAB Committee on Information Policy: S. WEIN USAB R&D Committee: A. C. SCHELL USNC/URSI : J. T. BERNHARDT Women in Engineering Representative: P. F. WAHID

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Eastern North Carolina: T. W. NICHOLS Egypt: H. M. EL-HENNAWY Finland: A. LUUKANEN Florida West Coast: K. A. O’CONNOR Foothill: F. G. FREYNE Fort Worth: S. TJUATJA France: M. DRISSI Fukuoda: M. TAGUCHI Hong Kong: K. W. LEUNG Houston: J. T. WILLIAMS Huntsville: H. SCHANTZ Israel: S. AUSTER Italy: G. VECCHI Japan: N. MICHISHITA Kansai: M. KOMINAMI Malaysia: E. MAZLINA Melbourne: A. M. JONES Montreal: K. WU

Nagoya: N. INAGAKI New South Wales: K. P. ESSELLE North Jersey: H. DAYAL Orlando: P. F. WAHID Ottawa: Q. YE Philadelphia: J. NACHAMKIN Poland: W. KRZYSZTOFIK Portugal: C. A. FERNANDES Queensland: D. V. THIEL Rio de Janeiro: J. R. BERGMANN Russia, Moscow: D. M. SAZONOV Russia, Nizhny: Y. I. BELOV St. Louis: D. MACKE San Diego: G. J. TWOMEY Santa Clara Valley/San Francisco: G. A. MANASSERO Seoul: H. J. EOM South Africa: P. W. VAN der WALT South Australia: B. D. BATES

Southern Alberta: S. BOUMAIZA Spain: L. J. DE-HARO Springfield: P. R. SIQUEIRA Sweden: A. RYDBERG Switzerland: C. O. FUMEAUX Syracuse: E. ARVAS Tainan: W.-S. CHEN Taipei: J-F. KIANG Thailand: P. AKKARAEKTHALIN Toronto: G. V. ELEFTHERIADES Turkey: I. TEKIN Ukraine, East: A. I. NOSICH Ukraine, Kharkov: N. N. KOLCHIGIN Ukraine, West: O. ZAMORSKA Vancouver: D. G. MICHELSON Victoria: K. GHORBANI Winnipeg: V. I. OKHMATOVSKI

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION Is the leading international engineering journal on the general topics of electromagnetics, antennas and wave propagation. The journal is devoted to antennas, including analysis, design, development, measurement, and testing; radiation, propagation, and the interaction of electromagnetic waves with discrete and continuous media; and applications and systems pertinent to antennas, propagation, and sensing, such as applied optics, millimeter- and sub-millimeter-wave techniques, antenna signal processing and control, radio astronomy, and propagation and radiation aspects of terrestrial and space-based communication, including wireless, mobile, satellite, and telecommunications. Author contributions of relevant full length papers and shorter Communications are welcomed. See inside back cover for Editorial Board.

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Vertically Multilayer-Stacked Yagi Antenna With Single and Dual Polarizations Olivier Kramer, Tarek Djerafi, and Ke Wu, Fellow, IEEE

Abstract—There are many applications such as local positioning systems (LPS) and wireless sensor networks that require high-directivity and compact-size or small footprint antennas. The classical Yagi-Uda antenna may be useful in meeting such demands, which however, becomes very large in size to achieve a high-gain performance due to a large number of directors as well as space required between those elements. In this paper, high-gain yet compact stacked multilayered Yagi antennas are proposed and demonstrated at 5.8 GHz for LPS applications. This structure makes use of vertically stacked Yagi-like parasitic director elements that allow easily obtaining a simulated gain of 12 dB. Two different antenna configurations are presented, one based on dipole geometry for single polarization, and the other on a circular patch to achieve dual polarization. The characteristics of these antennas with respect to various geometrical parameters are studied in order to obtain the desired performance. Measured results of the fabricated antenna prototypes are in good agreement with simulated results. The measured dipole Yagi antenna yields 11 dB gain over 14% bandwidth with a size of 80 80 29 mm3 . Radiation patterns of the dual-polarized Yagi antenna are nearly identical to those of the single-polarized antenna, which has a size of 50 50 60 mm3 , and also its two-port isolation is found to be as low as 25 dB over 4% bandwidth. The proposed antennas present an excellent candidate for compact and low-cost microwave and millimeter-wave integrated systems that require fixed or variable polarization capabilities and small surface footprint. Index Terms—Balun, circular patch, dipole, dual polarization, microstrip antenna, stacked antenna, Yagi-Uda.

I. INTRODUCTION HE Local positioning system (LPS) is a radio system used to search for and track down in real time objects of interest within a limited space range. This is a typical application of wireless sensor networks. With the LPS, a mobile object can be localized and can also collect information about its position at a precise time instant [1]. Applications of such a LPS system are multiple and diverse. Those systems are used for zone security with great efficiency, particularly indoor applications such as at airports where security is mandatory. Also location tracking can

T

Manuscript received March 12, 2009; October 19, 2009; accepted October 24, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and in part by Regroupement strategique of FQRNT. The authors are with the Département de Génie Électrique, Poly-Grames Research Center, École Polytechnique de Montréal, Montréal, QC H3T 1J4, Canada (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2041155

clearly improve maneuver of the business and logistics management, for example, hospitals can improve their healthcare services by keeping a constant track of the location of doctors, nurses, equipments, etc. LPS can also be used in many other applications such as construction and agriculture, leisure and sports, etc. For a full integration with GPS systems that are usually used for outdoor scenarios, LPS is expected to achieve the challenging task of being accurate, low-cost and autonomous at ISM frequencies such as 5.8 GHz. Therefore, compact-size, light-weight and low-cost antenna is required for such LPS design and implementation. The Yagi antenna, which has been very popular because of its simplicity as well as its customizable high gain (three-element Yagi antenna can reach 9 dB when optimized) [2], can be used for this type of applications. The basic unit of a three-element Yagi antenna consists of a half-wavelength driver dipole backed by a longer reflector and a director on the other side. Several microstrip-based Yagi or quasi-Yagi antenna structures have been reported in the literature [2]–[8]. Interesting approaches are related to the design of a microstrip Yagi array based on the microstrip patch antenna such as the array developed in [3] for mobile satellite system in the L-band. The antenna developed was based on patches, consisting of one reflector, one driving element and two directors. An array of four antenna elements was successfully used to achieve the required performances. The design reported in [4] was made on the basis of patches instead of dipoles as the driver element. An interesting printed Yagi antenna configuration was presented in [5], [6] where the Yagi-like printed dipole array antenna was fed by a microstrip-to-coplanar strip transition. In this case, a truncated microstrip ground plane was utilized as a reflecting element. On the other hand, an active quasi-Yagi version was proposed [7] for 5.8 and 60 GHz applications. In [8], the proposed antenna consisted of a dipole as a driving element, a parasitically coupled reflector and six directors. The antenna was designed by utilizing the same design rules as used in the conventional Yagi dipole antenna while taking into account the fact that the antenna was made on a planar substrate. The antenna was designed for 5 GHz band and has achieved a gain of 10 dB. To overcome the problem of size and footprint within the planar structure, two novel high-gain compact structures based on the Yagi-Uda antenna concept are presented for the first time in this work. These structures are constructed in a multilayer topology by stacking together the reflector, the driver, and the directors. Compared to the above-described uniplanar Yagi antennas, this design is able to provide a number of advantages. First of all, the usage of the third dimension (the vertical dimension) that has not been widely used in the design of microstrip

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KRAMER et al.: VERTICALLY MULTILAYER-STACKED YAGI ANTENNA WITH SINGLE AND DUAL POLARIZATIONS

antennas, allows an effective reduction in size and footprint. In fact, multilayer processing techniques have become more mature in integrated circuit design, fabrication and integration. Second, a high permittivity substrate can be used in this case, thereby reducing spacing between the directors, which is critical for a high-density integration between antenna and circuits. In fact, the use of special tunable substrates and/or materials such as ferrites or ferroelectric films allows a very easy realization of high-sensible frequency- or bandwidth-agile antenna systems. Third, the possibility of a dual polarization design based on the Yagi antenna concept is made possible, and the coupling-based feed mechanism can achieve wide bandwidth characteristics. Finally, the most important advantage can probably be visualized through the design of such antennas over millimeter-wave and terahertz ranges where the substrate spacing/thickness between Yagi antenna elements can naturally be made compatible with current three-dimensional circuit processing techniques such as the state-of-the-art 3-D through silicon via (TSV) techniques. Various stacked structures were investigated in [9] and [10]. The advantage of those topologies is that substrate thickness can be adjusted to achieve an optimized bandwidth performance. Unlike the Yagi antenna, which is a traveling-wave antenna [11], the structure is a resonant mode antenna. In the Yagi structure, the director is smaller than the driver and the distance between and . In [9] the thickness of the them is between , the bandwidth is 70% wider used substrate is about than the single patch without gain improvement and the upper layer patch is bigger. Li studied theoretically different patch shapes (square, circular, triangle, etc.) and investigated the optimal combination to achieve a lower cross polarization or circular polarization [10]. The ratio of substrate thickness to wavelength in the dielectric is close to 0.02. Stacked triangular microstrip antennas were also investigated experimentally in [12] to achieve a bandwidth about 17%-5% at the centre frequency of 3.407 GHz. To demonstrate the proposed concepts and design features, we will present two case studies in connection with the design of respective structures. The first design (see Fig. 1) is a multilayered printed-circuit version of the proposed Yagi antenna. It consists of a ground plane (as a reflector), a dipole (as a driver), and four directors. In the second design (see Fig. 3), the antenna presents a dual polarization using circular patches in the design of one driver element and four directors. In this paper, the configuration of the proposed Yagi-Uda antennas is described in detail. Design specifications are discussed, considering the effects of different dimensions on antenna performances. Both of the proposed antennas as described in Fig. 1 and Fig. 3 are fabricated. Simulated and measured results are then compared, and the work is finally concluded.

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Fig. 1. Proposed structure of design 1 (dipole stacked Yagi antenna).

Fig. 2. Layer II of design 1.

II. ANTENNA DESIGN CONSIDERATION Fig. 3. Proposed structure of design 2 (dual polarization circular patches).

A. Stacked Dipole Yagi Antenna (Design #1) The configuration of the dipole stacked Yagi antenna shown in Fig. 1, is based on the classical Yagi-antenna design principle. It consists of one dipole driver element, and four parasitic elements. The antenna is designed on the basis of the same design

rules that are used in the conventional Yagi dipole antenna, except that the antenna is made on a planar substrate. There have been many different design versions for printed dipole antennas and baluns, as well as coplanar strip dipoles

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fed by coplanar waveguide and stripline balun used to feed a printed quasi-Yagi antenna proposed in [5]. The proposed dipole allows the avoidance of a balun which becomes usually necessary when dipole antennas are fed by an unbalanced line. The dipole (layer II in Fig. 1) is printed on both sides of a dielectric substrate [13]–[15]. The bottom-tapered ground transition is designed to provide an impedance matching tuner with balanced output. The designed feed network of the dipole antenna with the tapered balun transition is tuned by the angle of the tapered ground plane for impedance matching and balanced output. This simplified feeding structure results in the reduction of transmission line length and in turn radiation loss. Moreover, it exhibits attractive wide bandwidth capabilities. The feeding dipole bandwidth (without any Yagi antenna elements) is relatively large; a bandwidth of 29% can be achieved at 5.8 GHz using a 0.762 mm thick substrate with a relative permittivity of 2.33 (Rogers RT/Duroïd5870). Director elements fabricated on substrates having a relative permittivity of 3.48 results in significantly reduced spacing between them. In the Yagi-antenna design, the gap between the elements is approximately given by

Fig. 4. Gain variation versus ground plane size for different driver-to-reflector spacing.

(1) where is the relative dielectric constant of the substrate (the constant of air in the classical Yagi-Uda antenna design) and is the free space wavelength. This equation clearly shows that if the dielectric constant is increased, the gap is reduced. In this design, the substrates should be stacked ideally without any air gaps. However, the layers are still very thick at 5.8 GHz, which are not feasible and affordable for fabrication. Of course, this is no longer a problem at very high frequencies such as millimeter-wave ranges. To some extent, this class of antennas becomes more attractive and easier-to-implement at lower cost when the operating frequency becomes higher. Antenna characteristics such as gain, front-to-back ratio, beamwidth and center frequency can be altered by changing the length of the driven element, the length of the parasitic elements, the spacing between reflector and dipole, the spacing between director and dipole, the spacing between directors or substrate thickness as well as the dielectric constant. It is shown that an array configuration is completely determined when any two of these constraints are specified [16]. The proposed stacked dipole Yagi antenna is simulated by using Ansoft Designer v2.0, a commercial simulator that can solve electric and magnetic fields via a method of moment. A reflector plane is added to the driver element, the size and the distance of the reflector are optimized. For different spacing between driver dipole and reflector, Fig. 4 shows the gain variation versus the reflector plane size. It is found that the optimum reflecting spacing for the maximum directivity is beand as in case of the standard Yagi structween ture. The gain increases as the size of reflector increases, and this . The dimenvariation becomes less pronounced beyond sional ratio of the reflector to the driven element can be somewhere between 2 and 2.6. Compared to the standard Yagi antenna, this ratio is doubled. The variation of the conductivity of

Fig. 5. Gain variation versus director length for different director-to-director spacing.

the reflector material has also influence, leading to the degradation of bandwidth and/or directivity as observed in [17]. Three directors are added to the dipole and reflector. Fig. 5 shows the variation of the gain with respect to the director length in dielectric for different director-to-director spacing. In this simulated result, the optimum director-to-director spacing is in the order of to , compared with typical 0.2 to 0.35 wavelengths in the design of a standard Yagi. The wavelength in the substrate thickness must be added to obtain the actual spacing. On the other hand, the gain increases with length, and the optimal length of director is around the dipole dimension where the coupling is maximized. A further increase of their size should reduce the array gain rapidly. The dimensional ratio of the director to the driven element can be between 0.8 and 0.95. As shown in Fig. 6, the gain enhancement is not significant when the fourth director is added, compared to the Yagi antenna with three stacked elements. The addition of an identical fourth director would increase the gain only by 0.25 dB. Upper planar substrate with a high permittivity is added in order to reduce the antenna size (footprint) and to increase the gain [19], [20]. Fig. 6 shows the effects of adding this substrate without directors on

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TABLE I DESIGN 1 FOR DIPOLE STACKED YAGI ANTENNA

Fig. 6. Gain variation as a function of director length for: staked Yagi with three and four identical directors; Stacked Yagi with three directors and high permittivity plan. TABLE II DESIGN 2 FOR PATCH STACKED YAGI ANTENNA

the antenna gain. An improvement of 2 dB can be achieved and the optimal length of the directors becomes shorter. The gain enhancement is given by the following equations [20]

(2) in which

(3) and

(4) with B being the thickness of the lower layer and , are and its relative permittivity and permeability, respectively. are the relative permittivity and permeability of the upper layer. is the refractive index. is defined as the frequency deviation parameter.

(5) where frequency is near by the center frequency . Composite planar dielectric structures can be used for increasing the directivity of a point source when a resonance condition is established. Particular attention has been given to the physical interpretation of this resonance gain effect in [18]. This is described in terms of leaky waves (LW) excited in the structure. Under certain resonance conditions, a pair of weakly attenuated TE/TM leaky waves becomes the dominant contribution to the antenna aperture field. Equation (2) applied to our specific configuration yields an approximate enhancement of 0.9 dB. A director is added on the upper substrate. Various parameters are optimized starting from the initial value defined in previous paragraphs. The entire structure of design#1 is optimized

in order to achieve high gain and large bandwidth at 5.8 GHz. Director lengths are perturbed to optimize the return loss with minimum gain loss. The distances between elements along the (8.58 mm) between reflector vertical axis are: layer I and driver layer II; (5.6 mm) between driver layers II, and director layers III, IV and V. Fig. 7 shows the influence of elements width on the bandwidth of the design #1. This design can achieve a bandwidth between 6% and 27% at 5.8 GHz. It was noticed that this parameter has the most significant effect when it is selected between and , and becomes much less influential outside this band. The return loss is more sensitive to the width of the elements (driver and director) rather than antenna directivity. The total height of the designed structure is 29 mm. Compared with (1), it can be observed that the spacing between elements can be reduced by a factor of two. The upper director layer is in direct contact with director V. The change of relative permittivity in the interface between the air gap and the substrate results in a higher equivalent relative permittivity, thus reducing . the air gap. The size of the stacked substrates is B. Circular Patch Stacked Yagi Antenna With Dual Polarization (Design #2) In this design demonstration, the classical dipole is replaced by a patch antenna. The proposed concept offers a possibility to use a wide range of patches instead of the classical dipole as driver. The circular patch has been used as duo, namely a dual polarization feeder, and an optimization facility, since that is only required to optimize one dimension (radius).

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Fig. 7. S11 as a function of element width.

The antenna consists of two substrate layers. The circular patch is etched on the top substrate and the feed lines and ground are etched on the bottom substrate. To achieve the dual polarization the patch is fed by two orthogonal lines. Such a feeding by coupling allows increasing the bandwidth. This patch has a bandwidth of 4.5% at 5.8 GHz. The feed layer (layer I in Fig. 3) is grounded on its bottom side and fed on the other side by 2.26 long strip lines. The feed lines mm width, 20.27 mm start from the middle of their respective side, and the width is calculated to match the antenna to 50 impedance. The circular (9.68 mm) and the spacing between patch has a radius of the layers I and II is 0.28 mm. The antenna is simulated using Ansoft HFSS v10, a simulator which makes use of the finite element method. The proposed circular patch stacked Yagi antenna is shown in Fig. 3. This antenna is designed with the geometry of several layers, and is very similar to the dipole Yagi antenna topology that was described in the above section. In this structure, the Rogers RF/Duroid 5870 substrate is used for all the layers with and thickness of 0.762 mm. As in the case of the first design, the antenna is optimized to achieve better gain and larger bandwidth, but also the isolation characteristics between the two feed lines must be taken into account. One constraint of the stacked dipole Yagi antenna is related to its requirement for a large ground plane. Fig. 8 shows the gain enhancement of the patch antenna in connection with the ground size, suggesting that the gain increases with the size of the ground plane. The optimal ratio of the reflector to the driven element is in the order of 1.6, which is comparable to the standard Yagi antenna topology. This antenna can be used to reduce overall dimensions of patch stacked Yagi antenna; a high gain with reduced ground plane can be obtained. Fig. 9 plots the gain versus the number of parasitic elements . Note that for adjacent director-to-director spacing of adding the sixth director increases the gain only by 0.1 dB. Similar to the first design case, the gain performances are generally controlled by the director spacing and director length. To define initial values, the array is optimized by varying the radius while maintaining the spacing parameters. Fig. 10 shows gain variation versus the element length on dielectric substrate for different director-to-director spacing. The ideal spacing is in

Fig. 8. Gain versus ground plane size.

Fig. 9. Gain versus the number of parasitic elements.

Fig. 10. Gain variation with director length for different director-to-director spacing.

the order of . The ideal radius of the directors should be and . The ratio to the driver patch defined between is the same as in the first design case. Optimization of the element spacing is followed by the perturbation of different radius starting from the initial value to op-

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Fig. 11. Effect of diameter variation on the gain and on the isolation between the two polarizations.

Fig. 13. Photograph of the second design example for patch-based Yagi antenna structure.

Fig. 12. Photograph of the first design example for dipole-based Yagi antenna structure.

timize the bandwidth with smaller gain losses. Maintaining the optimized spacing constant, the reflector spacing radius of the first director (R2) can be used to tune the return loss characteristics of antenna as well as the isolation between the two polarizations. The effect of variation of the radius on the gain is verified also. As shown in Fig. 11, the isolation between the two polarizations has the excursion about 10–15 dB for diameter variation . The effect of diameter variation on the in the order of gain is less pronounced as observed in Fig. 11. The optimized patch radius for layers III, IV, V and VI are, , , respectively, , and . . The size of the layers and the ground plane are Because of the effective permittivity of substrates that is lower than the previous configuration, the spacing between the directors has increased accordingly. Hence, the same substrate has been used throughout the antenna in order to simplify the fabrication process. The vertical spacing between the directors is while the total height of this designed close to 14.6 mm configuration is 60 mm. III. RESULTS AND DISCUSSION The fabricated prototypes of both antennas are shown in Figs. 12 and 13. The layers are aligned by using four threaded

Fig. 14. S-parameter characteristics of the prototyped antenna #1.

rods with nuts. With this assembly, the gaps between the layers can be easily and precisely controlled. A. The First Design Example 1) Transmission Characteristics: S-parameters are measured using Anritsu 37397C Vector Network Analyzer to characterize the transmission properties of the proposed structure over the frequency range of interest. Simulated and measured return losses versus frequency are presented in Fig. 14. It can be seen that the measured center frequency is shifted slightly from the designed target but still very close to 5.8 GHz. The simulation results obtained by HFSS show a good agreement 10 dB) with the measured results. The antenna bandwidth ( covers frequencies from 5.4 to 6.2 GHz or almost 14% at 5.8 GHz. 2) Radiation Pattern: Radiation pattern measurement is made in a MI Technology anechoic chamber. Fig. 15 presents calculated and measured co-polar (E plane) radiation patterns. From these results there is good agreement between the measured and simulated radiation patterns.

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Fig. 15. Radiation pattern performances of antenna #1 at 5.8 GHz.

Fig. 17. Radiation pattern performances of antenna #2 at 5.8 GHz.

the designed multilayer-stacked Yagi antenna is much more compact than the metallic horn antenna. B. The Second Design Example

Fig. 16. S-parameter characteristics of the prototyped antenna #2.

It is also observed that the radiation pattern is very directive and symmetric. The 3 dB beam width is approximately 70 for both measured and simulated patterns. The side lobes are very low. The oscillation of the right lobe is due to the mechanical constraints of our anechoic chamber. 3) Antenna Gain: As shown in Fig. 11, the peak antenna gain is about 12 dB based on our simulations and almost 11 dB can be obtained in practice according to our measured results. Although the gain is slightly lower than expected, both curves match very well. The 1 dB drop in measured gain is caused by the imperfections of the compact range anechoic chamber used and also by losses such as surfaces waves, dielectric losses, and connector losses (coaxial lines). The total height of the designed configuration is 29 mm compared to the designed structure described in [8], and uses two more directors which has a dimension of to achieve lower gain. Based on equations detailed in [2], the horn antenna is 125 mm in length with radiating aperture of 155 by 22.15 mm in order to get a gain of 12 dB, and a coaxial to waveguide transition must be added in this case. It is clear that

1) Transmission Characteristics: Measured and calculated -parameters of the second antenna are shown in Fig. 16. The measured center frequency for both polarizations (Tx and Rx) is at 5.8 GHz and agrees well with the simulated results. The bandwidth is from 5.6 GHz to 5.85 GHz or 4% at 5.8 GHz, for both polarizations. The isolation (S21 parameter) between the two ports is nearly 30 dB over the entire frequency band, instead of the 25 dB calculated isolation. The bandwidth of this antenna is narrower but sufficient for most LPS or wireless sensor applications at 5.8 GHz. This bandwidth can also be increased by using a wideband patch [21], [22]. The measured performance matches well the simulation prediction. A small difference between the measured and simulated results (S11 is not exactly the same as S22), can be attributed to the fabrication accuracy. 2) Radiation Pattern: Fig. 17 presents the simulated and measured co-polar (E plane) radiation patterns for the second design. The measured radiation pattern agrees well with the simulated ones. The radiation pattern is again very directive and symmetric. The measured beam width (3 dB definition) is approximately 60 degrees, compared to 70 degrees for the simulated pattern. Both polarization curves are very similar (that is normal because of the symmetry of this structure). 3) Antenna Gain: Fig. 17 shows the gain performance based on both simulated and measured results. The respective maximum gains for the simulation and the measurement are 11.66 dB and 9.76 dB. There is a loss of 1.9 dB between the two gains. This loss is due to a frequency shift of the fabricated antenna (a maximum gain of 10.28 dB can be reached at the frequency of 5.75 GHz), the fabrication tolerance and losses in the measurement circuits, in addition to the imperfections of the compact range anechoic chamber. To achieve a comparable gain, DeJean in [25] proposed a bi-Yagi and quad-Yagi which consist of seven planar patch elecompared to ments. The size of the ground plane is —the size of our demonstrated antenna.

KRAMER et al.: VERTICALLY MULTILAYER-STACKED YAGI ANTENNA WITH SINGLE AND DUAL POLARIZATIONS

IV. CONCLUSION In this paper, two classes of novel antenna based on the classic Yagi-Uda antenna concept, have been proposed and demonstrated for the first time theoretically and experimentally. By using multilayer-stacked substrates, these designs allow compact size realization and achieve good performance at the demonstrated frequency of 5.8 GHz. Two different antenna configurations are presented and showcased, one based on dipole for single polarization, and the other on circular patch for dual polarization. The characteristics of the two proposed antenna types with respect to various parameters such as reflector dimensions, director dimensions, and spacing between these elements have been studied. The measurement results of S-parameters show a reasonably good bandwidth (a bandwidth of 15% can be reached with the novel structures) and the measured radiation pattern performance is very similar to the simulated results. Both designs have a peak gain of about 10 dB. Compared to the microstrip array Yagi antenna, which makes use of planar patches, this novel design can yield high gain and also the entire structure is very compact in size by using the vertical dimension. In this case, a large number of directors can be implemented in such multilayered geometries. Based on such new design concepts, many innovative structures with interesting properties, including electronically tunable substrates, can be realized by using different planar patches in the driver layer, or stacked substrate with zero air gap (especially for high frequency applications such as millimeter-wave or even terahertz frequency ranges). This work suggests that the proposed concept provides light-weight, low-cost, high-performance and full integration solutions for local positioning platforms, wireless sensor systems and radar sensor applications. It can be expected that those new antennas will provide a very attractive design alternative for a wide range of microwave and millimeter-wave system applications. ACKNOWLEDGMENT The authors wish to thank J. Gauthier and S. Dube of PolyGrames Research Center, École Polytechnique de Montreal, in the fabrication and mounting of prototypes. REFERENCES [1] M. Vossiek, L. Wiebking, P. Gulden, J. Wieghardt, C. Hoffmann, and P. Heide, “Wireless local positioning,” Microw. Mag., vol. 4, pp. 77–86, Dec. 2003. [2] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 2nd ed. New York: Wiley, 1998. [3] J. Huang and A. C. Densmore, “Microstrip Yagi antenna for mobile satellite vehicle application,” Trans. Antennas Propag., vol. 39, pp. 1024–1030, Jul. 1991. [4] L. C. Kretly and C. E. Capovilla, “Patches driver on the quasi-Yagi antenna: Analyses of bandwidth and radiation pattern,” in Proc. Int. Microwave and Optoelectron. Conf., Iguazu Falls, Brazil, Sep. 2003, vol. 1, pp. 313–316. [5] Y. Qian, W. Deal, N. Kaneda, and T. Itoh, “Microstrip-fed quasi-Yagi antenna with broadband characteristics,” Electron. Lett., vol. 34, pp. 194–2196, 1998. [6] G. Zheng, A. Kishk, A. Yakovlev, and A. Glisson, “Simplified feeding for a modified printed Yagi antenna,” in Proc. Antennas and Propagation Society Int. Symp., Columbus, OH, Jun. 2003, vol. 3, pp. 934–937.

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[7] Q. Yongxi and T. Itoh, “Active integrated antennas using planar quasiYagi radiators,” in Proc. Int. Microwave and Millimeter Wave Technology Conf., Beijing, China, Sep. 2000, pp. 1–4. [8] M. Alsliety and D. Aloi, “A low profile microstrip Yagi dipole antenna for wireless communications in the 5 GHz band,” in Proc. Int. Conf. on Electro/Information Technology, East Lansing, MI, May 2006, pp. 525–528. [9] J. P. Damiano, J. Bennegueouche, and A. Papiernik, “Study of multilayer microstrip antennas with radiating elements of various geometry,” IEE Proc., vol. 137, no. 3, pp. 163–170, Jun. 1990. [10] R. Li, G. DeJean, M. Maeng, K. Lim, S. Pinel, M. M. Tentzeris, and J. Laskar, “Design of compact stacked-patch antennas in LTCC multilayer packaging modules for wireless applications,” Trans. Adv. Packag., vol. 27, no. 4, pp. 581–589, Nov. 2004. [11] H. W. Ehrenspeck and H. Poehler, “A new method for obtaining maximum gain from Yagi antennas,” IRE Trans. Antennas Propag., vol. AP-7, pp. 379–386, Oct. 1959. [12] P. S. Bhatnagar, J. P. Daniel, K. Mahdjoubi, and C. Terret, “Experimental study on stacked triangular microstrip antennas,” Electron. Lett., vol. 22, no. 16, pp. 864–865, Jul. 1986. [13] C. Guan-Yu and Jwo-Shiun, “A printed dipole antenna with microstrip tapered balun,” Microw. Opt. Technol. Lett., vol. 40, no. 4, pp. 344–346, Feb. 2004. [14] N. Michishita and A. Hiroyuki, “A polarization diversity antenna using a printed dipole and patch with a hole,” Electron. Commun. Jpn., vol. 86, no. 9, pp. 57–66, Mar. 2003. [15] S. Dey, C. K. Aanandan, P. Mohanan, and K. G. Nair, “Analysis of cavity backed printed dipoles,” Electron. Lett., vol. 30, no. 30, pp. 173–174, Feb. 1994. [16] L. C. Shen and G. W. Raffoul, “Optimum design of Yagi array of loops,” Trans. Antennas Propag., vol. 22, no. 11, pp. 829–830, Nov. 1974. [17] J. A. Nessel, A. Zaman, R. Q. Lee, and K. Lambert, “Demonstration of an X-band multilayer Yagi-like microstrip patch antenna with high directivity and large bandwidth,” in Proc. Antennas and Propagation Society Int. Symp., Washington, DC, Jul. 3–8, 2005, vol. 1B, pp. 227–230. [18] D. R. Jackson and A. A. Oliner, “A leaky-wave analysis of high-gain printed antenna configuration,” Trans. Antennas Propag., vol. 36, no. 7, pp. 905–910, Jul. 1988. [19] A. Hoorfar, “Analysis of a “Yagi-Like” printed stacked dipole array for high-gain application,” Microw. Opt. Technol. Lett., vol. 17, no. 5, pp. 317–381, Apr. 1998. [20] D. R. Jackson and N. G. Alexopoulos, “Gain enhancement methods for printed circuit antennas,” Trans. Antennas Propag., vol. 33, pp. 976–987, Sep. 1985. [21] S. K. Padhi, N. C. Karmakar, and C. L. Law, “An EM coupled dualpolarized microstrip patch antenna for RFID applications,” Microw. Opt. Technol. Lett., vol. 39, no. 5, pp. 354–360, Dec. 2003. [22] Y. M. M. Antar, D. Cheng, and G. Jiang, “Wide-band microstrip patch antenna for personal communication,” in Proc. 6th Int. Conf. on Electronics, Circuits and Systems, Pafos, Cyprus, Sep. 1999, vol. 3, pp. 1305–1308. [23] P. R. Grajek, B. Schoenlinner, and G. M. Rebeiz, “A 24 GHz high-gain Yagi-Uda antenna array,” Trans. Antennas Propag., vol. 52, no. 5, pp. 1257–1261, May 2004. [24] D. Gray, J. Lu, and D. V. Thiel, “Electronically steerable Yagi-Uda microstrip patch antenna array,” Trans. Antennas Propag., vol. 46, pp. 605–608, May 1998. [25] G. R. DeJean, T. T. Thai, S. Nikolaou, and M. M. Tentzeris, “Design and analysis of microstrip Bi-Yagi and Quad-Yagi antenna arrays for WLAN applications,” Antennas Wireless Propag. Lett., vol. 6, pp. 244–248, Jun. 2007.

Olivier Kramer was born in Lyon, France, in 1984. He received the B.Sc. degree (with distinction) from the Ecole nationale supérieure en Systèmes Avancés et Réseaux of the Institut National Polytechnique de Grenoble (ESISAR-INPG), France, in 2007. He is currently working toward the M.Sc.A. degree at Ecole Polytechnique, Montreal, QC, Canada. His current interests involve millimeter-wave multilayer structures.

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Tarek Djerafi was born in Constantine, Algeria, in 1975. He received the Dipl. Ing. degree from the Institut d’Aeronautique de Blida (IAB), Blida, Algeria, in 1998 and the M.A.Sc. degree in electrical engineering from Ecole polytechnique de Montreal, Montreal, QC, Canada, in 2005, where he is currently working toward the Ph.D. degree. His research deals with design of millimeter-wave antennas and smart antenna systems, microwaves, and RF components design.

Ke Wu (M’87-SM’92-F’01) received B.Sc. degree (with distinction) in radio engineering from Nanjing Institute of Technology (now Southeast University), China, in 1982 and the D.E.A. and Ph.D. degrees in optics, optoelectronics, and microwave engineering (with distinction) from the Institut National Polytechnique de Grenoble (INPG) and the University of Grenoble, France, in 1984 and 1987, respectively. He is a Professor of electrical engineering, and Tier-I Canada Research Chair in RF and millimeter-wave engineering at Ecole Polytechnique

(University of Montreal). He also holds a number of Visiting (Guest) and Honorary Professorships at various universities including the first Cheung Kong Endowed Chair Professorship at Southeast University, the First Sir Yue-Kong Pao Chair Professorship at Ningbo University, and honorary professorship at Nanjing University of Science and Technology and City University of Hong Kong. He has been Director of the Poly-Grames Research Center and the Founding Director of “Centre de recherche en électronique radiofréquence” (CREER) of Quebec. He has (co)-authored over 700 referred papers, a number of books/book chapters and patents. His current research interests involve substrate integrated circuits (SICs), antenna arrays, advanced CAD and modeling techniques, and development of low-cost RF and millimeter-wave transceivers. He is also interested in the modeling and design of microwave photonic circuits and systems. He serves on the Editorial Board of Microwave Journal, Microwave and Optical Technology Letters, and Wiley’s Encyclopedia of RF and Microwave Engineering. He is an Associate Editor of International Journal of RF and Microwave Computer-Aided Engineering (RFMiCAE).

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A Compact Tri-Band Monopole Antenna With Single-Cell Metamaterial Loading Jiang Zhu, Student Member, IEEE, Marco A. Antoniades, Member, IEEE, and George V. Eleftheriades, Fellow, IEEE

Abstract—A compact tri-band planar monopole antenna is proposed that employs reactive loading and a “defected” ground-plane structure. The reactive loading of the monopole is inspired by transmission-line based metamaterials (TL-MTM), which enables the loaded antenna to operate in two modes. The first resonance exhibits a dipolar mode over the lower WiFi band of 2.40 GHz – 2.48 GHz, and the second resonance has a monopolar mode over the 5.15–5.80 GHz upper WiFi band. Full-wave analysis shows that the currents of the two modes are orthogonal to each other, resulting in orthogonal radiation patterns in the far field. The feature of a “defected” ground-plane, formed by appropriately cutting an L-shaped slot out of one of the CPW ground-planes, leads to the third resonance that covers the WiMAX band at 3.30–3.80 GHz. Air bridges at the intersection between the antenna and the CPW feedline ensure a balanced current. A fabricated prototype has compact dimensions of 20.0 mm 23.5 mm 1.59 mm, and exhibits good agreement between the measured and simulated parameters and radiation patterns. The measured radiation efficiencies are 67.4% at 2.45 GHz, 86.3% at 3.50 GHz and 85.3% at 5.50 GHz. Index Terms—Defected ground plane, folded monopole antenna, metamaterials, multiband antenna.

I. INTRODUCTION

T

WO commonly used protocols for Wireless Local Area Networks (WLANs) based on access points to relay data, are WiFi and WiMAX, which promise higher data rates and increased reliability. A challenge in designing such multiple wireless communication protocol systems is to design compact, low cost, multiband and broadband antennas. The planar monopole is attractive for WLAN antenna design because it has a low profile, it can be etched on a single substrate and can provide the feature of broadband or multiband operation. The traditional approach is to use multibranched strips in order to achieve multiband operation [1], which generally leads to a large volume or requires a large ground-plane. Alternatively, the concept of the frequency-reconfigurable multiband antenna [2] has been proposed to develop multiband monopole antennas for WiFi and WiMAX applications [3]. Such reconfigurable antennas are reported to have the advantages of being able to switch to a desired service and to achieve good out-of Manuscript received June 18, 2009; revised September 25, 2009; accepted October 27, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. The work of J. Zhu was supported by an IEEE Antenna and Propagation Society Graduate Fellowship. The authors are with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (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.2010.2041317

Fig. 1. Tri-band monopole antenna with single-cell MTM loading and a “de,W :, fected” ground-plane. All dimensions are in mm: L L ,W : ,W : ,W : ,h ,W , W : ,L : ,L : ,L ,L ,W , : and slot width g . (a) Top view. (b) 3D schematic. h

= 11 = 23 5 = 90 = 25 = 55 = 55 = 1 59 =1

= 12 5 =9

=8 =1 =3

= 58 =3 =7

band noise rejection performance. However, this is traded off with an increased design complexity and an increased fabrication cost associated with switches and bias circuits. Transmission-line metamaterials (TL-MTM) [4]–[6] provide a conceptual route for implementing small resonant antennas [7]–[17]. TL-MTM structures operating at resonance were first proposed in order to implement small printed antennas in [8] and [9]. Furthermore, a dual-band MTM-inspired small antenna for WiFi applications was shown in [18] and multiband MTM resonant antennas were shown to exhibit several left-handed modes in [19]. However, typically TL-MTM antennas suffer from narrow bandwidths. Recently, [12] addressed the bandwidth problem by proposing a two-arm TL-MTM antenna resonating at two closely spaced frequencies. Another method to enhance the bandwidth consists of merging two resonances together in a TL-MTM printed monopole antenna [20]. In this work, a compact tri-band monopole antenna is proposed using reactive loading, that was inspired by previous TL-MTM work, and a “defected” ground-plane [21], in order to meet the specifications of the WiFi bands (lower WiFi band of 2.40 GHz – 2.48 GHz and upper WiFi band of 5.15 GHz – 5.80 GHz) and the WiMAX (3.30 GHz – 3.80 GHz) band while maintaining a small form factor. Herein, we thoroughly explain the operation of the proposed tri-band antenna and we fully characterize its performance both numerically and experimentally. As shown in Fig. 1, the proposed co-planar waveguide (CPW)-fed monopole antenna is loaded in a left-handed manner, inspired by the NRI-TL metamaterial unit cell [11].

0018-926X/$26.00 © 2010 IEEE

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Fig. 2. (a) Case 1: unloaded monopole antenna, (b) Case 2: dual-band monopole antenna with single-cell MTM loading and (c) Final: tri-band monopole antenna with single-cell MTM loading and a “defected” ground. The dimensions of these antennas are given in the caption of Fig. 1. (a) Case 1. (b) Case 2. (c) Final.

This loading consists of a single MTM cell which allows the monopole antenna to operate in two modes [20], covering both the WiFi bands. The first is a folded monopole mode, where the monopole together with the single-cell MTM loading forms a folded monopole around the frequency of 5.5 GHz [11]; The second mode is a dipole mode where the single-cell MTM loading forces horizontal currents to flow along the top edges of the ground-plane, thus rendering the entire ground a radiator at around 2.5 GHz. It will be shown in Section II, that the currents corresponding to the two modes are orthogonal to each other, as shown in Fig. 5. The third resonance covering the WiMAX band from 3.3 GHz to 3.8 GHz, is achieved by “defecting” the ground-plane by cutting an -shaped slot from one side of the CPW ground. Air bridges are added at the antenna terminals to ensure that only balanced currents flow on the CPW feedline. The resulting antenna is compact (including the ground-plane), completely uniplanar, low profile and “via-free”. Therefore, the proposed antenna is easy to fabricate using simple photolithography. A prototype antenna has been fabricated and tested. The measurements show good impedance matching at the WiFi and WiMAX bands, orthogonal pattern diversity in each of the WiFi bands and a reasonable radiation efficiency, all of which make it well suited for wireless LAN applications. II. ANTENNA DESIGN The antenna was designed on a low-cost FR4 substrate with mm, and . A rectheight angular patch was chosen as the monopole radiation element. The length of the patch was adjusted according to the general design guideline that the lowest resonance is determined when , is approximately . Therethe length of the monopole, monopole results in the lowest resfore, an onance occurring at 6.0 GHz, as can be seen in Fig. 4. This refers to the initial design where the metamaterial-inspired reactive loading and the “defected” ground-plane are not employed, which is shown as Case 1 in Fig. 2. In order to compare the performances with the proposed tri-band antenna shown in Fig. 1, the size of the monopole element and the width of the ground for Case 1 were kept the same as the proposed design. However, the length of the ground in Case 1 was adjusted to 20 mm in order to achieve good impedance matching. The antenna was fed by a CPW transmission-line, which can be easily integrated

Fig. 3. The equivalent circuit when the proposed tri-band antenna operates at the monopole mode. The dimension d refers to the size of the NRI-TL unit cell and it corresponds to roughly the length of the monopole L with reference to Fig. 1(a).

Fig. 4. Simulated jS j for Case 1: unloaded monopole antenna, Case 2: dual-band monopole antenna with single-cell MTM loading and Final: tri-band monopole antenna with single-cell MTM loading and a “defected” ground, as shown in Fig. 2.

with other CPW-based microwave circuits printed on the same substrate. The CPW feed was connected to the coaxial cable through a standard 50 SMA connector. All the structures were simulated in the finite-element method (FEM) based full-wave solver, Ansoft HFSS. The connector and coaxial cable were included in all simulations to characterize their effects on the antenna performance. Since the operating frequency of the initial design (unloaded monopole) was above the range of interest for existing wireless LAN applications, different approaches using single-cell MTM loading and a “defected” ground were pursued to create the corresponding second and third resonances, at a lower frequency range in order to meet the wireless LAN specifications. A. Single-Cell Metamaterial Reactive Loading In order to maintain the antenna’s small form-factor while decreasing the operating frequency, the CPW monopole was loaded with a single asymmetric negative-refractive-index unit cell. transmission-line (NRI-TL) metamaterial-based The equivalent circuit for the antenna of Fig. 1 is shown in Fig. 3 , (at the folded monopole mode). The series capacitance, was formed between the monopole on the top of the substrate and the rectangular patch on the bottom of the substrate (see Fig. 1(b)). The MTM cell was asymmetrically loaded with two and . was formed by the inductive shunt inductances, was formed by strip at the base of the monopole, while the thin inductive strip that joins the rectangular patch beneath

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Fig. 6. Simulated jS j of the equivalent dipole antenna representing the triband monopole antenna with single-cell MTM loading and a “defected” groundplane when operating under the dipole mode, as shown in Fig. 5(b).

Fig. 5. HFSS-simulated surface current distribution on the conductors of the tri-band monopole antenna with single-cell MTM loading and a “defected” ground-plane at the resonant frequencies of (a) 5.80 GHz and (b) 2.44 GHz. (a) Folded monopole mode (5.80 GHz). (b) Dipole mode (2.44 GHz).

the monopole to the rectangular patch beneath the right-hand ground plane. In order to simplify the fabrication, a via-free approach to implement the asymmetric unit cell was used, which can be realized using low-cost thin-film technology. In order to achieve a shunt RF short to ground at high frequencies, the capacitor was used, which connects the shunt inductor to ground (see was formed between the right-hand Fig. 1(b)). The capacitor ground plane and the rectangular patch beneath it. Case 2 in Fig. 2 refers to the dual-band monopole antenna with single-cell MTM loading. In Case 2, the feature of the

“defected” ground is temporarily removed, while the other geometrical parameters remain the same as the tri-band antenna shown in Fig. 1(a). From Fig. 4, it can be observed that the monopole antenna with this single unit-cell MTM loading exhibits dual-resonance characteristics [20]. The geometrical parameters, namely the size of the square patches underneath the monopole patch and the length of the thin strip determine , and shown in Fig. 3, respectively. They were adjusted in order to obtain in-phase currents along the top monopole and along the thin bottom strip at the resonant frequency around 6.0 GHz. When operating in the folded monopole mode, the antenna acts as a two-arm folded monopole along the -axis, similar to the four-arm folded monopole of [11]. As discussed in [11], it is possible to eliminate the odd-mode currents along the CPW feedline by adjusting the printed lumped elements, thus enabling the -directed even-mode currents on the antenna to radiate. This can be seen from the HFSS-simulated current distribution of Fig. 5(a). In addition to the monopole resonance at around 5.0 GHz – 6.0 GHz, the metamaterial loading introduces a second resonance around 2.4 GHz – 2.5 GHz, which is desired for WiFi applications. At this frequency, the antenna no longer acts as a folded monopole along the -axis, but rather as a dipole oriented along the -axis, as shown in Fig. 5(b), where the current path was sketched from HFSS. Since the currents along the right edge of the left ground-plane section are flowing against the currents along the left edge of the right ground-plane section, only the in-phase currents along the top edges of both the ground-plane sections contribute to the radiation, which renders the groundplane as the main radiating element. The length of the current , which is related to the size of the ground-plane, path determines the resonant frequency. This is verified by the simulation for a central-fed dipole shown in Fig. 6. In Fig. 6, the equivalent dipole was simulated with the same substrate and the . The resulting resonant frequency same length of is 2.42 GHz, which agrees with the dipole-mode resonance for of the tri-band antenna shown in Fig. 4. Bethe simulated sides, since the dipole-mode currents for the proposed design are flowing in a meandered path, an even larger miniaturization factor is achieved, compared to the loaded monopole antenna reported in [20] where the dipole-mode currents only flow along the top edges of the ground-plane.

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weakly excited. The length of this current path is approximately , which can be verified using an equivalent dipole simulation, similar to that shown in Fig. 6. The dual-mode operation due to the metamaterial loading is, indeed, verified by the HFSS simulation for Case 2 shown in for Fig. 2(b). As seen in Fig. 4, the simulated magnitude of the monopole antenna with single-cell MTM reactive loading (the dashed line) has a desired higher resonance at around 5.48 GHz which is lower than 6.0 GHz for Case 1 and covers the ) starting from 4.95 GHz to up to 7 GHz, bandwidth ( while the lower resonance is centered at 2.46 GHz and has a bandwidth of 90 MHz. Fig. 7. HFSS-simulated current distribution at the intersection between the antenna terminal and the CPW feedline under the dipole mode of operation.

B. The “Defected” Ground-Plane Antenna Ideally, the dipole mode would not be excited for the regular monopole antenna due to the symmetric current distribution along the central line of the CPW feedline. In order to excite this mode, a single unit cell of MTM reactive loading was utilized. At low frequencies, the host transmission-line sections are very short and can be considered negligible. Assuming that the feed is placed at the base of the shunt inductor , the entire circuit is simply transformed into a series resonator formed between the and and the loading inductors and loading capacitors , as shown in Fig. 1(b). If the resonant frequency of the series resonator is designed to overlap with the dipole mode resonance of the antenna, it forms a short circuit and therefore forces the in-phase currents along the -axis flowing from one side of the ground to the other, through the path shown in Fig. 5(b). This enables the ground-plane to radiate in a dipolar fashion, which is orthogonal to the radiation at the higher frequencies. Bearing in mind the design considerations discussed above, the optimized loading patches have dimensions of 3.0 3.0 and 2.5 2.5 , respectively, and the two sections of thin strips have the same length of 5.5 mm and width of 0.25 mm. In addition, the position of the central line of the CPW inner conductor is tuned and placed 1.7 mm off from the central symmetry line of the antenna, in order to obtain a good impedance match to 50 , which in turn results in asymmetric ground-planes, as shown in Fig. 1(a). Since the currents need a path from one ground to the other, an air bridge made of copper wires was placed at the intersection between the antenna terminal and the extended CPW feedline, in order to provide a shorting path. Additional air bridges were placed at the CPW feed side and parallel to the first air bridge, which ensure balanced currents and preserve the CPW mode. Fig. 7 shows the transition of the currents from unbalanced to balanced at the terminal, where one can observe that the majority of the unbalanced currents pass through the first air bridge but after the third air bridge, they are effectively suppressed. It is worth mentioning that in practice, the CPW feed with air bridges can be replaced with a conductor backed CPW with a ground-via fence which offers the additional advantage of lower EMI radiation. dip around 3.6 Lastly, it can been seen that there is a small GHz for Case 2 shown in Fig. 4. This corresponds to another resonant current path along the ground plane, which is nevertheless

In order to create the third resonance to meet the requirement for the WiMAX application (3.30 GHz – 3.80 GHz) in the responses of the monopole antenna with the proposed reactive loading (Case 2 in Fig. 2(b)), a slot was cut out of the antenna ground-plane, thus forming a defected ground-plane. Similar to [22], an -shaped slot was chosen to achieve a longer effective slot length, without having to increase the size of the ground-plane. However, unlike having the slot cut at the top edge of the ground [22], the proposed design has the slot cut at the bottom edge, to avoid the discontinuity of the current along the top edge of the ground-plane, which contributes to the while dipole-mode radiation. The width of the slot is and , were the vertical and horizontal length of the slot, adjusted in order to achieve a good impedance match throughout the WiMAX band. This leads to the final design topology as shown in Fig. 2(c). It can be seen from Fig. 4 (solid line) that inserting the -shaped slot provides the third resonance around 3.5 GHz for the WiMAX band, while the dual-mode operation for the WiFi bands at around 2.5 GHz and 6.0 GHz is preserved. The resonance due to the slot can be explained by observing the surface current distribution on the conductors of the antenna, as shown in Fig. 8. As can be seen, there is a strong concentration of the currents along the -shaped slot on the left ground-plane. The slot forces the current to wrap around it and thus creates an alternate path for the current on the left at its resoground-plane, whose length is approximately nance. It is also noted from Fig. 8 that the -shaped slot does not significantly affect the balanced CPW mode, since it is placed far enough away from the CPW. Even if there were a minimal amount of unbalanced current, it would be eliminated by the air bridges applied at the intersection between the antenna and the extended CPW feedline, as shown in Fig. 7. for a parametric Fig. 9 shows the simulated magnitude of . It can be observed study of the length of the horizontal slot that the horizontal cut has a large influence on exciting the slot , which refers to the case that there is mode. When only a vertical slot cut from the bottom, the slot mode is barely excited, compared with the simulated parameter characteristics in the case without a “defected” ground (Case 2) in Fig. 4. Moreover, it can be seen that the vertical slot cut from the bottom doesn’t affect the folded monopole and dipole modes. As is gradually increased, a better impedance match is achieved over

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Fig. 10. The fabricated prototype of the tri-band monopole antenna with singlecell MTM loading and a “defected” ground-plane. (a) Front side. (b) Back side.

Fig. 8. HFSS-simulated surface current distribution on the conductors of the tri-band monopole antenna with single-cell MTM loading and a “defected” ground-plane at the resonant frequency of 3.76 GHz.

Fig. 11. Measured and HFSS simulated jS j for the proposed tri-band monopole antenna with single-cell MTM loading and a “defected” ground.

Fig. 9. Simulated jS j of the tri-band monopole antenna with single-cell MTM loading and a “defected” ground-plane for the different lengths of the horizontal slot L .

the WiMAX band, while the performances of the monopole and dipole modes are both preserved. The final design of the tri-band monopole antenna with single-cell MTM reactive loading and a “defected” ground is shown in Fig. 1. The geometrical parameters were determined based on the previous discussion and were fine-tuned in order to meet the WiFi and WiMAX bands requirements. As shown in Fig. 1, the full size of the tri-band monopole antenna (including the ground-plane with the size ) is 20.0 mm 23.5 of , with respect to mm 1.59 mm, or the lowest resonant frequency of 2.45 GHz. The compact size and the tri-band performance of the antenna make it a good candidate for emerging WLAN applications. III. SIMULATION AND EXPERIMENTAL RESULTS The tri-band monopole antenna was fabricated and tested. The fabricated prototype is shown in Fig. 10, and the measured from HFSS are shown in versus the simulated magnitude of bandwidth of Fig. 11. The antenna exhibits a simulated 80 MHz for the lower WiFi band from 2.40 GHz to 2.48 GHz

and a bandwidth from 5.13 GHz to beyond 7 GHz for the higher bandwidth of 590 MHz WiFi band. It also exhibits a for the WiMAX band from 3.30 GHz to 3.89 GHz. The performances beyond 7 GHz are out of the scope of our interest for bandWiFi and WiMAX applications. The measured width is 90 MHz from 2.42 GHz to 2.51 GHz for the lower WiFi band, and from 5.20 GHz to beyond 7 GHz for the second WiFi band. The measured bandwidth for the WiMAX band is 620 MHz from 3.35 GHz to 3.97 GHz. The simulated results and the measured results show good agreement. The simulated and measured radiation patterns for the proposed tri-band monopole antenna with single-cell MTM reactive loading and a “defected” ground-plane are plotted in Figs. 12–14 for the three principle planes at the frequencies of 5.50 GHz, 2.45 GHz and 3.50 GHz, respectively, where good agreement between the simulations and measurements can be observed. Fig. 12 shows the radiation patterns at 5.50 GHz for the -plane ( -plane and -plane) and the -plane ( -plane). The fact that the antenna exhibits radiation patterns with a horizontal -directed linear E-field polarization, verifies that the antenna operates in a folded monopole mode around 5.50 GHz, due to the -directed in-phase currents along the monopole and the thin vertical inductive strip, as shown in Fig. 5(a). The -directed currents along the thin horizontal inductive strip have a contribution to the cross polarization in the -plane. It can also be seen that at this frequency, the slot on the left ground has a minimum contribution to the radiation since the currents are dominated by the -directed

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Fig. 12. Measured and simulated radiation patterns for the tri-band monopole antenna with single-cell MTM loading and a “defected” ground-plane at 5.50 GHz. Solid line: measured co-polarization, dashed black line: simulated co-polarization, solid blue line: measured cross-polarization, and dash-dot black line: simulated cross-polarization. (a) xz -plane. (b) yz -plane. (c) xy -plane.

Fig. 13. Measured and simulated radiation patterns for the tri-band monopole antenna with single-cell MTM loading and a “defected” ground-plane at 2.45 GHz. Solid line: measured co-polarization, dashed black line: simulated co-polarization, solid blue line: measured cross-polarization, and dash-dot black line: simulated cross-polarization. (a) xz -plane. (b) yz -plane. (c) xy -plane.

Fig. 14. Measured and simulated radiation patterns for the tri-band monopole antenna with single-cell MTM loading and a “defected” ground-plane at 3.50 GHz. Solid line: measured co-polarization, dashed black line: simulated co-polarization, solid blue line: measured cross-polarization, and dash-dot black line: simulated cross-polarization. (a) xz -plane. (b) yz -plane. (c) xy -plane.

ones along the monopole. At this frequency, the simulated radiation efficiency is 85.9%, which is in good agreement with the measured efficiency of 85.3%, using the Wheeler Cap method [23]. The Wheeler cap measurements were conducted according to the method described in [23], where the measured

data in free-space and within the Wheeler cap were rotated on the Smith chart in order to obtain purely real values for the input resistance at resonance. The sphere used in the measurements is shown in Fig. 8 of [11], together with the pertinent details of its size. For this size Wheeler cap, its resonant frequency was

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TABLE I SIMULATED AND MEASURED GAIN AND RADIATION EFFICIENCY FOR THE TRI-BAND MONOPOLE ANTENNA WITH SINGLE-CELL MTM LOADING AND A “DEFECTED” GROUND-PLANE

calculated to be 0.69 GHz, which is well below the operating range of the tri-band antenna. Therefore, during the efficiency measurements at the three distinct resonant frequencies, special attention was paid in order to avoid any of the cavity resonances by slightly re-adjusting the position of the antenna within the Wheeler cap. Since the antenna was completely enclosed by the Wheeler-cap sphere during the measurements, this eliminated any potential radiation from the feed cable. At 2.45 GHz, the -directed currents contribute to the radiation, as shown in Fig. 5(b). This is consistent with the radiation patterns measured at the same frequency shown in Fig. 13. The radiation patterns in the -plane and the -plane, which correspond to the two -planes of the ground-plane radiating mode, indicate that the structure radiates in a dipolar fashion at this frequency. Similar to [20], there is, however, a partial filling of the null around 90 in the -plane, that can be attributed to constructive interference from the -directed currents along the CPW and the vertical thin inductive strip, and also the -directed current around the slot on the left ground-plane. This additional current also manifests itself in the cross-polarization direction, as can be seen in data of the -plane in the Fig. 13(b). In the -plane, which corresponds to the -plane of the radiating ground-plane, the radiation pattern is as expected omnidirectional. Therefore, at 2.45 GHz, the antenna exhibits a -directed linear -field polarized radiation pattern. It is orthogonal to the one observed at 5.50 GHz, which verifies the in-phase -directed currents across the ground. The simulated and measured efficiencies are 69.8% and 67.4%, respectively, which show good agreement. Fig. 14 shows the radiation patterns at 3.50 GHz. Since the -shaped slot which is cut out of the left ground-plane results in meandered currents along both the -direction and the -direction, which have independent contributions to the radiation, it is observed from Fig. 14 that the antenna exhibits two linear electric fields that are orthogonally polarized in both the and directions. Additionally, the axial ratio attains values close to, but greater than, one around the broadside direction, indicating circular polarization behavior. The measured efficiency at this frequency is 86.3%, compared to a simulated efficiency of 88.2% at the same frequency. The measured and simulated gain and radiation efficiency values, at the frequencies of 2.45, 3.50, and 5.50 GHz, are summarized in Table I. IV. CONCLUSION A tri-band and compact monopole antenna is proposed, that can be used for WiFi and WiMAX applications. The antenna consists of a regular CPW-fed printed monopole antenna with the embedded features of metamaterial-based single-cell reactive loading and a “defected” ground-plane, which introduce an-

other two resonances at the lower frequencies, in addition to the monopole resonance. The theoretical performance is verified by full-wave simulations and experimental data. The fabricated provides a prototype with a size of 90 MHz ( ) bandwidth from 2.42 GHz to 2.51 GHz for the IEEE 802.11b/g/n standard (lower WiFi band) and a broad band from 5.20 GHz to beyond 7 GHz for the IEEE 802.11a/n standard (upper WiFi band), and also a bandwidth of 620 MHz from 3.35 GHz to 3.97 GHz for the WiMAX band. The antenna exhibits dipole-like and monopole-like radiation patterns within the lower and upper WiFi bands, respectively, which are orthogonal to each other. The radiation patterns at the WiMAX band exhibit two orthogonal linear E-field polarizations as expected. Reasonable radiation efficiencies, in the range of 70% – 90%, are obtained for all three bands. Fed by the CPW transmission-line, the proposed antenna can be easily integrated with CPW-based microwave circuits. These attributes make the proposed antenna well suited for emerging wireless applications. ACKNOWLEDGMENT The authors would like to thank T.V. C.T. Chan at the University of Toronto for assisting in the measurement of the prototype antenna. Financial support from Intel corporation is gratefully acknowledged. REFERENCES [1] Y. Ge, K. Esselle, and T. Bird, “Compact triple-arm multiband monopole antenna,” in Proc. IEEE Int. Workshop on: Antenna Technology Small Antennas and Novel Metamaterials, Mar. 2006, pp. 172–175. [2] S. Yang, C. Zhang, H. K. Pan, A. E. Fathy, and V. K. Nair, “Frequencyreconfigurable antennas for multiradio wireless platforms,” IEEE Microw. Mag., vol. 10, no. 1, pp. 66–83, Feb. 2009. [3] S. Yang, A. E. Fathy, S. El-Ghazaly, H. K. Pan, and V. K. Nair, “A novel hybrid reconfigurable multi-band antenna for universal wireless receivers,” presented at the Electromagnetic Theory Symp., Jul. 2007. [4] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [5] G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications. Hoboken/Piscataway, NJ: Wiley/IEEE Press, 2005. [6] C. Caloz and T. Itoh, “Novel microwave devices and structures based on the transmission-line approach of meta-materials,” in Proc. IEEE MTT-S Int. Microwave Symp., Jun. 2003, pp. 195–198. [7] G. V. Eleftheriades, “Enabling RF/microwave devices using negativerefractive-index transmission-line (NRI-TL) metamaterials,” IEEE Antennas Propag. Mag., vol. 49, no. 2, pp. 34–51, Apr. 2007. [8] G. V. Eleftheriades, A. Grbic, and M. Antoniades, “Negative-refractive-index transmission-line metamaterials and enabling electromagnetic applications,” in IEEE Antennas and Propagation Society Int. Symp. Digest, Jun. 2004, pp. 1399–1402. [9] A. Sanada, M. Kimura, H. Kubo, C. Caloz, and T. Itoh, “A planar zeroth order resonator antenna using a left-handed transmission line,” in Proc. 34th Eur. Microwave Conf. (EuMC), Amsterdam, The Netherlands, Oct. 2004, pp. 1341–1344.

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[10] F. Qureshi, M. A. Antoniades, and G. V. Eleftheriades, “A compact and low-profile metamaterial ring antenna with vertical polarization,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 333–336, 2005. [11] M. A. Antoniades and G. V. Eleftheriades, “A folded-monopole model for electrically small NRI-TL metamaterial antennas,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 425–428, 2008. [12] J. Zhu and G. V. Eleftheriades, “A compact transmission-line metamaterial antenna with extended bandwidth,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 295–298, 2009. [13] R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 2113–2130, Jul. 2006. [14] A. Erentok and R. W. Ziolkowski, “Metamaterial-inspired efficient electrically small antennas,” IEEE Trans. Antennas Propag., vol. 56, pp. 691–707, Mar. 2008. [15] R. W. Ziolkowski, “An efficient, electrically small antenna designed for VHF and UHF applications,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 217–220, 2008. [16] A. Lai, K. Leong, and T. Itoh, “Infinite wavelength resonant antennas with monopolar radiation pattern based on periodic structures,” IEEE Trans. Antennas Propag., vol. 55, pp. 868–876, Mar. 2007. [17] J.-G. Lee and J.-H. Lee, “Zeroth order resonance loop antenna,” IEEE Trans. Antennas Propag., vol. 55, pp. 994–997, Mar. 2007. [18] J. Zhu and G. V. Eleftheriades, “Dual-band metamaterial-inspired small monopole antenna for WiFi applications,” Electron. Lett., vol. 45, no. 22, pp. 1104–1106, Oct. 2009. [19] C.-J. Lee, K. Leong, and T. Itoh, “Composite right/left-handed transmission line based compact resonant antennas for RF module integration,” IEEE Trans. Antennas Propag., vol. 54, pp. 2283–2291, Aug. 2006. [20] M. A. Antoniades and G. V. Eleftheriades, “A broadband dual-mode monopole antenna using NRI-TL metamaterial loading,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 258–261, 2009. [21] J. Zhu, M. A. Antoniades, and G. V. Eleftheriades, “A tri-band compact metamaterial-loaded monopole antenna for WiFi and WiMAX applications,” presented at the IEEE Antennas and Propagation Society Int. Symp., Jun. 2009. [22] M. A. Antoniades and G. V. Eleftheriades, “A compact multi-band monopole antenna with a defected ground plane,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 652–655, 2008. [23] W. E. McKinzie, III, “A modified wheeler cap method for measuring antenna efficiency,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 1997, pp. 542–545. Jiang Zhu (S’06) received the B.Eng. degree in electrical engineering from Zhejiang University, Hangzhou, China, in 2003 and the M.A.Sc. degree in electrical engineering from McMaster University, Hamilton, ON, Canada, in 2006. He is currently working toward the Ph.D. degree at the University of Toronto, Toronto, ON, Canada. In August 2003, he joined the Positioning and Wireless Technology Center, Nanyang Technological University, Singapore. From 2004 to 2006, he was a Research Assistant with the Computational Electromagnetics Research Laboratory and the Simulation Optimization Systems Research Laboratory, Department of Electrical and Computer Engineering, McMaster University. He joined the Electromagnetics Group, University of Toronto, in September 2006, where he is currently a Research Assistant and Teaching Assistant. His research interests are electromagnetic metamaterials, small antennas, microwave computer-aided design and microwave imaging. Mr. Zhu is the recipient of IEEE Antenna and Propagation Society Doctoral Research Awards for 2008–2009 and the Chinese Government Award for Outstanding Self-Financed Student Abroad 2008.

Marco A. Antoniades (S’99–M’09) received the B.A.Sc. degree in electrical engineering from the University of Waterloo, ON, Canada, in 2001, and the M.A.Sc. and Ph.D. degrees in electrical engineering from the University of Toronto, ON, Canada, in 2003 and 2009, respectively. From 1997 to 2001, he worked for various telecom/engineering companies, including Spacebridge, Cisco, Honeywell, Ericsson, The Cyprus Telecommunications Authority and Westinghouse, as part of the work-experience program at the University of Waterloo. From 2001 to 2009, he was a teaching assistant at the University of Toronto, where he contributed to the design and teaching of undergraduate courses relating to electromagnetics. During the same period, he was a research assistant at the University of Toronto, where he was involved in the development of microwave devices and antennas based on negative-refractive-index transmission-line (NRI-TL) metamaterials. Currently, he is a Postdoctoral Fellow at the University of Toronto, pursuing his research interests in antenna design and miniaturization, RF/microwave circuits and devices, periodic electromagnetic structures and negative-refractive-index metamaterials. Dr. Antoniades was a recipient of the Edward S. Rogers Sr. Graduate Scholarship from the Department of Electrical and Computer Engineering at the University of Toronto for 2003/04 and 2004/05. He received the first prize in the Student Paper Competition at the 2006 IEEE Antennas and Propagation Society (AP-S) International Symposium in Albuquerque, NM. In 2009, he was the recipient of the Ontario Post-Doctoral Fellowship from the Ministry of Research and Innovation of Ontario.

George V. Eleftheriades (S’86–M’88–SM’02– F’06) received the Diploma in Electrical Engineering from the National Technical University of Athens, Greece in 1988 and the M.S.E.E. and Ph.D. degrees in electrical engineering from the University of Michigan, Ann Arbor, in 1989 and 1993, respectively. From 1994 to 1997, he was with the Swiss Federal Institute of Technology in Lausanne. Currently, he is a Professor in the Department of Electrical and Computer Engineering, University of Toronto, where he holds a combined Canada Research Chair/Velma M. Rogers Graham Chair in Engineering. His research interests include transmission-line and other electromagnetic metamaterials, small antennas and components for wireless communications, plasmonic and nanoscale optical components and structures, fundamental electromagnetic theory and electromagnetic design of high-speed interconnects. Prof. Eleftheriades received the Ontario Premier’s Research Excellence Award in 2001 and an E.W.R. Steacie Fellowship from the Natural Sciences and Engineering Research Council of Canada in 2004. He served as an IEEE AP-S Distinguished Lecturer during the period 2004–2009. Amongst his other scholarly achievements he is the recipient of the 2008 IEEE Kiyo Tomiyasu Technical Field Award “for pioneering contributions to the science and technological applications of negative-refraction electromagnetic materials.” He is an IEEE Fellow and has been elected a Fellow of the Royal Society of Canada in 2009. He serves as an elected member of the IEEE AP-S AdCom and as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He is a member of the Technical Coordination Committee MTT-15 (Microwave Field Theory). He is the general chair of the IEEE AP-S/URSI 2010 Int. Symposium to be held in Toronto Canada in July 2010.

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Investigation Into the Polarization of AsymmetricalFeed Triangular Microstrip Antennas and its Application to Reconfigurable Antennas Youngje Sung

Abstract—A single-feed equilateral-triangular microstrip antenna has been designed with truncated corners for diverse radiation polarizations. The proposed antenna is excited by a microstrip line placed parallel to the oblique side of an equilateral-triangular patch. The antenna exhibits linear polarization (LP), left-hand circular polarization, or right-hand circular polarization (RHCP) depending on whether the three identically sized corners of the triangular patch are cut. A novel reconfigurable microstrip antenna with selectable polarization is proposed based on this first antenna. This second antenna has a simple structure consisting of a corner-truncated triangular patch, a small triangular conductor, an asymmetrical microstrip feed line, and a biased PIN diode between the patch and the triangular conductor. It produces LP or RHCP depending on the bias voltage. The measured results show that this antenna has a low cross-polarization level when operated in the linear state and a good axial ratio in the circular state. Index Terms—Circular polarization, equilateral-triangular patch, reconfigurable antenna.

I. INTRODUCTION ICROSTRIP antennas are widely used due to their low profile, light weight, and easy fabrication [1]. While they are usually designed for linear polarization, circularly polarized antennas have attracted much attention in recent years. In some applications such as satellite communication systems, a microstrip antenna with circular polarization (CP) is more suitable because of its insensitivity to transmitter and receiver orientation [2]. Such antennas are also used in radar to reduce the clutter from spherically symmetric objects like raindrops or hail [3]. A microstrip antenna with CP operation is possible using two feed schemes. The dual-orthogonal feed method with an external power divider network is a double feed that yields a higher-order mode generation and undesirable spurious radiation from the feed structure, which adversely affect its axial ratio (AR) performance [4]. For the design of a single-feed circularly polarized microstrip antenna, the two required orthogonal modes with equal amplitude and 90 phase difference

M

Manuscript received July 29, 2008; revised February 01, 2009; accepted October 09, 2009. Date of publication November 10, 2009; date of current version April 07, 2010. The author is with the Department of Electronic Engineering, Kyonggi University, Suwon 443-760, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2036277

can be excited by truncated patch corners [5] and a protruding tuning stub [6]. The technique of truncating the corners to obtain single-feed circular polarization operation is well known and has been widely used in practical designs [5], [7], [8]. As described in [7], the CP operation was achieved simply by cutting one corner of an equilateral-triangular patch. In [8], the CP design was obtained by placing three triangular slots at appropriate locations below the equilateral-triangular radiating patch in the ground plane. In that case, one side of the triangular slots was slightly longer than the others. The antenna presented in this paper has different polarization characteristics depending on whether some or all of the three corners of the triangular patch are cut. The simulated and measured results show that of the eight combinations implemented using the proposed method, two produce left-hand circularly polarized (LHCP) characteristics and one produces right-hand circularly polarized (RHCP) characteristics. To facilitate the analysis, the parameters for the feed structure of the antennas were kept unchanged, and unlike the design in [8], the three corner-cuts were the same size. Simulated and measured results for the physical prototype are presented and discussed here. This paper proposes a reconfigurable antenna with polarization diversity based on the discussion above. The DC bias circuit in a reconfigurable microstrip antenna is complicated, and the size of the antenna becomes larger as the number of PIN diodes increases. In addition, an increased number of diodes may lead to more loss in the circuit or an increase in current consumption, possibly undermining the antenna performance. In this work, only one PIN diode was used to reconfigure the radiating patch, and by controlling the state of the diode, the polarization sense of the antenna could be switched between linear polarization (LP) and CP. Little information on the polarization diversity antenna with one PIN diode is currently available in the literature. A prototype using a PIN diode to switch the polarization electrically was constructed and the measured results are shown in this paper. II. CONFIGURATION Fig. 1 shows the proposed single-feed equilateral-triangular microstrip antenna. The equilateral-triangular patch has a side length of and is printed on a substrate of thickness with relative permittivity . The antenna is fed by a microstrip line by gap coupling where the gap for the feed structure is denoted by . A quarter-wave transformer is used as the matching circuit. Based on the simulated results, the dimensions and of the feed structure can be adjusted for good impedance matching.

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Fig. 1. Geometry of a single-feed equilateral-triangular microstrip antenna.

A conventional microstrip antenna is usually designed for LP. The radiating element should be asymmetrical to obtain a microstrip antenna with CP operation. Structures that provide asymmetric radiating elements generally include corner-cuts [5] and stubs [6], and these structures are referred to as perturbations. By introducing proper asymmetry into the equilateral-triangular microstrip antenna, the symmetry of the structure can be perturbed, which makes it possible to excite two degenerate orthogonal modes for CP operation. In this work, three corners, each with a side length of , were cut to provide the perturbation required to produce CP. In the proposed antenna, most of the radio-frequency (RF) power at coupling point A on the left oblique side is coupled from the feed structure into the antenna’s radiating element; a small amount of RF power is coupled at coupling point B. Since the CP radiation of the equilateral-triangular patch design is achieved by introducing proper asymmetry into the radiating and —based on coupling points element, both axes A and B, respectively—should be considered in designing the has no structure if it is to have CP characteristics. As axis coupling point, CP characteristics are not produced even when the two sides along this axis are asymmetrical. Section III shows from the simulated and measured results (e.g., reflection coefficient, AR, and radiation pattern) that the equilateral-triangular patch antennas have diverse polarization characteristics that depend on the combination of the three corner-cuts. Section IV presents a reconfigurable antenna with polarization diversity based on the simulated and measured results in Section III. Compared to a conventional reconfigurable antenna with polarization diversity [3], [5], [9], [10], the proposed reconfigurable antenna has the advantage of having a simple structure because a single PIN diode is used to obtain polarization diversity. The number of diodes is an important factor in the design of reconfigurable antennas as this governs the current consumption and total antenna size. III. EQUILATERAL-TRIANGULAR MICROSTRIP ANTENNA WITH CORNER-CUT The side length of the triangular patch was set at 50 mm. A conventional equilateral-triangular microstrip antenna of this

Fig. 2. Simulated and measured reflection coefficients for (a) antenna 1 and (b) antenna 2.

size without any corner-cuts would have a resonant frequency of 2.22 GHz. The antenna proposed here was fabricated on com, , mercially available FR4 substrate with . The quarter-wave transformer line was 17.3 and tan mm long and 1 mm wide, which corresponds to 86 . The side length of the three triangular patches was set to 4.2 mm. When the side length of the triangular corner was adjusted to be approximately 0.09 times that of the triangular patch, the dominant mode of the triangular patch split into two near-degenerate orthogonal modes of equal amplitude and 90 phase difference, which resulted in CP radiation [7]. The geometric dimensions of the gap-coupled structure were , , and . All as follows: antennas discussed here were designed to have feed structures with the same dimensions. With the variables related to feed structure fixed, eight combinations were obtained depending on whether the three corners were cut. Each antenna exhibited LP, LHCP, or RHCP characteristics based on these combinations. As most of the RF power is coupled at coupling point A on the left oblique side, the proposed antenna should be asymmetto have CP characteristics. In the rical and centered on axis case in which the triangular patch is symmetrical with respect to (i.e., corners 1 and 2 being either simultaneously elimaxis inated or retained), the equilateral-triangular microstrip antenna

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Fig. 4. Simulated axial ratio for antennas 3 and 4.

Fig. 3. Simulated and measured reflection coefficients for (a) antenna 3 and (b) antenna 4.

shown in Fig. 1 exhibits LP characteristics regardless of the state of corner 3. Four of the eight possible combinations fall into this category. In the two cases in which all three corners are either all removed or all retained, the antenna exhibits LP because of the symmetrical radiating element; these cases are thus not discussed further here. In this paper, the antenna with corners 1 and 2 removed is referred to as antenna 1, and the antenna with corner 3 removed is antenna 2. Fig. 2 shows the simulated and measured reflection coefficients for antennas 1 and 2. Because corner 3 has very a small effect on the current path, the resonant frequency remains unchanged. Therefore, the 2.27 GHz resonant frequency of antenna 2 is very close to that of the conventional equilateral-triangular microstrip antenna without any corners cut. However, the resonant frequency of the proposed design depends on cutting corners 1 and 2, and the 2.38 GHz resonant frequency of antenna 1 is slightly higher. Some RF power—although much less than that seen at coupling point A—is coupled at coupling point B, and hence the should also be asymmetry generated with respect to axis considered in achieving CP characteristics. The triangular patch of the antenna shown in Fig. 1 should be simultaneously asymand to achieve good metrical with respect to axes is the more important of these two CP performance. Axis axes of asymmetry. Thus, the shape of the asymmetry centered

Fig. 5. Simulated and measured reflection coefficients for (a) antenna 5 and (b) antenna 6.

on axis determines the polarization characteristics of the antenna. In this regard, the antenna with corner 1 removed exhibits LHCP while the antenna with corner 2 removed shows RHCP. One needs to remove only corner 1 for the antenna to be asymmetrical with respect to both axes of symmetry and achieve LHCP characteristics; corners 2 and 3 should be removed for the antenna to have RHCP characteristics. This paper defines

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Fig. 6. Simulated axial ratio for antennas 5 and 6.

Fig. 8. Simulated and measured reflection coefficient for different states of a PIN diode.

Simulated results show that antenna 6 has a minimum AR of 2.66 at 2.31 GHz and the 3 dB AR CP bandwidth obtained for the RHCP mode is about 0.2%. Therefore, antenna 6 has weak RHCP characteristics compared to antenna 3. Table I shows the summarized measured results of six different antenna configurations. IV. RECONFIGURABLE ANTENNA WITH POLARIZATION DIVERSITY

Fig. 7. The proposed reconfigurable equilateral-triangular microstrip antenna with polarization diversity.

antennas 3 and 4 as those that are asymmetrical with respect to both axes; they have LHCP and RHCP characteristics, respectively. Fig. 3 shows the simulated and measured reflection coefficients for antennas 3 and 4; the measured 10 dB bandwidths are 4.8% and 4.9%, respectively. Fig. 4 shows the simulated AR for antennas 3 and 4; the simulated CP bandwidths, determined from the 3 dB AR, is 1.07% and 0.94%, respectively. but symAn antenna that is asymmetrical with respect to will have weak CP or LP characteristics. In metrical to other words, an asymmetrical triangular radiating patch with reis a necessary condition for CP operation and spect to axis an asymmetrical radiating triangular patch with respect to axis is a sufficient condition for CP operation. In this work, the antenna with corner 2 removed is referred to as antenna 5 and the antenna with both corners 1 and 3 removed is antenna 6. Fig. 5 shows the simulated and measured reflection coefficients for antennas 5 and 6; the measured impedance bandwidths (10 dB reflection coefficient) are approximately 3.7% and 3.2%, respectively. Fig. 6 shows the simulated AR for antennas 5 and 6. The reflection coefficient shown in Fig. 5 and the AR performance shown in Fig. 6 indicate that antenna 5 produces LP while antenna 6 produces RHCP. Antenna 6 shows RHCP characteristics because it is missing corner 1, just like antenna 3.

This section presents a reconfigurable microstrip antenna with polarization diversity. The design concept of this antenna is based on the truncated equilateral-triangular patch that can generate diverse radiation polarizations (LP, LHCP, and RHCP) without changing the feed structure, depending on the cuts to the three identically sized corners of the triangular patch. A prototype was implemented and tested with a single PIN diode (HSMP-3864) to allow the antenna polarization to be switched electrically as shown in Fig. 7. The dimensions of the prototype refer to the geometry given in Fig. 1. The antenna polarization can be switched between LP and RHCP by controlling the connection state between the radiating triangular patch and its truncated corner. This antenna operates at the same frequency for different polarizations and is therefore suitable for frequency reuse. As the PIN diode is 2.5 mm 1.4 mm, the gap between the triangular patch and the triangular conductor was set to 1.4 mm. Only one PIN diode was inserted into the gap between the patch and the triangular conductor to control the antenna geometry, and a simple DC bias circuit was used to divide the antenna into DC-isolated areas [10] by simply turning the diode on or off. The dotted line represents the slot for the DC bias circuit on the opposite side. The DC bias circuit consists of two 200 nH inductors as a RF choke, one 47 pF capacitor located on the feed structure, and two 100 pF capacitors located on the ground plane. Fig. 8 shows the simulated and measured reflection coefficient for the physical prototype based on the design dimensions shown in Fig. 7. Note that due to the relatively small parasitic conductor, the shift in the resonant frequency for different states of the PIN diode was less than 10 MHz (less than 0.04% with respect to the resonant frequency). Theoretical simulation results

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TABLE I ANTENNA PERFORMANCE SUMMARY OF SIX DIFFERENT ANTENNA CONFIGURATIONS

obtained using the IE3D software agree closely with the measured result. The conductor and dielectric losses, which were not considered in the simulation, contribute to the small deviation between the simulation and the measurements. The case of the proposed antenna with the diode turned off was studied first. The radiation and polarization characteristics for this antenna were very similar to those of antenna 4. The PIN diode represents a capacitance of 0.7 pF in the reverse-biased state. The antenna exhibits RHCP due to its geometrical symmetry. Because of the , little coupling occurs and the resogap size of nant frequency is affected only slightly. The antennas with CP impedance bandwidth of 5.07%. With forward have a bias applied, the PIN diode appears as an ohmic resistance of 1.5 . Therefore, the gap is bridged and electric current flows

through the diode. When the diode is on, the antenna exhibits LP. Switching the PIN diode requires a bias current of 15 mA and a voltage of 0.74 V. The radiation and polarization characteristics for this antenna are very similar to those of antenna 5. impedance bandwidth of 3.44%. The antenna exhibits a Fig. 9 shows the simulated E-field distribution on the patch surface for antenna 5. The simulated frequency is 2.21 GHz. It is the resonant frequency of the antenna 5. Fig. 10 shows the simulated E-field distribution on the patch surface for antenna 4. The simulated frequency is 2.24 GHz. It is the frequency with minimal AR in the impedance bandwidth. Both plots are normalized at the minimum and maximum values. For both cases, the radiation patterns were measured at the resonant frequency. An open-ended waveguide antenna was

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Fig. 9. Simulated E-field vector distribution for antenna 5 (LP). (a) Phase 0 , (b) Phase = 45 , (c) Phase = 90 .

=

used as a test antenna and the electrical size of the ground 1.6 . Fig. 11 shows plane was kept to approximately 1.6 the measured radiation pattern for the proposed antenna with RHCP characteristics. The measured gain of the proposed antenna with LP characteristics was 1.8 dBi, lower than that of the RHCP antenna, which could have been due to the ohmic loss of the PIN diode. Measured results show that broadside radiation patterns with good CP characteristics were obtained at the resonant frequencies. Fig. 12 gives the measured AR for the proposed antenna with RHCP. The CP bandwidth, determined from the 3 dB AR, is 22 MHz, or about 0.98% with respect to the center frequency of 2.291 GHz, defined here to be the frequency with minimal AR in the impedance bandwidth. Typical far-field radiation patterns of the prototype

Fig. 10. Simulated E-field vector distribution for antenna 4 (RHCP). (a) Phase = 0 , (b) Phase = 45 , (c) Phase = 90 .

antenna operated in the LP mode are shown in Fig. 13. The measured results indicate that the proposed antenna with LP characteristics has almost the same quite stable co-polarized radiation pattern characteristics in both the - and -planes at the resonant frequency. The cross-polarization levels remained . Fig. 14 shows a photograph of the prototype below reconfigurable antenna. V. CONCLUSION The corner-cut method was used to produce diverse radiation characteristics of an equilateral-triangular microstrip antenna. The various polarization characteristics depend on whether the three corners of the triangular patch are cut. Prototypes of the

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Fig. 13. Measured radiation pattern for LP antenna (a) xz -plane and (b) yz -plane. Fig. 11. Measured radiation pattern for RHCP antenna. (a) xz -plane and (b) yz -plane.

Fig. 12. Measured axial ratio for RHCP antenna.

proposed design were successfully created. Based on this concept, a reconfigurable microstrip antenna with switchable polarization was also presented. Polarization diversity was achieved by controlling the state of one PIN diode. Measured results from prototype tests showed that the polarization of the antenna could be switched between LP and RHCP. The proposed antenna is suitable for applications in wireless and mobile satellite communications.

Fig. 14. Photograph of the prototype reconfigurable antenna. (a) front view and (b) bottom view.

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REFERENCES [1] K. C. Gupta and P. S. Hall, Analysis and Design of Integrated Circuit Antenna Modules. New York: Wiley, 2000. [2] Y. J. Sung and Y.-S. Kim, “Circular polarised microstrip patch antennas for broadband and dual-band operation,” Electron. Lett., vol. 40, no. 9, pp. 520–521, Apr. 2004. [3] M.K. Fries, M. Grani, and R. Vahldieck, “A reconfigurable slot antenna with switchable polarization,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 11, pp. 490–492, Nov. 2003. [4] J. R. James, P. S. Hall, and C. Wood, Microstrip Antenna Theory and Design. London, UK: Peter Peregrinus, 1981. [5] Y. J. Sung, T. U. Jang, and Y.-S. Kim, “Reconfigurable microstrip patch antenna for switchable polarization,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 11, pp. 534–536, Nov. 2004. [6] W.-S. Chen, C.-K. Wu, and K.-L. Wong, “Square-ring microstrip antenna with a cross strip for compact circular polarization operation,” IEEE Trans. Antennas Propag., vol. 47, pp. 1566–1568, Oct. 1999. [7] C.-L. Tang, J.-H. Lu, and K.-L. Wong, “Circularly polarized equilateral-triangular microstrip antenna with truncated tip,” Electron. Lett., vol. 34, no. 13, pp. 1277–1278, Jun. 1998. [8] J.-S. Kuo and G.-B. Hsieh, “Gain enhancement of a circularly polarized equilateral-triangular microstrip antenna with a slotted ground plane,” IEEE Trans. Antennas Propag., vol. 51, pp. 1652–1655, Jul. 2003.

[9] F. Yang and Y. Rahmat-Samii, “A reconfigurable patch antenna using switchable slots for circular polarization diversity,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 3, pp. 96–98, Mar. 2002. [10] Y. J. Sung, “A switchable microstrip patch antenna for dual frequency operation,” ETRI J., vol. 30, no. 4, pp. 603–605, Aug. 2008.

Youngje Sung was born in Incheon, Korea, on January 17, 1975. He received the B.S., M.S., and Ph.D. degrees from Korea University, Seoul, Korea, in 2001, 2002, and 2005, respectively. From 2005 to 2008, he was a Senior Engineer in the Antenna R&D Lab., Mobile Phone Division, Samsung Electronics, Korea. In 2008, he joined the Department of Electronic Engineering, Kyonggi University, Suwon, Korea, where he is currently an Assistant Professor. His research interests include reconfigurable antennas, cell-phone antennas, wideband slot antennas, EBG structures and their applications for microwave devices, compact circular polarized antennas, and compact dual-mode filters. Prof. Sung is a reviewer for the IEEE MICROWAVE WIRELESS COMPONENTS LETTERS, Progress in Electromagnetic Research (PIER), IET Electronics Letters, IET Microwaves, Antennas and Propagation, and ETRI Journal (Electronics and Telecommunications Research Institute, South Korea).

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Ultrawideband Dielectric Resonator Antenna With Broadside Patterns Mounted on a Vertical Ground Plane Edge Kenny Seungwoo Ryu, Student Member, IEEE, and Ahmed A. Kishk, Fellow, IEEE

Abstract—A novel portable dielectric resonator antenna (DRA) design with broadside radiation is presented for ultrawideband wireless applications and narrow pulse sensor for breast cancer detections. The antenna provides broadside radiation. By using a solid rectangular dielectric resonator mounted on a vertical ground plane edge, the total antenna volume is considered to be less than that with a perpendicular ground plane. A wide impedance matching bandwidth is achieved. In addition, a modified version of the rectangular dielectric resonator antenna, which has two dielectric pieces removed to have an A-shaped resonator, provides wider impedance matching bandwidth of around 93% with much better broadside radiation characteristics and lighter weight is also considered. Comparison between the two antennas is presented with parametric study. The modified antenna is built and tested and very good agreement between computed and measured results is obtained. Index Terms—Breast cancer sensor, dielectric resonator antenna (DRA), ultrawideband (UWB) antenna.

I. INTRODUCTION

B

ANDWIDTH enhancement techniques along with antenna miniaturization techniques are very important for antenna engineering design. For several applications, many antenna engineers have been tried to design compact size antennas with wide bandwidth performance. The DRA is an attractive option to achieve the above requirements. DRAs have been an active research area for the last two decades due to several striking characteristics such as high radiation efficiency, low dissipation loss, small size, light weight, and low profile, since the use of dielectric resonator as an antenna was originally proposed in 1983 [1]. Moreover, DRAs, which possess a high degree of design flexibility, have emerged as an ideal candidate for wideband, high efficiency, and cost-effective applications. Significant efforts for DRAs have been reported to achieve wide bandwidth enhancements in the past. The research of the wideband DRA with broadside radiation was first experimentally carried out in 1989 by Kishk et al. [2], who stacked two different DRAs on top of one another to obtain a dual-resonance operation with 25% impedance bandwidth. Since then,

Manuscript received April 01, 2009; revised October 02, 2009. Date of manuscript acceptance October 03, 2009; date of publication January 26, 2010; date of current version April 07, 2010. This work has was supported in part by the National Science Foundation under Grant No. ECS-524293. The authors are with the Department of Electrical Engineering, University of Mississippi, University, MI 38677 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.2010.2041160

different techniques were proposed to achieve bandwidth enhancement. Various geometries of DRAs such as conical, [3], elliptical [4], [5], tetrahedral [6], well [7], stair [8] and H-shaped [9] were proposed for bandwidth enhancement techniques for broadside radiation by using the advantage of DR structure flexibility. Also, the introduction of an air gap between the DRA and ground plane can further improve bandwidth [10]. A DRA of multiple layers can be used to enhance the bandwidth [11] as can be loaded dielectric resonators [12]. Since the coupling between the excitation mechanism and the DR significantly affects the resonant frequency and radiation Q-factor of a DRA, feeding techniques including T-strip-feed DRA [13], L-probe feed DRA [14], and vertical strip-fed [15], [16] have been proposed. By using the above bandwidth enhancement techniques, operating DRA bandwidth ranges from 25% to 67% have been reported for broadside radiation patterns. Wide bandwidth for monopole type radiation patterns have been reported [17], [18], which are much easier to achieve than the broadside type, and even wider bandwidth was previously achieved by loading a monopole with a suspended annular DR [19], [20]. Recently, several papers proposed DRAs with planar type vertical ground plane to obtain an omni-directional pattern [21]–[23]. Among them, however, there are no DRAs with broadside patterns and none of them achieved the bandwidth that is needed for the UWB. Here, we present a novel DRA design, in which the DR is mounted on a vertical ground plane edge for a broadside radiation pattern, which is different from the monopole type radiation pattern. Furthermore, we propose a new type of vertical ground plane to get a better radiation pattern then the planar type vertical ground plane. Generally using the ground plane edge resulted in a conceptual 75% volume reduction as compared to a perpendicular ground plane and in a lighter antenna weight. The analyses in this paper are performed using the commercial code HFSS [24]. II. CONFIGURATION OF ANTENNA Fig. 1 shows the evolution of the proposed DRA. Fig. 1(a) shows the rectangular DRA excited by a strip feed. DRAs excited by a strip-fed were reported in [16]. Some deformations in the E-plane patterns were observed at the upper frequency range of the antenna band because of the asymmetric structure and the higher order modes [16]. Fig. 1(b) shows the rectangular DRA mounted on a vertical ground plane edge. Mounting the DR in this way reduces the total volume of the antenna as compared to the structure in Fig. 1(a). The proposed structure provides much wider impedance matching bandwidth. However, this structure shows similar asymmetry with respect to the broadside direction

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Fig. 1. Evolution of the proposed DRA.

Fig. 3. E-plane cut radiation pattern for the rectangular DR with and without gap: (a) with no gap at 8 GHz, (b) with gap at 8 GHz, (c) with no gap at 9 GHz, (d) with gap at 9 GHz.

Fig. 2. Geometry of the rectangular DR on a vertical ground plane edge: (a) back, (b) isomitric, (c) front, and (d) side view.

in the E-plane patterns at the upper end of the frequency band as those observed from the structure in Fig. 1(a). The size of the DRA is 14 mm width, 18.3 mm length, and 5.08 mm thickness with a dielectric constant of 10.2, and it is supported by a 30 30 mm RT6002 substrate with a dielectric constant of 2.94 and a substrate thickness of 0.762 mm. The ground plane is partially printed on the substrate under the DRA. The size of ground plane is 11 30 mm on one side and a printed probe extends from the microstrip line of the same width that ends with the 50 line after certain length that is used as . The rest of the a matching transformer of length mm, antenna parameters are shown in Fig. 2 with mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, and mm. III. NUMERICAL INVESTIGATION The numerical investigation is important because it provides some understanding of the antenna characteristics to the antenna engineer since our proposed antenna has several contributions. First, we show how to improve the E-plane cut broadside radiation patterns. Even though the results of the Fig. 2 can be

found in the 2007 conference of IEEE Antennas and Propagation [25], it did not mention about the gap effect at all for improving E-plane cut patterns. In addition, we further improve radiation patterns using additional technique. Second, we investigate several parameters in terms of the bandwidth to understand our proposed DRA characteristics. Third, we compare the performance in terms of the H-plane cut pattern and cross polarized level between our proposed type vertical ground plane and the planar type vertical ground plane. It should be also indicated that the proposed type is mechanically strong and may not need any adhesive if planted in the dielectric substrate by using a little glue in the sides of the DRA to be housed in the substrate. The planar type has to be glued with adhesive over the substrate. Many wideband DR antennas with broadside patterns that are excited by electric current probes tend to have deformed broadside patterns, due to the existing higher-order modes, which deform the radiation patterns within the middle or the end of the frequency band. The E-plane radiation pattern peaks at the higher band show around a 40 tilted radiation pattern with a big dip from the antenna axis because of the asymmetric structure of the monopole electric probe excitation that also excites higher order modes. Fig. 3 shows the radiation patterns in the E-plane cut for the rectangular DR with and without gap at 8 and 9 GHz. Both of them obtain quite small cross polarized levels as compared to the planar type vertical ground plane (placing the DR just above the dielectric substrate), which will be presented later. As mentioned above, the E-plane radiation pattern of the rectangular DRA with no gap shows not only a 40 tilted radiation pattern peak from the antenna axis, but also a big dip at 30 at 8 GHz. The radiation patterns in the E-plane in the upper limit of the band have some deformation that makes them non-symmetric

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Fig. 4. Modified geometry. DR with two open tunnels (A shape DR) mounted on a vertical ground plane edge: (a) front and (b) isomitric view.

around the broadside direction to unacceptable degree. Therefore, we introduce a gap between the DR and the ground plane edge to strengthen the electric field at the opposite side of the feed probe and provide some balance between both sides. However, the strong dip still exists at around 50 degree mark due to other high order modes. We noticed that the gap has caused a rotation of the beam a few degrees in the desired direction, but still not enough to correct the problem because it only corrects for the monopole effect. We have observed the field distribution within the DR and found that there are some fields at two spots that could affect the radiation patterns. Therefore, we removed the dielectric materials from these spots aiming to shift the resonant frequency of these modes far away from our frequency band of interest and well structured higher order field distribution instead of the disordered higher order field distribution. In addition, the narrow portion of the center has strong field distribution at 8 GHz. It helps to improve the symmetry (balanced) E-plane cut. At 9 GHz, even though the narrow portion of the center still has strong field distribution, the position of the maximum field distribution is slightly moved to the right side due to moving spots. It affects slightly unbalanced E-plane cut. The modified geometry is shown in Fig. 4. The modified geometry has DR with two open tunnels mounted on a vertical ground plane edge. Using the same parameters stated with the rectangular DR with mm, mm, mm, mm, and mm, Fig. 5 illustrates the effect of the gap and the two opening in terms of the radiation patterns in the E-plane at 8 and 9 GHz. This figure can be compared with Fig. 3. The E-plane radiation pattern of the DR with two cuts provides corrections for the pattern deformation, since a strong electric field of the narrow portion of the upper and center part make a balanced electric field distribution of the DR as can be seen from Fig. 6. Also, it can be easily noticed that a small gap contributes to the high directivity of the E-plane radiation patterns for the A-shaped DRA. Fig. 7 compares reflection coefficients of the rectangular DRA and the modified DRA, which has two cuts, mounted on a vertical ground plane edge. While the rectangular DRA mounted on a vertical ground plane edge provides an 84% impedance matching bandwidth, the DRA with two cuts and mounted on a vertical ground plane edge provides much wider impedance matching bandwidth of around 93%. It can be noticed that the higher order modes that we eliminated or shifted

Fig. 5. E-plane cut radiation pattern for the A shape DRA with and without gap: (a) with no gap at 8 GHz, (b) with gap at 8 GHz, (c) with no gap at 9 GHz, (d) with gap at 9 GHz.

Fig. 6. Electric field distribution at 8 GHz and 9 GHz: (a) rectangular DRA at 8 GHz, (b) A-shaped DRA at 8 GHz, (c) rectangular DRA at 9 GHz, (d) A shape DRA at 9 GHz.

by the dielectric cut were acting as a resonator providing a band notch at the upper band (8.5 GHz to 9.5 GHz). Figs. 8 and 9 show the ground effect of the proposed DRA mounted on a vertical ground plane edge. It is well known that printed ultrawideband (UWB) planar monopole antennas are essentially an unbalanced design, where electric currents are dis-

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Fig. 7. Reflection coefficients for rectangular DRA and modified DRA.

Fig. 9. Reflection coefficient for various gap.

Fig. 10. Reflection coefficient for various holder sizes. Fig. 8. Reflection coefficient for various ground plane height.

tributed on both the ground plane and radiator, so that the performance of the antenna is significantly affected by the size of the ground plane and the gap between ground plane and radiator. Fig. 8 shows the effect of the ground plane height on the reflection coefficients for the proposed DRA. Our proposed UWB DRA shows minimal variations in the reflection coefficient regardless of the size of the ground plane. The size of the ground plane Lg slightly affects the lower frequency range. Fig. 9 shows the effect of the gap between the ground plane and DR while maintaining the total size of the DRA. The gap between the DR and the ground plane is not much sensitive in achieving a good impedance match if compared with the printed UWB antenna. It was shown that DRAs made of a combination of different materials could enhance the antenna bandwidth [2]. As the substrate holder, with dielectric constant of 2.94, for the DR is considered part of the proposed DRA, we study the effect of the holder size. From Fig. 10, no effect is noticed for holder size . Another important point to note is that efficient coupling between the DR and the feed can be achieved by optimizing the width and length of the matching transformer. Figs. 11 and 12 show the effect of the matching transformer for the proposed DRA mounted on vertical ground plane edge. The impedance matching is sensitive to the length and width of the matching transformer, especially at higher frequencies.

Fig. 11. Reflection coefficient for various width of delay line (F

= 13 mm).

We have noticed from above that the effect of the ground plane and holder size can be ignored, and the gap between the ground plane and the DR is not sensitive to achieve impedance matching. However, optimized values of the length and width of the matching transformer are quite important to achieve good impedance matching compared with other parameters. Several DRAs use planar type vertical ground plane [21]–[23]. None of them have achieved the UWB requirements or broadside radiation patterns. The DRA structures, which have vertical ground planes, are the planar mounting

RYU AND KISHK: UWB DRA WITH BROADSIDE PATTERNS MOUNTED ON A VERTICAL GROUND PLANE EDGE

Fig. 12. Reflection coefficient for various length of delay line (F = 1 mm). 18:4 mm, F

+F

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=

Fig. 14. Reflection coefficients for DRA with the planar type vertical ground plane and proposed type vertical ground plane.

Fig. 13. Planar type vertical ground plane: (a) isomitric view (b) side view.

type (placing the DR just above the dielectric substrate), which require adhesive to glue the DR above the substrate. The side view and isomitric view of DRA with the planar type vertical ground plane and the proposed type vertical ground plane are shown in Fig. 13. Fig. 14 shows the comparison of reflection coefficients of DRA with the planar type vertical ground plane and proposed vertical ground plane. As it can be seen from Fig. 13, the DRA with planar type vertical ground plane is slightly thicker by the substrate thickness than the proposed one. The resonant frequency is shifted the frequency up as compared with the proposed one. It should also be mentioned that the results presented for the planar DRA type are our results that have not been achieved by others for this type as the broadside radiation characteristics have not been presented by others in the past. Fig. 15 illustrates the maximum cross polarized level of the E-plane with in the operating frequency range. As one can see from the figures, the cross polarized levels of the planar type vertical plane are higher by around 5 to 10 dB than the present proposed type mounted on a vertical ground plane edge. In addition, the H-plane radiation pattern of the planar type does not have a symmetric pattern due to the asymmetric structure of the antenna, while the H-plane radiation pattern of the present proposed type has a symmetrical pattern. IV. EXPERIMENTAL RESULT Fig. 16 shows the photo of the prototype DRA with two openings mounted on the vertical ground plane edge. The measurements are performed using the HP8510c Network Analyzer.

Fig. 15. Maximum cross-polarization levels of E plane within the operating frequency band.

Fig. 16. Photo of a DRA with two open tunnels. (a) A-shaped DR, (b) A shaped DRA housed in the substrate from the feeding side view, (c) side view of A-shape DRA.

Fig. 17 shows the simulated and measured matching frequency band of the proposed antenna for dB reflection coefficient from 3.53 GHz to 9.675 GHz with a bandwidth of 93% in simulation and the frequency band of 3.685 GHz to 9.96 GHz with a bandwidth of 92 % in measurement. It can also be noticed that the reflection coefficient for dB level provides a bandwidth of about 77%. The far field radiation patterns of the proposed DRA are also measured and verified with the simulated results. Fig. 18 plots the simulated and measured radiation patterns at three different

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Fig. 17. Measured and simulated impedance matching bandwidth.

frequencies (4 GHz, 6.5 GHz, and 9 GHz). The antenna radiates in a broadside direction. We can observe the good agreement between the computed and measured radiation patterns. Also we can see the good symmetry of the radiation patterns along the broadside direction. Fig. 19 shows the estimated and measured antenna efficiency with the gain of the antenna versus frequency for the proposed DRA. It should be mentioned that the measured antenna efficiency, which is performed in the reverberation chamber at the Chalmers University of Technology, has been smoothed through averaging and using small number of frequencies than what was experimentally provided. The antenna efficiency is computed by . The estimated anradiation efficiency multiplied by tenna efficiency is based on the HFSS computation from lossless and lossy antennas. The antenna radiation efficiency is computed as the ratio of gain to directivity of the antenna. This is supposed to be 100% for the lossless case. However, we found that it is higher because of numerical convergence problem that we could not achieve. Therefore, we consider the error in the lossless case and calibrate the error in the lossy case based on the losses case. The estimated antenna efficiency is in good agreement with the measured antenna efficiency within the frequency range that is available. The antenna achieved more than 95% antenna efficiency within most of the band. In addition, the gain of the antenna is constantly slightly increasing within the operating frequency band.

Fig. 18. Measured and simulated E plane (left) and H plane (right) radiation patterns at (a) 4 GHz (b) 6.5 GHz (c) 9 GHz; Solid lines for computed and dashed lines for measured. Dark color for co-polar and light color for cross-polar.

V. CONCLUSION A rectangular DRA mounted on a vertical ground plane edge for wireless portable applications and narrow pulse sensors was designed to provide broadside radiation patterns. The structure provided smaller antenna volume as compared with the DRA over a horizontal ground plane. The present design of an A-shaped DR overcame the problem of deformed E-plane radiation patterns. The proposed DRA has very high antenna efficiency with stable gain over the frequency band. The parametric study showed that the proposed DRA has excellent characteristic that make it a good candidate for sensor applications. This design has further extended the use of DRA as thin antennas suitable to be easily integrated with portable wireless devices.

Fig. 19. Estimated and measured antenna efficiency and the gain of the antenna versus frequency.

ACKNOWLEDGMENT The authors would like to thank Prof. P.-S. Kildal and Mr. C. Xiaoming for providing the efficient measurements performed by the reverberation chamber Antenna Group, Department

RYU AND KISHK: UWB DRA WITH BROADSIDE PATTERNS MOUNTED ON A VERTICAL GROUND PLANE EDGE

of Signals and Systems Chalmers University of Technology, Gothenburg, Sweden. REFERENCES [1] S. A. Long, M. W. McAllister, and L. C. Shen, “The resonant cylindrical dielectric cavity antenna,” IEEE Trans. Antennas Propag., vol. 31, pp. 406–412, 1983. [2] A. A. Kishk, B. Ahn, and D. Kajfez, “Broadband stacked dielectric resonator antennas,” Electron. Lett., vol. 25, no. 18, pp. 1232–1233, Aug. 1989. [3] A. A. Kishk, Y. Yin, and A. W. Glisson, “Conical dielectric resonator antennas for wideband applications,” IEEE Trans. Antennas Propag., vol. 50, pp. 469–474, Apr. 2002. [4] A. A. Kishk, “Elliptic dielectric resonator antenna for circular polarization with single feed,” Microw. Opt. Technol. Lett., vol. 37, no. 6, pp. 454–456, Jun. 2003. [5] P. V. Vijumon, S. K. Menon, M. N. Suma, B. Lehakumari, M. T. Sebastian, and P. Mohanan, “Broadband elliptical dielectric resonator antenna,” Microw. Opt. Technol. Lett., vol. 48, no. 1, pp. 65–67, Jan. 2006. [6] A. A. Kishk, “Wideband dielectric resonator antenna in a truncated tetrahedron form excited by a coaxial probe,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2907–2912, Oct. 2003. [7] T.-H. Chang and J.-F. Kiang, “Broadband dielectric resonator antenna with an offset well,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 564–567, 2007. [8] R. Chair, A. A. Kishk, and K.-F. Lee, “Wideband stair-shaped dielectric resonator antennas,” IET Microw., Antennas Propag., vol. 1, no. 2, pp. 299–305, Apr. 2007. [9] X. L. Liang and T. A. Denidni, “H-shaped dielectric resonator antenna for wideband applications,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 163–166, 2008. [10] G. P. Junker, A. A. Kishk, A. W. Glisson, and D. Kajfez, “Effect of an air gap on a cylindrical dielectric resonator antennas operating in the TM01 mode,” Electron. Lett., vol. 30, no. 2, pp. 97–98, 1994. [11] W. Huang and A. A. Kishk, “Apertures-coupled multi-layer cylindrical dielectric resonator antennas and modal analysis,” presented at the Applied Computational Electromagnetics Society (ACES) 2007 Conf., Verona, Italy, Mar. 19–23, 2007. [12] A. A. Kishk, A. W. Glisson, and G. P. Junker, “Bandwidth enhancement for split cylindrical dielectric resonator antennas,” Progr. Electromagn. Res., pp. 97–118, 2001, PIER 33. [13] P. V. Vijumon, S. K. Menon, M. N. Suma, B. Lethakumari, M. T. Sebastian, and P. Mohanan, “T-strip-fed high-permittivity rectangular dielectric resonator antenna for broadband applications,” Microw. Opt. Technol. Lett., vol. 47, no. 3, pp. 226–228, Nov. 2005. [14] A. A. Kishk, R. Chair, and K.-F. Lee, “Broadband dielectric resonator antennas excited by L-shaped probe,” IEEE Trans. Antennas Propag., vol. 54, no. 8, pp. 2182–2189, Aug. 2006. [15] B. Li and K. W. Leung, “Strip-fed rectangular dielectric resonator antennas with/without a parasitic patch,” IEEE Trans. Antennas Propag., vol. 53, no. 7, pp. 2200–2207, Jul. 2005. [16] A. Al-Zoubi and A. Kishk, “Wide band strip-fed rectangular dielectric resonator antenna,” in Proc. EuCAP’09, Berlin, Germany, Mar. 23–27, 2009, pp. 2379–2382. [17] S. H. Ong, A. A. Kishk, and A. W. Glisson, “Wideband disc-ring dielectric resonator antenna,” Microw. Opt. Technol. Lett., vol. 35, no. 6, pp. 425–428, Dec. 2002. [18] S. H. Ong, A. A. Kishk, and A. W. Glisson, “Rod-ring dielectric resonator antenna,” Int. J. RF Microw. Comput.-Aided Eng., vol. 14, no. 5, pp. 441–446, Sep. 2004. [19] M. Lapierre, Y. M. M. Antar, A. Ittipiboon, and A. Petosa, “Ultra wideband monopole/dielectric resonator antenna,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 1, pp. 7–9, Jan. 2005. [20] D. Guha and Y. M. M. Antar, “New half-hemispherical dielectric resonator antenna for broadband monopole-type radiation,” IEEE Trans. Antennas Propag., no. 12, pp. 3621–3628, Dec. 2006. [21] M. N. Suma, P. V. Bijumon, M. T. Sebastian, and P. Mohanan, “A compact hybrid CPW fed planar monopole/dielectric resonator antenna,” J. Eur. Ceramic Society, vol. 27, pp. 3001–3004, 2007. [22] J. M. Ide, S. P. Kingsley, S. G. O’Keefe, and S. A. Saario, “A novel wide band antenna for WLAN applications,” in Proc. IEEE Antenna and Propagation Society Int. Symp., Washington, DC, Jul. 2005, vol. 4A, pp. 243–246. [23] T. Chang and J. Kiang, “Broadband DR-loaded planar monopole,” in Proc. IEEE Antenna and Propagation Society Int. Symp., Honolulu, HI, Jun. 2007, pp. 553–556.

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[24] HFSS: High Frequency Structure Simulator Based on Finite Element Method 2007, v. 11.0.2, Ansoft Corporation. [25] K. S. Ryu and A. A. Kishk, “UWB dielectric resonator antenna mounted on a vertical ground plane edge,” in Proc. IEEE Antenna and Propagation Society Int. Symp., North Charleston, SC, Jun. 2009, pp. 1–4.

Kenny Seungwoo Ryu (S’08) received the B.S. degree in radio and communication engineering from Yonsei University, Korea, in 1999 and the M.S. degree in electrical engineering from Mississippi State University, in 2004. He is currently working toward the Ph.D. degree at the University of Mississippi. His research interest includes the area of design and analysis of ultrawideband antenna, dielectric resonator antenna, dual-polarized antenna, computational electromagnetic, artificial magnetic conductors, microwave passive and active circuit design, sensor design for detection, spectral estimation, biomedical applications of signal processing and microwave imaging.

Ahmed A. Kishk (S’84–M’86–SM’90–F’98) received the B.S. degree in electronic and communication engineering from Cairo University, Cairo, Egypt, in 1977, the B.S. degree in applied mathematics from Ain-Shams University, Cairo, Egypt, in 1980, and the M. Eng. and Ph.D. degrees from the University of Manitoba, Winnipeg, Canada, in 1983 and 1986, respectively. From 1977 to 1981, he was a Research Assistant and an Instructor at the Faculty of Engineering, Cairo University. From 1981 to 1985, he was a Research Assistant at the Department of Electrical Engineering, University of Manitoba. From December 1985 to August 1986, he was a Research Associate Fellow at the same department. In 1986, he joined the Department of Electrical Engineering, University of Mississippi, as an Assistant Professor. He was on sabbatical leave at Chalmers University of Technology, Sweden during the 1994–1995 academic years. He is now a Professor at the University of Mississippi (since 1995). He was the chair of Physics and Engineering Division of the Mississippi Academy of Science (2001–2002). He was an Associate Editor of Antennas and Propagation Magazine from 1990 to 1993 and is now an Editor. He was a Co-Editor of the Special Issue on Advances in the Application of the Method of Moments to Electromagnetic Scattering Problems in the ACES Journal. He was also an Editor of the ACES Journal during 1997. He was an Editor-in-Chief of the ACES Journal from 1998 to 2001. He was a Guest Editor of the Special Issue on Artificial Magnetic Conductors, Soft/Hard Surfaces, and Other Complex Surfaces of the January 2005 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. His research interest includes the areas of design of millimeter frequency antennas, feeds for parabolic reflectors, dielectric resonator antennas, microstrip antennas, EBG, artificial magnetic conductors, soft and hard surfaces, phased array antennas, and computer aided design for antennas. He has published over 210 refereed journal articles and 27 book chapters. He is a coauthor of the Microwave Horns and Feeds book (London, UK, IEE, 1994; New York: IEEE, 1994) and a coauthor of chapter 2 in the Handbook of Microstrip Antennas (Peter Peregrinus Ltd., U.K., 1989). Dr. Kishk received the 1995 and 2006 Outstanding Paper Awards for papers published in the Applied Computational Electromagnetic Society Journal. He received the 1997 Outstanding Engineering Educator Award from Memphis section of the IEEE. He received the Outstanding Engineering Faculty Member of the Year on 1998 and 2009, Faculty Research Award for Outstanding Performance in Research on 2001 and 2005. He received the Award of Distinguished Technical Communication for the entry of IEEE Antennas and Propagation Magazine, 2001. He also received The Valued Contribution Award for outstanding Invited Presentation, “EM Modeling of Surfaces with STOP or GO Characteristics—Artificial Magnetic Conductors and Soft and Hard Surfaces” from the Applied Computational Electromagnetic Society. He received the Microwave Theory and Techniques Society Microwave Prize 2004. He is a Fellow member of IEEE since 1998 (Antennas and Propagation Society and Microwave Theory and Techniques), a member of Sigma Xi society, a member of the U.S. National Committee of International Union of Radio Science (URSI) Commission B, a member of the Applied Computational Electromagnetics Society, a Fellow member of the Electromagnetic Academy, and a member of Phi Kappa Phi Society.

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 4, APRIL 2010

Transparent Dielectric Resonator Antennas for Optical Applications Eng Hock Lim, Member, IEEE, and Kwok Wa Leung, Senior Member, IEEE

Abstract—The transparent dielectric resonator antenna (DRA) for optical applications is proposed for the first time. For demonstration, a dual function transparent hemispherical DRA made of Borosilicate Crown glass (Pyrex) is investigated. The dual function DRA simultaneously works as an antenna and a focusing lens for an underlaid solar cell. The system is very compact because no extra footprint is needed for the solar cell. A conformal strip is used to excite the hemispherical DRA in its fundamental broadside 111 mode. Due to the focusing effect of the DRA, higher voltage and current outputs of the solar cell can be obtained. In this paper, the transparent rectangular DRA was also studied, and it was found that the rectangular DRA does not provide the focusing function. It was also found that the proposed transparent DRAs can provide a ) than for the state-of-the-art transparent mihigher gain ( crostrip antennas ( to 0 dBi). The reflection coefficients, input impedances, antenna gains, and radiation patterns of the two transparent DRAs are studied, and reasonable agreement between the simulated and measured results was observed. The proposed configurations can potentially be used for applications that need a self-sustaining power.

TE

4 dBi

5 dBi

Index Terms—Dielectric resonator antennas (DRAs), optical components, solar energy.

I. INTRODUCTION

I

N 1983, Long et al. [1] showed that the dielectric resonator (DR) can be used as an effective radiator, which is now commonly known as the DR antenna (DRA). In the past two decades, the DRA has been studied extensively due to a number of advantages such as its small size, low loss, low cost, light weight, and ease of excitation [2], [3]. By using the DR, the an, tenna size can be scaled down by roughly a factor of where is the dielectric constant of the DR. This can be very useful to reduce the antenna size. Today, compactness has become one of the topmost priorities [4], pushing up the development of multifunction components to miniaturize the system.

Manuscript received June 09, 2009; revised August 11, 2009; accepted October 24, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. This work was supported by a GRF Grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project CityU 116007). E. H. Lim was with the Wireless Communications Research Center and Department of Electronic Engineering, Kowloon, Hong Kong. He is now with the Faculty of Engineering and Science, Universiti Tunku Abdul Rahman, 53300 Setapak, Kuala Lumpur, Malaysia (e-mail: [email protected],my). K. W. Leung is with the Wireless Communications Research Center and Department of Electronic Engineering, Kowloon, Hong Kong (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.2010.2041315

As a result, it has been a trend to bundle several microwave functions into a single module [5], [6]. For example, Yeung and Wu combined several microwave resonators to provide multiple functions [7]. Jung [8] designed a microstrip single-resonator balun-filter. Recently, Lim et al. [9], [10] and Hady et al. [11] showed that the antenna and filter can be designed using a single DR. Also, it was demonstrated that the DRA can be integrated with an oscillator [12]–[14]. With the advent of the ultrawideband and millimeter-wave era, it has become increasingly popular to combine microwave and optical circuits in modern communication systems [15], [16]. The transparent microstrip antenna has been studied for optical applications [17], [18], but having a highly conducting transparent film is still a challenging problem. As the conductivity of the conductive transparent film ( [19], [20]) is relatively low as compared with that of metals, most of the transparent planar antennas reported thus far have an antenna gain of lesser than 0 dBi. Song et al. [20] proposed to apply conductive paste to the slot edge for improving the radiation efficiency, at the cost of reducing the transparency of the antenna. Using this technique, the antenna gain has been increased from in the past to . In our paper, the transabout parent DRA is proposed to circumvent the problem for it does not need any conducting parts to resonate. More importantly, it can provide an antenna gain of more than 4 dBi across the entire passband, which is a new height for the transparent antenna. Several studies on the integration of planar antennas and solar cell panels have been reported in the past decade [21]–[23]. The integration of the microstrip antenna and solar cell panel usually causes the antenna gain to degrade significantly. Very recently, Shynu et al. [22] have advanced the technology to increase the antenna gain of the solar-cell-integrated (metallic) microstrip lower than that of antenna to 1.05 dBi, but it is still metallic microstrip antennas. Moreover, the effective illumination area of the solar cell panel is somewhat reduced because of introducing the non-transparent microstrip antenna. In order to solve this problem, Vaccaro et al. [23] proposed to use a slot antenna, but it requires a removal of part of the solar cell panel. In this paper, it will be demonstrated that the gain of our proposed antenna is virtually not affected by the solar cell panel. Also, it does not need to remove any parts of the panel. The idea proposed in this paper should be very useful for integrating the solar cell panel with the antenna. For optical systems, it is well known that the lens is a very important component. It is of great interest to develop a dual function antenna that additionally provides the function of a lens. In this paper, the dual function transparent hemispherical DRA that simultaneously functions as an antenna and a focusing lens

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Fig. 2. Photograph of the transparent hemispherical DRA and the solar cell panel. The wires are for the power output of the solar cell panel. In the actual configuration, the transparent hemispherical DRA is placed on top of the panel.

Fig. 1. The proposed dual function transparent hemispherical DRA with an underlaid solar cell. (a) Front view. (b) Top view.

for a solar cell is investigated for the first time. To make the system compact, the solar cell is placed beneath the DRA to save the footprint. It is worth mentioning that the DRA can also serve as a protective cover for the solar cell. A conformal strip is used to excite the transparent hemispherical DRA in its dommode. It was found that due to its focusing effect, inant the hemispherical DR can increase the output voltage and current of the solar cell. Apart from the hemispherical DRA, the transparent rectangular DRA is also investigated for applications that do not need the focusing function. The results will be used to verify the focusing ability of the hemispherical counterpart. The reflection coefficients, input impedances, radiation patterns, and antenna gains of the two transparent DRAs are investigated. Ansoft HFSS was used to simulate the DRAs, and reasonable agreement between the simulated and measured results was obtained.

II. CONFIGURATIONS OF THE TRANSPARENT DRAS Fig. 1 shows the proposed configuration. The transparent hemispherical DRA [24] was made of Borosilicate Crown Glass, which is commonly known as K9 glass or Pyrex. It has and is placed above the solar cell. A a radius of and a length conformal strip [25] with a width of of was used to feed the DRA, which was excited in mode at 1.87 GHz. By using the Agilent its fundamental 85070D Dielectric Probe Kit, the dielectric constant of the round 1.9 GHz. glass was measured and it is equal to

It should be mentioned that at optical frequencies, the glass , which was has a much lower dielectric constant of [26], [27]. calculated from its refractive index of A square solar cell was immediately available in our laboratory and was therefore used in our experiment. But since the dielectric properties of the solar cell are not available, the values of and reported for a solar cell [28] were used in our HFSS simulations. The solar cell has a side length and a and 1.8 mm, respectively, whereas thickness of its left and right output pins at the back have a height of 0.2 mm. Because of the thickness of the solar cell (1.8 mm) and the height of the output pins (0.2 mm), the DRA has a displacement of 2 mm from the substrate. This information was also input in our HFSS simulations. In this paper, the substrate has a dielec, a thickness of , and a tric constant of size of 16 16 . It can serve as an additional insulator between the solar cell and the ground plane. The output pins of the solar cell were connected to a voltmeter and an ammeter for measurements of the voltage and current, respectively. To study the focusing effect of the DRA, the solar cell is masked with a . A very circular exposure area having a radius of thin and dark hard paper was used as the mask, which was not included in the simulations. Fig. 2 shows the photo of the prototype. Since the rectangular DRA is mechanically easier to fabricate, it is of great interest to the antenna engineers. In this paper, the transparent rectangular DRA was also investigated for nonfocusing applications. For ease of comparison, the rectangular DRA was designed to resonate at the resonance frequency of the hemispherical counterpart. The DRA was excited in its fundamode using a vertical excitation strip. mental broadside Fig. 3 shows the configuration, with , , , , , , and . The same masked solar cell was used again in this part.

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Fig. 5. Simulated and measured normalized radiation patterns of the hemi. There is no underlaid solar cell. (a) E -plane. spherical DRA with g (b) H -plane.

= 0 mm

Fig. 3. The transparent rectangular DRA with an underlaid solar cell. (a) Front view. (b) Top view.

Fig. 6. Simulated and measured reflection coefficients of the hemispherical DRA with the underlaid solar cell (Fig. 1). Their corresponding input impedances are shown in the inset.

Fig. 4. Simulated and measured reflection coefficients of the hemispherical DRAs for g and g . The results were obtained with no underlaid solar cell.

= 0 mm

= 2 mm

III. SIMULATED AND MEASURED RESULTS Ansoft HFSS was used to simulate the antenna part of the configuration, and measurements were carried out using the Agilent 8753 to verify the results. The effect of the airgap between the DRA and substrate is studied first without considering the solar cell. Fig. 4 shows the measured and simulated reflection coefficients of the DRA for and 2 mm, and reasonable agreement between the measured and simulated results is observed for each case. With reference to the figure, the airgap causes the measured resonant frequency and impedance ) to increase from 1.92 GHz to bandwidth (

2.26 GHz and from 14% to 19%, respectively. The trends are expected because similar results were obtained for the cylindrical and ring DRAs [29]–[31]. The antenna gains of the two and ) were measured and found to DRAs ( be in the range of 4 – 6.8 dBi across their passbands. It was observed that the DRA with the airgap has a wider gain bandwidth, which is, again, to be expected. It is found that the antenna gain around the resonance for each case, which is typical is for the DRA. The simulated and measured radiation patterns for are shown in Fig. 5. As can be observed from the figure, the co-polarized fields of both the - and -planes are stronger than the cross-polarized fields by more than 20 dB in the boresight direction ( ). The radiation pattern for was also simulated and measured and very similar results were obtained. Next, the characteristics of the DRA with the underlaid solar cell (Fig. 1) are investigated. Fig. 6 shows the simulated and measured reflection coefficients of the configuration. As can be observed from the figure, the measured and simulated resonant frequencies of the DRA are 1.94 GHz and 1.89 GHz, respectively, with an error of 2.65%. The measured and simulated

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Fig. 9. The top-down view of the Coherent Sabre Innova Argon Laser system for generating parallel blue light beams.

Fig. 7. Measured antenna gains of the hemispherical DRA with and without ) the solar cell. (g

= 0 mm

Fig. 8. Simulated and measured normalized radiation patterns of the hemispherical DRA with the underlaid solar cell. (a) E -plane. (b) H -plane.

impedance bandwidths are given by 16.5% and 22.8%, respectively. Although the DRA in this case also has a displacement of 2 mm from the substrate as for the previous one with the airgap, its measured resonant frequency (1.94 GHz) is lower than that of the airgap case (2.26 GHz). This is because the solar cell increases the effective dielectric constant of the DRA. It is interesting to note that the measured resonant frequency (1.94 GHz) is quite close to that of the previous DRA (1.92 GHz) resting ). With reference to the figure, a on the ground plane ( small resonant mode was measured at 2.25 GHz. This mode is caused by the solar cell, which can be verified by the fact that it was still observed when a rectangular DRA was used. The simulated result does not predict this resonance mode, which is not surprising because the exact dielectric parameters of the solar cell are not known. Fig. 7 shows the measured antenna gains of the DRA with and without the underlaid solar cell. With reference to the figure, the two antenna gains are very close to each other around the resonance of the DRA. This is a very positive result, as it implies that the loss due to the solar cell is negligibly small. It is observed around the resonance. from the figure that the gain is Fig. 8 shows the measured and simulated - and -plane radiation patterns. As can be seen from the figure, the co-polarized fields are stronger than their cross-polarized counterparts by more than 22 dB in the boresight direction.

Fig. 10. Output voltages and currents of the solar cell with and without the hemispherical DRA: R .

= 15 mm

The optical part of the system is now discussed. A Coherent Sabre Innova Argon Laser was used in the optical measurements, and parallel blue light beams at a wavelength of 488 nm were generated. The laser was tuned and provided an even light power of 130 mW to the DRA. To measure the outputs of the solar cell at different illumination angles ( ), the DRA was placed on a rotator, as shown in Fig. 9. Fig. 10 shows the measured output voltage and current of the solar cell as a function of . Also shown in the figure are the outputs without the DRA. With reference to the figure, larger outputs can be obtained for by using the DRA because of its focusing effect. With the DRA, the output voltage and current are increased by 13.5% , respectively. A smaller exposure radius of and 27.2% at was also used for the mask. Again, larger outputs when the DRA is present. In this were obtained for case, the voltage and current outputs at are increased by 11% and 21.4% with the use of the DRA, respectively. The curves, however, are not included here for brevity. In practical applications, the solar cell panel can be associated with a mechanical rotator so that it can track the light source if needed. In this case, the proposed DRA can be designed into a phased array so that it can scan the beam as the solar cell panel rotates. The results of the transparent rectangular DRA (Fig. 3) are discussed. Fig. 11 shows the simulated and measured reflection coefficients, whereas their corresponding input impedances are

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Fig. 11. Simulated and measured reflection coefficients of the transparent rectangular DRA with the underlaid solar cell (Fig. 3). Their corresponding input impedances are shown in the inset.

Fig. 13. Output voltages and currents of the solar cell with and without the . rectangular DRA: R

= 15 mm

should be mentioned that the inclusion of the solar cell and the dielectric substrate may decrease the efficiency of the radiating structure by making surface waves possible along the ground plane. IV. CONCLUSION

Fig. 12. Simulated and measured normalized radiation patterns of the rectangular DRA with the underlaid solar cell. (a) E -plane. (b) H -plane.

shown in the inset. With reference to the figure, the measured and simulated resonance frequencies of the DRA are given by 1.91 GHz and 1.86 GHz, respectively, with an error of 2.7%. For the impedance bandwidth, the measured and simulated values are 17.6% and 15.8%, respectively. The resonance due to the solar cell is observed again in the measured result. Its resonance frequency slightly shifts from 2.25 GHz to 2.23 GHz due to the changes of the dielectric and excitation-strip loadings. The antenna gain of the transparent rectangular DRA was around the resoalso measured. It was found to be nance. The simulated and measured radiation patterns are shown in Fig. 12. As can be observed from the figure, the crosspolarized fields are weaker than the co-polarized ones by more than 25 dB in the boresight direction, showing that the rectangular DRA has a very good polarization purity. Fig. 13 shows the measured output voltage and current using the same solar cell with . With reference to the figure, the rectangular DRA does not increase the outputs of the solar cell, suggesting that the rectangular DRA can be used for applications that do not require the focusing function. From the result, the focusing ability of the hemispherical DRA can be verified. Although the rectangular DRA does not provide the focusing function, its angular range for light reception is wider than its hemispherical counterpart, and the drop in its transparency above the solar cell panel is much smaller than the hemispherical version. Finally, it

The dual function transparent hemispherical DRA made of Borosilicate Crown Glass has been investigated for the first time. It simultaneously functions as a radiating element and an optical focusing lens. It can also serve as a protective cover for its underlaid solar cell. Since the DRA is transparent, the light can pass through it and illuminate on the underlaid solar cell. Because of the focusing effect of the DRA, the voltage and current outputs of the solar cell can be increased. The system is very compact for the solar cell does not need any extra footprint. The weight of the DRA can be a concern in practical applications, but it can be possibly reduced by the advancement of future material technologies. The DRA was mode using a conexcited in its fundamental broadside formal excitation strip. Ansoft HFSS was used to simulate the configuration and reasonable agreement between the measured and simulated results was obtained. A second configuration that uses a transparent rectangular DRA has also been investigated. This DRA is more suitable for applications that do not want the focusing effect. It is hoped that the proposed configurations would be useful for wireless systems that need a self-sustaining power. Finally, it should be mentioned that the transparency feature of the DRA can find other optical applications. For example, a light source (e.g., an LED) can be placed inside a hollow transparent DRA to give a lamp that also works as an antenna [32]. ACKNOWLEDGMENT The authors are grateful to the reviewers for their very useful comments, particularly for the suggestion of using the transparent DRA in a phased array with a mechanical rotation system that tracks the sunlight. The authors are thankful to Dr. A. H. P. Chan and Mr. R. Lai of City University of Hong Kong and Dr. K. K. Chong of Universiti Tunku Abdul Rahman for their useful discussions. The work of E. H. Lim was performed during his

LIM AND LEUNG: TRANSPARENT DIELECTRIC RESONATOR ANTENNAS FOR OPTICAL APPLICATIONS

visit to the Wireless Communications Research Centre and Department of Electronic Engineering, City University of Hong Kong. REFERENCES [1] S. A. Long, M. W. McAllister, and L. C. Shen, “The resonant cylindrical dielectric cavity antenna,” IEEE Trans. Antennas Propag., vol. 31, pp. 406–412, May 1983. [2] K. M. Luk and K. W. Leung, Eds., Dielectric Resonator Antennas London, U.K., Research Studies Press, 2003. [3] A. Petosa, Dielectric Resonator Antenna Handbook. Norwood, MA: Artech House, 2007. [4] A. K. Skrivervik, J. F. Zurcher, O. Staub, and J. R. Mosig, “PCS antenna design: The challenge of miniaturization,” IEEE Antennas Propag. Mag., vol. 43, no. 4, pp. 13–27, Aug. 2001. [5] Y. P. Zhang, X. J. Li, and T. Y. Phang, “A study of dual-mode bandpass filter integrated in BGA package for single-chip RF transceivers,” IEEE Trans. Adv. Packag., vol. 29, pp. 354–358, May 2006. [6] E. H. Lim and K. W. Leung, “Novel application of the hollow dielectric resonator antenna as a packaging cover,” IEEE Trans. Antennas Propag., vol. 54, pp. 484–487, Feb. 2006. [7] L. K. Yeung and K. L. Wu, “A dual-band coupled-line balun filter,” IEEE Trans. Microw. Theory Tech., vol. 55, pp. 2406–2411, Nov. 2007. [8] E. Jung and H. Hwang, “A balun-BPF using a dual mode ring resonator,” IEEE Microw. Wireless Compon. Lett., vol. 17, pp. 652–654, Sep. 2007. [9] E. H. Lim and K. W. Leung, “Experimental study on the dual function dielectric-resonator-antenna-filter,” presented at the Asia-Pacific Microwave Conference, Bangkok, Thailand, Dec. 2007. [10] E. H. Lim and K. W. Leung, “Use of the dielectric resonator antenna as a filter element,” IEEE Trans. Antennas Propag., vol. 56, pp. 5–10, Jan. 2008. [11] L. K. Hady, D. Kajfez, and A. A. Kishk, “Triple mode use of a single dielectric resonator,” IEEE Trans. Antennas Propag., vol. 57, pp. 1328–1335, May 2009. [12] E. H. Lim and K. W. Leung, “Novel dielectric resonator antenna oscillator,” presented at the IEEE TENCON, Hong Kong, Nov. 14–17, 2006. [13] E. H. Lim and K. W. Leung, “Novel utilization of the dielectric resonator antenna as an oscillator load,” IEEE Trans. Antennas Propag., vol. 55, pp. 2686–2691, Oct. 2007. [14] L. K. Hady, D. Kajfez, and A. A. Kishk, “Dielectric resonator antenna in a polarization filtering cavity for dual function applications,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 3079–3085, Dec. 2008. [15] E. Hamidi and M. Weiner, “Post-compensation of ultra-wideband antenna dispersion using microwave photonic phase filters and its applications to UWB systems,” IEEE Trans. Microw. Theory Tech., vol. 57, pp. 890–898, Apr. 2009. [16] A. Das, A. Nkansah, N. J. Gomes, I. J. Garcia, J. C. Batchelor, and D. Wake, “Design of low-cost multimode fiber-fed indoor wireless networks,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 3426–3432, Aug. 2006. [17] M. Wu and K. Ito, “Basic study on see-through microstrip antennas constructed on a window glass,” in IEEE Proc. Antennas and Propagation Symp., 1992, pp. 499–502. [18] R. N. Simons and R. Q. Lee, “Feasibility study of optically transparent microstrip patch antenna,” in Proc. IEEE Antennas and Propagation Symp., 1997, pp. 2100–2103. [19] A. Katsounaros, Y. Hao, N. Collings, and W. A. Crossland, “Optically transparent ultra-wideband antenna,” Electron. Lett., vol. 45, no. 14, pp. 722–723, Jul. 2009. [20] H. J. Song, T. Y. Hsu, D. F. Sievenpiper, H. P. Hsu, J. Schaffner, and E. Yasan, “A method for improving the efficiency of transparent film antennas,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 753–756, 2008. [21] M. Tanaka, Y. Suzuki, K. Araki, and R. Suzuki, “Microstrip antenna with solar cells for microsatellites,” Electron. Lett., vol. 31, no. 1, pp. 5–6, Jan. 1995. [22] S. V. Shynu, J. Maria, P. McEvoy, M. J. Ammann, S. McCormack, and B. Norton, “Integration of microstrip patch antenna with polycrystalline silicon solar cell,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3969–3972, Dec. 2009.

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[23] S. Vaccaro, P. Torres, J. R. Mosig, A. Shah, J. F. Zurcher, A. K. Skrivervik, P. de Maagt, and L. Gerlach, “Stainless steel slot antenna with integrated solar cells,” Electron. Lett., vol. 35, no. 25, pp. 2059–2060, Dec. 2000. [24] A. A. Kishk, G. Zhou, and A. W. Glisson, “Analysis of dielectric resonator antennas with emphasis on hemispherical structures,” IEEE Antennas Propag. Mag., vol. 36, no. 2, pp. 20–31, Apr. 1994. [25] K. W. Leung, “Conformal strip excitation of dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 48, pp. 961–967, Jun. 2000. [26] Company Website [Online]. Available: www.camglassblowing.co.uk/ gproperties [27] Datasheet: Borosilicate Glass Properties. [28] J. Dheepa, R. Sathyamoorthy, A. Subbarayan, S. Velumani, P. J. Sebastian, and R. Perez, “Dielectric properties of vacuum deposited Bi2Te3 thin films,” Solar Energy Mater. Solar Cells, vol. 88, no. 2, pp. 187–198, Jul. 2005. [29] G. P. Junker, A. A. Kishk, A. W. Glisson, and D. Kajfez, “Effect of an air gap around the coaxial probe exciting a cylindrical dielectric resonator antenna,” Electron. Lett., vol. 30, no. 3, pp. 177–178, 1994. [30] G. P. Junker, A. A. Kishk, A. W. Glisson, and D. Kajfez, “Effect of an air gap on a cylindrical dielectric resonator antennas operating in the TM01 mode,” Electron. Lett., vol. 30, no. 2, pp. 97–98, 1994. [31] S. M. Shum and K. M. Luk, “Characteristics of dielectric ring resonator antenna with an air gap,” Electron. Lett., vol. 30, pp. 277–278, Feb. 1994. [32] A. H. P. Chan, Personal Communication. Eng Hock Lim (S’05–M’08) was born in Selangor, Malaysia. He received the B.Sc. degree in electrical engineering from National Taiwan Ocean University in 1997, the M.Eng. degree in electrical and electronic engineering from Nanyang Technological University in 2000, and the Ph.D. degree in electronic engineering from City University of Hong Kong, in 2007. Since 2008, he has been an Assistant Professor at Universiti Tunku Abdul Rahman, Malaysia. His current research interests include dielectric resonator antennas, microstrip antennas, multi-functional antennas, and mobile communications.

Kwok Wa Leung (S’90–M’93–SM’02) was born in Hong Kong on April 11, 1967. He received the B.Sc. degree in electronics and the Ph.D. degree in electronic engineering from the Chinese University of Hong Kong, in 1990 and 1993, respectively. From 1990 to 1993, he was a Graduate Assistant with the Department of Electronic Engineering, the Chinese University of Hong Kong. In 1994, he joined the Department of Electronic Engineering at City University of Hong Kong as an Assistant Professor 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. Prof. Leung 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. He was the Chairman of the IEEE AP/MTT Hong Kong Joint Chapter for the years of 2006 and 2007. He was the Chairman of the Technical Program Committee, 2008 Asia-Pacific Microwave Conference, the Co-Chair of the Technical Program Committee, IEEE TENCON, Hong Kong, Nov. 2006, and the Finance Chair of PIERS 1997, Hong Kong. His research interests include RFID tag antennas, dielectric resonator antennas, microstrip antennas, wire antennas, guided wave theory, computational electromagnetics, and mobile communications. He serves as an Editor for HKIE transactions. He also serves as an Associate Editor for 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. Also, he is an Associate Editor for IEEE ANTENNAS WIRELESS PROPAGATION LETTERS and a Guest Editor of IET Microwaves, Antennas and Propagation.

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A Low-Profile Linearly Polarized 3D PIFA for Handheld GPS Terminals Andrea Antonio Serra, Paolo Nepa, Member, IEEE, Giuliano Manara, Fellow, IEEE, and Riccardo Massini

Abstract—A 3D planar inverted F-antenna (PIFA) is proposed for a Global Positioning System (GPS) receiver. The antenna is designed to be integrated in a portable device and must meet compact size requirements. The 3D PIFA consists of a single metal sheet properly cut and bent in order to minimize costs for realization, materials and series production. The linearly polarized antenna exhibits good impedance matching performance, wide beam radiation patterns and a relatively high gain, which allow the reception from a wide angle toward the satellite constellation. An antenna prototype has been embedded in a commercial GPS receiver and an experimental measurement campaign has been carried out to evaluate the most important performance parameters. Measurements have been performed out by collecting time sweeps signal samples along urban and suburban routes. Time to first fix, carrier to noise ratio distribution and horizontal dilution of precision have been calculated; these parameters are shown and discussed. Comparisons with respect to some reference circularly polarized antennas (a nearly square patch and a quadrifilar helix antenna) and to a simpler inverted-F antenna printed on the receiver PCB have been performed and are shown in the paper. Index Terms—Global Positioning System, planar inverted F-antenna (PIFA).

I. INTRODUCTION

G

LOBAL Positioning System (GPS) applications are becoming very popular in everyday life because of recent advances in electronic technology. First, the miniaturization of electronic components allows the integration of circuits and components in compact devices and this results in small and light handheld terminals. Secondly, costs of electronic circuitry are gradually reducing so making the expense for these modern and technologic tools affordable. The resulting mass production and distribution of such devices push companies to design and outsell GPS receivers while reducing production and material costs. Antennas play a critical role in the miniaturization and saving process because of their cost space occupation and the expenses deriving from the cost for the production of efficient radiators. The GPS signal is transmitted in the L1 (1.575 GHz)

Manuscript received April 16, 2009; revised July 25, 2009; accepted October 01, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. This work was supported in part by R.I.CO srl, 60022 Castelfidardo (Ancona). A. A. Serra is with the Department of Information Engineering, University of Pisa, Pisa, Italy (e-mail: [email protected]). P. Nepa, G. Manara, and R. Massini are with the Department of Information Engineering, University of Pisa, Pisa, Italy and also with Consortium Ubiquitous Technologies (CUBIT), Navacchio, Pisa, Italy (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2041162

and L2 (1.227 GHz) frequency bands and it is right-hand circularly polarized (RHCP). Among the wide range of extensively studied GPS antennas it is worth mentioning, as the most successful and performing, the conical spiral antennas (CSA), the quadrifilar helix antennas (QFHA), and microstrip patches. CSA exhibits a broad main lobe, a wide frequency band and good front-to-back radiation ratio [1], [2]; additionally, QFHA is relatively insensitive to mutual coupling effects [2]. On the other hand, microstrip patches are light, low profile, relatively low cost, although inherently narrowband [3], and a stacked configuration is required to cover both the L1 and L2 band [4]. The three previously mentioned categories of radiators are circularly polarized (CP) antennas and then suitable to satisfy polarization matching condition with the GPS signal. The received signal is generally susceptible to multipath effects that arise when the signal comes through different paths and resulting in a significant amplitude and phase distortion [5], [6]. This signal degradation can be mitigated with the design of antennas with high cross polarization rejection, typically greater than 15 dB, and sharp drop-off, larger than 1 dB/deg, of the radiation patterns at low elevation angles where multipath is much effective [1]. However, such radiation characteristics are often difficult to be obtained especially for integrated radiators which are affected by the presence of circuitry and metallic parts of the printed circuit board (PCB), as well as that of the user’s hands and body, which modify the radiation pattern properties. In addition to this, the integration of GPS functionality in consumer products, especially in mobile handsets, causes the customers to use the GPS-enabled device in urban environments where multipath is more effective. Since a circularly polarized signal is highly susceptible to distortions in the presence of multipath, and also accounting for the deformation of the radiation patterns due to interaction with the user, the above designs often result to be inefficient. A linearly polarized (LP) signal is less susceptible to distortion caused by the environment and by the presence of the user [6]. Consequently, by taking into account the above comments on signal degradation, for inexpensive devices it could result more convenient to design a simple LP antenna rather than a more complex and expensive CP one. In this paper the design of a 3D linearly polarized planar inverted F-antenna (PIFA) for L1 band GPS portable devices is presented. To reduce production efforts needed for soldering, assembling and mounting the antenna on the PCB, the antenna is realized by a single metal sheet, which is cut and properly bent. The radiating element presents a wide beam radiation pattern and, at the same time, a relatively high gain because of the low losses deriving from the absence of a dieletric layer under the radiating plate. An experimental comparative study has been

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conducted among the designed PIFA, a RHCP microstrip patch and a commercial QFHA antenna when they are mounted on the PCB and enclosed in a commercial GPS handheld terminal. Time to first fix (TTFF), carrier to noise ratio (CNR) and horizontal dilution of precision (HDOP) have been measured when the receiver is carried through a real urban environment. This investigation trial shows that the 3D PIFA performance is comparable to that of the most acknowledged RHCP GPS antennas. Furthermore, the designed PIFA is a simpler antenna, as it can be produced and installed within the same manufacturing process of all other electronic circuitry. II. ANTENNA DESIGN AND PERFORMANCE The 3D PIFA is described in this section. Before analyzing each part of the proposed antenna it is worth summarizing the design procedure that led to the optimized structure. A space volume was available in the terminal chassis where the antenna had to be embedded, thus forcing the design towards the research of a compact antenna. Miniaturized antennas, as for instance resonator antennas, satisfy this requirement, but they usually show a very low gain which, as it will be shown in the following, represents a key parameter for a good reception of the GPS signal. PIFA antennas guarantee higher gain values and they are more valuable space saving solutions with respect to traditional patches because of their size that is usually about a quarter of a wavelength (traditional patches present a resonant dimension of about half of a wavelength) [7], [8]. Starting from this point of view, a preliminary PIFA was first designed, as shown in Fig. 1. The radiating element is at a distance of 3.59 mm from a 0.41 mm thick FR4 laminate on the first ground plane of a multilayer PCB. The antenna, rectangular in shape, is 42 mm long (L) and 20 mm wide (W) and it is composed of a single radiating plate and two 5 mm wide shorting plates at one of its ends. A microstrip line printed on the FR4 substrate feeds the antenna and it is connected to the radiating plate through a vertical metallic pin. This antenna needs to be mounted on a ground plane which is represented, in this specific case, by the upper ground layer of the PCB of the GPS device. This PIFA prototype shows a good impedance matching and wide beam radiation patterns in free space but it exhibited unstable performance when mounted inside the device. The most important drawback is the absence of any support to the radiating plate whose distance from the ground plane varies significantly when the device is carried by the user. This results in poor impedance matching and degraded radiation patterns, as widely studied for portable devices interacting with humans [9]. To avoid any radiating plate fluctuation, low permittivity plastic supports could be used but they represent additional parts to be manufactured and they would increase production costs. The idea to solve this inconvenient was to add two vertical shorting plates (in the z-y plane, elements H and I in Fig. 2(a)) provided with two horizontal support bases, (in the x-y plane, elements C and D in Fig. 2(a)) at the open end of the PIFA, as well as two support bases (in the x-y plane, elements A and B) for the shorting plates F and G. The element J is a pin that connects the antenna to the microstrip feeding line E. The position of elements C, D, H and I was accurately chosen not to modify the current distribution on the radiating element

Fig. 1. Side view (stackup) and 3D layout of a preliminary PIFA for GPS reception.

Fig. 2. 3D layout and dimensions of the proposed PIFA antenna for GPS reception (dashed lines indicate where the sheet needs to be bent to obtain the 3D version of the antenna, dimensions are in mm).

with respect to the preliminary design, and to tune the antenna by taking into account the resonant frequency downshift introduced by their additional length. All the dimensions of the properly cut metallic sheet are reported in Fig. 2(a). Fig. 2(b) shows a 3D view of the final design. Dashed lines in Fig. 2(a) indicate where the sheet needs to be 90 bent to obtain the 3D version of the antenna, allowing the correct positioning of any element on the PCB. Elements C and D need to be glued to the upper side of the FR4 laminate, without soldering or any electric contact to the ground plane. Elements A and B are soldered to two metallic pitches on the substrate and then grounded through foiled vias. As in the preliminary version, a microstrip line (element E in Fig. 2) feeds the antenna through a vertical narrow plate (element J in Fig. 2), which needs to be 90 folded as done for the

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Fig. 3. Prototype of the designed antenna when it is mounted on a simplified PCB (GPS-controller chip, low noise amplifier under a metallic screen and two SMA connectors for testing purposes). In the inset, the detail of the metallization voids is shown.

Fig. 5. Simulated (HFSS data) radiation patterns of the antenna in the principal planes.

Fig. 4. Simulated (HFSS data) and measured reflection coefficients for the antenna mounted on the receiver PCB.

shorting and supporting plates. In Fig. 2(b) a dashed square is highlighted around element D, which represents a metallization void in the antenna ground plane. This shrewdness is necessary to eliminate the capacitive coupling effect between elements C and D and the ground plane, which would introduce impedance mismatch and frequency detuning. Fig. 3 shows a prototype of the designed antenna when it is mounted on a simplified PCB. In the figure, the RF module can be seen as well as a detail of the reverse side of the simplified PCB, where the small square voids in the ground plane under elements C and D of Fig. 2(a) are visible. Fig. 4 shows the simulated and measured reflection coefficients. A good agreement is observed between numerical and experimental results. Bandwidth requirements are fully satisfied with a percentage . Fig. 5 shows the simubandwidth of 2.5% lated radiation patterns of the PIFA in the principal planes. It can be seen that in the x-z and x-y planes two dips are present toward the direction of the PCB while an almost omnidirectional pattern is achieved in the y-z plane. The simulated gain of the PIFA antenna is about 2.8 dB. III. GPS RECEPTION EVALUATION A comparative analysis on the 3D PIFA and other three antenna types has been carried out, when each of them is embedded in a device based on the STA2051 “VESPUCCI” GPS controller by ST Microelectronics [10]. The radiating elements

are a high performance QFHA GeoHelix-P2 by Sarantel [11], a RHCP microstrip patch and a simpler printed IFA. Examples of these radiating elements are shown in Fig. 6. The QFHA is a RHCP antenna with a percentage bandwidth equal to 0.3% ), it exhibits a gain of 2.8 around 1.575 GHz ( dB. The microstrip patch is printed on a 3.2 mm Taconic CER10 . It is a nearly square patch laminate with permittivity fed on the diagonal to radiate a RHCP electric field. Its perand the centage bandwidth is around 1.2% gain is greater than 1.3 dB. Both the Sarantel GeoHelix-P2 and the microstrip patch are surface mountable antennas that just need to be attached to the PCB and soldered at the low noise amplifier (LNA) input port. The third antenna is a meandered inverted F-antenna printed on a 0.41 mm thick FR4 laminate. The radiating element is 34 mm long and it is at a distance of 10 mm from the upper border of the metallic ground plane of the PCB. The antenna has been meandered in order to fit the available space of the device case. The antenna meets the bandwidth standard requirements with a percentage bandwidth of 12.6% . The wider bandwidth with respect to the previous configurations is due to the high losses of the FR4 laminates which it is printed on and it consequently results in a lower gain of about 6.1 dB. This antenna shows a lower gain with respect to the QFHA and the patch antenna but it has the advantage to be extremely simple as it can be printed together with the PCB traces. In order to evaluate the antenna performance, four GPS receivers including the above antennas have been carried along urban and suburban routes in the city of Pisa. Samples of the routes are shown in Fig. 7. Performance of a GPS handheld device is usually evaluated through the analysis of some parameters known as time to first fix (TTFF), horizontal dilution of precision (HDOP) and carrier

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TABLE I WARM AND HOT TIME TO FIRST FIX FOR THE GPS PERFORMANCE TEST

Fig. 6. Examples of the three reference antennas used in the measurement campaign: (a) QFHA; (b) nearly square patch; (c) meandered printed IFA.

Fig. 7. Suburban (a) and urban (b) routes along the Arno riverside for the GPS antenna testing.

to noise ratio (CNR) [12]. They have been measured and collected when each antenna is embedded in the device. TTFF is the time needed for a GPS receiver to determine its actual position. When trying to acquire a lock, it needs to rely on both almanac and ephemeris data that are transmitted by each satellite every 12.5 minutes and 30 seconds, respectively. Almanac data is course orbital parameters for all satellites in the GPS constellation which isn’t very accurate information but is usually valid for up to several months. Ephemeris data is very precise orbital information and it can be considered actual up to 4 hours. TTFF is commonly broken down into three, more specific scenarios as follows.

• Cold or Factory: The receiver has no information on the almanac nor on the ephemeris. As the almanac is transmitted repeatedly over 12.5 minutes, manufacturers typically claim the Cold TTFF to be 15 minutes. • Warm or Normal: The receiver has estimates of almanac data but it must acquire ephemeris data. • Hot or Standby: The receiver has valid almanac and ephemeris and the time required for acquisition is very short. Warm and Hot TTFF are the most common parameters for a daily use of GPS devices, and it could be relevant to evaluate them in order to quantify the receiver’s performance. Warm TTFF occurs when the GPS terminal is turned off for more than 4 hours, i.e. at night for example, and Hot TTFF occurs when the devices is switched off for a short period, usually a couple of hours, i.e. a short break while driving or walking. Warm and Hot TTFF have been evaluated for the proposed PIFA and for the three reference antennas. While Hot TTFF was always less than 3–5 seconds, including the time needed for the GPS devices to be switched on, different behaviors have been noticed for the Warm TTFF. Several measured samples of the Warm TTFF have been collected for each antenna after erasing ephemeris data from the device memory and they have been averaged. The QFHA GeoHelix-P2 shows the shortest average warm TTFF that is around 9 seconds. A similar value has been found for the microstrip RHCP patch that is around 11 seconds. The proposed 3D PIFA exhibits an average warm TTFF of 14 seconds. The fix procedure was a little harder for the printed IFA whose average TTFF is calculated to be 18 or 48 seconds. These two values could depend on the fact that sometimes the receiver has been able to connect within 30 seconds, that is by using the first available transmission of ephemeris from the detected satellites, and sometime within 60 seconds, that is by using the second available transmission of the satellites ephemeris. This different behavior with respect to the previous antenna configurations, which are able to make the receiver fix its position within 30 seconds, is due to the reduced value of the antenna gain that determines an apparent signal weakness. TTFF values are summarized in Table I. Horizontal dilution of precision (HDOP) is a GPS parameter used to describe the geometric strength of satellite configuration on GPS accuracy. When visible GPS satellites are close together in the sky, the geometry is said to be “weak” and the HDOP value is high; whereas they are far apart, the geometry is

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Fig. 9. CNR level distributions of the received satellite signals for the four antennas used in the GPS performance tests.

Fig. 8. Horizontal dilution of precision for the four antennas used in the GPS performances test.

“strong” and the HDOP value is low. Thus a low HDOP value represents a better GPS positional accuracy due to the wider angular separation between the satellites used to calculate a GPS unit position. The latter follows mathematically from the positions of the usable satellites. The effect of geometry of the satellites on position error is called geometric dilution of precision and it is roughly interpreted as ratio of position error to range error. Now let’s imagine that a tetrahedron is formed by lines joining four satellites and a receiver. The larger the volume of the tetrahedron, the better the value of HDOP. In other words, the greater the angle between the received satellite signals (that correspond to a wide beam radiation pattern), the better the value of HDOP is. Fig. 8 shows the HDOP values measured for the proposed 3D PIFA and a comparison with the three reference antennas used in the measurement campaign. It can be noticed that the Sarantel QFHA GeoHelix-P2, the RHCP nearly square patch and the proposed 3D PIFA exhibit very low values of HDOP, always less than 2, proving a good coverage of the satellite constellation. The printed IFA is not so performing and its HDOP reaches values up to 6; alternatively, or it makes the receiver lose the fix . The last parameter analyzed in this study is the carrier to noise ratio. The carrier-to-noise ratio (CNR) is a measurement of the GPS signal quality. CNR is the ratio between carrier power and noise power at the receiver input. This parameter is usually available for each satellite signal also for commercial GPS devices and we collected its values in the measurement campaign described above. Fig. 9 shows the distributions of all the CNR samples collected in the measurement campaign. It is apparent that the Sarantel QFHA GeoHelix-P2 receives a large number of high CNR samples (35–40 dB) but the nearly square patch and the proposed 3D PIFA are able to get also a significant number of samples with a CNR up to 45–47 dB. As expected the printed IFA has a lower mean CNR. Fig. 10 shows the average CNR level of the satellite signals versus time.

Fig. 10. Average CNR level of the received satellite signals for the four antennas used in the GPS performance tests.

TABLE II AVERAGE AND MAXIMUM CNR VALUE FOR THE GPS PERFORMANCE TEST

It is apparent that the printed IFA has a much lower average CNR level with respect to the other three antenna configurations and, as expected, it confirms previous results on TTFF and CNR distribution. The Sarantel QFHA GeoHelix-P2, the nearly square RHCP microstrip patch and the proposed PIFA show higher CNR values and similar performance. Table II summarizes the average and maximum CNR value for the four antennas used in the GPS performance tests.

SERRA et al.: A LOW-PROFILE LINEARLY POLARIZED 3D PIFA FOR HANDHELD GPS TERMINALS

The best average CNR is obtained with the Sarantel QFHA GeoHelix-P2 and it is equal to 31.6 dB, whereas the maximum CNR is obtained with the RHCP microstrip patch and it is equal to 48 dB. The proposed 3D PIFA shows an average CNR of 29.6 dB and a maximum of 44 dB. Measured results shown in this session indicate that even a linearly polarized antenna can provide performance comparable to those of a more complex RHCP antenna. An accurate design process results in a reduced size element, which is also easy to manufacture and suitable for series production. Finally, the proposed antenna exhibits a gain comparable or higher than those of microstrip antennas of the same size. This relatively high gain contributes to good performance of the receiving GPS device. IV. CONCLUSION In this paper, a 3D PIFA has been designed to be embedded in a GPS handheld terminal, and its properties have been shown through experimental data. Experimental results have shown that a linearly polarized 3D PIFA antenna can reach relatively high performance, comparable with that of right hand circularly polarized antennas. Although the proposed PIFA exhibits a linear polarization, the relatively high gain allows the antenna to obtain suitable performance. Indeed, the measured time to first fix (TTFF) and horizontal dilution of precision (HDOP) from the signal samples collected during an outdoor measurement campaign exhibit very low values whereas the carrier to noise ratio (CNR) results to be as high as that attainable by the most used RHCP radiating elements. ACKNOWLEDGMENT The authors wish to acknowledge the support of Dr. E. Franchi of R.I.CO srl for his helpful technical support and stimulating discussions.

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[10] [Online]. Available: http://www.st.com [11] [Online]. Available: http://www.sarantel.com/ [12] D. N. Aloi, M. Alsliety, and D. M. Akos, “A methodology for the evaluation of a GPS receiver performance in telematics applications,” IEEE Trans. Instrum. Meas., vol. 56, no. 1, pp. 11–24, Feb. 2007.

Andrea Antonio Serra received the Laurea degree in telecommunications engineering and the Ph.D. degree from the University of Pisa, Pisa, Italy, in 2003 and 2007, respectively. In 2006, he was a Visiting Ph.D. Researcher at the Electronic, Electrical and Computer Engineering, University of Birmingham. In September 2003, he joined the Microwave and Radiation Laboratory, Department of Information Engineering, University of Pisa. He is currently conducting his postdoctoral research at the University of Pisa. His research interests include antenna design and diversity for mobile terminals and on-body communication systems.

Paolo Nepa (M’08) received the Laurea (Doctor) degree in electronics engineering (summa cum laude) from the University of Pisa, Italy, in 1990. Since 1990, he has been with the Department of Information Engineering, University of Pisa, where he is currently an Associate Professor. In 1998, he was at the ElectroScience Laboratory (ESL), The Ohio State University (OSU), Columbus, as a Visiting Scholar supported by a grant of the Italian National Research Council. At the ESL, he was involved in research on efficient hybrid techniques for the analysis of large antenna arrays. His research interests include the extension of high-frequency techniques to electromagnetic scattering from material structures and its application to the development of radio propagation models for indoor and outdoor scenarios of wireless communication systems. He is also involved in the design of wideband and multiband antennas, mainly for base stations and mobile terminals of communication systems. More recently he is working on channel characterization and wearable antenna design for body-centric communication systems. Dr. Nepa received the Young Scientist Award from the International Union of Radio Science, Commission B, in 1998.

REFERENCES [1] N. Padros, J. I. Ortigosa, J. Baker, M. F. Iskander, and B. Thornberg, “Comparative study of high-performance GPS receiving antenna designs,” IEEE Trans. Antennas Propag., vol. 45, no. 4, pp. 698–706, Apr. 1997. [2] J. M. Tranquilla and S. R. Best, “A study of the quadrifilar helix antenna for Global Positioning System (GPS) applications,” IEEE Trans. Antennas Propag., vol. 38, no. 10, pp. 1545–1550, Oct. 1990. [3] L. Boccia, G. Amendola, and G. Di Massa, “A shorted elliptical patch antenna for GPS applications,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 1, pp. 6–8, 2003. [4] L. Boccia, G. Amendola, and G. Di Massa, “A dual frequency microstrip patch antenna for high-precision GPS applications,” IEEE Antennas Wireless Propag. Lett., vol. 3, no. 1, pp. 157–160, 2004. [5] T. Kos, R. Filjar, and S. B. Rimac-Drlje, “GPS signal deterioration in urban environment: A winter case-study,” in Proc. Int. Conf.. on Information and Communication Technologies, Apr. 2004, pp. 317–318. [6] V. Pathak, S. Thornwall, M. Krier, S. Rowson, G. Poilasne, L. E. Desclos, and H. Miller, “Mobile handset system performance comparison of a linearly polarized GPS internal antenna with a circularly polarized antenna,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 2003, vol. 3, pp. 666–669. [7] P. Nepa, G. Manara, A. A. Serra, and G. Nenna, “Multiband PIFA for WLAN mobile terminals,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 349–350, 2005. [8] B. K. Fankem and K. L. Melde, “Nested PIFAs for dual mode of operation: GPS and global communications,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 701–705, 2008. [9] M. A. Jensen and Y. Rahmat-Samii, “EM interaction of handset antennas and a human in personal communications,” IEEE Proc., vol. 83, no. 1, pp. 7–17, 1995.

Giuliano Manara (F’04) was born in Florence, Italy, on October 30, 1954. He received the Laurea (Doctor) degree in electronics engineering (summa cum laude) from the University of Florence, Italy, in 1979. Currently, he is a Professor at the College of Engineering of the University of Pisa, Italy. Since 2000, he has been serving as the President of the Bachelor and the Master Programs in Telecommunication Engineering at the same University. Since 1980, he has been collaborating with the Department of Electrical Engineering of the Ohio State University, Columbus, where, in the summer and fall of 1987, he was involved in research at the ElectroScience Laboratory. His research interests have centered mainly on the asymptotic solution of radiation and scattering problems to improve and extend the uniform geometrical theory of diffraction. In this framework, he has analyzed electromagnetic wave scattering from material bodies, with emphasis on the scattering from both isotropic and anisotropic impedance wedges. He has also been engaged in research on numerical, analytical and hybrid techniques (both in frequency and time domain), scattering from rough surfaces, frequency selective surfaces (FSS), and electromagnetic compatibility. More recently, his research has also been focused on the design of microwave antennas with application to broadband wireless networks, and on the development and testing of new microwave materials (metamaterials). Prof. Manara was elected an IEEE Fellow in 2004 for “contributions to the uniform geometrical theory of diffraction and its applications.” Since 2000, he has been serving as the Secretary/Treasurer of the Italian Society on Electromagnetics (Societ` Italiana di Elettromagnetismo, SIEm). Since 2002, he has been working as a member of the IEEE Italy Section Executive Committee. In

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May 2004, he was the Chairman of the Organizing Committee for the International Symposium on Electromagnetic Theory of Commission B of the International Union of Radio Science (URSI). He also served as a Convenor for several URSI Commission B international conferences, and URSI General Assemblies. In August 2008, he has been elected Vice-Chair of the International Commission B of URSI.

Riccardo Massini received the M.Sc. degree in electronic engineering from the University of Pisa, Pisa, Italy, in 2005. Since 2005, he is a member of the technical staff in the RF/Microwave Laboratory, Electronics Department, University of Pisa, as an RF/Microwave Engineer. His activities concern the design, development and characterization of millimeter-wave planar passive circuits, MIC hybrid circuits and multichip modules (MCM), in frequency ranges up to 40 GHz. His areas of expertise include also electromagnetic simulations, high-frequency interconnections and packaging and characterization of monolithic RF integrated circuits. He has designed and built many microwave devices such as millimeter-wave filters, X-band radar modules, RF front-end and antennas, phased array subsystems and broadband microwave modules for 10 and 40 Gbps optical communications systems.

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Analysis of Strong Coupling in Coupled Oscillator Arrays Venkatesh Seetharam and L. Wilson Pearson, Fellow, IEEE

Abstract—Significant improvement in the mutual injection locking range (MILR) and phase noise can be obtained by strengthening the coupling between the oscillator elements in a coupled oscillator array (COA). An analysis of the array properties in the strongly coupled regime is provided in this paper. Previous analyses of COAs have employed a so-called broadband condition that substantially simplifies analysis. The broadband condition is observed to break down in the strongly coupled regime, a feature that is central in the understanding of the behavior of strongly coupled arrays. The observed improvement in the phase noise performance can be attributed to the highly resonant nature of the coupling network in the strongly coupled regime. The theory is verified using a five-element linear COA operating at 3.75 GHz. Significant improvements in the MILR and phase noise is reported. The beam steering capabilities of strongly coupled arrays are also presented. Index Terms—Beam steering, phase noise, phased arrays, radiation patterns, voltage controlled oscillators.

I. INTRODUCTION

I

T is well known that sensitivity to component variation from cell-to-cell in a coupled oscillator array (COA) is problematic in array performance. This sensitivity is minimized if one employs wide-locking bandwidth oscillators. It has been shown that oscillator design can be optimized for wide-locking bandof the oscillators [1]. Operating on width by lowering the their own, low Q oscillators lead to poor phase noise performance in the array. However, this phase noise can be controlled by injection locking the array to an external source that exhibits phase stability commensurate with the eventual application of the array. The mutual injection locking range (MILR) between two oscillators in an array is directly proportional to the coupling strength [2]. In the weakly coupled regime, the phase noise performance of the array improves with increase in the coupling strength. Thus strong coupling can possibly be utilized to widen the mutual injection locking range of the oscillators and improve the phase noise performance intrinsic to a given COA

Manuscript received October 09, 2008; revised September 28, 2009; accepted October 13, 2009. Date of publication January 22, 2010; date of current version April 07, 2010. V. Seetharam is with Ansoft LLC, San Jose, CA 95132 USA (e-mail: [email protected]; [email protected]). L. W. Pearson is with the Holcombe Department of Electrical and Computer Engineering, Clemson University, Clemson, SC 29634 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.2010.2041141

(prior to external injection locking). However, strong coupling between the elements complicates analysis of array interaction. The amplitude variation along the array is negligible for the case of weak or loose coupling between the oscillator elements. This variation becomes significant for strongly coupled arrays. Reducing the value of the coupling resistors to achieve strong coupling between the oscillator elements results in increasing the Q of the coupling network thereby making the network highly resonant at higher coupling strengths. For these reasons the analysis of strongly coupled oscillator arrays is not straightforward. Strongly coupled arrays have been investigated by Nogi et al. [3]. They analyzed the issue of multiple modes in such arrays and showed that all but one of the resulting modes has amplitude variation across the array. They also present a technique to suppress the undesired modes. Lynch and York [4] investigated the effects of narrowband coupling networks on the synchronization properties of an array. They employed resistive coupling networks and performed analysis on weakly and strongly coupled oscillators for broadband and narrowband coupling. They concluded that narrowband coupling decreases the probability for the oscillators in an array to lock with each other. Georgiadis et al. [5] performed a stability analysis on one-dimensional coupled oscillator array systems under weak and strong coupling conditions to understand the factors that limit the achievable values of constant inter-oscillator phase difference. In this paper, an analysis of strongly coupled oscillator arrays is performed. The analysis focuses on obtaining a detailed understanding of the amplitude and phase distribution along the array for strongly coupled oscillators. Section II provides a brief introduction to COA theory. The oscillator array dynamics in the strongly coupled regime is presented. Section III provides computed and the measured amplitude and phase distributions. The measured mutual injection locking ranges are also presented in this section. The array amplitude variations result in undesired radiation patterns. Section IV shows a means of reducing the side lobe levels by suitably adjusting the oscillator amplitudes. The beam steering capabilities of strongly coupled arrays are also evaluated. Section V discusses the phase noise characteristics of strongly coupled arrays. The phase noise theory developed by Chang et al. [6], [7] assuming weak coupling, is used to understand the effect of increased coupling strength on the phase noise performance of a COA. Then, the improvement in the phase noise performance of strongly coupled arrays is explained by computing the quality factor of the coupling network for various coupling strengths. The measured array free-running and injection locked phase noise characteristics are presented.

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arrays. The amplitude and phase dynamic equations can be derived from (1) and are given as

(5a) and Fig. 1. An N-element coupled oscillator array.

(5b) II. AMPLITUDE AND PHASE DYNAMICS A. Array Dynamics Assuming Coupling Network is Broadband Previous analysis of coupled oscillator arrays like that in Fig. 1 have demonstrated that the amplitude and phase of the individual oscillators can be expressed as [2], [8]:

(1) where , and frequency of the

In (2), (4) and (5) wherever sign is used, the upper sign applies to series resonant oscillators while the lower sign applies to parallel resonant oscillators. Equation (5) is valid provided the oscillator and the coupling network satisfy the condition [8] (6) Applying this to a typical bilateral, nearest neighbor cou, where pling network (see Fig. 1) and setting can be expressed as [8]

are the amplitude, phase and free-running oscillator respectively, and

(2) and comprise the elements of the vectors The variables [A] and . The parameters in (2) are , the amplitude satuoscillator; ration factor; , the uncoupled amplitude of the , oscillator quality factor; , the load conductance; is and oscillators and , the coupling phase between the the admittance of the coupling network from port to port . as the coupling coefficient, (2) can Defining be simplified when the frequency dependence of the coupling network is much weaker than that of the oscillator, or (3)

and reduces (2)

(4) Condition (3) is the broadband condition defined in [8]. The left member of (3) appears in the denominator in (2), and neglecting it greatly simplifies the analysis of coupled oscillator

(7) is the magnitude of the coupling coefficient and is where termed the coupling strength. The coupling strength can be varied by changing the value of . The mutual injection locking range, which is defined as the maximum range by which the free-running frequencies of the oscillators can deviate collectively and still maintain a mutually locked state, is expressed as (8) results in a wider mutual injection locking From (8), a large causes an undesirable change in the range. However, a large array amplitudes [9] and presents a significant load to the oscilbetween the oscillator lator [10]. Strong coupling elements is desirable as long as other coupling features are not compromised. For in-phase synchronization, series resonant oscillators re, where n is odd and parallel quire a coupling phase, , where n is even [11]. The resonant oscillators require amplitude and phase dynamic equations (5) would specialize to

(9a) and (9b)

SEETHARAM AND PEARSON: ANALYSIS OF STRONG COUPLING IN COUPLED OSCILLATOR ARRAYS

Equation (9) is traditionally used to ascertain the amplitude and phase dynamics of COAs. These equations break down with large coupling factor, consistent with exceeding the limitations of the broadband condition.

VARIATION OF

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Q

TABLE I BROADBAND FACTOR WITH

AND THE



B. Array Dynamics Assuming Coupling Network is Not Broadband When the broadband condition (3) does not hold, a COA must be modeled with (2) including the resonant denominator term that the broadband condition obviates. This term bring the resonances of the coupling networks into play. Viz: assuming

otherwise (10) is the resonant frequency of the coupling network. where is a useful The quality factor of the coupling network parameter through which to view the networks’ influence. can be derived by applying the analogy of a resonant cavity [10]. Alternatively, can also be obtained from the denominator terms of (2) as

(11) For in-phase synchronization of the oscillators, using (10), (11) simplifies to

(12) Clearly, is directly proportional to the coupling strength. and the broadband factor Table I shows the variation of with for and . The is to introduce loss main purpose of the coupling resistor, . Decreased increases the and and thereby reduce thereby the inequality (3). The coupling resistors also act as a mode killer and suppress the undesired modes that arise in a strongly coupled array [3]. From Table I one observes that as the value of resistor is reduced to increase the coupling strength, also increases. The lowered loading causes the coupling network to become more highly resonant resulting in the breakdown of the broadband condition. When the broadband factor value is of order 1 and larger, (3) does not hold. From Table I, and is one observes that (3) starts to break down around . clearly violated for

Fig. 2. Photograph of the five-element COA. The top ports are the outputs, and the lower ports are injection points.

The resonant nature of the coupling network complicates the interaction of strongly coupled oscillators. Therefore, (9) does not present an accurate representation of the amplitude and phase of the oscillator elements when the broadband condition is invalid. An accurate estimate of the behavior of the oscillator array under strong coupling conditions can be obtained by deriving the amplitude and phase dynamics equations without using (3). We use (7) and (12) in (2) and note that for in-phase synchro, where is even for parallel resonant oscilnization lators and odd for series resonant oscillators, the amplitude and phase dynamics equations result from (1) as in (13a) and (13b) shown on the following page. III. EXPERIMENTAL VERIFICATION The theoretical amplitude and phase distributions are verified on a five-element coupled oscillator array, which is built based on the oscillator cell design presented in [12]. The oscillators are coupled with 50- transmission lines resistively loaded with chip resistors (see Fig. 1) and a coupling phase of . The array is fabricated on 0.635-mm thick Rogers TMM 10 board. A photograph of the array is shown in Fig. 2. Injection ports introduced at each element are utilized to measure the inter-element phase difference. Output signals are obtained by using a vector network analyzer configured for S21 measurement. The given injection port is taken as “port 1” in the measurement and supplied with the signal from the network analyzer. An oscillator output port is taken as “port 2.” All other output ports are terminated with matched loads. Each output is measured in

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succession by changing the network analyzer connection and the terminations. The amplitude distribution across the array is measured for 10 , 20 and 30 inter-element phase differences using a network analyzer. Fig. 3 shows the computed and for measured array amplitude variation for various values of 20 inter-element phase difference. The oscillator amplitudes are normalized with respect to the measured amplifor each tude of the center element. Equation (13a) is used to compute the array amplitude variation. The computed and measured amplitudes are within a dB of each other. No amplifiers or attenuators were connected to the oscillators for these measurements. The mutual injection locking range exhibited by two nearest neighboring elements for various coupling strengths is determined using the procedure detailed in [12]. Table II presents the

measured MILR exhibited by the oscillators for various coupling strengths. One observes that the MILR increases proportionately with the coupling strength. For , the MILR exhibited by oscillator #5 is about 750 MHz or 10% of 3.75 GHz. To the authors’ knowledge, this is the largest MILR reported in literature to date. IV. RADIATION PATTERNS AND BEAM STEERING The amplitude variations exhibited in Fig. 3 lead to poor radiation patterns. Since the end oscillators have greater amplitudes than the interior oscillators, when such an array is employed in a phased array system, the side lobe level is undesirably large. A better radiation pattern can be obtained by introducing an amplitude taper wherein the end oscillator elements have the lowest

(13a)

(13b)

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TABLE II MUTUAL INJECTION LOCKING RANGE EXHIBITED BY THE OSCILLATORS FOR VARIOUS COUPLING STRENGTHS

=05

=1

=15

Fig. 3. Amplitude variation with  (a)  : , (b)  , (c)  : , , (e)  : . The solid and dashed lines represent measured and (d)  computed data respectively.

=2

=25

amplitude in the array. In our experimental work, this is accomplished by amplifying each VCO output with a Mini-Circuits ERA-1SM+ buffer amplifier and coaxial attenuators were used to trim the output power. Phase relationships were maintained by using equal numbers of attenuators in all paths. In practice, one might employ amplifiers with trimmable gain so that efficiency is maintained. The trimmed power levels are delivered to

a patch antenna, 17.8 mm wide and 12.5 mm long, which employed a quarter wave transformer matching section to present a 50- load to the oscillator. The antenna spacing is 29.5 mm at the oscillation frequency. The buffer which is about 0.4 amplifier and patch elements are realized on a separate Rogers TMM 10 board. Fig. 4 shows the photograph of the oscillator array connected to the patch antennas. Figs. 5–9 show the measured array broadside and steered rawith and without the amplitude diation patterns for various taper. The buffer amplifiers are driven by the oscillator outputs (see Fig. 4) and the radiative coupling from the antennas disturbs the symmetry that is present in Fig. 3. Table III displays the radiated amplitudes with and without the taper. Table IV shows the maximum beam steering from broadside exhibited by the COA for different coupling strengths. The beam steering is limited by the mutual injection locking range. It is noteworthy that the MILR limit was not encountered. The tuning for

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Fig. 4. Photograph of the oscillator elements connected to the patch antennas. Oscillator cells, with amplifiers reside in the box below. Attenuators of chosen value connect the outputs of the cells to the patches through flexible cables.

Fig. 5. Radiation patterns before and after amplitude tapering for 

Fig. 6. Radiation patterns before and after amplitude tapering for 

= 0:5.

Fig. 7. Radiation patterns before and after amplitude tapering for 

= 1:5.

Fig. 8. Radiation patterns before and after amplitude tapering for 

= 2.

Fig. 9. Radiation patterns before and after amplitude tapering for 

= 2:5.

= 1.

range of the end oscillators (#1 and #5) was saturated before lock was broken. Consequently, the last numbers in Table IV do not relate monotonically to the remainder of the table.

V. PHASE NOISE CHARACTERISTICS OF STRONGLY COUPLED OSCILLATOR ARRAYS Assuming weak coupling between the oscillator elements element COA injection locked to an the phase noise of an

SEETHARAM AND PEARSON: ANALYSIS OF STRONG COUPLING IN COUPLED OSCILLATOR ARRAYS

TABLE III MEASURED AMPLITUDES WITH AND WITHOUT AMPLITUDE TAPER. THE AMPLITUDES ARE MEASURED AT A DISTANCE OF 2.67 m FROM THE RADIATING PATCHES

TABLE IV MEASURED BEAM STEERING OFF BROADSIDE FOR VARIOUS 

is the power spectral density of the oscillator’s where is the phase noise of the external quadrate noise source, is a vector of the relative strengths of the injection source, is a matrix representing the coupling topology. signals and For the free running COA, the second term on the right hand side for the free running and of (14) becomes zero. The elements injected arrays are computed from the inverse of matrix given in (15) and (16), respectively, shown at the bottom of the page, and , [6], [7] where . see (16) where Using (15) and (16) in (14) it is clear that within the weakly coupled regime, the phase noise performance of a COA imincreases. Analysis of the phase noise of strongly proves as coupled arrays on similar lines is extremely complicated due to the resonant nature of the coupling network. However the phase noise performance of strongly coupled oscillator arrays can be . The increase in with understood by observing (see Table I) can be interpreted as an increase in the overall of the COA system, which results in the improvement in the phase noise performance of the COA. The phase noise characteristics of the five-element COA is measured using an Aeroflex PN9000B phase noise measurement system. An Agilent E4433B, ESG-D signal generator is used as the external injection source. The oscillators’ output power was 10 dBm, and for injection signal strength of 45 dBm, the injection ratio is calculated as

external source is given by [7]

(14)

.. .

..

.

..

.

..

.

..

.

.. .. .. .

..

.

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.

.

..

.

..

.

.. .

(15)

.. . (16)

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the phase noise performance is usually obtained at the expense of mutual injection locking range and vice versa. Tapering of the array amplitude distribution is utilized to achieve reduction in the side lobe level at the expense of beam directivity and effective isotropic radiated power. REFERENCES

Fig. 10. Variation of array free running phase noise with  .

Fig. 11. Variation of array phase noise with  for 

= 045 dBm.

. is the loss associated with the cables and the injection port. The array was set for zero phase difference. Fig. 10 and Fig. 11 show the measured phase noise characteristics of the free running and injected array respectively for in the strongly coupled regime. It is clear different values of from Fig. 10 and Fig. 11 that the phase noise performance of the array improves with increase in . A 25 dB improvement in the free running array phase noise is observed on increasing from 0.5 to 2.5. Moreover, a higher causes the injected array to operate closer to the noise floor. Hence, strongly coupled oscillator arrays can operate almost at the noise floor with less injection signal power than weakly coupled arrays. VI. CONCLUSION In this paper, the behavior of coupled oscillator arrays in the strongly coupled regime has been analyzed. For large coupling strengths, the resonant nature of the coupling network increases resulting in the breakdown of the broadband condition. A five-element array is fabricated and significant improvement in the mutual injection locking range and phase noise performance has been shown. This is important since improvement in

[1] X. Wang and L. W. Pearson, “Design of coupled-oscillator arrays without a posteriori tuning,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 410–413, Jan. 2005. [2] R. A. York, “Nonlinear analysis of phase relationships in quasi-optical oscillator arrays,” IEEE Trans. Microw. Theory Tech., vol. 41, pp. 1799–1807, Oct. 1993. [3] S. Nogi, J. Lin, and T. Itoh, “Mode analysis and stabilization of a spatial power combining array with strongly coupled oscillators,” IEEE Trans. Microw. Theory Tech., vol. 41, pp. 1827–1837, Oct. 1993. [4] J. J. Lynch and R. A. York, “Synchronization of oscillators coupled through narrow-band networks,” IEEE Trans. Microw. Theory Tech., vol. 49, pp. 237–249, Feb. 2001. [5] A. Georgiadis, A. Collado, and A. Suarez, “New techniques for the analysis and design of coupled-oscillator systems,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 11, pp. 3864–3877, Nov. 2006. [6] H. C. Chang, X. Cao, U. K. Mishra, and R. A. York, “Phase noise in coupled oscillators: Theory and experiment,” IEEE Trans. Microw. Theory Tech., vol. 45, pp. 604–615, May 1997. [7] H. C. Chang, X. Cao, M. J. Vaughan, U. K. Mishra, and R. A. York, “Phase noise in externally injected-locked oscillator arrays,” IEEE Trans. Microw. Theory Tech., vol. 45, pp. 2035–2042, Nov. 1997. [8] R. A. York, P. Liao, and J. J. Lynch, “Oscillator array dynamics with broadband N-port coupling networks,” IEEE Trans. Microw. Theory Tech., vol. 42, pp. 2035–2042, Nov. 1994. [9] J. Shen, “A Study of the Design of Coupled Oscillator Phased Arrays,” Ph.D. dissertation, Dept. Elect. Comp. Eng., Clemson University, , SC, 2002. [10] R. J. Pogorzelski, “On the design of coupling networks for coupled oscillator arrays,” IEEE Trans. Antennas Propag., vol. 51, pp. 794–801, Apr. 2003. [11] H. C. Chang, E. S. Shapiro, and R. A. York, “Influence of the oscillator equivalent circuit on the stable modes of parallel-coupled oscillators,” IEEE Trans. Microw. Theory Tech., vol. 45, pp. 1232–1239, Aug. 1997. [12] C. Tompkins, V. Seetharam, and L. W. Pearson, “Improved mutual injection locking range for VCOs in a coupled oscillator system,” in IEEE Aerospace Conf., Mar. 2006. Venkatesh Seetharam received the B.S. degree in electrical engineering from Shanmugha College of Engineering, India, in May 2002, and the M.S. and Ph.D. degrees from Clemson University, Clemson, SC, in 2005 and 2007, respectively. He is currently an Application Engineer at Ansoft LLC, San Jose, CA. His research interests include coupled oscillator array systems, phased array antenna systems, signal integrity, EMI and EMC. Dr. Seetharam was the recipient of the 2007 Harris Outstanding Researcher Award from the Holcombe Department of Electrical and Computer Engineering, Clemson University.

L. Wilson Pearson (F’91) received the B.S. and M.S. degrees from the University of Mississippi, University, and the Ph.D. degree from the University of Illinois, Urbana. Currently, he is the Samuel R. Rhodes Professor of Electrical and Computer Engineering at Clemson University, Clemson, SC, and Director of the University’s Center for Research in Wireless Communications. He has served on the faculties of the University of Kentucky, the University of Mississippi, as an Affiliate Professor at Washington University, and as Senior Scientist, then Principal Scientist in Electromagnetic Signatures at the McDonnell Douglas Research Laboratories, St. Louis, MO. He has also held a position at the Naval Surface Weapons Center. His research interests include antennas, millimeter wave systems, coupled oscillator arrays, asymptotic methods

SEETHARAM AND PEARSON: ANALYSIS OF STRONG COUPLING IN COUPLED OSCILLATOR ARRAYS

in electromagnetics, and software defined radio. Application areas include communications, spatial power combining, and low-cost phased-array antennas. Dr. Pearson is a Fellow of the IEEE and a member of Commissions B and D of the U.S. National Committee of the International Union for Radio Science (URSI). He has, in the past, served on the Administrative Committee for the IEEE Antennas and Propagation Society, as Secretary of the Electromagnetics Society, as the Technical Program Committee Chair for the 1998 International Symposium on Antennas and Propagation/URSI National Radio Science Meeting, and as Head of the Department of Electrical and Computer Engi-

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neering Department at Clemson University. He has served as Editor-in-Chief for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and on the Editorial Board of the IEEE PROCEEDINGS. He served as Vice President, then President, of the IEEE Antennas and Propagation Society in 2002 and 2003, respectively. He served as Chair of USNC-URSI Commission D between 2005–2008. He is a recipient of an IEEE Third Millennium Medal, the (Clemson University) Provost’s Award for Scholarly Excellence, and the McQueen Quattlebaum Faculty Achievement Award.

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Compact Elongated Mushroom (EM)-EBG Structure for Enhancement of Patch Antenna Array Performances Martin Coulombe, Sadegh Farzaneh Koodiani, Member, IEEE, and Christophe Caloz, Senior Member, IEEE

Abstract—A compact elongated mushroom electromagnetic band-gap (EM-EBG) structure, exploiting the thickness of the substrate to achieve higher isolation compared to the case of the conventional mushroom EBG (CM-EBG), is proposed for the enhancement of the performances of patch antenna arrays. Guidelines, based on fairly accurate formulas, are provided for the design of the stop-band of the structure. The compactness of the EM-EBG is investigated and shown to be superior to that of the CM-EBG in practical cases. The superior reduction of mutual coupling of the EM-EBG is demonstrated by full-wave and experimental results both for a pair of patches and for a 4-element patch array. Finally, the benefits of the EM-EBG in terms of array processing, specifically side-lobe level (SLL) control and direction of arrival (DOA) estimation, are presented. Index Terms—Antennas arrays, array processing, direction of arrival (DOA), electromagnetic band-gap (EBG) structures, microstrip patch antennas, mutual coupling, side-lobe level (SLL).

I. INTRODUCTION LECTROMAGNETIC-bandgap (EBG) structures consist of periodic dielectric or metal elements. Their major characteristic is to exhibit a frequency band structure with passbands and stop bands (or band gaps), analogous to the energy band structure of crystals in solid state physics. They can be 2.5D structures, as the mushroom EBG presented in [1] that we will here refer to as the conventional mushroom EBG (CMEBG), 2D structures, as the uniplanar compact EBG (UC-EBG) presented in [2], or even 3D structures, as summarized in [3].

E

Manuscript received March 06, 2009; revised August 04, 2009; accepted October 01, 2009. Date of publication January 22, 2010; date of current version April 07, 2010. M. Coulombe was with École Polytechnique de Montréal, Poly-Grames Research Center, Centre de Recherche en Électronique Radiofréquence (CREER), Montréal, QC H3T 1J4, Canada. He is now with Infield Scientific Pointe-Claire, Quebec H9R 1A3, Canada (e-mail: [email protected]; coulombe. [email protected]). S. Farzaneh Koodiani was with École Polytechnique de Montréal, Poly-Grames Research Center, Centre de Recherche en Électronique Radiofréquence (CREER), Montréal, QC H3T 1J4, Canada. He is now with SDP Components Inc., Dorval, QC H9P 1J1, Canada (e-mail: [email protected]). C. Caloz is with École Polytechnique de Montréal, Poly-Grames Research Center, Centre de Recherche en Électronique Radiofréquence (CREER), Montréal, QC H3T 1J4, 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.2010.2041152

Typical applications of EBGs include high-impedance surface ground planes [1] and mutual coupling suppressors in antenna arrays [4], [5]. Mutual coupling degrades the performances of an antenna array in array processing applications, such as side-lobe-level (SLL) control and direction of arrival (DOA) estimation. Some methods are available to partially compensate for the effects of mutual coupling, but they imply additional processing with subsequent increase in processor complexity and decrease in data rate. Moreover, such compensation methods require several preliminary measurements, which may be unpractical in real-time applications. It is therefore generally desirable to reduce mutual coupling at the level of the antenna structure prior to the signal processor. In general, suppression of mutual coupling strongly enhances the performances of antenna arrays, as demonstrated for instance in [5]–[9]. Mutual coupling reduction may be achieved by inserting an EBG between the elements of the antenna array. However, the CM-EBG inherent drawback that its unit cells tend to be electrically too large to be really efficient in antennas array applications. Several efforts have been made to reduce the size of the CM-EBG unit cell more [11]–[13]. However, there is still room for improvement. As a further contribution to this effort, this paper presents a novel 2.5D-EBG structure: the elongated mushroom (EM)EBG. The EM-EBG exploits the vertical dimension of the substrate, assumed electrically relatively thick, to strongly enhance the capacitance per unit cell, and henceforth reduce the resonance (gap) frequency of the structure. After rescaling the EBG to the initial design frequency, the unit cell size is reduced, which allows to fit a larger number of unit cells between the antenna array elements in order to provide stronger mutual coupling reduction. Section II describes and motivates the proposed EM-EBG. Section III presents a prototype, its design procedure, and the corresponding dispersion diagram. Section IV compares the EM-EBG with the CM-EBG in terms of center gap frequency (i.e., compactness of the unit cell) versus the period and height of the structures. Section V demonstrates, by full-wave analysis and experiment, the mutual coupling reduction achieved by the EM-EBG between two patch antennas, in comparison with the cases of the CM-EBG and of a conducting block occupying the volume of the EM-EBG between the patches. Section VI presents a 4-element patch array with both CM-EBG and EM-EBG mutual coupling reduction. The benefits of the

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compactness. Specifically, it is crucial in the common application of mutual coupling mitigation in antenna arrays [4]–[9], as it will be demonstrated in Sections VI and VII. In this case, the maximal allowed inter-element spacing is typically set to , where is the free space wavelength, to avoid grating lobes [18], [19]. This constraint limits the number EBG cells which can be inserted between the array elements. The CM-EBG structure’s unit cell size, for a microstrip structure, , where is the effective (or guided) is in the order of wavelength. The number of EBG unit cells fitting within the available spacing between two elements directly depends on the effective refractive index of the substrate. In fact, is significantly smaller than , because of the space used by the patches within the array period and a minimal distance required between the EBG and the patches on each side of the EBG. Under the reasonable assumption that this distance must . For be at least one EBG period, we have a given EBG whose unit cell size is times smaller than , , the maximal allocable number of unit cells is

(1) Fig. 1. Proposed elongated-mushroom (EM)-EBG structure. (a) Perspective view. (b) Cross sectional view of the unit cell. (c) Top view of the unit cell. (d) Simplified circuit model.

EM-EBG for array processing applications in terms of scanning and SLL control and DOA estimation are presented in Section VII. Finally, conclusions are provided in Section VIII. II. STRUCTURE DESCRIPTION AND MOTIVATION Fig. 1 shows the proposed EM-EBG structure with its design parameters. It consists of a periodic array of metal via holes within a host dielectric grounded substrate. These vias have a smaller diameter at their lower part, which is connected to a ground plane, and a larger diameter at their higher part, which extends up to the top of the substrate. The terminology EM-EBG is motivated by the shape of these double-diameter vias, which look like “elongated mushrooms” (EM) in reference to CM-EBG mushroom elements [1], [4]. The main parameters are the small and large via diameters, and , the and , leading to a total substrate corresponding heights, , and the unit cell period . Compared thickness of to the CM-EBG, the EM-EBG includes the additional degrees of freedom and and it offers therefore more flexibility for the design of the band-gap location and bandwidth. The idea with the proposed EM-EBG is to exploit the height of the substrate to compress the lateral size of the unit cells, so as to fit a larger number of them within a given electrical area, and thereby achieve higher stop-band attenuation per guided wavelength. Such a compression is beneficial in general for circuit

, we have therefore for typical In a CM-EBG, where , 2.2, 10.2, (i.e., substrates permittivities (i.e., also no EBG), (i.e., no EBG), EBG with at most 3 cells), respectively, where is the effective permittivity of a microstrip line with the width of the patch antenna. Clearly, an EBG with a sufficient number of cells—3 as a strict minimum—requires a high substrate permittivity. In any case, increasing (i.e., compressing the unit cell size) is required to accommodate an EBG with sufficient number of cells for sufficiently strong attenuation to mitigate mutual coupling. For instance, if is doubled in the above examples , we have (i.e., no EBG), (i.e., no EBG), (i.e., 8 cells at most), respectively. In this case, a high permittivity substrate is still required, but up to 8 cells may fit between the elements, leading to a much stronger mutual coupling reduction. This benefit will be demonstrated in Sections V and VI. Fig. 1(d) shows the approximate circuit model for the EM-EBG. This model is identical to that of the CM-EBG. It elements, which resonate, so as to consists of cascaded open up a Bragg stop-band, at the frequency [1], [4]. The difference between the EM-EBG and the CM-EBG is mostly quantitative, since the latter may be seen as the limit with . However, by case of the former for using elongated mushroom tops, the EM-EBG benefits of a dramatic capacitance enhancement, because the top surfaces provide much higher electric flux compared to the patch edges is kept the of the CM-EBG. If the total substrate height same as in the CM-EBG, the inductance is slightly reduced “mushroom bottom”. However, because of the shorter this inductance reduction is minor compared to the capacitance

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Fig. 2. Photograph of the fabricated EM-EBG (Fig. 1) prototype. The param: ,h h : : ,p ,D eters are: h : : : . (a) Top view (from the side and d , and " of the large hole and interface with air). (b) Bottom view (from the side of the small hole and ground plane).

1 296 mm

= 2 54 mm = = 0 508 mm

= 1 27 mm = 10 2

= 1 55 mm

=

increase. Thus, the product is strongly increased, which leads, for a reasonably thick substrate, to significant reduction in , or equivalently, in the size of the unit cell. Quantitative results will be provided in Section IV. III. PROTOTYPE, DESIGN AND DISPERSION DIAGRAM Fig. 2 shows an EM-EBG prototype. This prototype has the , following parameters (Fig. 1): , , and , , which are used in all forthcoming experimental and results. This EM-EBG is an integrated and monolayer structure, which is fabricated by a simple and low-cost process. This process, starting from a grounded substrate, is as the following: are drilled from the top of i) First, the large holes of height the substrate by a computer numerical control (CNC) machine; using a ii) next, the CNC drills the bottom holes of height smaller drill bit; iii) finally, the resulting double-diameter holes are copper-plated from both sides using electro-deposition on the walls of the holes. If the process is done for the first time or is not fully mastered, it is recommended to check all the connections between the via tops and the ground. The inductance and capacitance of the EM-EBG’s circuit model, shown in Fig. 1(d), are approximately given by [4], [14]

(2) and [20]

(3) respectively. The band-gap center frequency of the structure is then determined, as mentioned in Section II, as

. The band-gap frequency bandwidth is [1], [4], approximately given by . [14], and is thus proportional to the impedance The EM-EBG structure is initially designed to exhibit a stopband at a frequency of around 5.8 GHz, for the sake of comparison with available literature results for the CM-EBG [4]. The design procedure starts by setting , which sets the product. For optimal plating per, which sets the according formances, we use . The corresponding values are to (2), and therefore from and . This value for sets by (3) . From this initial design, based on the approximate formulas (2) and (3), full-wave analysis using the eigen-mode solver of CST Microwave Studio is carried out for fine tuning. At this stage, anticipation for the application of mutual coupling suppression, which is presented in the forthcoming sections, is in order. The band-gap for an EBG of finite size is slightly different from that of the same EBG with infinite size. Empirical full-wave simulations in Section V reveal that a finite-size gap center frequency of corresponds to an infinite-size . This yields , which completes the design with , where is set to 0.254 mm following fabrication constraints, leading to the final parameter values given in Fig. 2. The corresponding dispersion diagram is shown in Fig. 3, where the EM-EBG gap extends over the frequency range from 5.34 to 5.92 GHz. The EM-EBG exhibits a band structure which is qualitatively identical to that of the CM-EBG, with a lower TM band and a higher TE band [1]. The EM-EBG has a fractional band. This gap corresponds width of 10.3% region of the dispersion diagram, the guided-wave excluding leaky-wave region of the TE mode, where the wave radiates and is therefore strongly attenuated along the structure. In comparison, the CM-EBG gap extends over the frequency range from 5.21 to 6.06 GHz, yielding a fractional band, which is larger than width of 15.1% ratio. that of the EM-EBG, as expected from the larger that, by strongly increasing We see from , in addition to slightly decreasing , the EM-EBG decreases the bandwidth compared to the CM-EBG. Nevertheless, the resulting bandwidth, which is of 10.3% here, is still largely sufficient for the vast majority of applications. Specifically, in the case of microstrip patch antenna arrays, the bandwidth of the system is limited by the bandwidth of the patch antennas, which typically does not exceed 5%. There is thus a comfortable margin in terms of the bandwidth. (For the case of the relatively thick substrate used here, the patch antenna 10-dB bandwidth is of 4.1%, from Fig. 8(a) and (b)). IV. COMPARISON WITH CONVENTIONAL MUSHROOM EBG The CM-EBG structure is composed of square metal patches connected to vertical conducting vias and may be seen as a parand ), as pointed ticular case of the EM-EBG ( out in Section III. The values of and for this structure are approximately given by [4], [14] (4)

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Fig. 4. Comparison of gap center frequency f versus unit cell period p for the CM-EBG (unit cell with square patch connected to a single via) and the : . EM-EBG structures (Figs. 1 and 2) with a substrate thickness h

= 2 54 mm

Fig. 3. Dispersion diagram of the proposed EM-EBG structure (Figs. 1 and 2) compared with that of the CM-EBG structure computed by CST Microwave Studio for the parameters of the prototype shown in Fig. 2. (a) First two modes. (b) Zoom on the region of the guided-wave band-gap between these two modes.

and

(5) respectively, where is the width of the metal patch and is the gap between the patches. In all forthcoming results, the substrate and total height are kept the same for permittivity the CM-EBG and EM-EBG structures for proper comparison. Fig. 4 presents a parametric comparison for the gap center frequency versus the unit cell period for the constant sub( for the strate thickness EM-EBG). As expected from (2) and (4), decreases when is increased for both structures. The graph shows that for a given , is always (full-wave results) smaller for the EM-EBG or, equivalently, that for a given , is always smaller, which proves that the EM-EBG unit cell is more compact than the conventional one. For a central band-gap frequency of 5.63 GHz, (point B in the the CM-EBG requires a period of is found for graph), while the smaller period of the EB-EBG (point A in the graph), which corresponds to a size compression of 2.18 with the EM-EBG (close to the factor of 2 used in the examples of Section II). The formulas (2)–(5) are

Fig. 5. Comparison of gap center frequency f versus substrate thickness h for : and the EM-EBG structure the CM-EBG structure with period p (Figs. 1 and 2) with p , where the respective periods have been : chosen to provide the same center gap frequency, f : , for a thickness h of 2.54 mm, according to Fig. 4.

= 1 55 mm

= 3 38 mm

= 5 63 GHz

seen to provide relatively good approximations, which justifies their utilization as design initial guesses. Fig. 5 presents a parametric comparison for the gap center versus the substrate thickness for fixed periods frequency at , which corresponds to the yielding the same for the EM-EBG). As expected prototype thickness ( from (2), (3) and (4), decreases when is increased for both structures. The graph shows the following: i) the unit cell size reduction in the EM-EBG requires a thickness larger than a given threshold (here 2.54 mm for 5.63 GHz, where the EM-EBG period is 2.18 times smaller); ii) the size compression gain of the EM-EBG increases as is increased. Fig. 6 presents a more general perspective to show the versus simultaneous variations trends for the evolution of of (Fig. 4) and (Fig. 5). In order to better assess the unit cell compression ratio achieved by the EM-EBG compared , to the CM-EBG, this figure displays the ratio which corresponds to the unit cell size compression ratio (still for the EM-EBG). Values higher than one pairs for which the EM-EBG structure is corresponds to more compact than the CM-EBG structure. This graph shows parameters were that there exists a large area of realizable the EM-EBG is more compact. If thicker EM-EBGs, compared

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Fig. 7. Setups of the microstrip patch antennas with EM-EBG separation to determine the level of reduction of mutual coupling achieved by the EM-EBG structure. The antennas are assumed to be distant by 0:5 as in a realistic antenna array scenario. The patches are on the side of the large holes. (a) E-plane coupling. (b) H-plane coupling. Fig. 6. Ratio between the CM-EBG and the EM-EBG structures gap center frequencies (f =f ) versus substrate thickness h and unit cell period p using (2)–(5). Values higher than 1 indicate (h; p) parameter regions where the EM-EBG is more compact (smaller operation frequency) than the CM-EBG, and the plotted ratio is a measure of the compactness superiority of the EM-EBG compared to the CM-EBG.

to the fabricated one with , can be engineered, a very significant additional size compression gain could be achieved, still for thicknesses negligible compared to the later dimensions of the overall EM-EBG structures within a given circuit board. V. REDUCTION OF MUTUAL COUPLING The main factors impacting on mutual coupling in microstrip structures are the substrate dielectric constant, the substrate thickness, and the distance between the strips, which are typically patches in antenna array applications [15]–[17]. Mutual coupling occurs both in terms of surface waves (the , has a zero cutoff frequency and is therefirst mode, fore always present) and in terms of space (radiated) waves, where the former become increasingly dominant over the latter when the electric thickness of the substrate is increased. In antenna arrays, mutual coupling deteriorates the performance in different ways, including SLL and beam shape degradation, input impedance mismatch, grating lobes generation, and scan blindness. When the substrate is electrical thin, only the surface wave is excited, and mutual coupling is relatively moderate. On the other hand, when the substrate is relatively thick and has a relatively high permittivity, mutual coupling becomes very significant and must be reduced. In this case, it is stronger in the E-plane than in the H-plane, since surface waves

are launched mostly along the E-plane direction, as shown in [4] for a substrate with and . Fig. 7 shows the setups used for the evaluation of the mutual coupling in the E-plane [Fig. 7(a)] and in the H-plane [Fig. 7(b)] between two patch antennas separated by an EM-EBG structure. Corresponding full-wave results, compared with the cases without EBG, with the CM-EBG and with a PEC block completely occupying the EBG volume, are shown in Figs. 8 and 9 . for a center-to-center antenna elements spacing of Consistently with previously reported results [4], it is observed that mutual coupling without EBG is stronger in the E-plane than in the H-plane [Figs. 8(c) and 9(c)]. We will therefore focus on the E-plane case in the experimental results. For the E-plane case (Fig. 8), the antennas are matched at the frequency where the two EBGs provide maximum coupling reduction, i.e., at 5.87 GHz [Fig. 8(c)]. The slight difference [Fig. 8(a)] and [Fig. 8(b)] are due observed between to the asymmetry in the excitations of the two patches. The double resonance observed in Fig. 8(a) and (b) is explained as follows. The field distributions on the patches at both resonances were observed to be essentially identical, both corresponding to the regular patch antenna operation. The field distributions at the maxima of the reflection coefficient between the resonances were observed to also correspond to the patch antenna operation. However, coupling is much stronger at the local maximum, as seen in the coupling peak of Fig. 8(c). Since the antenna was matched for the situation of minimal mutual coupling (in fact only at the first resonance point), it cannot be matched at the same time for the situation of strong coupling, where the EBG

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Fig. 8. Comparison of full-wave simulated (Ansoft HFSS) scattering parameters for an E-plane coupled antenna pair as shown in Fig. 7(a) for four cases: without EBG, with a PEC block occupying the entire volume of the EM-EBG region (w l = 10:85 20:25 = 219:71 mm ), with CM-EBG (3 7 unit cells w l = 9 21 = 189 mm , p = 3 mm), and with the EM-EBG (7 15 unit cells w l = 10:85 20:25 = 219:71 mm , p = 1:55 mm). The overall thickness is always set to h = 2:54 mm, and " = 10:2. The sizes of the PEC block, CM-EBG and EM-EBG areas are almost identical, except that the CM-EBG structure can fit only 3 unit cells within the antenna spacing of s =  =2. (a) Reflection coefficient S . (b) Reflection coefficient S . (c) Mutual coupling S .

2

2

2

2 2

2

2

2

structure loading alters the input impedance. The second resonance is less pronounced than the first one due to higher coupling. The coupling levels without EBG, with PEC block, with CM-EBG and with EM-EBG are of 16.1 dB, 18.3 dB, 23.4 dB and 31.1 dB, respectively. The first observation is that both EBGs perform better than the PEC block. Intuitively,

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Fig. 9. Compared full-wave simulated (Ansoft HFSS) scattering parameters for an H-plane coupled antenna pair as shown in Fig. 7(b) for the same cases and for the same parameters as in Fig. 8. (a) Reflection coefficient S . (b) Reflection coefficient S . (c) Mutual coupling S .

this may be understood from the fact that that the PEC block, with its sharp discontinuities, diffracts a significant part of the energy from the excited patch over the block to the other patch, whereas the EBGs progressively attenuate the energy within its cells like kinds of “electromagnetic sponges,” as clearly apparent in Fig. 10. The EM-EBG provides a mutual coupling reduction of 15 dB compared to the case without EBG. Moreover, it provides a mutual coupling reduction larger by around 7.7 dB compared to the CM-EBG, as expected from its much larger number of unit cells (7 versus 3 along the propagation direction). For the H-plane (Fig. 9), the differences in mutual

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Fig. 10. Full-wave HFSS simulated electric near-field (just on top of the substrate) magnitude for the setup of Fig. 7(a) (E-plane) at 5.87 GHz, corresponding to the mutual coupling of Fig. 8(c). (a) Without EBG. (b) With PEC block. (c) With EM-EBG structure.

coupling are less pronounced, with still a better isolation by 3 dB for the EM-EBG compared to the CM-EBG. However, as previously pointed, the coupling levels are much lower, and therefore, H-plane coupling is much less critical for the overall antenna performances. A strong transmission peak, producing a coupling even stronger than in the case without EBG (by 10 dB), is observed at 6.1 GHz in Fig. 9(c). This peak is located at the upper limit of the EM-EBG stop-band, as seen in Fig. 3, on the TE mode of the structure. Its presence is thus explained by the excitation of this mode, as confirmed by full-wave simulations (not shown). More specifically, this peak is located at the intersection with the air line, and where the TE mode behaves as a leaky-wave , where is the with grazing angle, angle measured from the normal to the structure (broadside). So, at this point, coupling occurs via the TE leaky-mode, but inside the EBG substrate and via the air, which explains the maximum. At higher frequencies, the TE mode is purely guided (Fig. 3) and therefore air propagation is mitigated, which explains why the coupling level gets smaller. The reason why the transmission peak is much stronger in the H-plane than in the E-plane coupled antennas is the following. For a TE to propagate along the direction (Fig. 7), the wave . electric field must be perpendicular to , and therefore This corresponds to the co-polarization of the patch antenna in the H-plane and to its cross-polarization in the E-plane, which explains why the coupling is much stronger in the former case. Fig. 11 shows photographs of the E-plane coupled 2-antenna array corresponding to Fig. 7(a) for both the CM EBG [Fig. 11(a)] and EM EBG [Fig. 11(b)]. The corresponding experimental results are shown in Fig. 12. Compared to the full-wave results of Fig. 8, these experimental reflection coefficients have their two resonance frequencies, which correspond to the quasi-square patches, closer to each other, probably due to some fabrication tolerances. Nevertheless, satisfactory matching is achieved (over 15 dB) at the resonance of interest, which is 5.63 GHz. This frequency is slightly smaller than the simulated frequency in Fig. 8 (5.87 GHz), which may be explained by tolerance in the permittivity of substrate and by

Fig. 11. Photograph of the microstrip patch antennas with EBG separation in the case of E-plane coupling, as in Fig. 7(a), designed at 5.8 GHz, within the gap of the EBG structure (Fig. 3). All the parameters are identical to those in Fig. 8. (a) CM EBG. (b) EM EBG.

fabrication imperfections. The coupling levels without EBG, with CM-EBG and with EM-EBG are of 14.2 dB, 21.1 dB, and 27.3 dB, respectively. The EM-EBG provides a mutual coupling reduction of 13.1 dB [15 dB in full-wave results, Fig. 8(c)] compared to the case without EBG. Moreover, it provides a mutual coupling reduction larger by 6.2 dB [7.7 dB in full-wave results, Fig. 8(c)], which confirms the benefit of accommodating a larger number of unit cells between the patches offered by the EM-EBG structure. VI. ANTENNA ARRAY WITH EBG 1 elements), with Fig. 13 shows the array antenna (4 E-plane aligned elements and isolating EM-EBG structures, which will be considered in forthcoming experimental results. The scanning angle, which will be used in Section VII, is indicated as the elevation angle in the plane of the array. Fig. 14 shows the experimental prototypes, which include the array without EBG [Fig. 14(a)], with CM-EBG [Fig. 14(b)], and with EM-EBG [Fig. 14(c)]. The measured results for the three prototypes of Fig. 14 are presented in Figs. 15 (reflection coefficient) and 16 (mutual coupling). The distance between the antennas are 25.85 mm for the three cases, which corresponds the half a free space wavelength for the EM-EBG and without EBG cases, and (precisely 0. for the CM-EBG case) to avoid grating lobes for all the scanning angles [15]. Fig. 15 shows that the four antennas could be reasonably matched at the stop-band of the EBGs. Fig. 16 shows that in most cases the EM-EBG provides more isolation than the CM-EBG, except for the lowest coupling levels (between antennas at the extremities of the array), where the levels are negligible for both structures (below 30 dB), compared to

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2

Fig. 13. Array of 4 1 E-plane aligned microstrip antenna elements with isolating EM-EBG structures.

Fig. 14. Photography of fabricated prototypes corresponding to Fig. 13, with the parameters of Fig. 8. (a) Without EBG structure. (b) With CM-EBG structure. (c) With EM-EBG structure.

Fig. 12. Compared measured scattering parameters for the E-plane coupled antenna pair shown in Fig. 8 for the cases without and with EM-EBG with the same dimensions as in Fig. 11. (a) Reflection coefficient S . (b) Reflection coefficient S . (c) Mutual coupling S .

the case without any EBG (around 18 dB). The EM-EBG provides isolation better than 25 dB in all cases. VII. IMPACT OF EBG MUTUAL COUPLING REDUCTION IN ARRAY PROCESSING Mutual coupling degrades the performance of an antenna array used for array processing, since it alters the fields (phase and magnitude) in the radiating elements compared to the ideal case without coupling. This section investigates the benefits of the mitigation of mutual coupling achieved by the EM-EBG in terms of SLL control and direction of arrival (DOA) estimation.

Fig. 15. Compared measured reflection coefficients for the antennas arrays of Fig. 14. The operation frequencies (indicated by arrows) are 5.62 GHz for the CM-EBG, and 5.67 for the EM-EBG and the array without EBG.

The effect of mutual coupling is modeled by the coupling matrix approach of [21], which was initially proposed for an antenna aperture waveguide array and later applied to a microstrip

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Fig. 16. Compared (selected as representative) measured mutual couplings for the antennas arrays of Fig. 14, with the same legend as in Fig. 15. The difference between the EM-EBG and CM-EBG isolations at the operation frequencies are indicated as .

1

antenna array [22]. In this method, the coupling matrix is calculated from the scattering parameters matrix of the array at , where is the identity the aperture reference plane as matrix. Since it is not possible to measure the scattering matrix at aperture plane of the antennas in practice, the measured matrix is transferred to the aperture plane by compensating the phase difference between the measurement plane and the aperture plane, taking into account the connectors [23]. Fig. 17 shows two examples of beam steering with 20 dB Chebyshev SLL control for the arrays of Fig. 14, compared to the ideal case without any mutual coupling. According to Fig. 17(a), the SLL increase over the ideal case for the arrays without EBG, with CM-EBG, and with EM-EBG are around 5.5 dB, 4 dB, and 2 dB, respectively. Fig. 17(b) shows the pattern scanned to 30 . The SLL for the array without EBG and with CM-EBG are further increased to around 9 dB, and 6.5 dB, respectively. Thus, in general, EM-EBG provides better SLL control compared to the CM-EBG. Fig. 18 investigates the performance improvement of the MUSIC DOA estimation algorithm [24], [25] with the EM-EBG structure. The MUSIC spatial spectrum is calculated by

(6) is the steering vector where is the differential phase between of the array and the signals on adjacent antenna elements for a signal incident under an angle measured from broadside (Fig. 13). In (5), is a matrix formed by the noise eigenvectors of the array autocorrelation matrix , where is the array signal vector. The effect of mutual coupling is taken into account by

Fig. 17. Comparison of the beam steering with Chebyshev SLL control for the four-element array without mutual coupling (ideal), without EBG, with CM-EBG, and with EM-EBG. (a) When the beam is steered to broadside and . (b) When the beam is steered to 30 and .

SLL = 020 dB

SLL = 020 dB

multiplying the array signal vector by the coupling matrix . Ideally (in the absence of coupling), the maxima of indicate the DOA of the signal incident on the array. Fig. 18(a) shows the spatial spectrum computed by (6) when , , and 0 with a signal-tosignals are impinging from noise ratio (SNR) of 10 dB. The covariance matrix is calculated by averaging over 5000 samples. As can be seen in Fig. 18(a), without EBG and with the CM-EBG, the DOA of signals imand are not differentiated. For the pinging from signal impinging from 0 the DOA error for the cases without EBG and CM EBG are around 2 while for the EM EBG it is around 1 . Therefore, using an EBG, and more specifically an EM-EBG, improves the MUSIC performance in this scenario. Fig. 18(b) shows another example when there are three direc, 0 , and 40 . tional signals in the channel impinging from , 0 , and 40 are 2.8 , 1.5 , and 3 The DOA errors at for the case without EBG, 0.2 , 1 , and 0.2 for the case with CM-EBG, and, 0.7 , 0.1 , 0.5 for the case with EM-EBG, respectively. In this case again using an EBG structure improves the MUSIC performance. The performance of CM-EBG and EM-EBG are close based on DOA error. However, the spectrum in the DOA directions for CM-EBG and EM-EBG are [25 dB,

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By exploiting the substrate thickness, the EM-EBG structure reduces the transverse dimensions of the EBG for a given number of cells or improves the isolation between antenna elements by fitting more cells between them for the same transverse dimensions. This benefit is critical in antenna arrays, where the maximal allowable distance between the antenna elements is set to half a free-space wavelength to avoid grating lobes. By further increasing the thickness of the substrate, using proper multilayer and via-plating manufacturing techniques, further improvement in isolation, with subsequent further improvement in array processing, could be obtained in principle. ACKNOWLEDGMENT The authors thank J. Gauthier, S. Dubé and T. Antonescu, Poly-Grames Research Center, École Polytechnique de Montréal, QC, Canada, for fabrication of the prototypes. They also thank Ansoft Corporation for the donation of HFSS licenses and CST for the donation of Microwave Studio lensises. REFERENCES

Fig. 18. The MUSIC spatial spectrum calculated using (6) for the four-element array without mutual coupling (ideal), without EBG, with CM-EBG and with EM-EBG. (a) Three directional signals impinging from 60 , 30 and 0 . (b) Three directional signals impinging from 50 , 0 and 40 .

0

0

0

20.5 dB, 17 dB] and [23.5 dB, 24.5 dB, 23.4 dB], respectively. The higher spatial spectrum reduces the possibility of disappearance of the peaks due to other errors in the system and increased SNR. Thus, in general, the EM-EBG provides better DOA estimation compared to the CM-EBG. VIII. CONCLUSION A compact elongated mushroom electromagnetic band-gap (EM-EBG) structure, exploiting the thickness of the substrate to achieve higher isolation compared to the case of the conventional mushroom EBG (CM-EBG), has been proposed for the enhancement of the performances of patch antenna arrays. Guidelines, based on fairly accurate formulas, have been provided for the design of the stop-band of the structure. The compactness of the EM-EBG has been investigated and shown to be superior to that of the CM-EBG in practical cases. The superior reduction of mutual coupling of the EM-EBG has been demonstrated by full-wave and experimental results both for a pair of patches and for a 4-element patch array. Finally, the benefits of the EM-EBG in terms of array processing, specifically side-lobe level (SLL) control and direction of arrival (DOA) estimation, have been presented.

[1] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [2] R. Cocciolo, F. R. Yang, K. P. Ma, and T. Itoh, “Aperture-coupled patch antenna on UC-PBG substrate,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2123–2130, Nov. 1999. [3] Y. Rahmat-Samii and H. Mosallaei, “Electromagnetic band-gap structures: Classification, characterization, and applications,” in Proc. Inst. Elect. Eng.-ICAP Symp., Apr. 2001, pp. 560–564. [4] 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. [5] Z. Iluz, R. Shavit, and R. Bauer, “Microstrip antenna array with electromagnetic band-gap substrate,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1446–1453, Jun. 2004. [6] Y. Fu and N. Yuan, “Elimination of scan blindness in phased array of microstrip patches using electromagnetic bandgap materials,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 63–65, Oct. 2004. [7] L. Zhan, J. Castaneda, and N. G. Alexopoulos, “Scan blindness free phased array design using PBG materials,” IEEE Trans. Antennas Propag., vol. 52, no. 8, pp. 2000–2007, Aug. 2004. [8] N. Lombart, A. Neto, G. Gerini, and P. de Maagt, “1-D scanning arrays on dense dielectrics using PCS-EBG technology,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 26–35, Jan. 2007. [9] A. Neto, N. Lombart, G. Gerini, M. D. Bonnedal, and P. de Maagt, “EBG enhanced feeds for the improvement of the aperture efficiency of reflector antennas,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2185–2193, Aug. 2007. [10] L. Yang, M. Fan, F. Chen, J. She, and Z. Feng, “A novel compact electromagnetic-bandgap (EBG) structure and its applications,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 183–190, Jan. 2005. [11] M. F. Abedin, M. Z. Azad, and M. Ali, “Wideband smaller unit-cell planar EBG structures and their application,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 903–908, Mar. 2008. [12] Q. R. Zheng, Y. Q. Fu, and N. C. Yuan, “A novel compact electromagnetic band-gap (EBG) structure,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1656–1660, Jun. 2008. [13] E. Rajo-Iglesias, O. Quevedo-Teruel, and L. Inclán-Sánchez, “Mutual coupling reduction in a patch antenna arrays by using a planar EBG structure and a multilayer dielectric substrate,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1648–1655, Jun. 2008. [14] D. F. Sievenpiper, “High-Impedance Electromagnetic Surfaces,” Ph.D dissertation, UCLA, Los Angeles, CA, 1999. [15] D. M. Pozar, “Input impedance and mutual coupling of rectangular microstrip antenna,” IEEE Trans. Antennas Propag., vol. AP-30, no. 6, pp. 1191–1196, Nov. 1982.

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[16] D. M. Pozar, “Considerations for millimeters wave printed antennas,” IEEE Trans. Antennas Propag., vol. AP-31, no. 5, pp. 740–747, Sept. 1983. [17] D. M. Pozar and D. H. Schaubert, “Analysis of an infinite array of rectangular microstrip patches with idealized probe feeds,” IEEE Trans. Antennas Propag., vol. AP-32, no. 10, pp. 1101–1107, 1984. [18] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 2nd ed. New York: Wiley, 1998. [19] H. J. Visser, Array and Phased Array Antenna Basics. New York: Wiley, 2005. [20] F. Gardiol, Électromagnétisme Presses polytechniques et universitaires romandes, 2002. [21] H. Steyskal and J. Herd, “Mutual coupling compensation in small array antennas,” IEEE Trans. Antennas Propag., vol. 38, pp. 1971–75, Dec. 1990. [22] I. Salonen, A. Toropainen, and P. Vainikainen, “Linear pattern correction in a small microstrip antenna array,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 578–586, Feb. 2004. [23] I. Salonen and P. Vainikainen, “Optimal virtual element patterns for adaptive arrays,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 204–210, Jan. 2006. [24] L. C. Godora, “Application of antenna arrays to mobile communications, part II: Beamforming and direction of arrival considerations,” Proc. IEEE, vol. 85, no. 8, pp. 1195–1254, Aug. 1997. [25] L. C. Godara, Smart Antennas. Boca Raton, FL: CRC Press, 2004.

Martin Coulombe was born in Montréal, Canada, in 1980. He received the B.A.Sc., D.E.S.S., and M.A.Sc. degrees in electrical engineering from the École Polytechnique de Montréal, Montréal, QC, Canada, in 2004, 2006, and 2009. He is currently working at Infield Scientific as an Electromagnetic Engineer. His current interests include artificial dielectric substrates, electromagnetic bandgap structures, metamaterials, electromagnetic compatibility and electromagnetic interference.

Sadegh Farzaneh Koodiani (S’05–M’08) received the B.S. and M.S. degrees (with honors) from Shiraz University, Iran, in 1996 and 1999, respectively, and the Ph.D. degree from Concordia University, Canada, in 2008, all in electrical engineering. From 1997 to 2000, he worked with Iran Components Industries on the design of microwave circuits. Between 2000 and 2003, he was with the Electrical Engineering Department, Kazeroun Azad University, as a Lecturer. From 2008 to 2009, he worked at Concordia University and Ecole Polytechnique de Montreal as a Research Associate and Postdoctoral Fellow. He is currently working at SDP Components Inc., as a Senior RF Engineer. His research interests include smart antennas, phased array antennas, microwave circuits, and wireless communications. Dr. Farzaneh was the recipient of the Concordia University School of Graduate Studies Doctoral Teaching Assistantship, International Fee Remission Award in 2005, and the France and André Desmarais Graduate Fellowship in 2006.

Christophe Caloz (S’99–M’03–SM’06) received the Diplôme d’Ingénieur en Électricité and Ph.D. degrees from the École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, in 1995 and 2000, respectively. From 2001 to 2004, he was a Postdoctoral Research Engineer at the Microwave Electronics Laboratory, University of California at Los Angeles (UCLA). In June 2004, he joined École Polytechnique of Montréal, where he is now an Associate Professor, a member of the Microwave Research Center Poly-Grames, and the holder of a Canada Research Chair (CRC). He has authored and coauthored 350 technical conference, letters and journal papers, seven book and book chapters, and he holds several patents. is research interests include all fields of theoretical, computational and technological electromagnetics engineering, with strong emphasis on emergent and multidisciplinary topics. Prof. Caloz is a Member of the Microwave Theory and Techniques Society (MTT-S) Technical Coordinating Committee (TCC) MTT-15, a Speaker of the MTT-15 Speaker Bureau, and the Chair of the Commission D (Electronics and Photonics) of the Canadian Union de Radio Science Internationale (URSI). He is a member of the Editorial Board of the International Journal of Numerical Modelling (IJNM), of the International Journal of RF and Microwave Computer-Aided Engineering (RFMiCAE), of the International Journal of Antennas and Propagation (IJAP), and of the journal Metamaterials of the Metamorphose Network of Excellence. He received the UCLA Chancellor’s Award for Postdoctoral Research in 2004 and the MTT-S Outstanding Young Engineer Award in 2007.

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Low-Profile PIFA Array Antennas for UHF Band RFID Tags Mountable on Metallic Objects Horng-Dean Chen, Senior Member, IEEE, and Yu-Hung Tsao

Abstract—Two PIFA array antennas, designed for UHF band RFID tags mountable on metallic objects, are presented in this paper. The proposed array antennas are fabricated on a very thin substrate with a thickness of 0.8 mm. The first array that has shorting pins placed at the outer edges of PIFAs is fed by two quarter-wavelength microstrip lines. This array has a simple feeding structure, which is its main advantage. Meanwhile, the second array has shorting pins located at the inner edges of PIFAs, and incorporating additional matching stubs into the quarter-wavelength microstrip lines is needed to achieve complex impedance matching, which can offer a broad bandwidth. By properly selecting the spacing between the PIFAs, an enhanced antenna gain and a good reading-range performance can be obtained for both array antennas. In addition, the two array antennas proposed are studied mounted on differently sized metallic plates. It is found that there is a negligible influence of the metal-plate size on the reading range is found for the second array antenna, which is superior to the first one. Index Terms—Array antenna, PIFA, radio frequency identification (RFID), tag antenna.

I. INTRODUCTION HE use of radio frequency identification (RFID) system has increased noticeably in recent years. For RFID systems, the desirable frequencies are 125 KHz (LF), 13.56 MHz (HF), and 860–960 MHz (UHF). At present, an increasing number of research studies have been done in the field of UHF band RFID system owing to forceful demands for long reading range, high data rate, and small antenna size. Tag antenna is one of the essential components for an RFID system. Many applications require tag antennas to be of low profile and to be mounted on electrically metallic objects such as vehicles and notebooks. Among tag antennas in UHF band RFID applications, label-type dipoles are mostly used since they are printed on a very thin film at low cost [1]–[3]. However, this type of antenna does not work efficiently when mounted on a metallic object. To overcome this problem, several designs have been developed on the basis of PIFAs or microstrip antennas [4]–[12]. Although these reported antennas give good reading-range performance, they are limited by their inconvenient mounting

T

Manuscript received March 27, 2009; revised August 13, 2009. Date of manuscript acceptance September 19, 2009; date of publication January 26, 2010; date of current version April 07, 2010. This work was sponsored by the National Science Council of Taiwan under Contract NSC 97-2221-E-017-006. The authors are with the Department of Optoelectronics and Communications Engineering, National Kaohsiung Normal University, Kaohsiung 824, 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.2010.2041158

on metallic objects because of their high profile (thickness mm). However, it is inherently obvious for a PIFA or microstrip antenna that lowering the antenna profile would degrade its antenna gain. In order to improve the gain of a low-profile antenna and provide applicable reading range, array antennas are considered to be effective candidates. However, to date, little research has been done on the development of the RFID tag in an array antenna. In this paper, we propose two different designs of low-profile PIFA array antennas for UHF band RFID tags that are mounted on metallic objects. In the first design [see Fig. 1(a)], with shorting pins placed at the outer edges of the patches, two quarter-wavelength microstrip lines without matching stub are applied for the excitation of a two-element PIFA array antenna. Note that the two microstrip lines acts not only as an impedance transformer to achieve conjugate impedance matching between the antenna and tag chip, but also as a phase-reversal device that makes the surface currents of the two PIFAs in phase, thus achieving the enhanced antenna gain. As for the second design (see Fig. 7), with shorting pins placed at the inner edges of the patches, two quarter-wavelength microstrip lines with two additional matching stubs are needed in order to achieve conjugate impedance matching, and bandwidth improvement is obtained, as compared to the first design. Both proposed tag array antennas were fabricated on thin FR4 substrates, and the effects of the spacing between the PIFAs on RFID tag reading range in the presence of metal plates are investigated. The details of the proposed designs and the obtained experimental results are presented and discussed. II. ARRAY ANTENNA FED BY TWO QUARTER-WAVELENGTH MICROSTRIP LINES WITHOUT MATCHING STUB Fig. 1(a) shows the first PIFA array antenna proposed, which is fed by two quarter-wavelength microstrip lines without any additional matching stub. The antenna is designed for operating in the 902 to 928 MHz frequency band that is used in UHF band RFID system in several countries such as North America. It consists of two identical PIFAs and two identical quarter-wavelength microstrip lines that make the structure symmetrical. A mm and very thin FR4 substrate with a thickness of is used to fabricate the antenna. dielectric constant of has a shorting pin with a Each PIFA with dimensions radius of 0.2 mm located at the outer edge of the patch to reduce the patch size, and it is designed to roughly resonate as a quarterwavelength structure. The spacing between the centers of the two PIFAs is . The two quarter-wavelength microstrip lines with length and width are used to connect the tag chip, which is attached to a plastic strap with metallic pads, to the antenna.

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Fig. 2. Simulated surface current distribution at 915 MHz for the antenna shown in Fig. 1(a); L = 34 mm, W = 15 mm, h = 0:8 mm, " = 4:4; ` = 45 mm, t = 1:2 mm, S = 66 mm, G = 130 mm, and G = 45 mm.

design of impedance matching. The tag chip used in this design is Alien Higgs, which exhibits an impedance of at 915 MHz. In order to deliver the maximum power between the antenna and chip, the input impedance of the antenna needs . From symmetry, the impedance at to be as shown in Fig. 1(a) should be . In point order to obtain the desired impedance value, three adjustable parameters are employed: the width of the quarter-wavelength of the PIFA. By microstrip line, the length and the width mm of the microstrip line selecting the proper width with characteristic impedance , the impedance at point is found theoretically as follows: Fig. 1. (a) Geometry of the proposed first PIFA array antenna fed by two quarter-wavelength microstrip lines without additional matching stub. (b) Reference antenna A1. (c) Reference antenna A2.

Note that the quarter-wavelength microstrip lines are bent into meander-line sections, which facilitates the spacing adjusting and the optimal antenna gain obtaining. The ground-plane size . To verify gain-enhanced characteristic, reference is that has only a PIFA structure and reference antenna antenna that comprises two PIFAs and is directly fed by tag chip are compared with the proposed first array antenna. Their design parameters are shown in Figs. 1(b) and (c), respectively. There are two primary considerations that affect the successful design of the first array antenna: the excitation of the surface currents on the two PIFAs, and the impedance matching between the antenna and chip. First, we consider the excitation of the surface current for the antenna. Fig. 2 shows the simulated surface current distributions of the antenna at the frequency of 915 MHz, obtained using the commercial software Ansoft HFSS. A sinusoidal curve that indicates the intensity of the current on the PIFAs and microstrip lines is also superimposed on this figure. We can see that since the total paths of the two quarter-wave microstrip lines result in 180 phase shift, the excited surface currents in the two PIFAs are in the same phase. This result can lead to constructive radiation and enhanced gain for the antenna. Next, we consider the

At the same time, the length is selected to be 34 mm so that the real part of the impedance at point seen by looking into the PIFA is 7 at 915 MHz, and then the width is adjusted to 15 mm so that its imaginary part is . This completes the conjugate impedance matching between the antenna and chip. Fig. 3 shows the measured input impedance for the first array antenna with various spacings , and Fig. 4 presents the corresponding measured and simulated return loss. Here, the length mm and corresponds of the microstrip line is fixed at at 915 MHz. For comparison, the results of the reference to antennas and are also shown in the Fig. 4. The bandwidth performances are given in Table I. As can be seen from Fig. 3, the input impedances at 915 MHz for all the antennas are near , and they are well matched the desired value of with chip impedance. Good agreement between the measurement and the simulation is also observed in Fig. 4. A slight increase in the impedance bandwidth for the first array design is seen when the spacing decreases, as shown in Table I. The bandwidth determined from 10 dB return loss for the case of mm is about 15 MHz (907–922 MHz) only. However, a wider impedance bandwidth is required for UHF band RFID system in North America.

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TABLE I BANDWIDTH PERFORMANCE FOR THE ANTENNAS STUDIED IN FIG. 4. THE BANDWIDTH IS DETERMINED WITH 10-dB RETURN LOSS.  IS THE WAVELENGTH IN FREE SPACE AT 915 MHZ

Fig. 3. Measured input impedance against frequency for various spacings S ; = S + L + 30 mm. Other parameters are the same as those studied in Fig. 2.

G

Fig. 5. Simulated radiation patterns at 915 MHz in free space and on a 400 400 mm metal plate for the antenna studied in Fig. 2.

2

Fig. 4. Measured and simulated return loss against frequency for the antennas studied in Fig. 3.

The effect of an electrical metal platform on the radiation pattern was studied by attaching the present array antenna on differently sized copper plates. Since the input impedance of the tag antenna cannot match a 50 coaxial line, it is difficult to measure the radiation patterns of the tag antenna accurately by using a conventional far-field measurement system in an anechoic chamber. Thus, only the simulated radiation patterns obtained from the software Ansoft HFSS are presented. Fig. 5 shows the simulated radiation patterns for the array antenna placed in free space (without metal plate) and on a 400 400 mm metal plate, where all the maximum radiations have been normalized to 0 dB. It is seen that the direction of the maximum radiation does not vary with the size of the metal plate, and it . It is also found that the back radiation in the occurs at direction is decreased when the antenna is mounted on the metal plate. The RFID tag was fabricated by attaching the tag chip to the antenna. To find the optimal spacing of the first array antenna, the reading range of the fabricated tag was first measured in free space for various spacing. Reading range is the maximum dis-

tance at which the tag can be recognized by an RFID reader. A commercial RFID reader (Symbol XR440) was used in the measurement. Table II lists the results of the reading ranges for the first array antennas with various spacings . It is seen that changing the spacing significantly affects the reading range performance of the array antenna, and the optimal spacing mm (0.2 ). In this case, the is found to be about reading range in free space reaches 3.1 m, and it is about 1.8 and 1.9 m longer m longer than that of reference antenna . This is a simple verification than that of reference antenna of the gain enhancement of the proposed PIFA array antenna, as and . Furthermore, the compared to reference antennas reading range was measured for the array antenna mounted on metal plates, and the measured data are also given in Table II. It is seen that the reading range of the array antenna decreases by increasing the metal-plate size. It should be mentioned that the operating frequencies and impedance characteristics of the array antenna are insignificantly affected by varying the metal-plat size. The degradation in the reading range can be explained by the result of the surface current distribution for the array antenna mounted on the 400 400 mm metal plate as illustrated in Fig. 6, in which the current intensity on the metal plate is exaggeratedly plotted for clarity. It is seen that strong currents are excited on both left and right regions of the metal plate near the shorting pins of the antenna, and their current directions are opposite to that on the center region of the metal plate. This behavior may cause some canceling effects in the radiation patterns, resulting in the degradation of the reading range. III. ARRAY ANTENNA FED BY TWO QUARTER-WAVELENGTH MICROSTRIP LINES WITH TWO MATCHING STUBS To improve bandwidth performance and overcome the degradation of the reading range due to the use of a metal platform in

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Fig. 7. Geometry of the proposed second PIFA array antenna fed by two quarter-wavelength microstrip lines with two matching stubs.

Fig. 6. Simulated surface current distribution on a 400 at 915 MHz for the antenna studied in Fig. 2.

2 400 mm

metal plate

TABLE II INFLUENCE OF METAL-PLATE SIZE ON THE READING RANGE FOR THE ANTENNAS STUDIED IN FIG. 4

Fig. 8. Simulated surface current distribution at 915 MHz for the antenna mm, W mm, h : mm, " :; shown in Fig. 7; L : mm, ` mm, t : mm, S ` : mm, t mm, d : mm, G mm, and G mm.

= 46 5 =25

the first PIFA array antenna in Section II, a second PIFA array antenna is proposed, as shown in Fig. 7. The main differences between the second and first array structures are the locations of shorting pins and the inclusions of the matching stubs. In the second array design, each PIFA now has two shorting pins, and the locations of the two shorting pins are changed to the inner edge of the patch with a distance away from the center line ( -axis) of the patch. It is then expected that when the second array antenna is mounted on the metal plate, the excited current in the metal plate can be significantly lowered and thus the degradation of the reading range can be improved, as compared to the first array design. However, since the feed point (at point in Fig. 7) of the PIFA is now located close to the shorting pins, the current has a maximum value at point , and is found there. This condition a low impedance value of to a desired makes it difficult to transform the impedance impedance value at point by selecting a suitable characteristic impedance of a quarter-wavelength microstrip line without matching stub. Thus, in the second array

= 31 = 06 = 127

= 22 = 08 = 70 = 06 = 52

= 44 = 66

is design, a meander-line stub of width and total length considered to be connected in shunt at point . The design procedure of the second array antenna can be at point is developed as follows. First, the impedance designed to be about 17 at 915 MHz when the parameters are selected at mm, mm, and mm, which were obtained from the software Ansoft HFSS. The mm quarter-wavelength microstrip line of the length mm with a characteristic impedance of and width 80.7 is used to transform to an impedance value of 383 at its other end. In order to obtain a matched condition, the mewill be needed in shunt ander-line stub with a reactance of at point . The length of the stub that gives this reactance can be estimated by the following formula:

where and are the guided wavelength at 915 MHz and characteristic impedance of the stub, respectively. Owing to the existence of mutual coupling between the adjacent segments of the stub, the actual length of the stub should be slightly longer than the calculated one. From the above discussion, the resulting ; that impedance at point is is, the total input impedance of the second PIFA array antenna . This impedance is very close to is

CHEN AND TSAO: LOW-PROFILE PIFA ARRAY ANTENNAS FOR UHF BAND RFID TAGS MOUNTABLE ON METALLIC OBJECTS

Fig. 9. Measured input impedance against frequency for various spacings S; S L mm. Other parameters are the same as those studied in Fig. 8.

G

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= + + 30

Fig. 11. Simulated radiation patterns at 915 MHz in free space and on a 400 400 mm metal plate for the antenna studied in Fig. 8.

2

TABLE IV INFLUENCE OF METAL-PLATE SIZE ON THE READING RANGE FOR THE ANTENNAS STUDIED IN FIG. 9

Fig. 10. Measured return loss against frequency for the antennas studied in Fig. 9. TABLE III BANDWIDTH PERFORMANCE FOR THE ANTENNAS STUDIED IN FIG. 10. THE BANDWIDTH IS DETERMINED WITH 10-dB RETURN LOSS

of the conjugated impedance of the chip, resulting in a matched condition. Fig. 8 shows the simulated surface current distributions of the second array antenna. It is clearly seen that the excited surface currents in the two PIFAs are in the same direction. It would also be expected that constructive radiation and enhanced gain can be obtained for the antenna. Figs. 9 and 10 respectively show the measured input impedance and corresponding return loss for various spacings . The bandwidth performances are presented in Table III. As can be seen from Fig. 9, the input impedances over for all the antennas are around the impedance a wide frequency range, which results in a wideband characteristic. The 10 dB return loss bandwidth for the case of mm reaches 35 MHz (901–936 MHz), which covers the required bandwidth (902–928 MHz) of UHF band RFID system in North America. In this case, the bandwidth is about 2.3 times that of the first array antenna with the same spacing . The simulated radiation patterns of the second array antenna mm placed in free space and on a 400 400 mm with metal plate are plotted in Fig. 11. It is seen that the back radiation direction is decreased when the antenna is attached on in the

Fig. 12. Simulated surface current distribution on a 400 at 915 MHz for the antenna studied in Fig. 8.

2 400 mm

metal plate

the metal plate. Also, the antenna has a high cross-polarization radiation. Table IV summarizes the results of the reading range for the second array antennas with various spacings placed on different metal plates. It is first seen that the optimal spacing mm of the second array antenna also occurs at about , and its reading range in free space is 2.5 m, which is . This beabout 1.3 m longer than that of reference antenna havior also verifies that the enhanced gain for the second array

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antenna can be achieved. Most importantly, no severe degradation in the reading range was found when the second array antenna is mounted on the metal plate. This result indicates very good tolerance to metal platform. The little effect of the metal plane on the reading range is largely because in this array design, the shorting pins are moved to the inner edges of the patches, and the excited surface current is less distributed in the metal plate close to the outer edges of the patches, as shown in Fig. 12. However, probably because the inclusions of the stubs in the second array design cause a large standing wave on the stub line, the optimal reading range in free space is 0.6 m less than that mm. of the first array antenna with the same spacing In conclusion, the second PIFA array antenna is selected as a promising antenna that is suitable for achieving a broad bandwidth and high gain in very low profile and metal proximity use. IV. CONCLUSION Two novel PIFA array antennas fed by quarter-wavelength microstrip lines with or without matching stubs have been successfully implemented. The results show that an enhanced gain and optimal reading range can be obtained for both array antennas by selecting the spacing between the PIFAs to be about 0.2 free-space wavelength. For the first array without matching stub, the optimal reading range reaches about 3.1 m, but it is degraded when the antenna is mounted on a metal object. For the second array with matching stubs, the optimal reading range is about 2.5 m, and less performance degradation is achieved when the antenna is mounted on a metal object. Also, the obtained impedance bandwidth of the second array antenna covers the bandwidth (902–928 MHz) of the UHF band RFID system in North America. It is therefore concluded that the second PIFA array antenna is a promising antenna for achieving a broad bandwidth and high gain in very low profile and metal platform use.

[4] M. Hirvonen, P. Pursula, K. Jaakkola, and K. Laukkanen, “Planar inverted-F antenna for radio frequency identification,” Electron. Lett., vol. 40, no. 14, pp. 848–850, 2004. [5] H. Kwon and B. Lee, “Compact slotted planar inverted-F RFID tag mountable on metallic objects,” Electron. Lett., vol. 41, no. 24, pp. 1308–1310, 2005. [6] L. Ukkonen, L. Sydanheimo, and M. Kivikoski, “Effects of metallic plate size on the performance of microstrip patch-type tag antennas for passive RFID,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 410–413, 2005. [7] S. J. Kim, B. Yu, Y. S. Chung, F. J. Harackiewicz, and B. Lee, “Patchtype radio frequency identification tag antenna mountable on metallic platforms,” Microw. Opt. Technol. Lett., vol. 48, no. 12, pp. 2446–2448, 2006. [8] M. Hirvonen, K. Jaakkola, P. Pursula, and J. Saily, “Dual-band platform tolerant antennas for radio-frequency identification,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2632–2637, 2006. [9] H. W. Son, G. Y. Choi, and C. S. Pyo, “Design of wideband RFID tag antenna for metallic surfaces,” Electron. Lett., vol. 42, no. 5, pp. 263–265, 2006. [10] B. Yu, S. J. Kim, B. Jung, F. J. Harackiewicz, and B. Lee, “RFID tag antenna using two-shorted microstrip patches mountable on metallic objects,” Microwave Opt. Technol. Lett., vol. 49, no. 2, pp. 414–416, 2007. [11] C. Cho, H. Choo, and I. Park, “Design of planar RFID tag antenna for metallic objects,” Electron. Lett., vol. 44, pp. 175–177, 2008. [12] J. Y. Park and J. M. Woo, “Miniaturised dual-band S-shaped RFID tag antenna mountable on metallic surface,” Electron. Lett., vol. 44, pp. 1339–1341, 2008. Horng-Dean Chen (SM’05) was born in Kaohsiung, Taiwan, in 1961. He received the B.S. degree in electric engineering from National Cheng Kung University, Tainan, Taiwan, in 1984, and the M.S. and Ph.D. degrees in electric engineering from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 1992 and 1995, respectively. From 1995 to 1999, he was an Associate Professor with the Department of Electronic Engineering, Nai-Tai Institute of Technology. From 1999 to 2004, he was an Associate Professor with the Department of Electronic Engineering, Cheng-Shiu University. He is currently an Associate Professor with the Department of Optoelectronics and Communications Engineering, National Kaohsiung Normal University. His current research interests include small, planar and broadband antennas for wireless communications, and tag antennas for RFID applications.

REFERENCES [1] K. V. S. Rao, P. V. Nikitin, and S. F. Lam, “Antenna design for UHF RFID tags: A review and a practical application,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 3870–3876, 2005. [2] C. Cho, H. Choo, and I. Park, “Broadband RFID tag antenna with quasi-isotropic radiation pattern,” Electron. Lett., vol. 41, no. 20, pp. 1091–1092, 2005. [3] Z. Fang, R. Jin, J. Geng, and J. Sun, “Dual band RFID transponder antenna designed for a specific chip without additional impedance matching network,” Microw. Opt. Technol. Lett., vol. 50, no. 1, pp. 58–60, 2008.

Yu-Hung Tsao was born in Tainan, Taiwan, in 1984. He received the M.S. degree from National Kaohsiung Normal University, Kaohsiung, Taiwan, in 2009. His main research interests are in tag antennas for RFID applications.

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CMOS Phased Array Transceiver Technology for 60 GHz Wireless Applications Mohammad Fakharzadeh, Member, IEEE, Mohammad-Reza Nezhad-Ahmadi, Behzad Biglarbegian, Student Member, IEEE, Javad Ahmadi-Shokouh, Member, IEEE, and Safieddin Safavi-Naeini

Abstract—Based on the indoor radio-wave propagation analysis, and the fundamental limits of CMOS technology it is shown that phased array technology is the ultimate solution for the radio and physical layer of the millimeter wave multi-Gb/s wireless networks. A low-cost, single-receiver array architecture with RF phase-shifting is proposed and design, analysis and measurements of its key components are presented. A high-gain, two-stage, low noise amplifier in 90 nm-CMOS technology with more than 20 dB gain over the 60 GHz spectrum is designed. Furthermore, a broadband analog phase shifter with a linear phase and low insertion loss variation is designed, and its measured characteristics are presented. Moreover, two novel beamforming techniques for millimeter wave phased array receivers are developed in this paper. The performance of these methods for line-of-sight and multipath signal propagation conditions is studied. It is shown that one of the proposed beamforming methods has an excess gain of up to 14 dB when the line of sight link is obstructed by a human. Index Terms—Beam-forming, CMOS, millimeter wave, phased array antenna, 60 GHz.

I. INTRODUCTION

I

N the endless pursuit of higher bandwidth for wireless communications, researchers and industries are becoming more and more interested in millimeter wave (MMW) spectrum [1]–[7]. Recently, 60 GHz frequency band has been released and proposed for short-range wireless applications such as wireless personal area network (WPAN) [8], [9], and wireless multimedia/high-definition (HD) streaming. IEEE 802.15.3 task group 3c (TG3c) is working on standardization of this frequency band for short range wireless applications. At the same time, Wireless HD Consortium is defining a wireless protocol to create a 60 GHz wireless video network for consumer electronic audio and video devices [10]. High-volume markets for 60 GHz systems are promising if compact, low cost, high performance transceivers become available.

Manuscript received July 16, 2008; revised August 16, 2009; accepted November 10, 2009. Date of publication January 22, 2010; date of current version April 07, 2010. This work was supported in part by the National Science and Engineering Research Council (NSERC) of Canada, RIM (Research In Motion), Ontario Center of Excellence (OCE), and in part by Nortel. M. Fakharzadeh, M.-R. Nezhad-Ahmadi, B. Biglarbegian, and S. SafaviNaeini are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, Canada (e-mail:[email protected]; [email protected]; [email protected]; [email protected]; [email protected]). J. Ahmadi-Shokouh is with the Electrical Engineering Department, University of Sistan and Baluchestan, Zahedan, Iran (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.2010.2041140

Complementary metal-oxide semiconductor (CMOS) is the dominating technology for most wireless products below 10 GHz. This dominance has been achieved by reliability, low cost, and high device count advantages of CMOS compared to the other semiconductor technologies such as SiGe and GaAs. Today, with the aggressive scaling of gate length, CMOS technology is pushing further into the MMW region. Moreover, CMOS is the most promising technology for system-on-chip design, because it enables integration of the analog RF circuits as well as the digital signal processing and baseband circuits in the lowest possible chip area, which leads to a lower cost and more compact solution. Therefore, the nano-scale CMOS technology, such as 90 nm, 65 nm and 45 nm, offers commercial MMW solutions for short range and high data rate applications. However, several system and circuit level challenges must be met, such as lack of the efficient and low cost antenna and packaging solutions, low output power and nonlinearity of power amplifiers, severe path loss, shadowing loss, limited gain of the low noise amplifier (LNA) and its high noise figure. For a wide range of emerging applications in the 60 GHz spectrum, as will be discussed in this paper, use of multiple antennas with beam steering capabilities is a key enabling technology to address most of these challenges. However the efficiency of a phased array depends on the performance of power amplifier or LNA as well as phase shifter. Furthermore, to lower the overall cost, all microwave components must be implemented in a low-cost technology and possibly on a single chip. Therefore one objective of this paper is to present a high-performance phase shifter and LNA in CMOS technology. So far, the reported MMW systems at 52, 60 and 77 GHz bands in CMOS and SiGe are either simple receiver [4], [5], [11], two-element receiver [12], transmitter only [13], [14], or transceivers which are not able to meet some of the above mentioned challenges [2], [15]. Moreover, the proposed phased array solutions are costly and complex requiring large chip area and high power consumption [6], [12]. The situation is worsened if a large number of array elements are needed to meet the link budget and network coverage specification. In this paper, feasibility, system architecture and the key components of a low-cost and low-noise CMOS phased array transceiver for a broadband wireless network at 60 GHz, are analyzed. Another objective of this paper is to show that the key microwave components, i.e. high-gain LNA and linear phase shifter, can be implemented in CMOS technology at 60 GHz. Furthermore, we will show that a 9 or 16 element CMOS phased array can achieve the required signal to noise ratio for multi-Gb/s wireless communication in a picocell (such as a regular office), if a powerful beamforming algorithm is used.

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Fig. 1. (a) Simulated 3D view, and (b) top view of the propagation environment (CAD Lab at the University of Waterloo).

The organization of this paper is as follows. In Section II the indoor 60 GHz propagation channel is studied and a test set-up for measuring the shadowing loss of a human body is described. A low-cost architecture for 60 GHz phased array transceiver is presented in Section III and the design, analysis and measurements of its key components are presented in Section IV. Finally, novel beamforming algorithms for the phased array receiver are introduced in Section V and Section VI concludes this paper. II. INDOOR 60 GHz PROPAGATION CHANNEL: MODELING AND MEASUREMENT A. Modeling Indoor propagation at MMW frequencies can be modeled using geometrical optics (GO) ray-tracing method enhanced by uniform asymptotic diffraction theories and experimental models. The reliability of channel characterization results obtained by ray-tracing method at microwave frequencies has been confirmed by different measurements [16]–[18]. For indoor applications at 60 GHz, the high penetration loss of the material isolates adjacent rooms and significantly limits the received interference, so only those objects inside the room need to be included in ray-tracing simulation. In this in-room propagation channel, both line of sight (LOS) and non line of sight (NLOS) rays must be considered. The NLOS rays are caused by reflections from the objects inside the room as well as some significant first order diffracted rays. Furthermore, propagation loss in the ray-tracing modeling consists of free-space loss according to Friis formula, gaseous loss, and reflection, transmission and diffraction losses. In this work, a 3D ray-tracing modeling (GO plus diffraction) is employed to assess the signal coverage at 60 GHz for a regular office area. The CAD Laboratory, located in the EIT building at the University of Waterloo, is used as a typical indoor wireless environment (see Fig. 1(a)). The lab is furnished with tables, chairs and shelves mostly constructed of wooden

TABLE I MEASURED PERMITTIVITY OF INDOOR MATERIALS AT 60 GHz

and plastic material. The walls consist of layers of different material such as plasterboard, concrete, and wood. Moreover, various electronic equipments such as computers, printers and test devices are placed in this lab. The empirical data reported in [19] and [20] is used to calculate the reflection coefficients of the material. Moreover, to evaluate the human body shadowing effect the measured permittivity data for biological tissues in [21] is used. Table I summarizes the measured permittivity data at 60 GHz used in this work. The transmitter antenna in Fig. 1 is located 10 cm below the center of the ceiling (facing down) which is 3.45 m above the floor. The receiver is located on a wooden table 75 cm above the floor. To simulate the shadowing effect, a human-body blocks the LOS path between the transmitter and the receiver as shown in Fig. 1(b). To model a mobile user (portable end-device), the receiver antenna moves within a 2 m 1.75 m grid located 1 m above the floor (25 cm above the table). The resolution of the to . grid-cells varies from Fig. 2 demonstrates the ray-tracing results at 60 GHz for the rectangular grid in front of the human body in Fig. 1. The human body model is 1.8 m tall centered at ( 0.5 m, 1.4 m, 2.54 m) relative to the transmitter antenna. The white area in Fig. 2(a) illustrates the relative opaqueness of the human body at 60 GHz. The transmitter antenna was a 2 2 microstrip patch array with a maximum gain of 10 dBi (see Section IV-A for the radiation pattern of the antenna) and the input power to the antenna was 2 dBm. An isotropic antenna was used as the receiver. Fig. 2(b)

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Fig. 2. (a) Received power over the rectangular grid in Fig. 1(b) around the human body. (b) Received power on Path A and Path B shown in Fig. 1(b).

shows the received power on two horizontal lines, named Path A and Path B, in the region shown in Fig. 2(a). Path A and Path B are respectively 20 cm and 80 cm in front of the human body and 1.7 m and 2.3 m away from the projected transmitter position in Fig. 1(b). The shadowing effect attenuates the received power level by 10 to 40 dB for Path A and by 15 to 30 dB for Path B. The other objects in the room cause up to 5 dB fluctuations in the received signal level. In calculating the received power the time-delay and phase of all rays, which their magnitudes are above 120 dBm, have been considered. This threshold is almost 40 dB below the thermal noise power with 2.16 GHz equivalent bandwidth.

to . The maximum shadowing loss is around 40 dB which occurs when the human body blocks the LOS path completely. Ray-tracing results show that at deep shadowing region all LOS rays are absorbed by the human body, so the measured received power is the combination of NLOS rays. This result has been verified by the measurements in Fig. 3(d). A phased array receiver, as will be discussed in Section V-C, has the potential of steering the array beam to the direction of the strongest NLOS ray when the receiver is inside the deep shadowing region. Table II shows the range of parameters used in the link budget design of a MMW wireless network which uses CMOS phased arrays at its front ends.

B. Measurements Fig. 3(a) demonstrates the developed test set-up to measure the shadowing loss of human body over the frequency range of 50–75 GHz. Two rectangular horn antennas with 24 dBi gain at 60 GHz and 10 beamwidth were used as the transmitter (Tx) programmable signal genand receiver (Rx). Agilent MMW source module was erator connected to used to generate source signal. On the receiver side, Agilent spectrum analyzer connected to preselected mixer was used to measure the spectrum of the received signal. The transmitted power was around 12 dBm. Wave absorbers with 40 dB attenuation were used to weaken the reflections from the source module and mixer. The Tx and Rx antennas were installed at the height of 135 cm and 130 cm, respectively. The horizontal distance between the Tx and Rx antennas in Fig. 3(b) was 3 m. The Tx antenna was fixed, but the Rx antenna was moved along a horizontal line in steps of 5 cm (the orientation of the Rx antenna was kept unchanged). At each point the received power spectrum was measured (after calibration) at three frequencies, i.e. 57, 60 and 64 GHz. For each spectral measurement, the average of 100 successive frames was taken to smooth the instantaneous fluctuations. Fig. 3(c) shows the LOS (no shadowing) measured spectrum at one of these Rx locations. It is seen that the measured power level reduces as the frequency increases, due to the larger path loss at the higher frequencies. Fig. 3(d) compares the measured shadowing loss with ray-racing results for the same room. There is a good agreement between simulation and measurement from

III. PROPOSED 60 GHz PHASED ARRAY TRANSCEIVER ARCHITECTURE To steer the main beam of the phased array antenna, phase shifters can be incorporated in different stages of a receiver or transmitter. Hence, different phased array configurations have been developed. These configurations, which are shown in Fig. 4, are known as RF phase-shifting [22], Local Oscillator (LO) phase-shifting [23], IF phase shifting [24] and digital beamforming phased arrays [25]. In the RF phase shifting architecture, depicted in Fig. 4(a), different RF paths are phase shifted and then combined at RF frequency. The combined signal is then down-converted to the IF or baseband. In this architecture the spatial filtering of the strong undesired signals is performed at the combination point prior to the mixer. Hence, the upper dynamic range requirement of the mixer is relaxed and the level of unwanted in-band inter-modulations after mixer decreases. The design of the phase shifter on silicon, however, remains a challenge in this architecture. In addition, the insertion loss variation with phase shift should be small; otherwise, the array factor is deteriorated [26]. LO phase shifting architecture is displayed in Fig. 4(b). The main advantage of this architecture over RF phase shifting is that the phase shifter loss, non-linearity, and noise performance do not directly affect the receiver performance. However, as compared with the RF phase-shifting architecture, the number of components is larger. This leads to more silicon chip area and therefore higher cost. Besides, since the combining of signals

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Fig. 3. Measurement environment and results. (a) Test set-up with transmitter and receiver. (b) Tx and Rx antenna locations in the room. (c) One sample of the measured spectrum at 57, 60 and 64 GHz. (d) Comparison of the measured and simulation results.

TABLE II LINK-BUDGET DESIGN FOR A 60 GHz WIRELESS NETWORK USING CMOS TECHNOLOGY

and beamforming are performed after mixers, in-band intermodulations are stronger. Also the upper dynamic range of the mixer must be high enough to stand strong interference signals. Fig. 4(c) shows the IF phase shifting architecture. The phase shifters are placed at the first IF stage. The phase-shifted IF signals are combined before downconversion to baseband. As compared to RF phase shifting architecture, some of the challenges

in phase shifter design are relaxed. However, since it needs multiple mixers, this architecture is not a proper option for low cost and low power phased array transceiver. Fig. 4(d) illustrates digital array architecture. Down-converted to a suitable IF frequency, each RF path is digitized by an analog-to-digital converter (ADC) and all outputs are passed to a digital signal processing (DSP) unit, which executes all tasks of beamforming and recovering the desired signal from the undesired interferences. The dynamic ranges of mixers and ADCs must be high enough to withstand the probable strong interferences. In case of WPAN since the data rate may exceed 2 Gb/s, very high-speed ADC’s are required and to accommodate the required dynamic range each ADC must have a large number of bits which increases the ADC cost and power consumption extensively. Table III summarizes the comparison of different phased array architectures in terms of power consumption, chip area and design challenges. To overcome the high path loss and shadowing loss at 60 GHz as well as CMOS output power and noise figure limitations multiple antennas and phase shifters are required. Considering Table III, the most appropriate configuration to lower the cost and power consumption and achieve a compact CMOS phased array transceiver is the RF phase shifting architecture. However, designing an efficient front-end as well as developing fast, efficient beamforming algorithms are the keys to overcome the phase shifter non-idealities and challenges in RF path. Fig. 5 demonstrates the proposed block diagram of an RF phase shifting 60 GHz phased array trans-

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Fig. 4. Different phased array configurations (a) RF phase shifting, (b) LO phase shifting (c) IF phase shifting, and (d) digital beamforming array.

TABLE III OVERALL COMPARISON OF DIFFERENT PHASED ARRAY ARCHITECTURES

ceiver. The receiver has a dual conversion architecture. The first mixer downconverts the combined RF signal to the first IF of about 8 GHz and the second I and Q down-conversion transforms the IF signal directly to the baseband. The image of the first down-conversion is 16 GHz far from the desired signal and can be easily rejected by an on-chip band pass filter before the first down-conversion. The analog baseband signal is amplified and after filtering is converted to a digital signal by two high speed A/D converters in I and Q channels. A fraction of the IF signal goes to a power detector and provides an estimate of the power level for the beamforming algorithm. The algorithm which will be described in Section V adjusts the phase shifters (and gain of each LNA if required). In the transmit section, both I and Q digital signals are converted to analog domain by two high speed D/A converters. Harmonics of converted signals and spurious signals are rejected by two low pass filters. The filtered signal is then up-converted to the 8 GHz IF by the first I and Q up-conversion stage. The LO feed-through at the first up-conversion stage is minimized by calibration techniques implemented in the DSP. After another stage of filtering and amplification the signal is up-converted to

Fig. 5. Block diagram of the proposed RF phase shifting phased array transceiver at 60 GHz band.

60 GHz. LO feed-through and generated spurious at the output of the second mixer are filtered out by the band pass filter centered at 60 GHz. The 60 GHz signal is then divided and applied to a number of paths. The signal in each path passes through a

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Fig. 6. Configuration and radiation pattern of two designs for 2 Reflection coefficients of the patch arrays.

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2 2 patch arrays. (a) The maximum gain design. (b) The maximum beamwidth design. (c)

phase shifter block. The phase shifted signals are then amplified by power amplifier (PA) stages and applied to the transmit antenna array. The saturated output power of a 90 nm CMOS PA is around 10 dBm but to accommodate sufficient linearity for the complex amplitude sensitive modulations such as QAM, the output power of power amplifier is set to 2 dBm to keep the PA in its linear operation region. IV. DESIGN AND MEASUREMENT OF 60 GHz PHASED ARRAY COMPONENTS In this section the design and analysis of the key components of a 60 GHz phased array receiver, namely antenna element, LNA and phase shifter, are described, and measured results for LNA and phase shifter are presented. A. Off-Chip Wide-Beam Antenna Design A substantial body of research on millimeter-wave antenna has been conducted [2], [27]–[30]. For fixed wireless access (FWA) applications, a high gain antenna is preferred to relax the performance requirements of the front-end elements. A high gain antenna has a narrow beam. Thus, for mobile applications, where a wide antenna coverage is required (see Table II), a single high-gain antenna is not an appropriate choice. For , MMW wireless networking applications assuming the base station has been located at the center of the ceiling, almost 2.5 m above the user, the antenna beam to cover the whole area coverage should be greater than of a room or office. A wide beam coverage and a high radiation gain cannot be achieved at the same time unless a phased array antenna with beam steering capability is used. Even in this case the 3 dB beamwidth of each array element must be more than 65 to limit the beam steering loss. For this beamwidth the element gain is limited to 10.2 dBi [31]. Thus, designing the

appropriate antenna element for 60 GHz WPAN is a delicate task. A single rectangle patch antenna which is excited by its fundamental mode (TM10) has about 7 dBi gain and more than 100 beamwidth [32]. Patch antenna is considered as a planar radiator that can be integrated easily with the rest of the system; hence, it is widely used for wireless applications. To obtain higher gains one can use an array of the patches. In this section, two different 2 2 arrays of patch antennas are designed and compared. The first antenna is designed for the maximum gain and the second one for the maximum beamwidth. A very low loss substrate (RT/duroid 5880) is chosen with . Fig. 6 shows the structure of both antennas. far from In the first design, the patches are placed about each other. The overall gain of the 2 2 array is 13.1 dBi, and the HPBW of the structure is only 36 . In the second design, the gain is scarified by 3 dB to achieve a wider beamwidth. The gain of the second structure is 10.1 dBi while the HPBW is 65 . Fig. 6 also shows the polar plot of the gain of both structures for both E-plane and H-plane. Further analysis shows that the radiation efficiency of the wide-beam antenna excluding the matching network, is around 90% over the frequency range of 57–63 GHz. Such high values for efficiency have been reported for MMW antennas before [33]. B. Optimum Low Noise Amplifier Design Recently, different CMOS topologies which are mostly inspired by low frequency designs are utilized and implemented at 60 GHz band such as common-gate [34], common-source [35] and cascode [5], [36]. Compared to the common-gate (CG) or common-source (CS) topologies, the cascode topology shows a better isolation and higher gain. Although the noise figure of cascode configuration might be larger than that of CC and CB,

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Fig. 8. Noise figure and gain of the designed LNA versus frequency. Fig. 7. The designed two-stage cascade amplifier and the die micrograph of the realized high gain LNA at 60 GHz band in 90 nm CMOS technology.

but its larger gain lowers the total noise figure of the system. Fig. 7 depicts the schematic of a two-stage amplifier using the proposed cascode amplifier design. It consists of a cascode topology with lumped inductors or inductive transmission lines in the source of the lower transistor, gate of the upper transistor and in between the two transistors. Also biasing of drain and gate has been provided through the lumped elements that are part of the matching circuit. The MOS transistors for the LNA are sized for maximum available gain in 90 nm-CMOS technology with a 1 V supply voltage. The device size is . Each cascode stage consumes about 9 mA, so the total dissipated power is about 18 mW. All the lumped inductors and interconnectors for biasing, interconnecting and matching are simulated and optimized in ADS. Furthermore, ADS Momentum is utilized to model the interaction between inductors, T-junctions, bends and transmission lines. For the transistor, the RF model provided by foundry (STM) is used. Optimum cascode LNA described in [37] was utilized to improve the LNA gain and reduce the noise figure. For the off-chip antenna case, pad and ribbon bonding parasitics were incorporated into the LNA input matching. Fig. 8 shows the simulated LNA gain and NF. About 25 dB gain is achieved is about in this design. Also, the minimum noise figure 6.1 dB. Measured LNA Gain: Fig. 8 also shows the measured gain of the two-stage LNA (fabricated in 90 nm CMOS technology) shown in Fig. 7. The maximum gain of the LNA is above 20 dB over the frequency range of 56 GHz to 61 GHz. This amount of gain is sufficient to diminish the noise generated by those stages of phased array receiver following the LNA. The difference between the simulated and measured values for LNA gain is due to the approximate transistor models at millimeter wave and extra loss in interconnects and matching elements. C. Phase Shifter Design The objectives of the phase shifter design are minimizing the insertion loss and its variation, and maximizing the phase lin-

Fig. 9. (a) General block diagram of the reflective-type phase shifter. (b) Die micrograph of the fabricated analog phase shifter in 90 nm CMOS technology.

earity versus control voltage. The authors have reported a beamforming technique which compensates for the remaining insertion loss variation of the phase shifters [26]. In general, phase shifters can be classified into digital and analog types. Although, the linearity of digital phase shifter is fairly well, the loss associated with the switches is still a challenge [38]. Moreover, digital phase shifters do not provide continuous phase shifting, which causes high side-lobe level in the radiation pattern of the antenna, and beam pointing errors. In contrast, analog phase shifters vary the phase continuously. Two well-known analog structures for phase shifters are vector summing and reflective-type phase shifter (RTPS). The linearity of the vector summing phase shifter is limited due to the active phase shifting. Furthermore, a large amount of power must be consumed to achieve a high dynamic range [39], [40]. Hence, in this work the focus is placed on RTPS type to provide a continuous, low-power phase shift over the desired frequency band. Fig. 9 shows the general block diagram of the RTPS, which employs a 4-port 90 -hybrid and two similar purely imaginary (reflective) loads. The through and coupled ports of the hybrid are terminated to the reflective loads and the isolated port is used as the output. The reflective loads are varied electronically by changing the control voltage. Thus, the phase of the reflection coefficient at the through and coupled ports changes, which results in the phase shift of the output signal. The amount of the phase shift depends on the load reactance. Several passive terminations are proposed for the reflective loads. A single varactor cannot provide a phase shift more than 75 in practice [41].

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Fig. 10. Measured characteristics of the phase shifter over the frequency range of 50–65 GHz for the DC tuning voltage of 0–0.8 V, (a) phase shift. (b) Insertion loss (S ) and return loss (S ).

Adding a series inductor can increase the phase shift up to 180 . To achieve a complete 360 phase shift one should use dual resonant loads [41]. The RTPS, being fundamentally passive, shows linear input-output characteristics [42]. The main challenge in the integrated CMOS based RTPS design is the loss. The main sources of loss in an RTPS are the transmission-line loss in the 90 -hybrid and the loss in the reflective terminations. To reduce phase shifter loss, active, negative resistance circuits have been used [43]. This, however, limits the linearity and noise performance of the RTPS. The insertion-loss variation can also be minimized by using an equalization resistance and modification of the 90 -hybrid [44]. In this work, an RTPS with dual resonant loads is designed in 90 nm CMOS. For a relative broadband 90 degree hybrid a broadside coupler is used. The broadside structure is implemented on the thick metal layers of the 90 nm CMOS process. The coupler has been meandered to have a compact size. The varactor is implemented in CMOS by using a regular CMOS transistor in which the Source and Drain terminals are connected. This structure is similar to a diode which can be biased reversely to generate a variable capacitance. Circuit simulation demonstrates that by varying the applied DC voltage from 0 to and can work 1 V, a transistor with as a varactor with a minimum capacitance of 49 fF and a tuning ratio of 3. The typical Q of the varactors in this technology is not higher than 10. The proposed reflective load consists of two inductors, a capacitor and the varactor. The die micrograph of the fabricated RTPS is depicted in Fig. 9(b). The area of this device is 0.35 mm 0.2 mm which is smaller than other CMOS phase shifters in this band (see Table I in [42]). Fig. 10(a) and (b) depict the phase shift and insertion loss variation of the phase shifter over the frequency range of 50–65 GHz. When the input DC voltage is changed between 0 and 0.8 V, the phase shift changes from 0 to 90 . Furthermore, the measured insertion loss varies from 4.5 dB to 8 dB at 60 GHz. By using parallel resonant loads, the maximum phase shift can be enhanced. Fig. 11(a) and (b) demonstrate the characteristics of the simulated RTPS with parallel resonant loads versus , 60 and 62 GHz. The phase sweep for bias voltage for is more than 370 for the whole fre-

Fig. 11. Simulated phase shift and insertion loss of the designed phase shifter for 360 phase shift.

TABLE IV SUMMARY OF THE PARAMETERS USED IN NOISE FIGURE CALCULATIONS

quency range. The maximum insertion loss variation is 2.75 dB, which occurs at the lowest frequency. D. Noise Figure of the Phased Array Receiver If the LNA gain is sufficiently high temperature simplifies to

, the system noise

(1) denote the antenna efficiency, front end where , , and loss, and noise figure of the LNA, respectively. Assuming the antenna temperature is equal to the room temperature , noise figure of the designed CMOS phased array receiver is between 6 to 10 dB as shown in Table IV.

FAKHARZADEH et al.: CMOS PHASED ARRAY TRANSCEIVER TECHNOLOGY FOR 60 GHz WIRELESS APPLICATIONS

V. BEAMFORMING In this section, two novel beamforming algorithms for the MMW receiver phased array antenna are proposed, and the achieved improvement in the signal to noise ratio at the array output is presented.

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the total received power by the array. The voltage dependent characteristics of the phase shifters form the constrains of the optimization problem. Hence, the beamforming problem can be stated as

A. Beamforming Algorithms for MMW Receiver Phased Array The goal of the beamforming algorithm is to increase the array factor and consequently provide the SNR determined by Bit Error Rate (BER) constrains. The ideal limit of the array factor is equal to the number of array elements; however, as it will be shown, the practical maximum array factor is smaller than that due to the variable insertion loss of phase shifter. denote 1) Signal Model: Let the received signals by all elements of the array. Then it con, interference , and sists of three parts: source signal , background noise

(2) The background noise is assumed to be spatially white. Assume the source (transmitter) is located at direction in the receiver array coordinate system, transmitting RF signals at frequency . The RF signal received by an element of the array is given by located at

(3) where , and are respectively the RF wave number . and the source waveform. The path loss is included in The array output for a single receiver array structure shown in Fig. 4(a) is

(4) where is the array weights vector and denotes the Hermitian operator. If analog phase shifters, such as the one shown in Fig. 11, were used to adjust the array weights for beamforming the wight vector would be [45]

(7) In [45], [46], the authors have shown that an efficient way to solve this problem is to use a gradient estimation approach such as zero-knowledge beamforming algorithm. In this case the control voltages are updated in an iterative manner (8) where is an internal algorithm parameter called the step size, is the gradient of power with respect to . Since and the exact calculation of the gradient is not practical it is replaced by an estimated vector: (9) where each component is the approximate partial derivaw.r.t. . tive of 3) Reverse-Channel Aided Beamforming: The size and cost constrains of the 60 GHz receiver do not allow for incorporating a complex processor in the portable node. However, the access point (fixed node) can handle more elaborate signal processing tasks. Moreover, in MMW networking standards such as WPAN, 50 MHz of the spectrum is reserved for the reverse channel to carry the control signals between the access points and mobile nodes. Access point can be equipped with direction-of arrival (DOA) estimation unit. This unit can calculate the relative position of the mobile nodes and send this information to them. The mobile node can use this information to adjust its beam. Although this method is very fast, in the case of shadowing it is not efficient. In this case the beamformer must be able to maintain the array beam on the direction of the strongest component of the multipath signal. B. Beamforming Results for LOS Propagation

(5) where is the control voltage of the phase shifter, and and denote the amplitude (insertion loss) and phase-shift functions of the phase shifter. The total received power by the array is then

(6) 2) Statement of the Problem: In the absence of co-channel interference, beamforming for a MMW receiver array is a constrained optimization problem with the objective of maximizing

Fig. 12 demonstrates the results of the Aided Beamforming algorithm for a typical case, where it is assumed that the equivalent noise bandwidth is 2.16 GHz, the receiver NF is 6 dB, the output power of the power amplifiers is 2 dBm, and the antenna element is the 2 2 patch array shown in Fig. 6(b). The 9-element phased array is a 3 3 square array with (5 mm) spacing. In Fig. 12 it is assumed that the user is moving away from the transmitter while both receiver and transmitter antenna axes are parallel. Fig. 12 compares the results of ideal, fast and slow beamforming for a 3 3 square array. Slow beamforming is the case where during one iteration of the algorithm user moves more than 1 cm, while in fast beamforming the user’s displacement is less than 1 mm. The difference between ideal and fast beamforming when the user is close to the transmitting node

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Fig. 12. Received SNR by a 3 beamforming scenarios.

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2 3 square phased array antenna for different

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Fig. 13. Beamforming results for a 4 4 square array with  spacing. The dashed and solid curves correspond to the receiver location on Path A and Path B in Fig. 2. (a) SNR at the array output after beamforming. (b) The improvement in SNR due to using phased array for the region shown by a rectangle in Fig. 13(a).

m is 2 dB. As the distance increases to this difference raises to more than 3.5 dB. Another important result of Fig. 12 is that the beamforming gain depends on the received power. So, as the distance between the portable receiver and the transmitting node increases, the performance of the beamforming degrades. Moreover, the beamforming speed affects the beamforming gain. A similar version of this algorithm has been successfully applied to a 34 element Ku-band phase array antenna [46]. The duration of each beamforming iteration was measured to be less than 5 ms with a 125 KHz Digital-to-Analog convertor. C. Beamforming Results for NLOS Propagation In this case, user location information is not available and the receiver seeks for the strongest signal by running a beam-search mode (acquisition phase). A method has been proposed for direction finding in [47]. Fig. 13 shows the results of beamforming with a 4 4 square array (with spacing), when the receiver moves along Path A and B shown in Fig. 2. When the receiver is close to the human body (Path A) the width of the fading region is larger to ) and the shadowing loss varies from ( 10 dB to 40 dB. Fig. 13(a) illustrates that the array output SNR is always above 12 dB. Fig. 13(b) shows the improvement in SNR after applying the proposed beamforming algorithm to a 16-element phased array. This improvement is due to three factors: using a 16-element array at the access point to increase the effective radiated power, using a 2 2 patch antenna element at the receiver (instead of an isotropic antenna), and the beamforming gain (array factor). While for the LOS section the improvement is 28 dB, it increases up to 42 dB in the shadowing region, implying that the proposed beamforming algorithm has an excess gain (up to 14 dB) when shadowing occurs. Increasing the number of elements in the array improves this gain. Fig. 14 illustrates how the beamforming algorithm for NLOS propagation works. In this figure the strongest received rays for two receiver positions on Path B, namely Point M and Point N shown in Fig. 13(b), are depicted. Point M is in the deep shadowing region, hence its strongest ray has an excess delay of 21 ns and is 26 dB weaker than that of the point N. Fig. 14(b) and (c)

Fig. 14. Characteristics of the strongest received ray for two positions.

show the directions of the two strongest rays which are resolvable in their coordinates. If the signal level drops suddenly, the beamforming algorithm finds the strongest ray and steers the array beam to its direction. VI. CONCLUSION CMOS is the promising technology for low-cost high-volume 60 GHz transceivers. However, the fundamental limitations of current CMOS technology, such as low amplifier gain and output power as well as the fairly high noise figure restrict the performance of the single-antenna 60 GHz CMOS systems. It was shown that an adaptive (intelligent) integrated phased array antenna and radio system (at both ends) is the most viable approach to provide the required SNR for reliable MMW network operation. Moreover, the RF-phase shifting array configuration is a practical and low-cost architecture for 60 GHz applications. The efficiency of phased array system depends on the performance of LNA and phase shifter; hence, one objective of this paper was to present the design of a high-performance phase shifter and LNA in CMOS technology.

FAKHARZADEH et al.: CMOS PHASED ARRAY TRANSCEIVER TECHNOLOGY FOR 60 GHz WIRELESS APPLICATIONS

To design the link budget for 60 GHz network, the propagation of the MMW signal in an indoor environment was studied for LOS and NLOS scenarios, and verified by measurements. It was found that the shadowing loss of a human body can be as high as 40 dB. In this case all LOS rays are absorbed by the human body and only NLOS rays are received. A low noise amplifier using cascode topology, was designed and fabricated in 90 nm CMOS technology. The measured gain of this two stage LNA exceeded 20 dB at the frequency range of 56–61 GHz. Furthermore, to meet the beamforming requirements, a broadband reflective type phase shifter in 90 nm CMOS with a linear phase and low insertion loss, was designed, fabricated and successfully tested. Finally, a fast beamforming algorithm was developed to realize the potentials of phased array for both LOS and multipath signal propagation. The imbalanced insertion loss of phase shifters results in a margin between the ideal and practical array gain, which reduces the effective range or transmitted/received power by the array. Moreover, the beamforming gain is proportional to the input SNR. In the case of shadowing, the proposed beamforming algorithm seeks for the strongest ray. It was shown an excess gain up to 14 dB can be obtained by this method. ACKNOWLEDGMENT The authors would like to acknowledge CMC (Canadian Microelectronics Corporation) for device fabrication. The authors are thankful to Mr. J. Dietrich at CMC Advanced RF Lab at the University of Manitoba for on-wafer measurements, and Mr. H. Mirzaei. REFERENCES [1] K. Ohata et al., “Sixty-GHz-band ultra-miniature monolithic T/R modules for multimedia wireless communication systems,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2354–2360, Dec. 1996. [2] S. Reynolds et al., “A silicon 60 GHz receiver and transmitter chipset for broadband communications,” IEEE J. Solid-State Circuits, vol. 41, no. 12, pp. 2820–2830, Dec. 2006. [3] S. E. Gunnarsson et al., “60 GHz single-chip front-end MMICs and systems for multi-Gb/s wireless communication,” IEEE J. Solid-State Circuits, vol. 42, no. 5, pp. 1143–1157, May 2007. [4] T. Mitomo et al., “A 60-GHz CMOS receiver front-end with frequency synthesizer,” IEEE J. Solid-State Circuits, vol. 43, no. 4, pp. 1030–1037, Apr. 2008. [5] C. H. Doan, S. Emami, A. M. Niknejad, and R. W. Brodersen, “Millimeter-wave CMOS design,” IEEE J. Solid-State Circuits, vol. 40, no. 1, pp. 144–155, Jan. 2005. [6] A. Babakhani et al., “A 77-GHz phased-array transceiver with on-chip antennas in silicon: Receiver and antennas,” IEEE J. Solid-State Circuits, vol. 41, no. 12, pp. 2795–2806, Dec. 2006. [7] B. Razavi, “Gadgets gab at 60 GHz,” IEEE Spectrum, vol. 45, no. 2, pp. 46–58, Feb. 2008. [8] P. Smulders, “Exploiting the 60 GHz band for local wireless multimedia access: Prospects and future directions,” Commun. Mag., IEEE, vol. 40, no. 1, pp. 140–147, Jan. 2002. [9] C. Park and T. Rappaport, “Short-range wireless communications for next-generation networks: UWB, 60 GHz millimeter-wave WPAN, and ZigBee,” IEEE Wireless Commun., vol. 14, no. 4, pp. 70–78, Aug. 2007. [10] Wirelesshd Specification Version 1.0 Overview 2007 [Online]. Available: http://www.wirelesshd.org/WirelessHD-Full-Overview071009.pdf [11] B. Razavi, “A 60-GHz CMOS receiver front-end,” IEEE J. Solid-State Circuits, vol. 41, no. 1, pp. 17–22, Jan. 2006. [12] K. Scheir, S. Bronckers, J. Borremans, P. Wambacq, and Y. Rolain, “A 52 GHz phased-array receiver front-end in 90 nm digital CMOS,” IEEE J. Solid-State Circuits, vol. 43, no. 12, pp. 2651–2659, Dec. 2008.

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[13] S. R. Alberto Valdes-Garcia and J.-O. Plouchart, “60 GHz transmitter circuits in 65 nm CMOS,” in Proc. IEEE Radio Frequency Integrated Circuits Symp., Jun. 2008, pp. 641–644. [14] A. P. M. Boers and N. Weste, “A 60 GHz transmitter in 0.18  silicon germanium,” in Proc. Wireless Broadband and Ultra Wideband Communications, Aug. 27–30, 2007, p. 36. [15] B. Razavi, “CMOS transceivers for the 60-GHz band,” presented at the IEEE Radio Frequency Integrated Circuits Symp., Jun. 2006. [16] T. Manabe, Y. Miura, and T. Ihara, “Effects of antenna directivity and polarization on indoor multipath propagation characteristics at 60 GHz,” IEEE J. Selec. Area Commun., vol. 14, no. 3, pp. 1441–1448, Apr. 1996. [17] C.-P. Lim, M. Lee, R. J. Burkholder, J. L. Volakis, and R. J. Marhefka, “60 GHz indoor propagation studies for wireless communications based on a Ray-tracing method,” EURASIP J. Wireless Commun. Net., 2007, article ID 73928. [18] P. F. M. Smulders, C. F. Li, H. Yang, E. F. T. Martijn, and M. H. A. J. Herben, “60 GHz indoor radio propagation—Comparison of simulation and measurement results,” presented at the IEEE 11th Symp. on Commun. and Veh. Technol., 2004. [19] B. Langen, G. Lober, and W. Herzig, “Reflection and transmission behaviour of building materials at 60 GHZ,” in Proc. IEEE Int. Symp. Personal, Indoor and Mobile Radio Commun., 1994, pp. 505–509. [20] L. M. Correia and P. O. Frances, “Estimation of materials characteristics from power measurements at 60 GHz,” in Proc. IEEE Int. Symp. Personal, Indoor and Mobile Radio Commun., 1994, pp. 510–513. [21] C. M. Alabaster, “Permittivity of human skin in millimetre wave band,” Electron. Lett., vol. 39, no. 21, pp. 1521–1522, Oct. 2003, article ID 73928. [22] R. J. Mailloux, Phased Array Antenna Handbook—Chapter 1, 2nd ed. Boston, MA: Artech House, 2005. [23] H. Hashemi, X. Guan, A. Komijani, and A. Hajimiri, “A 24-GHz SiGe phased-array receiver LO phase-shifting approach,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 2, pp. 614–626, Feb. 2005. [24] S. Raman, N. Barker, and G. Rebeiz, “A w-band dielectric-lens-based integrated monopulse radar receiver,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2308–2316, Dec. 1998. [25] R. Miura, T. Tanaka, I. Chiba, A. Horie, and Y. Karasawa, “Beamforming experiment with a DBF multibeam antenna in a mobile satellite environment,” IEEE Trans. Antennas Propag., vol. 45, no. 4, pp. 707–714, Apr. 1997. [26] M. Fakharzadeh, P. Mousavi, S. Safavi-Naeini, and S. H. Jamali, “The effects of imbalanced phase shifters loss on phased array gain,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 192–196. [27] T. Zwick, D. Liu, and B. P. Gaucher, “Broadband planar superstrate antenna for integrated millimeterwave transceivers,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2790–2796, Oct. 2006. [28] E. Ojefors, H. Kratz, K. Grenier, R. Plana, and A. Rydberg, “Micromachined loop antennas on low resistivity silicon substrates,” IEEE Trans. Antennas Propag., vol. 54, no. 12, pp. 3593–3601, Dec. 2006. [29] C. Karnfelt, P. Hallbjorner, H. Zirath, and A. Alping, “High gain active microstrip antenna for 60-GHz WLAN/WPAN applications,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pt. 2, pp. 2593–2603, Jun. 2006. [30] A. Lamminen, J. Saily, and A. R. Vimpari, “60-GHz patch antennas and arrays on LTCC with embedded-cavity substrates,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2865–2874, Sep. 2006. [31] R. C. Johnson and H. Jasik, Antenna Engineering Handbook. New York: McGraw-Hill, 1961. [32] J. R. James and P. S. Hall, Handbook of Microstrip Antennas. London, U.K.: Inst. Elec. Eng., 1989. [33] R. A. Alhalabi and G. M. Rebeiz, “High-efficiency angled-dipole antennas for millimeter-wave phased array applications,” IEEE Trans. Antennas Propag., vol. 56, no. 10, pp. 3136–3142, Oct. 2008. [34] B. Razavi, “A 60-GHz direct-conversion CMOS receiver,” in IEEE ISSCC. Proc., Feb. 2005, pp. 400–401. [35] B. Heydari, P. Reynaert, E. Adabi, M. Bohsali, B. Afshar, M. A. Arbabian, and A. M. Niknejad, “A 60-GHz 90-nm CMOS cascode amplifier with interstage matching,” IEEE J. Solid-State Circuits, vol. 42, no. 12, pp. 2893–2904, Dec. 2007. [36] C.-M. Lo, C.-S. Lin, and H. Wang, “A miniature V-band 3-stage cascode LNA in 0.13  CMOS,” in IEEE ISSCC Dig. Tech. Papers, Feb. 2006, pp. 322–323. [37] M. R. Nezhad Ahmadi, B. Biglarbegian, H. Mirzaei, and S. SafaviNaeini, “An optimum cascode topology for high gain micro/millimeter wave CMOS amplifier design,” in Proc. Eur. Microwave Integrated Circuit Conf. (EuMIC), Netherlands, Oct. 2008, pp. 394–397.

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[38] B.-W. Min and G. M. Rebeiz, “Ka-band BiCMOS 4-bit phase shifter with integrated LNA for phased array T/R modules,” in Proc. IEEE/ MTT-S Int. Microwave Symp., Jun. 2007, pp. 479–482. [39] S. Alalusi and R. Brodersen, “A 60 GHz phased array in CMOS,” in Proc. IEEE Custom Integrated Circuits Conf., Sep. 2006, pp. 393–396. [40] K.-J. Koh and G. M. Rebeiz, “An X- and Ku-band 8-element linear phased array receiver,” in Proc. IEEE Custom Integrated Circuits Conf., Sep. 2007, pp. 761–764. [41] R. V. F. Ellinger and W. Bächtold, “Ultracompact reflective-type phase shifter MMIC at C-band with 360? Phase-control range for smart antenna combining,” IEEE J. Solid-State Circuits, vol. 37, no. 4, pp. 481–486, Apr. 2002. [42] B. Biglarbegian, M. R. Nezhad-Ahmadi, M. Fakharzadeh, and S. Safavi-Naeini, “Millimeter-wave reflective-type phase shifter in CMOS technology,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 9, pp. 560–562, Sep. 2009. [43] H. Zarei and D. J. Allstot, “A low-loss phase shifter in 180 nm CMOS for multiple-antenna receivers,” in Proc. IEEE Int. Solid-State Circuits Conf., Feb. 2004, pp. 382–534. [44] C. S. Lin, S. F. Chang, C. C. Chang, and Y. H. Shu, “Design of a reflection-type phase shifter with wide relative phase shift and constant insertion loss,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 9, pp. 1862–1868, Sep. 2007. [45] M. Fakharzadeh, S. H. Jamali, P. Mousavi, and S. Safavi-Naeini, “Fast beamforming for mobile satellite receiver phased arrays: Theory and experiment,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1645–1654, Jun. 2009. [46] P. Mousavi, M. Fakharzadeh, S. H. Jamali, K. Narimani, M. Hossu, H. Bolandhemmat, and S. Safavi-Naeini, “A low-cost ultra low profile phased array system for mobile satellite reception using zero-knowledge beam-forming algorithm,” IEEE Trans. Antennas Propag., vol. 56, no. 12, pp. 3667–3679, Dec. 2008. [47] H. Bolandhemmat, M. Fakharzadeh, P. Mousavi, S. H. Jamali, G. Rafi, and S. Safavi-Naeini, “Active stabilization of vehicle-mounted phased-array antennas,” IEEE Trans. Veh. Technol., vol. 58, no. 6, pp. 2638–2650, Jul. 2009. Mohammad Fakharzadeh (S’05–M’09) received the B.Sc. degree (honors) from Shiraz University, Shiraz, Iran and the M.Sc. degree from Sharif University of Technology, Tehran, Iran, in 2000 and 2002, respectively, all in electrical engineering. From 2004 to 2008, he was a Ph.D student at the Intelligent Integrated Radio and Photonics Group, University of Waterloo, ON, Canada, where he is currently a Postdoctoral Researcher and Coordinator of the Millimeter-wave Group. From January 2003 to September 2004, he was a Researcher and Instructor at the Electrical Engineering Department, Chamran University of Ahvaz, Iran. Since June 2005, he has been a consultant to Intelwaves Technologies, developing the beamforming, signal processing and tracking algorithms for mobile satellite receiver phased array antennas. His areas of interest include phased array design, beamforming and signal processing, millimeter wave systems for short range wireless networks, integrated antennas, and miniaturized optical delay lines. Dr. Fakharzadeh is the recipient of the University of Waterloo Outstanding Graduate Studies Award, and the 2008 Khwarizimi International Award for his Ph.D. research.

Mohammad-Reza Nezhad-Ahmadi received the B.Sc. degree in electrical engineering, from Isfahan University of Technology, Isfahan, Iran, and the M.Sc. degree in electrical engineering from Sharif University of Technology, Tehran, Iran in 1998 and 2000, respectively. He is currently working toward the Ph.D. degree at the University of Waterloo, Waterloo, ON, Canada. He joined ON semiconductor Canada in 2006, where he is now a senior RF/Analog IC and System Engineer and involved with ultra low power radio

circuit and architecture for wireless medical applications. From 1999 to 2003, he was a Design Engineer with Unistar-Micro Technology, Tehran, where he was involved with the design of a full CMOS RF chip for GPS and a BiCMOS WLAN chipset. From 2003 to 2005, he was a senior RF Design Engineer with Ameri-Tech Co., Tehran, where he was involved with the design and development of integrated digital microwave radio systems. His research interests are silicon millimeter phased array systems, miniaturized on-chip antennas, and ultra low power radios.

Behzad Biglarbegian (S’07) was born in Tehran, Iran, in August 1980. He received the undergraduate and Master’s degree in electromagnetics and antennas from the University of Tehran and Iran University of Science and Technology, Tehran, in 2002 and 2005, respectively. He then joined AmeriTech telecommunication Co., Tehran, where he was a Senior Engineer in designing digital microwave radios for point to point wireless links. In 2007, he joined University of Waterloo, Waterloo, ON, Canada, to pursue his academic career as a Ph.D. program in the area of intelligent integrated wireless systems. He is currently investigating novel ideas for the implementation of low cost integrated millimeter-wave phased array wireless devices. His research is mainly to design and implement smart antennas in CMOS technology considering the packaging issues. Mr. Biglarbegian is the recipient of IEEE Antennas and Propagation Society predoctoral/doctoral research award in 2009.

Javad Ahmadi-Shokouh (S’04–M’08) received the B.Sc. degree in electrical engineering from Ferdowsi University of Mashad, Mashad, Iran, in 1993, the M.Sc. degree in electrical engineering from the University of Tehran, Tehran, Iran, in 1995, and the Ph.D. degree in electrical engineering from the University of Waterloo, Waterloo, ON, Canada, in 2008. From 1998 to 2003, he was with the Department of Electrical Engineering, University of Sistan and Baluchestan, Zahedan, Iran, as a Lecturer, where he is currently an Assistant Professor . He was also with the Department of Electrical Engineering University of Manitoba, Winnipeg, MB, Canada (2008–2009) as a Postdoctoral Fellow. His research interests are RF-baseband co-design for wireless communication systems, millimeter wave and ultrawideband (UWB) systems, smart antennas, statistical and array signal processing, optimal and adaptive MIMO systems, and Underwater-acoustic hydrophone-array signal processing.

Safieddin Safavi-Naeini received the B.Sc. degree in electrical engineering from the University of Tehran, Tehran, Iran, 1974 and M.Sc. and Ph.D. in electrical engineering both from the University of Illinois at Urbana-Champaign, in 1975 and 1979 respectively. He joined the University of Waterloo, Waterloo, ON, Canada, in 1996 where he is now a Professor in the Department of Electrical and Computer Engineering and holds the RIM/NSERC Industrial Research Chair in Intelligent Radio/Antenna and Photonics. He has more than 30 years of research experience in antenna, RF/microwave technologies, integrated photonics, and computational electromagnetics. He has published more than 70 journal publications and 200 conference papers in international conferences. He has led several international collaborative research programs with research institutes in Germany (DAAD fund), Finland (Nokia), Japan, China (BVERI, Institute of Optics), and USA, which have resulted in novel technologies and efficient design methodologies. He has been a scientific and technical consultant to many North American, European, and Asian international companies.

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Direction of Arrival Estimation in Time Modulated Linear Arrays With Unidirectional Phase Center Motion Gang Li, Shiwen Yang, Senior Member, IEEE, and Zaiping Nie, Senior Member, IEEE

Abstract—A novel approach for estimating the direction of arrivals (DOAs) in time modulated linear arrays (TMLAs) with unidirectional phase center motion (UPCM) scheme is proposed in this paper. Based on the fact that the main beams of the patterns at different sidebands can be directed at different directions, the corresponding received signals can be used to compose a received data space. Thus, the spatial locations of the far-field sources can be estimated by using multiple signal classification (MUSIC) algorithm. Simulation results of the DOA estimation in an 8-element TMLA with the UPCM scheme validate the proposed approach, where the performance such as the accuracy and resolution of the DOA estimation is obtained through Monte-Carlo simulations. As compared to the DOA estimation based on conventional uniform linear arrays (ULAs), a much better resolution performance is obtained. Index Terms—Antenna arrays, direction of arrival (DOA), moving phase center, time modulation.

I. INTRODUCTION

T

HE direction of arrival (DOA) estimation of multiple narrowband sources has been receiving much attention over the years. Among the techniques proposed to realize the DOA estimation, it has been found that subspace algorithms definitely have many advantages due to their superior performance, such as the MinNorm [1], multiple signal classification (MUSIC) [2], and the estimation of signal parameters via rotational invariance technique (ESPRIT) [3], [4]. By using eigenvalue decomposition or singular value decomposition (SVD), the subspace methods can be used to decompose the sample covariance matrix into two orthogonal spaces, which are commonly referred to as the signal and noise subspaces. Then, the DOAs can be estimated from one of these spaces. However, these eigenvalue decomposition-based methods usually have high computational complexity, especially for those cases arrays with a large number of elements. An efficient approach to solve this problem is to reduce the dimension of the received array data, by mapping the full dimension of element-space Manuscript received May 17, 2009; revised September 11, 2009. Date of manuscript acceptance October 01, 2009; date of publication January 26, 2010; date of current version April 07, 2010. This work was supported in part by the Natural Science Foundation of China under Grant 60971030, the New Century Excellent Talent Program in China under Grant NCET-06-0809, and in part by the 111 project of China under Grant B07046. The authors are with the Department of Microwave Engineering, School of Electronic Engineering, University of Electronic Science and Technology of China (UESTC), Chengdu 611731, China (e-mail: [email protected]) Digital Object Identifier 10.1109/TAP.2010.2041313

into lower dimension of the beam-space through beamforming [5]–[7]. Other approaches such as the unitary root-MUSIC [8] and unitary ESPRIT [9] using real-valued SVD were also proposed to reduce the computational complexity. So far, DOA estimation techniques have been widely applied to various types of antenna arrays, e.g., uniform linear arrays (ULAs), uniform circular arrays, and uniform rectangular arrays. However, as far as the authors know, there has been no study on the DOA estimation in time modulated antenna arrays (TMAAs). The TMAAs were firstly proposed in 1960’s. By using highspeed RF switch to control the array elements, low sidelobe array patterns can be synthesized [10]–[12]. By introducing the fourth dimension—“Time”—Into the antenna design, the dynamic range ratios of the amplitude excitations in the TMAAs are usually much lower as compared to those of the conventional antenna arrays. Due to the periodic time modulation, there are many sideband signals spaced at multiples of the time modulation frequency, which may cause the energy losses at the center frequency. Thus, it is generally agreed that sideband signals may not be desirable and should be suppressed to improve the efficiency of the antenna array [12]–[18]. However, by designing appropriate time sequences of each element, sidebands are not always harmful to the TMAAs. Instead, the sidebands can be used in some special applications. For example, a simultaneous scanning operation based on the time modulation technique was proposed in [19], where the beams at different sidebands are directed to different directions. More recently, an electronic null scanning was realized at the first sideband in a two-element TMAA by adjusting switching times of the both elements [20]. A hybrid analog-digital beamforming scheme based on time modulated linear arrays (TMLAs) was proposed in [21], where the adaptive beamforming can be achieved at the first sideband. However, the DOA estimation based on TMAAs are rarely seen. Thus, the study on the DOA estimation in TMAAs becomes the motivation of our study. However, there is a problem in the DOA estimation based on TMAAs. Due to the fact that high-speed RF switches periodically switch on and off according to a specific time sequence to realize the time modulation in TMAAs, the received signals in some channels are forced to be zero during a certain time interval within one modulation period, which will deteriorate or invalidate conventional DOA estimation algorithms. To solve this problem, a novel approach for estimating DOAs in TMLAs with unidirectional phase center motion (UPCM) scheme is proposed in this paper. With the UPCM scheme, the beams at different sidebands in TMLAs are capable of pointing at different

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directions and the corresponding received signals can be used to compose a received data space. After down converting these signals into the same IF stage, the DOA estimation can be performed by using the MUSIC algorithm. The organization of this paper is as follows: Section II introduces the theory of the TMLA with UPCM scheme. The approach for DOA estimation in the TMLA is presented in Section III. Simulation results of DOA estimation in the TMLA are provided in Section IV. Finally, some conclusions are drawn in Section V.

Fig. 1. Scheme of an

N -elements linear array with UPCM scheme.

II. PRINCIPLE OF TMLA WITH UPCM SCHEME The UPCM scheme was firstly proposed by Lewis et al. in 1983 to reduce the signals received from the sidelobes of antenna arrays [22]. By moving the phase center of a phased array antenna, the signals received through the antenna sidelobes can be shifted out of the passband of the electronic system receiver due to the Doppler shift effect. An -element linear array of equally spaced isotropic elements is considered (Fig. 1). Suppose that each element is controlled by a high-speed RF switch, and the moving phase center technique can be implemented by high speed RF switches in the feed line of each element. The array elements are numbered 1 elements to from the left to the right. Firstly, the left most are turned on for a time step , given by

Fig. 2. Time sequences of an 8-element TMLA with

. The

M = 4.

th order Fourier component can be

written as (6)

(1)

,

where

, and

is given by

where is the time modulation period. Then, consecutive elements numbered from 2 to are switched on in the next time step. The array factor of the TMLA is given by [23] (7) (2) is the center operating frequency, and are the where static excitation amplitude and phase of the th element, is the , is the velocity of element spacing of the array, light in free space, denotes the angle measured from the broadrepresents the ON-OFF side direction of the array, and time switching function for the th element. According to the is given by UPCM scheme,

(8) For simplicity, the static excitation amplitude and phase are selected as uniform. Without loss of generality, let and , . Then, (6) becomes

(9) otherwise

(3)

where otherwise, otherwise.

In (9), the terms

and can be considered as the amplitude and phase excitations of the th element, respectively. Based on the analysis above, the approach for the DOA estimation in the TMLA with UPCM scheme will be presented in Section III.

(4) III. DOA ESTIMATION IN TMLA WITH UPCM (5)

in detail, an example of a TMLA with To illustrate the and is shown in Fig. 2. Due to the fact that is a periodic function of time, the space and frequency response of (1) can be obtained by decomposing it into Fourier series, and each frequency component has a frequency of

Suppose that there are far-field narrowband noncoherent sources with the same carrier frequency and are impinging on the TMLA, which is excited with uniform static amplitude and phase excitation. The received signal is given by (10)

LI et al.: DOA ESTIMATION IN TMLAs WITH UNIDIRECTIONAL PHASE CENTER MOTION

where is the signal emitted by the th far-field source lodenotes the zero-mean cated at the direction of , and white Gaussian noise with variance of , which is statistically independent to the incident signals. Based on the previous received by the analysis, each signal TMLA will generate many sideband signals at . The th sideband signal can be expressed as

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in (18), According to (7), in order to guarantee that , must be satisfied. Then, the covariance matrix of can be simplified as

(19) where

denotes the complex conjugate transpose, and

(20)

(11) According to Nyquist theorem, when the time modulation frequency is equal to or greater than the bandwidth of the signal , there is no overlap between sidebands. Consequently, each sideband signal and the center frequency signal can be separated using band-pass filters. After down converting the signals at into the same IF stage, the output signals can be obtained and is given by

(12) denotes the IF signal of , is a positive inwhere teger which stands for the maximum order of sideband. Thus, the total number of sidebands including the center frequency utilized for the DOA estimation is . Supsignal . pose that the number of available data points (snapshots) is For convenience, (12) can be rewritten as a matrix form, given by (13) where

Thus, we can represent the eigenvalues of by and , the corresponding eigenvectors by , . Based on the supposing that beamspace-MUSIC algorithm [5], the eigenvectors corresponding to the first largest eigenvalues are referred to as the signal eigenvectors spanning the signal subspace, and those corresponding to the minimum eigenvalues are referred to as the noise eigenvectors spanning the noise subspace. Moreover, it is well known that the signal subspace and the noise subspace are orthogonal, where the null-spectrum can be obtained by

(21) where

is the noise subspace and given by

(22) Thus, the DOAs can be determined by the locations in the , referred to as the spatial spectrum and written peaks of by

denotes the transpose, and

(23) (14)

the matrix

is defined as

(15)

IV. SIMULATION RESULTS

(16)

In this section, the resolution and accuracy of the proposed approach for the DOA estimation in the TMLA with UPCM scheme is evaluated and compared to the DOA estimation in conventional ULAs with the MUSIC algorithm. In all statistic results, 1000 times Monte-Carlo simulations are performed.

with (17)

and is an given by

-dimensional complex matrix and is

.. .

.. .

..

.

.. .

(18)

A. Patterns in TMLA With UPCM Let us consider a TMLA of isotropic elements with half a wavelength uniform spacing, based on the UPCM scheme. For a constant , can be determined by using (1) when , 2, 3, and 4. The corresponding patterns at with , 2, 3, and 4 can be obtained by (9), which are plotted in Fig. 3(a)–(d), respectively. It is observed that the patterns at the center frequency always point at

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Fig. 4. Spatial spectrum versus SNR (

M = 2, Q = 2, and NN = 100).

hand, the patterns at different sideband are capable of pointing th order at different directions, and the patterns at sidebands are symmetrical with respect to the normal axis of the TMLA. Moreover, the sideband levels are found to be further reduced with the increase of . B. Spatial Spectrum Suppose that there are three far-field narrowband uncorrelated binary phase shift keying (BPSK) modulated signals with and random codes, equal power, the same carrier frequency , and , rebandwidth, arriving from spectively. The 8-element TMLA with UPCM scheme is used to receive the signals. The number of snapshots is , the spatial spectrums can be ob100. By using (23) with tained. Fig. 4 plots the spatial spectrums versus signal-to-noise ratio (SNR). It can be seen that the resolution performance is getting better with the increase of SNR. Meanwhile, the DOAs of the three signals can be successfully determined by the locations of the peaks in the spatial spectrum at higher SNRs. C. The Resolution Performance

M=3 M=4

Fig. 3. Patterns in 8-element TMLA with UPCM scheme. (a) ; (c) ; (d) .

2

M = 1; (b)M =

, and the SLLs are , , and for , 2, 3 and 4, respectively. On the other

In this subsection, two far-field narrowband BPSK modulated and are considered. signals closely located at Similarly, the two signals are also uncorrelated with each other and have random codes, equal power, the same carrier frequency and bandwidth. Firstly, the number of snapshot is 100 and . Fig. 5 , 2, shows the probability of resolution versus SNR ( 3 and 4). In order to make comparisons, an 8-element conventional ULA with half a wavelength element spacing is considered. The DOAs of the two signals are estimated by the MUSIC algorithm [2]. It can be seen that the resolution performance in the TMLA with UPCM scheme is better than that in the ULA. In addition, with the decrease of , better performance of resolution can be obtained due to the increase of sideband levels. Moreover, the proposed approach can be used to separate the two signals successfully with the probability of 100%, when the SNR is greater than 8 dB. is selected to be 100, Secondly, the number of snapshots . The performance of resolution of the two signals and versus SNR with different is shown in Fig. 6. Obviously, the performance of resolution in the TMLA with UPCM scheme

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Q=1

7. Probability of resolution versus SNR with different number of snapshots M (NN = 100 Fig. (M = 1 and Q = 1).

M=1

8. RMSEs of the DOA estimation versus SNR with different M ( = 5 , Q (NN = 100 Fig. Q = 1, and NN = 100).

Fig. 5. Probability of resolution versus SNR with different and ).

Fig. 6. Probability of resolution versus SNR with different and ).

is also better than that in the ULA, and the lowest threshold of , which is identical to the resolution can be obtained when results in [5]. According to [5, (93)], the lowest SNR threshold of resolution for two equal power signals can be obtained when . the column dimension of is 3 are set to be 1. The probaThirdly, both parameters and bility of resolution versus SNR with different numbers of snapis shown in Fig. 7. It can be seen that the performance shots . Moreover, of resolution gets better with the increase of with the same number of snapshots, the SNR threshold of resolution in the TMLA with UPCM scheme is also lower than that in the ULA. D. The Accuracy Performance Suppose that there is a single signal arriving from far-field . The accuracy performance of the DOA estimation of the signal is evaluated in the case of different parameters. Firstly, is 100 and . The root mean the number of snapshots square errors (RMSEs) of the DOA of the signal estimated by are shown in Fig. 8. the proposed approach with different Obviously, the RMSEs are decreased with the increase of SNR, has the best accuracy and the proposed approach with performance as compared to other values of . Meanwhile, it

is also found that the conventional ULA with MUSIC algorithm gives slightly better accuracy performance than of the TMLA with UPCM scheme. and are set to be Secondly, the number of snapshots 100 and 1, respectively. The RMSEs of the DOA estimation versus SNR in the case of different are shown in Fig. 9. It is observed that the parameter has little influence on the accuracy performance of the DOA estimation in the TMLA with UPCM scheme. and are selected as 1. The RMSEs of the Thirdly, both DOA estimation versus SNR with different numbers of snapare shown in Fig. 10. It is found that the larger the shots number of data involved in the analysis, the better accuracy performance of the DOA estimation is obtained. Furthermore, with the same number of snapshots, the RMSEs obtained by the proposed approach in the TMLA with UPCM scheme are only a little higher than those in the conventional ULA. of the single signal varies Finally, assume that the DOA to 25 , , , and . The from RMSEs of the DOA estimation versus are shown in Fig. 11. , there are three beams at , and When utilized for the DOA estimation. It can be seen that the region is covered by the three beams shown in Fig. 3(a) to 20 , which leads to the results that approximately from

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V. CONCLUSION

Fig. 9. RMSEs of the DOA estimation versus SNR with different ). and

M=1

NN = 100

Q ( = 5

,

In this paper, a novel approach for DOA estimation based on the sideband signals in TMLA is proposed. Using the UPCM scheme, the patterns of the TMLA at different sidebands can be used to point at different directions, and the corresponding received signals can be used to compose a received data space. Similar to the beamspace-based DOA estimation algorithm, the DOAs can be estimated in the TMLA with UPCM scheme by using the MUSIC algorithm. The statistic performance of the resolution and accuracy of DOA estimation in the TMLA with UPCM scheme is presented. The lowest resolution threshold of two closely far-field signals with equal power can be obtained and . Moreover, and also when provide the best accuracy performance of DOA estimation in the TMLA with UPCM scheme, and there is only a little degradation of the accuracy performance as compared to the conventional ULA. The simulation results demonstrate the validity of the proposed approach for DOA estimation in the TMLA with UPCM scheme. REFERENCES

Fig. 10. RMSEs of the DOA estimation versus SNR with different number of , and ). snapshots (

 =5 M =1

Q=1

Fig. 11. RMSEs of the DOA estimation versus directions of the far-field signal ( , , , and .

M = 1 Q = 1 NN = 100

the DOA estimations for .

SNR = 10 dB)

are better than those for

[1] R. Kumaresan and D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-19, pp. 134–139, Jan. 1983. [2] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antenna Propag., vol. AP-34, no. 3, pp. 276–280, Mar. 1986. [3] R. Roy, A. Paulrajand, and T. Kailath, “Direction-of-arrival estimation by subspace rotation methods-ESPRIT,” in Proc. Int. Conf. Acoust., Speech, Signal Process., (ICASSP), 1986, vol. 11, pp. 2495–2498. [4] R. Roy and T. Kailath, “ESPRIT-estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal Process., vol. 37, no. 7, pp. 984–995, Jul. 1989. [5] H. B. Lee and M. S. Wengrovitz, “Resolution threshold of beamspace MUSIC for two closely spaced emitters,” IEEE Trans. Acoust., Speech, Signal Process., vol. 38, no. 9, pp. 1545–1559, Jul. 1990. [6] M. D. Zoltowski, G. M. Kautz, and S. D. Silverstein, “Beamspace rootMUSIC,” IEEE Trans. Signal Process., vol. 41, no. 1, pp. 344–364, Jan. 1993. [7] G. Xu, S. D. Silverstein, R. H. Roy, and T. Kailath, “Beamspace ESPRIT,” IEEE Trans. Signal Process., vol. 42, no. 2, pp. 349–356, Feb. 1994. [8] M. Pesavento, A. Gershman, and M. Haardt, “Unitary root-MUSIC with a real-valued eigendecomposition: A theoretical and experimental performance study,” IEEE Trans. Signal Process., vol. 48, no. 5, pp. 1306–1314, May 2000. [9] M. Haardt and J. A. Nossek, “Unitary ESPRIT: How to obtain increased estimation accuracy with a reduced computational burden,” IEEE Trans. Signal Process., vol. 43, no. 5, pp. 1232–1242, May 1995. [10] H. E. Shanks and R. W. Bickmore, “Four-dimensional electromagnetic radiators,” Canad. J. Phys., vol. 37, pp. 263–275, 1959. [11] R. W. Bickmore, “Time versus space in antenna theory,” in Microwave Scanning Antennas, R. C. Hansen, Ed. New York: Academic, 1966, vol. III. [12] 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. AP-11, no. 6, pp. 633–639, Nov. 1963. [13] 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. [14] 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. [15] 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. [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.

LI et al.: DOA ESTIMATION IN TMLAs WITH UNIDIRECTIONAL PHASE CENTER MOTION

[17] 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. [18] Y. Chen, S. Yang, and Z. Nie, “Synthesis of optimal sum and difference patterns from time modulated hexagonal planar arrays,” Int. J. Infrared Millim. Waves, vol. 29, no. 10, pp. 933–945, Oct. 2008. [19] H. E. Shanks, “A new technique for electronic scanning,” IEEE Trans. Antennas Propag., vol. 9, no. 2, pp. 162–166, Mar. 1961. [20] A. Tennant and B. Chambers, “A two-element time-modulated array with direction-finding properties,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 64–65, 2007. [21] G. Li, S. Yang, Y. Chen, and Z. Nie, “An adaptive beamforming in time modulated antenna arrays,” in Proc. 8th Int. Symp. Antenna, Propag., and EM Theory, (ISAPE), Kunming, China, 2008, pp. 221–224. [22] B. L. Lewis and J. B. Evins, “A new technique for reducing radar response to signals entering antenna sidelobes,” IEEE Trans. Antennas Propag., vol. 31, no. 6, pp. 993–996, Nov. 1983. [23] S. Yang, Y. B. Gan, and A. Qing, “Moving phase center antenna arrays with optimized static excitations,” Microw. Opt. Technol. Lett., vol. 38, no. 1, pp. 83–85, Jul. 2003.

Gang Li was born in Tianjin, China, in 1981. He received the B.S. degree in information countermeasure technology from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2005, where he is currently working toward the Ph.D. degree. His current research interests include antennas, antenna arrays and array signal processing.

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Shiwen Yang (M’00–SM’04) received the B.Sc. degree in electronic science from East China Normal University, Shanghai, the M.Eng. degree in electromagnetics and microwave technology and the Ph.D. degree in physical electronics from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 1989, 1992 and 1998, respectively. From 1994 to 1998, He was a Lecturer at the Institute of High Energy Electronics, UESTC. From 1998 to 2001, He was a Research Fellow at the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He joined the Temasek Laboratories, National University of Singapore, as a Research Scientist in 2002. He is currently a Full Professor with the Department of Microwave Engineering, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu. His research interests include antennas, antennas arrays and computational electromagnetics.

Zaiping Nie (SM’99) was born in Xi’an, China, in 1946. He received the B.Eng degree in 1968 and the M.Eng degree in 1981 from the University of Electronic Science and Technology of China (UESTC, formerly know as the Chengdu Institute of Radio Engineering), respectively. He was a Visiting Scholar in the Electromagnetic Laboratory, University of Illinois at Urbana-Champaign, from 1987 to 1989. He is currently a Full Professor with the Department of Microwave Engineering, School of Electronic Engineering, UESTC. He has authored or coauthored over 350 journal and conference papers and coauthored three technical books. His research interests include electromagnetic radiation, scattering, inverse scattering, wave and field in inhomogeneous media, computational electromagnetics, smart antenna technique in mobile communications and MIMO wireless communications. Prof. Nie was the first winner of the 2nd class of China National Science and Technology Progress Award in 2002.

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Reactive Energies, Impedance, and Radiating Structures

Q Factor of

Guy A. E. Vandenbosch, Senior Member, IEEE

Abstract—New expressions are derived to calculate the reactive energy stored in the electromagnetic field surrounding an electromagnetic device. The resulting expressions are very simple to interpret, completely general, explicit and without approximations in terms of the currents flowing on the device. They are also fast since they involve integrals solely over the device generating the field. The new technique is very feasible to be used in cases where the electric and magnetic reactive energies are important in practice, especially in the case of radiating structures. Used there, they allow to study the effect of the shape of the device on the amount of reactive energy, and thus on the of the device. The implementation of the new expressions in numerical CAD tools is extremely simple and straightforward.

Q

Index Terms—Poynting theorem, energy.

Q factor, radiation, reactive

I. INTRODUCTION

T

HE reactive energy stored in the electromagnetic field surrounding the device generating the field is an important parameter. From this parameter, important characteristics can be derived, for example the factor of the system, linked to the system bandwidth. Many authors have considered the problem of determining the factor of an antenna. For the general case, this goes back to the paper of Collin [1], where a spherical mode decomposition is used to calculate the reactive energies. For the special case of an electrically small antenna, the paper of Chu [2], using ladder networks, is a basic paper. Many other authors have followed the same paths. Fante [3] extended the results of Collin. McLean [4] re-examined the case of small antennas. Sten studied the case of a small antenna near a ground plane [5]. The classical spherical mode approach actually calculates the reactive energies, both electric and magnetic, stored in the space outside a sphere with radius . There are two important disadvantages linked to this approach: 1. the reactive energy within this sphere is neglected, and, related to this, 2. the exact shape of the volume of the radiating source is not taken into account. The result is that the reactive energies calculated are only approximations and that the effect of the topology of the source is hard to investigate. The well-known Chu limit was challenged by Thal recently [6]. In his paper the internal reactive energy Manuscript received May 04, 2009; revised August 26, 2009. Date of manuscript acceptance October 01, 2009; date of publication January 26, 2010; date of current version April 07, 2010. The author is with the Department of Electrical Engineering, Division ESAT-TELEMIC (Telecommunications and Microwaves), Katholieke Universiteit Leuven, B-3001 Leuven, Belgium (e-mail: guy.vandenbosch@ esat.kuleuven.be) Digital Object Identifier 10.1109/TAP.2010.2041166

is incorporated for spherical wire antennas. Relatively recently, Geyi [7] published a technique to calculate the reactive energies taking into account the exact topology of the, in this case, small radiator considered. He used a combination of the Poynting theorem in frequency and time domain to separate electric and magnetic reactive energy. Shlivinski performed a study of the reactive energy completely in the time domain, aiming at applications involving pulsed fields [8]. A brute force technique is used in [9], where the authors calculate the reactive energy using the FDTD method. A very complete paper is [12]. This paper gives a state-of-the-art overview of techniques and formulas to calfactors of antennas. culate impedances, bandwidths, and However, no method is given to calculate the reactive energies factor is based on the explicitly. The calculation of the knowledge of the derivative of the impedance. The main goal of this paper is to propose new rigorous, general, and efficient expressions for the reactive energies surrounding a completely arbitrary source (or device). The goal is also to illustrate what can be achieved with these expressions. The traditional technique to rigorously calculate the electric and magnetic energies works as follows. — First, the currents are solved for the structure, in many cases numerically. — Second, the fields are calculated in every point of space in terms of these currents. — Third, the energy densities linked with the fields are calculated and integrated over space. This is the procedure for example followed in [9], [12]. In [12], the formulas for reactive energies are indeed based on the traditional technique. Further in [12], since the main goal of the paper is to study bandwidth and factor, the lower efficiency of this traditional technique becomes a non-issue, because the authors derive approximate expressions for the factor, not making use any more of the electric or magnetic energy. The technique followed in this paper works as follows. — First, it is assumed that all the currents are solved for the structure: conductor currents, polarization currents, and magnetization currents. This can be done using the wide range of available techniques, such as MoM. This step is not explicitly considered in this paper. — Second, the newly found expressions are used to calculate the energies directly in terms of these currents. It will be shown that the calculation of the fields over entire infinite space is not necessary any more. In fact, the integration over entire space is performed analytically. It is obvious that in most cases this is computationally much more effective. This last point is the key novelty in this paper. The derivation is performed entirely in the frequency domain.

0018-926X/$26.00 © 2010 IEEE

VANDENBOSCH: REACTIVE ENERGIES, IMPEDANCE, AND

FACTOR OF RADIATING STRUCTURES

Sections II – IV contain the procedure to derive the new expressions. These expressions are formulated in terms of currents assumed to be known. The derivation requires a good understanding of vector field theory and is quite involved. However, it has to be emphasized that no approximations are made. The derivation is general and rigorous. Readers that are interested in the main result only, the expressions (63) and (64), can easily skip Sections II – IV. The resulting expressions are discussed in detail in Section V. Examples and numerical results are given in Section VI. Amplitudes are used throughout the paper. This explains the factor 2 difference for energy and power quantities with formulas using root-mean-square values sometimes found in literature. Vectors are in bold notation.

III. POYNTING’S THEOREM AND MATTER In this section, first the classical energies linked to Poynting’s theorem are derived. However, these energies will be split in a vacuum term and a term present inside matter only. This step is crucial for the rest of the paper. Multiplying the complex conjugate of (2) with , then using and inserting (1) in the result, introducing the factor 1/2 because time averages have to be considered, and finally integrating over the entire space volume yields Poynting’s theorem (8) where electric and magnetic energy are given by power

II. MAXWELL’S EQUATIONS AND MATTER

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and

and Consider an arbitrary material with permittivity , depending on coordinates , in a volume permeability . The permittivity and permeability may be complex to represent losses. Maxwell’s laws become

, and radiated

(9) (10)

(1) (2) is the exciting source current, flowing inside the where differs from . The total radiating volume . Note that device thus fills a volume . Since inside a , with magnetic material the magnetization vector [13], [14], (2) can also be written as

(11) (12) (13)

(3) with

(4) (5)

This means that, obviously, any system is a system in the homogeneous medium vacuum. Conductors, dielectrics, and magnetic materials can be modeled by the introduction of conduction currents (linked to the imaginary part of the permittivity: ), polarization currents (linked to the real part of the permittivity), and magnetization currents (linked to the permeability), respectively [13], [14]. These currents are the electrical currents that are physically flowing inside the material to provide the material with its macroscopic electromagnetic characteristics. Equation (4) is considered a boundary equation that has to be satisfied within the material (dielectric and/or metal). The huge advantage of (3) is that the solution is known in closed form in terms of the total current . It is given by (6) (7) where

is the free space Green’s function and . These equations

will be used explicitly further in this paper.

(14) The last step is based on the fact that, since , integrates to 0 over .

is zero outside

(15) These equations show that the electric and magnetic energies can be calculated as the sum of the energies in vacuum, and energies linked to the polarization and magnetization currents, only present inside matter. These latter energies consist of two and on the one hand, components, the real parts representing stored energies in the fields, apart from the vacuum and on the other terms, and the imaginary parts hand, representing dissipated energies or in other words losses. It has to be emphasized that, since matter is present in a finite volume only, the polarization and magnetization energies are

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integrals over the volume , whereas the vacuum terms are integrals over entire space. Note that in case frequency dispersion is included, (9) and (12) have to be generalized, following [15].

The dot product of the complex conjugate of (17) with gives the dot product of (1) with

(19)

IV. ENERGIES AND THE FREQUENCY DERIVATIVE The first goal of this section (Section IV-A) is to derive an equation relating the change with frequency of the power provided by the source to other energy and power quantities of the system: the radiated power, the electric and magnetic losses, the electric and magnetic energies stored in the fields, and two newly defined energy quantities. This equation will further be called the frequency derivative equation. The followed line of reasoning is very similar to the one given in [12]. The difference is that here the relation will be generalized to cover complex quantities, and not the reactive part only [12]. After the proper equation has been stated, expressions will be worked out for power and energy quantities occurring in it (Sections IV-B – IV-E), directly in terms of the total current flowing in the device, and the free space Green’s function. By using the vacuum terms of the energies, the complication of matter being present is efficiently circumvented. It is evident that the observations made at the end of Section II are crucial in this respect. The final goal of this section is to show that the frequency derivative equation shows a term that can be rigorously combined in a very natural way with the vacuum terms of the energies (Section IV-F). This combination does three things: — It provides a more mathematical background to the traditional subtraction technique originally introduced by Collin [1] to cope with the infinite electric and magnetic energies “stored” in the fields. — It shows how the technique has to be applied in the near field. The application in the far field is straightforward and well-known. — It eliminates the problem reported in [12]: the fact that the reactive energies are to some degree depending on the coordinate system chosen. This leads to modified vacuum terms of the stored energies and , which are finite, uniquely defined, independent of the coordinate system, and explicit in terms of the currents. A. The Frequency Derivative Equation Deriving (1) and (2) with respect to yields

(indicated by the prime)

(16) (17) The dot product of (16) with complex conjugate of (2) with

minus

minus the dot product of the gives

Subtracting (19) from (18) yields

(20) Summing (19) to (18) yields

(21) Note that the first two terms on the right hand side of (20) and (21) can easily be related to the electric and magnetic energies. The four terms on the second line of (20) can be related to the derivatives of these energies. The left hand side of (20) can be related to the derivative of the radiated power. Integrating over space yields for (20)

(22) and for (21)

(23) Note that on the right hand side the remaining integrands are or . Choosing a source current which does zero outside not change with frequency, similar as in [12], so , and combining the real part of (22) with the imaginary part of (23), and rearranging the terms yields

(24) where two new energies are defined, in [12]

and

, similar as

(25) (18)

(26)

VANDENBOSCH: REACTIVE ENERGIES, IMPEDANCE, AND

FACTOR OF RADIATING STRUCTURES

Using the energies defined in (10), (11), (13), and (14), we obtain

(27) Up to now, the line of reasoning is valid for permittivity and permeability dyadics. We already have expressions in terms of the , , , and . In the following currents for sections general closed form expressions in terms of the currents , , , , and . For will be derived for this will be done for the case that the permittivity and permeability are (complex) scalars.

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homogenous medium vacuum was not explicitly used. This will be done in this section. is defined in (22) as . Deriving (6) with reyields three contributions, the first due to the spect to derivative of the frequency dependent factor in front of the integrals, the second due to the derivative of the Green’s function, and the third due to the derivative of the current itself (32) (33) (34)

B. The Energy For a scalar permittivity and permeability without frequency dispersion

(35) The shorthand notation is introduced for easy reference. The derivative of the Green’s function and its gradient is given by

(28) Then writing the electric field as tive we obtain

with derivaand using

(36) Deriving (7) with respect to

yields

(29) (37)

Using (4) this finally yields

(38) (30) Using the same technique for the magnetic field, we obtain

(31) The energy is thus related to losses (through the imaginary parts of permittivity and permeability) and to the speed of the phase change of the polarization current and magnetization vector with frequency. Note that the change of the amplitude of the magnetization vector and polarization current with frequency has no effect on this term. C. The Energies

(39) Here, there are only two contributions, due to the derivative of the Green’s function and the derivative of the current itself. This means that there are five terms in the integrand and can be split up as shown in (40) at the bottom of the page. Let us first . Since concentrate on the term , in the far field . The second term of in (6) and of in (33), the divergence terms, are thus directed along , while is normal to . These second terms thus do not contribute to and , respectively. However, it is easily checked that the first term of in (33) is times the first term of in (6). This means that

and

Up to now, the fact that matter can be replaced by an electromagnetically completely equivalent system of currents in the

(41)

(40)

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Now, let us look at (42) The procedure starts with inserting (6), (7), (34), and (38) in (42). Straightforward manipulation, involving reordering the terms and changing the order of integration, yields (43), shown at the bottom of the page, where the subscripts 1 and 2 both replace the subscript d and indicate source coordinates, linked to the evaluation of (6) and (34) for the electric field, and the evaluation of (7) and (38) for the magnetic field. This means . Using (99) and (101) of the appendix thus that (A.2) the result can be written as (44), shown at the bottom of the page. Using the following asymptotic expansions

(45a)

the correct places, and taking the imaginary part. Taking the . Note that the integral imaginary part ensures that corresponding to the 3rd line of (44) is clearly real, for corresponding to the 2nd line of (44) is and the integral for pure imaginary (proof by asymptotic expansion), so these terms do not contribute. The result is given in (49) at the bottom of and are terms the page. The remaining terms involving the current derivative, through the formulas (35) and (39). Since these formulas (35) and (39) for the evaluation of the fields in terms of these currents are identical to (6) and (7), and we can simplify the notation and write . D. The Radiated Power Inserting (6) and (7) in (15) and basically following the same line of reasoning as in the previous section, but without the complication of having to deal with derivatives, nor the fact that we have to sum/subtract two terms, it can be proven that

(45b) and (analytical) evaluation of the following integrals (50) (46) E. The Vacuum Terms of Electric and Magnetic Energies (47) yields (48) at the bottom of the page. For , defined in (26), the procedure is almost identical, but with the proper – sign at

The vacuum term of the electric energy is (51)

(43)

(44)

(48)

(49)

VANDENBOSCH: REACTIVE ENERGIES, IMPEDANCE, AND

FACTOR OF RADIATING STRUCTURES

where is entire space. Inserting (6) and straightforward manipulation, involving reordering the terms and changing the order of integration yields

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F. Modification of the Vacuum Terms The second term on the right hand side of both (53) and (56) is identical. This means that this term disappears in (8), when a subtraction is made. However, in (27), the energies are summed, which doubles this term. Let us consider the following sum term occurring in (27)

(57)

(52) where the subscripts 1 and 2 both replace the subscript d and indicate source coordinates, linked to the first evaluation of (6), for the electric field itself, and the second evaluation of (6), for the complex conjugate of the electric field. This means thus . Using the identities (A.1) and (A.3) of that the appendix , (52) can be transformed into (53), shown at the the distance between points 1 and 2. bottom of the page, with The vacuum term of the magnetic energy is (54) Inserting (7) into (54) gives

(55) Using the identity (A.5) of the Appendix, this can be transformed into (56), shown at the bottom of the page.

Using (53), (56), and (49) this term can be manipulated into (58) at the bottom of the page. It is extremely important to emphasize that since the Green’s functions are depending only on or , (58) as a whole and thus also the factor between brackets on the 2nd line, consisting of the subtraction of a surface integral from a volume integral, are independent of the choice of the coordinate system. This means the coordinate system in principle can be chosen freely. Asymptotic evaluation leads to of the Green’s functions for

(59) and Choosing the origin in the middle between the points ensures that . The term containing in this asymptotic evaluation results in an imaginary contribution, so taking the real part yields 0. The remaining term containing gives the integral

(60)

(53)

(56)

(58)

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Using (60) in (58) then delivers (61) at the bottom of the page. The last integral of (61) can be calculated analytically, as given in the appendix and yields

source current does not change with frequency, we can write

, and

(62) Analyzing (53), (56), and (61) it is seen that it is possible to introduce modified electric and magnetic energies in such a way that both Poynting’s theorem, given in (8), and the derivative equation, given in (27), are still satisfied. The new energies are (65) where the dependency of the fields on the current or its derivative are explicitly mentioned. V. DISCUSSION

(63)

(64) Note that Taylor series expansion of the cosine and sine and keeping only the leading term yields the expressions as formulated by Geyi for small devices [7]. For the total electric and magnetic energies, including the term in matter, and , given by the real parts in (11) and (14), have to be added. G. Impedance Derivative Equation At this moment we have established closed-form expressions for all energies in terms of all currents flowing within the structure. Following the same line of reasoning as in [12], in terms of the exciting source current, the final equation becomes a relation between energies and the impedance derivative. Since the

The procedure followed, resulting in (63) and (64), solves two fundamental problems well-known in literature. The first problem is that the radiation field gives an infinite contribution to the stored electric and magnetic energies and in practical cases does not have to be considered. A discussion on this phenomenon can be found in [1], [3], [4], [7], [12]. In [7] and [12], this issue is solved by subtracting the term from both energies. is the speed of light in vacuum. The physical interpretation is that if the source has , the energy in the radiated been radiating for a time given by field is just the product of this time and the power radiated by the source. Indeed, in case the source has been radiating for a time , the field outside the sphere with radius is zero, because the wave did not reach these points yet, and there is no contribution to any integral outside this sphere. However, no explicit procedure is given in [7] and [12] to realize this in practice. Geyi [7], puts is very clearly “it should be notified that the direct numerical calculation is very delicate since the integrations have to be performed over an infinitely large domain. It can be shown by numerical calculations that the rounding errors become increasingly significant when the integration region is very large.” It is thus not possible to calculate numerically the stored energy in a finite sphere, subtracting the term given, with the corresponding , and taking the limit for the finite sphere going to infinity. Other authors treat the radiation field based on a spherical mode decomposition [1], [3], [4]. In the case that only one mode is considered, thus neglecting higher order modes, this gives rise to unique closed form expressions for the radiation field at all

(61)

VANDENBOSCH: REACTIVE ENERGIES, IMPEDANCE, AND

FACTOR OF RADIATING STRUCTURES

-distances. It is then easy to subtract the corresponding energy density. The approach used in this paper is essentially the same as in [7] and [12]. The problem pointed out by Geyi is avoided by carefully following the procedure explained above. It leads to a subtraction within the integrand at the end of the last line of (61). This is physically equivalent to subtracting the radiation energy at every point in space, thus in far and near field. Equation (61) is thus an explicit formula, which can easily be used to actually do the calculation in practice. The second problem is reported in [12]. The reactive energies defined there are dependent on the coordinate system. It is mentioned: “nonetheless, we ultimately have to live with the fact that our defined reactive and internal energies of an antenna depend to some degree on the choice of the origin of the coordinate system relative to the antenna.” The problem was analyzed by giving a measure of how much the energies would shift with a change of coordinate system. For small antennas, this shift turns out to be small. Also it is pointed out that for radiation patterns with a specific symmetry, no change is observed, and thus no dependency on coordinate system. In our procedure a similar phenomenon is introduced in (59), where we by the asymptotic expansion for go from an expression independent of the coordinate system to an expression depending on it. In other words going to infinity in a different way yields a different value. By choosing , which actually means that a different coordinate system is chosen for each couple of points 1 and 2, our procedure leads to expressions clearly independent of the coordinate system. A deeper analysis of (58) explains how this is possible. The subtraction of the surface integral from the volume integral on the last line of (58) is clearly independent of the coordinate system. The asymptotic expansion makes the surface integral depending on it. Clearly, the integration domain of the volume integral has to go to infinity in the same way as the surface integral, and this actually cancels out again the dependency in the sum. This is the implementation of the solution that was mentioned in [12]. The final expressions (63), (64) (for the vacuum terms of the reactive electric and magnetic energies), are the key novelty in this paper. These expressions exhibit the following properties. — Since no approximations have been used anywhere in the procedure, they are rigorous in terms of the original energy definitions. — Since they were derived from Maxwell’s equations themselves, they are completely general. They can be applied to any structure. — Although the procedure to derive them is quite involved, the expressions themselves are extremely easy to understand and interpret. They only involve series of double integrations over the source volume of the device. The inte-

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grations concern either the current on the device, or its divergence (which corresponds to the charge on the device). Only the free space Green’s function is involved. In general, the integrals are 3D, but for most practical radiating structures, where the skin depth of the conductors is small, they reduce to 2D integrals, and in case of wire structures, even to 1D integrals. — The expressions are extremely easy to implement in software tools. In MoM tools, where the current is automatically given as a result, this involves only the numerical evaluation of the necessary integrals. To the knowledge of the author, this is the first time that the unique relation between currents and reactive energies has been rigorously formulated in a direct and explicit way, without going through the fields, which is the traditional technique to express these energies. VI. APPLICATION: CALCULATION OF

FACTORS

In this section, it is illustrated how the new expressions can be used in practice to calculate factors. The basic definition , where is the timeof factor in this work is averaged energy stored and the radiated and dissipated power. A. Numerical Calculation of Energies and Using our terminology, the

Factors

factor given in [12] is (66)

, see In [12] also an approximate formula is given (67) at the bottom of the page, with the resistance and the reactance of the structure. Comparing (66) with (67), this means that

(68) It can be proven that this relation is exact for any lossless circuit. For a parallel and a series RLC circuit, the circuits actually used in [12] to derive (67), (68) is not an approximation, as suggested in [12], appendix D, but it is also exact. The proof is straightforward, but beyond the scope of this paper. Equation (67) is a very good approximation if resonances and anti-resonances are sufficiently separated in frequency domain. However, if two anti-resonances are close to each other, as was already observed in [12], (67) deteriorates. This will be illustrated further in this section.

(67)

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Fig. 2. Dipole of length 1 m: factor Q

=

2

Fig. 1. Dipole of length 1 m. Top: radiated power P RjI j = and difX jI j = ; ference between magnetic and electric energies ! W 0 W bottom: relation between total reactive energy and the derivatives of the energies depicted in the top figure.

2 (

)=

2

The energy expressions are extremely easy to implement in software tools. In this section, two examples are given where the current is obtained numerically: a dipole antenna and a planar antenna involving an inhomogeneous dielectric. The currents are obtained using our in-house developed software tool MAGMAS [10]. MAGMAS solves the integral equations describing the structure using the method of moments. The first antenna considered is the dipole described in [12]. The length is 1 m and the diameter is 1 mm. For the analysis the dipole was subdivided into 200 segments along its length. A feeding current of 1 A is imposed in the middle of the antenna. The antenna is analyzed over a large frequency band, including the “small antenna” regime and several resonances. The energies, numerically calculated with the new formulas, are studied in Fig. 1. The agreement with the energies directly derived from the impedance is excellent. Since resonances are further apart, the approximation of (68) is accurate. The factor is derived from these energies, using the definition of [12], and is given in Fig. 2. The agreement with the results of [12] is again excellent. The difference is due to the fact that a strip model was used for the dipole, compared to a wire model in [12], and to the fact that no detailed info was available on the feeding gap.

= !(W + W )=P

.

The second antenna considered is a microstrip antenna with the substrate reduced to two supporting dielectric spacers, which also make the antenna electrically smaller compared to a structure in vacuum. The square patch is 8.25 cm. The height of the two spacers is 7 mm. They support the patch over its whole length at both sides. Their width is 16.5 mm. The permittivity is 2. The patch is fed by a coaxial probe in the middle of the two spacers at 22 mm from the center of the patch. The patch is subdivided into 900 square segments and the dielectric spacers in 720 square blocks in the numerical calculations. The energies are given in Fig. 3, the factor in Fig. 4. This example illustrates the validity of the expressions in case of the presence of finite dielectrics, where the energies in matter have to be added, in this case given by (11). Two observations can be made: 1. the suggested in [12] deteriorates in case resoapproximation nances are close to each other, and 2. since a derivative has to be taken for this approximation, it is more affected by small numerical noise, due to the limited numerical accuracy of the computational tool used. This is especially true near sharp resonances. The factor calculated with the new expressions does not suffer from this, because no derivatives have to be taken. B. Analytical Calculation of Reactive Energies and

Factors

In the previous section the current used in (63) and (64) was obtained from a computational tool. However, in case a good estimate is available for the current, which is feasible for simple topologies, (63) and (64) can be applied directly. In this section, an example is given where the new expressions give rise to full wave series expressions for the reactive and radiation energies. Consider a loop with radius , made of wire with diameter . The loop is located in the plane and the origin is in the center of the loop. In cylindrical coordinates, the current on the loop is given by (69) This source current is well-chosen since for , actually a a magnetic dipole or loop antenna is obtained. For ring type electric dipole configuration is obtained. Using the

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following expressions are obtained for the electric dipole type .

(70)

(71) (72) For very small wire diameters, only the logarithmic terms have to be kept. In this case it can be seen that goes to . zero, yielding a reactance also equal to zero, for Using the expressions, the deterioration from this simple result in terms of growing wire diameters can be studied. The of the antenna, as defined in [1], [2], becomes

(73) For the magnetic dipole

=

2 )=

Fig. 3. Microstrip antenna with spacers. Top: radiated power P R jI j = and difference between magnetic and electric energies ! W 0 W X jI j = ; bottom: relation between total reactive energy and the derivatives of the energies depicted in the top figure.

2

2 (

(74)

(75) (76) In this case, the reactance cannot be made zero for wire diameters going to zero. The of the magnetic dipole, as defined in [1], [2], becomes

(77)

Fig. 4. Microstrip antenna with dielectric spacers: frequency.

Q factor as a function of

thin wire approximation and after Taylor series expansion of the sine and cosine, all integrals can be evaluated analytically. The

This example illustrates that the new expressions allow the study of antenna in terms of antenna size. C.

of Lossless Small Antenna The factor of a small antenna is a parameter of prime importance. Numerous papers have been published on this topic [2], [4], [6], [7], [16]–[18]. In this section, it is shown that the expressions (50), (63), and (64) can be used to study the lower

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bounds for radiation in terms of the topology and the current flowing within this topology. Consider a lossless small antenna without dispersion. Keeping only the two most important terms in the Taylor series , and using for

(78)

we obtain (79)–(81) at the bottom of the page. Assuming tuning to zero imaginary part of the impedance, by adding an external reactance, as explained in [1], [2], the is then

2) Loop Type Current on Spherical Surface: The second cur. The rent assumed to be flowing on the sphere is result is (84) Also this result agrees perfectly with Thal [6]. 3) Dipole Type Current Within Spherical Volume: The third current is flowing not only at the surface of the sphere, but it is filling its entire volume. This example is chosen in order to check whether filling the volume of an antenna with current could reduce the amount of reactive energy, and thus the factor. In cylindrical coordinates the assumed current is given by

(85) (82)

Several specific cases are investigated. 1) Dipole Type Current on Spherical Surface: Consider a sphere with radius . Using a spherical coordinate system, the . first current assumed to be flowing on the sphere is All integrals in (79), (80), and (81) can be evaluated numerically to a very high accuracy, which yields

Since in this case real volume integrals are involved, the numerical evaluation of (79), (80), and (81) is computationally much more expensive than in the previous cases. A resolution of 1 for angular coordinates and taking 40 subdivisions along the radial coordinate yields

(83)

(86)

As expected, (83) is above the classical limit as defined by Chu [2], McLean [4], etc. However, it agrees perfectly with Thal [6]. with , the For higher order currents, i.e., factor was checked and yields much higher values.

It is seen that this volume current dipole yields a higher , compared to the surface current dipole. 4) Dipole Type Current on Cylindrical Surface: Consider a cylinder enclosed within a sphere with radius , i.e., the radius of the cylinder and its height combine in such

(79)

(80)

(81)

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TABLE I THE Q RATIO FOR BASIC ANTENNAS

Fig. 5. Coefficients of 1=(k a) and 1=(k a) terms of Q factor for small cylindrical dipole antenna enclosed in a sphere with radius a, for a sinusoidal or a linear current.

that in [16] the antennas are of a finite size 0.1 , while our results are in the limit for size going to zero. The examples clearly illustrate the fact that the new expressions can be used to study factors very efficiently. For many basic antennas, our results correspond very well with the results available in literature, see Table I. VII. CONCLUSION

Fig. 6. Coefficient of 1=(k a) term of Q factor for cylindrical inductor with constant current and disc dipole with two different radial charge distributions, enclosed in a sphere with radius a.

a way that . Using a cylindrical coordinate system, the current assumed to be flowing on the cylinder is sinusoidal, , or linear, . All integrals in (79), (80), and (81) can be evaluated numerically as a . The resulting coefficients in function of the ratio and are presented in Fig. 5. This example illustrates the fact that the factor is depending on the current itself. The sinusoidal current distribution gives a slightly lower than the linear current. 5) Cylindrical Inductor and Disc Dipole: The cylindrical inductor and the disc dipole of [16] are considered next. The cylindrical inductor consists of a constant loop current flowing on a cylinder with diameter and height . The disc dipole consists of two circular capacitor plates of diameter with opposite charge distribution, at a distance , fed by a wire in the center. term of the factor for both anThe coefficient of the tennas is depicted in Fig. 6. For the inductor a minimum of 4.39 . For the disc dipole the minimum is is reached at . The difference with [16] is due to the fact 2.37 at

General and rigorous expressions are derived for the electric and magnetic energy stored in the electromagnetic field around an arbitrary source or device. The expressions do not suffer from the disadvantages linked with the spherical mode decomposition technique, traditionally used to calculate these energies. Also, no approximation based on the impedance derivative is needed. They remove the unphysical phenomenon that the internal reactive energies depend to some degree on the choice of the origin of the coordinate system. The new expressions are very feasible to be used in practice and easy to implement in software tools. A straightforward application is the study of the energies and factors of radiating structures in terms of the topology of the device. By assuming well-chosen currents on a specific topology, even radiation limits can be determined. Using this technique, the results available in literature for four basic antennas have been reproduced. However, the expressions can be used for much more general topologies, an issue of high importance in the small antenna community. APPENDIX Consider the free space Green’s function with the distance and an observation point between a source point . Consider the current distributions and inside and , so ,2. indicates entire infinite finite volumes space. Green’s Functions Identities: The following identities are well-known or can easily be verified

(87) with the Dirac impulse function. The prime indicates the derivative with respect to .

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Current – Charge Integral Identity: Using we can write

the distance between the two source points. Using (87) with and inserting (92) in (90) yields

(93)

(88) The last step makes use of the fact that the current component normal to the surface is zero. Identity (A.1):

is the function evaluated at , i.e., for the distance . Identities (A.2): The first integral to evaluate is presented in (94) at the bottom of the page. Using and the property that the integral over a closed surface of a curl equals zero, we obtain (95) at the bottom of the page. and the fact that Using outside the source volume, based on (87)

(89) The Green’s function subscripts 1 and 2 indicate the two source points considered, in volume 1 and 2, respectively. Using this can be written as

(96) and also using the identity, based on (88), that

(90) (97) It is easily calculated that in the far field (91) so that the first term in (90) can be calculated analytically as

(92)

and a similar expression for we obtain (98) at the bottom of for the page. Since in the third term going to infinity, using (88) for current 2 in the third term, and also for current 1 in the second term, but in the opposite sense, we finally get (99) at the bottom of the next page. The second integral to evaluate is (100) at the bottom of the next page. Since for any combination of Green’s functions 1 and 2 and their derivatives, , we can and using write (101) at the bottom of the next page.

(94)

(95)

(98)

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Identity (A.3): The integral worked out is (102)

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Comparing the integral with its complex conjugate shows that the first term is imaginary and the second term is real. Identity (A.4): The integral worked out is shown as (106) at the bottom of the page. Using we get

Using (91) it is easily proven that (107) It is easily proven that

(103)

(108)

Using this equation we get (104) at the bottom of the page. Using (88) we obtain

because the curl of a divergence is always zero. Since the leading is directed in the -direction for going to infinity, term of the integral goes to zero. This proves that

(105)

(109)

(99)

(100)

(101)

(104)

(106)

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Identity (A.5): The integral worked out is

Using the property that be written as

(110) , this can

Fig. 7. Auxiliary coordinate system.

(111) Using now that , we can write (112) at the bottom of the page. Using the identity proven in (109), this becomes

entire space is actually the distance between the two source points. As explained in the paper, we have to choose an auxiliary coordinate system with the origin in the middle of the points 1 and 2 and the -axis directed from point 1 to point 2. Scaling of the through a coordinate transforcoordinates with respect to mation yields then

(113) Using (88) for both 1 and 2 we get

(117) with (114)

and with (93) we finally obtain (115) at the bottom of the page. Analytical Integration Over Space: This section evaluates the integral

(118) and can be considered dimensionless distances. Another transformation to cylindrical coordinates around the -axis and evaluation of the resulting elementary integrals yields

(116) The configuration of the source points 1 and 2 and the observaand are the two tion point is illustrated in Fig. 7, where unit vectors pointing in the direction of the observation point from the two points where the current is considered, and and are the distances between these two points and the obover servation point. The absolute maximum value of

(119)

(120)

(112)

(115)

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FACTOR OF RADIATING STRUCTURES

Since the exponents are odd functions in , this can be simplified to

(121) The next step is using the Taylor series expansion. This gives

(122)

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[8] Shlivinski and E. Heyman, “Time-domain near-field analysis of shortpulse antennas – Part II: Reactive energy and the antenna ,” IEEE Trans. Antennas Propag., vol. 47, pp. 280–286, Feb. 1999. [9] S. Collardey, A. Sharaiha, and K. Mahdjoubi, “Calculation of small antennas quality factor using FDTD method,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 191–194, 2006. [10] [Online]. Available: www.esat.kuleuven.be/telemic/antennas/magmas [11] R. F. Harrington, Time-Harmonic Electromagnetic Fields. Piscataway, NJ/New York: IEEE Press/Wiley-Interscience, 2001. [12] D. Yaghjian and S. R. Best, “Impedance, bandwidth and of antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 1298–1324, Apr. 2005. [13] R. Feynman, The Feynman Lectures on Physics: Commemorative Issue. Boston, MA: Addison-Wesley, 1989, vol. II. [14] J. Van Bladel, Electromagnetic Fields. Piscataway/Hoboken NJ: IEEE/Wiley-Interscience, 2007. [15] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media. London, U.K.: Pergamon Press, 1960. [16] A. R. Lopez, “Fundamental limitations of small antennas: Validation of Wheeler’s formulas,” IEEE Antennas Propag. Mag., vol. 48, no. 4, pp. 28–36, Aug. 2006. [17] G. A. Thiele, P. L. Detweiler, and R. P. Penno, “On the lower bound of the radiation for electrically small antennas,” IEEE Trans. Antennas Propag., vol. 51, pp. 1263–1269, Jun. 2003. [18] S. R. Best and D. Yaghjian, “The lower bounds on for lossy electric and magnetic dipole antennas,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 314–316, Dec. 2004.

Q

Q

Q

A series of integrals has to be evaluated,

Q

(123) It has to be emphasized that the results of these integrals are just numbers. This means that they have to be evaluated only once, after which they can be tabled. This has been done numerically . to a very high accuracy. The result is The final result for the integral is (124)

ACKNOWLEDGMENT The author would like to thank Dr. S. Best for fruitful discussions on the topic. His input has lead to important corrections and improvements in this paper. The author would like to thank two anonymous reviewers. Their remarks have led to the removal of ambiguities in the followed line of reasoning and several formulas. REFERENCES [1] R. E. Collin and S. Rothschild, “Evaluation of antenna ,” IEEE Trans. Antennas Propag., vol. AP-12, pp. 23–27, Jan. 1964. [2] L. J. Chu, “Physical limitations on omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, Dec. 1948. [3] R. L. Fante, “Quality factor of general ideal antennas,” IEEE Trans. Antennas Propag., vol. AP-17, pp. 151–155, Mar. 1969. [4] J. S. McLean, “A re-examination of the fundamental limits on the radiation of electrically small antennas,” IEEE Trans. Antennas Propag., vol. 44, pp. 672–676, May 1996. [5] J. C.-E. Sten, A. Hujanen, and P. K. Koivisto, “Quality factor of an electrically small antenna radiating close to a conducting plane,” IEEE Trans. Antennas Propag., vol. 49, pp. 829–837, May 2001. [6] H. L. Thal, “New radiation limits for spherical wire antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2757–2763, Oct. 2006. [7] W. Geyi, “A method for the evaluation of small antenna ,” IEEE Trans. Antennas Propag., vol. 51, pp. 2124–2129, Aug. 2003.

Q

Q

Q

Q

Guy Vandenbosch (M’92–SM’08) was born in Sint-Niklaas, Belgium, on May 4, 1962. He received the M.S. and Ph.D. degrees in electrical engineering from the Katholieke Universiteit Leuven, Leuven, Belgium, in 1985 and 1991, respectively. He is the holder of a certificate of the postacademic course in Electro-Magnetic Compatibility at the Technical University Eindhoven, The Netherlands. He was a Research and Teaching Assistant from 1985 to 1991 with the Telecommunications and Microwaves Section, Katholieke Universiteit Leuven, where he worked on the modeling of microstrip antennas with the integral equation technique. From 1991 to 1993, he held a postdoctoral research position at the Katholieke Universiteit Leuven, where, since 1993, he has been a Lecturer, and since 2005, a Full Professor. He has taught or teaches courses on “electrical engineering, electronics, and electrical energy,” “wireless and mobile communications, part antennas,” “digital steer- and measuring techniques in physics,” and “electromagnetic compatibility.” His research interests are in the area of electromagnetic theory, computational electromagnetics, planar antennas and circuits, electromagnetic radiation, electromagnetic compatibility, and bio-electromagnetics. His work has been published in ca. 100 papers in international journals and has been presented at ca. 160 international conferences. Prof. Vandenbosch has been a member of the Management Committees of the consecutive European COST actions on antennas since 1993, where he is leading the working group on modeling and software for antennas. Within the ACE Network of Excellence of the EU (2004–2007), he was a member of the Executive Board and coordinated the activity on the creation of a European antenna software platform. He has convened and chaired numerous sessions at many conferences. He was Co-Chairman of the European Microwave Week 2004 in Amsterdam, and chaired the TPC of the European Microwave Conference. He was a member of the TPC of the European Microwave Conference in 2005, 2006, 2007, and 2008. Since 2001, he has been President of SITEL, the Belgian Society of Engineers in Telecommunication and Electronics. Since 2008, he is a member of the board of FITCE Belgium, the Belgian branch of the Federation of Telecommunications Engineers of the European Union. In the period 1999-2004, he was Vice-Chairman, and in the period 2004-2009, Secretary of the IEEE Benelux Chapter on Antennas en Propagation. He currently holds the position of Chairman of this Chapter. In the period 2002-2004, he was Secretary of the IEEE Benelux Chapter on EMC.

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Electromagnetic Boundary Conditions Defined in Terms of Normal Field Components Ismo V. Lindell, Life Fellow, IEEE, and Ari H. Sihvola, Fellow, IEEE

Abstract—A set of four scalar conditions involving normal components of the fields and and their normal derivatives at a planar surface is introduced, among which different pairs can be chosen to represent possible boundary conditions for the electromagnetic fields. Four such pairs turn out to yield meaningful boundary conditions and their responses for an incident plane wave at a planar boundary are studied. The theory is subsequently generalized to more general boundary surfaces defined by a coordinate function. It is found that two of the pairs correspond to the PEC and PMC conditions while the other two correspond to a mixture of PEC and PMC conditions for fields polarized TE or TM with respect to the coordinate defining the surface.

D

B

Index Terms—Boundary conditions, electromagnetic theory, metamaterials.

I. INTRODUCTION

W

HEN representing electromagnetic field problems as boundary-value problems the boundary conditions are generally defined in terms of fields tangential to the boundary surface. A typical example is that of impedance boundary conditions which can be expressed in the form [1], [2] (1)

for some surface impedance dyadic which may have infinite components. Here, is the unit vector normal to the boundary surface. Special cases for the impedance boundary are the perfect electric conductor (PEC) boundary, (2)

is the PEMC admittance [8]. In fact, for and (4) yields the respective PMC and PEC boundaries. A different set of boundary conditions involving normal components of the vectors and at the boundary surface,

where

(5) has been recently introduced in [9], [10] in the context of electromagnetic cloaking and, independently, by these authors [11]–[14], in which a boundary defined by the conditions (5) was dubbed DB boundary for brevity. The corresponding conditions for the and vectors depend on the medium in front of the boundary. In this study we assume a simple isotropic medium with permittivity and permeability , whence (5) are equivalent to the conditions (6) A more recent study of literature has revealed that the DB conditions either in the form of (5) or (6) have been considered much earlier. In fact, in 1959 V. H. Rumsey proved uniqueness for problems involving a DB boundary and discussed its realization in terms of the interface of a uniaxially anisotropic medium [15]. The uniqueness and existence problems were subsequently considered with added mathematical rigor in [16]–[20]. The DB-boundary conditions (5) were introduced by these authors in [13] as following from the interface conditions for the of a certain exotic material called uniaxial IB half space medium or skewon-axion medium [21], [22]. However, a simpler realization for the planar DB boundary appears possible in of a uniaxially anisotropic medium terms of the interface defined by the permittivity and permeability dyadics [13], [15]

and the perfect magnetic conductor (PMC) boundary [3], [4], (7) (3) with the transverse unit dyadic defined by also known as high-impedance surface [5]–[7]. A generalization of these is the perfect electromagnetic conductor (PEMC) boundary defined by (4)

Manuscript received February 16, 2009; revised August 04, 2009. Date of manuscript acceptance October 01, 2009; date of publication January 22, 2010; date of current version April 07, 2010. This work was supported in part by the Academy of Finland. The authors are with the Department of Radio Science and Engineering, Helsinki University of Technology, Espoo, Finland (e-mail: [email protected]; [email protected]; http://www.tkk.fi). Digital Object Identifier 10.1109/TAP.2010.2041149

(8) , , the Assuming vanishing axial parameters, satisfy and fields in the uniaxial medium , whence because of the continuity, the DB conditions . Such a uniaxial medium was (5) are valid at the interface dubbed zero axial parameter (ZAP) medium in [23]. Obviously, the same principle applies for curved boundaries as well, when the medium is locally uniaxial with vanishing normal components of permittivity and permeability. Zero-valued electromagnetic parameters and their applications have been studied recently together with their realizations in terms of metamaterials [24]–[27].

0018-926X/$26.00 © 2010 IEEE

LINDELL AND SIHVOLA: ELECTROMAGNETIC BOUNDARY CONDITIONS DEFINED IN TERMS OF NORMAL FIELD COMPONENTS

Basic properties of the DB boundary for electromagnetic fields have been recently studied. In [13] it was shown that, since the Poynting vector has only the normal component, the DB boundary is an isotropic soft surface in the definition of Kildal [28]–[30]. Thus, coupling between aperture antennas on a DB plane is smaller than on a PEC plane. Further properties of a planar DB boundary were analyzed in [14]. It was shown that the DB plane can be replaced by a PEC plane for fields with respect to the normal direction and, by polarized fields. Thus, radiation from a current a PMC plane for source in front of the DB plane can be found by splitting radiating a field and the source in two parts, radiating a field, whence the DB plane can be replaced by the images of the two source components, PEC image for the component and PMC image for the component. In [31] the resonator defined by a spherical DB boundary was studied. Splitting all modes in two sets, those polarized and with respect to the radial coordinate , it was shown and the modes equal those of the respective that the PEC and PMC resonator. Thus, the number of modes with the same resonance frequency is twice of that of the PEC or PMC resonator which means that there is more freedom to define the resonance field. Finally, the circular waveguide defined by the DB boundary was analyzed in [32]. Since general modes cannot be decomand parts with respect to the polar radial coposed in ordinate , the modes had to be computed in the classical way. It was shown that there may exist backward-wave modes in such a waveguide although there is no dispersion or any periodic structure involved. The objective of the present paper is to extend the concept of DB boundary by adding another set of conditions involving normal derivatives of the normal field components. First, basic properties of the plane-wave reflected from planar boundaries satisfying different boundary conditions are derived, after which the more general curved boundary defined by a coordinate function will be studied. II. PLANAR BOUNDARY CONDITIONS In the following we consider a planar boundary defined by and fields in the half space . The DB condition (5) can be expressed as vanishing of the components of the two fields,

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boundary for and let us call such a boundary by the name boundary, the fields brevity. If there are no sources at the and are solenoidal and they satisfy

(12) Thus, the conditions (11) can be alternatively expressed in terms of the transverse components of the fields as (13) In an isotropic medium the equivalent to

conditions (11) and (13) are

(14) (15) There are two other combinations of the four conditions in (9) and (11) which may appear useful. According to the previous pattern, let us call the conditions (16) as those of the

boundary and the conditions (17)

as those of the be replaced by

boundary. In an isotropic medium (16) can

(18) while the conditions (17) can be replaced by (19) in From (18) it follows that there exists a scalar potential at the boundary terms of which we can express the field and satisfies the Laplace equation (20) Similarly, we can write for the field terms of a potential as

at the

boundary in

(9)

(21)

at the boundary. If there are no sources at the boundary, the components of the Maxwell equations yield conditions for the transverse components of the fields as

Assuming a localized source, the tangential components of the and the radiation fields are known to decay in the infinity as [36]. Following the argumentation radial components as of the Appendix, we must then have and at the boundary surface. Thus, under the assumption of localized conditions equal the PMC condition (3) and the sources the conditions equal the PEC condition (2). However, this is not valid for all non-localized sources. For example, a constant planar surface current gives rise to a TEM field with and everywhere, whence (16) and (17) are satisfied identically. This effect was discussed for the DB boundary in [14], [23].

(10) Obviously, the conditions (10) are equivalent to (9). Let us now introduce another possible set of boundary conditions involving the normal derivatives of the field components at the planar surface: (11)

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III. REFLECTION OF PLANE WAVE

From (29), (30) we see that the tangential components of the reflected field vectors satisfy

A. Plane-Wave Relations boundary is to The basic problem associated with the find the reflection of a plane wave from the planar boundary in the isotropic half space defined by the parameters , . Assuming time dependence and choosing the axis so that the wave vector of the incident and reflected waves plane, the field and wave vectors have the form are in the (22) (23) (24) Because the fields of a plane wave are divergenceless, they satisfy the orthogonality relations (25) (26) Assuming , i.e., excluding the normal incidence case, the fields can be expressed in terms of their components as (27) (28) (29) (30) Here, (31)

(36) Combining this with (33) the condition for the total tangential field at the boundary becomes (37) which is of the same form (32) as for the components of the incident field. This leads us to the following conclusions. , (37) yields • For the TE incident wave with which corresponds to the PEC condition (2). • For the TM incident wave with , (37) yields which corresponds to the PMC condition (3). , (37) yields which • Denoting corresponds to the PEMC condition (4). C.

Boundary

At the boundary tions (14) yield

the fields satisfying the condi(38) (39)

whence, again, the reflected field components satisfy a relation of the same form (32) as the incident field components . From (29), (30) we now see that the tangential components of the reflected field vectors satisfy

denote the respective wavenumber and wave-impedance quantities. Let us consider an incident plane wave whose components satisfy the condition

Combining with (34) the condition for the total tangential fields at the boundary becomes

(32)

(41)

Actually, any plane wave satisfies a condition of the form (32) for some parameter . (The TEM case , is ex.). The special cases of cluded because of the assumption TE and TM waves correspond to the respective cases and . From (27), (28) the incident field vectors tangential to the boundary can be shown to satisfy the two conditions

This condition leads us to the following conclusions. • For the TE incident wave with , (41) yields which corresponds to the PMC condition (3). , (41) yields • For the TM incident wave with which corresponds to the PEC condition (2). • Denoting , (41) yields which corresponds to the PEMC condition (4). boundary do not Since these properties of the DB and depend on the vector of the incident plane-wave field (ex), they are valid for any plane waves. Being cept that linear conditions, they are equally valid for general fields which do not have sources at the boundary. It is interesting to noconditions appear complementary tice that the DB and in showing PEC and PMC properties to TE and TM polarized and correfields. Moreover, the two PEMC admittances boundary conditions satisfy the sponding to the DB and simple condition

(33) (34)

B. DB Boundary The DB boundary conditions (9) at

require (35)

whence the reflected field components satisfy a relation of the same form (32) as the incident field components , .

(40)

(42)

LINDELL AND SIHVOLA: ELECTROMAGNETIC BOUNDARY CONDITIONS DEFINED IN TERMS OF NORMAL FIELD COMPONENTS

TABLE I BOUNDARY CONDITIONS INVOLVING NORMAL FIELD COMPONENTS CAN BE REPLACED BY EFFECTIVE PEC AND PMC CONDITIONS FOR FIELDS WITH TE AND TM POLARIZATIONS

This result has a close connection to the duality transformation [2] (43) which induces a transformation of media and boundary conditions. In particular, the DB and conditions are invariant in the transformation but the PEMC admittance is transformed as . D.

and

Boundaries

Let us finally consider the two other boundary conditions (16) and (17) for the incident plane wave (27), (28) satisfying the boundary the reflected fields satisfy condition (32). For the (44) From (30) the reflected transverse magnetic field component becomes (45) which compared with (28) can be seen to equal . Because boundary condition equals the PMC this is valid for any condition (3) for any fields except when . Similarly, the condition (17) leads to

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waves but, assuming a passive surface, it can be ruled out by a power consideration. So the anomaly is associated with TEM waves, only, and we are free to define the reflection coefficient for the TEM wave (it may depend on the polarization of the wave). The additional information comes naturally if the surface is taken as the limiting case of a given physical interface. For example, for the DB boundary studied in terms of its realization by a ZAP-medium interface [14], [23], [33] the reflection of the TEM wave depends on the transverse medium parameof the medium. For the special case when the wave ters impedance of the TEM wave is equal in both media, there is no reflection. In this case we can define the corresponding DB boundary by requiring no reflection for the TEM wave. Another ZAP medium leads to another definition. Similar consideration -boundary conditions requires a corresponding mafor the and terial realization which is yet to be found. For the boundaries it appears quite natural to assume that the respective PMC and PEC conditions are valid for the TEM wave as well. For a localized source giving rise to a continuous spectrum of plane waves, the normally incident component has zero measure, i.e., it corresponds to zero portion of the total radiated power. Thus, the TEM component plays no role and can be neglected, as was already pointed out by Rumsey [15]. Also, in the case of a non-planar boundary an incident plane wave has a component appearing as a local TEM wave with zero energy which also can be omitted. It was shown through numerical computations for plane-wave scattering from spherical and cubic objects conthat there is no back-scattered wave when the DB or ditions are satisfied at the scatterer as predicted by the theory [34], [35]. F. Other Possibilities For completeness, the two remaining possible combinations of boundary conditions,

(46)

(48) (49)

(47)

appear to be of little use. As an example, imposing (48) on the previous plane wave yields

and from (29) to

which when compared with (27) equals . This corresponds to the PEC condition (2). As a summary we can compare the four boundary conditions for TE and TM polarized incident fields in Table I. E. Reflection of TEM Wave corAt this point one should consider the omitted case responding to the TEM wave incident normally to the boundary. Since in this case the incident wave does not have normal field components, the preceding conditions are already satisfied by the incident field and there appears to be no reflected field. However, since any reflected TEM wave can be added to the field, one may conclude that the conditions involving normal field components are not enough for defining the TEM fields. This anomaly, which was already pointed out by Rumsey [15], can be removed by adding the missing information. One may argue that such a reflected TEM wave could also be associated to any TE and TM

(50) This restricts the freedom of choice of the incident field. Thus, launching a TM incident field creates a contradiction, whence (48) appears to be of improper form. IV. MORE GENERAL BOUNDARY SURFACES Let us generalize the previous analysis by replacing the planar as boundary surface by one defined by a function . Defining two other functions and so that they make a system of orthogonal coordinates satisfying (51) allows us to express various differential operators in the form given in the Appendix.

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Expressing the fields as

as can be easily checked. Thus, (60) and (61) can be compactly expressed in vector form as (66) (67)

(52) and similarly for the and tions are expressed by

fields, the DB-boundary condi-

On the other hand, outside of sources, the divergence of the fields vanishes, whence from (94) we can write

(53)

(68) (69)

. The conditions for the boundary are somewhat for more complicated. Writing the expansion of the divergence (54)

Let us apply these on the different combinations of boundary conditions (56)–(59). B. DB Boundary

for the Cartesian coordinates with

as (55)

gives us a reason to anticipate that a boundary condition of the form for the planar surface should take the form for the more general surface. A. Boundary Conditions Assuming an isotropic medium bounded by the surface , we can propose four possible boundary conditions on involving only the normal field components and and their normal derivatives as (56) (57) (58) (59) Let us first consider consequences of these conditions. For on the boundary one of the Maxwell equations yields

the fields on can be Starting from (56) expressed in terms of two potential functions as given in (66), (67). Let us now consider two special cases. Fields satisfying everywhere will be called fields and fields will be called fields. satisfying field satisfies everyObviously, a where, including the boundary surface . From (67) and (68) we conclude that on the potential satisfies (70) From the discussion in the Appendix we conclude that this implies (71) i.e., PEC condition on the boundary surface . The result can be generalized to problems where is not closed but extends to infinity, provided the sources are localized so that the fields vanish in the infinity. field the potential must satisfy Similarly, for the (72)

(60) while for

the other Maxwell equation yields (61) such

(60) is satisfied if there exists a scalar function that on we can write

(62) (63) Similarly, (61) is satisfied for a function

and

field sees the boundary deon the surface , whence the fined by the DB conditions (56) as a PMC boundary. This condition can also be obtained from the duality transformation which swaps electric and magnetic fields and, hence, PEC and PMC conditions. The DB boundary is invariant to the duality transformation. It must be pointed out that there is no guarantee that a given field can be expressed as a sum of partial fields TE and TM in polarized with respect to the coordinate function the general case. Such a decomposition is known to be valid with respect to Euclidean coordinates and the radial spherical coordinate. C.

Boundary

(64)

Considering now the boundary conditions (59), from (68), (69) the fields must satisfy

(65)

(73)

LINDELL AND SIHVOLA: ELECTROMAGNETIC BOUNDARY CONDITIONS DEFINED IN TERMS OF NORMAL FIELD COMPONENTS

at the boundary. From (66) a field can be represented in which from (73) satisfies (72). terms of a potential field sees a From (98) we again conclude that a boundary as a PMC boundary. Similarly a field sees the same boundary as a PEC boundary. Thus, in this respect DB and boundaries show complementary properties. conditions (59) represent a generalIt is clear that the ization of the conditions (14) for boundary surfaces more general than the planar surface. Comparing (59) and (94) whose last , we see that the proper form term can be written as -boundary conditions is for the

and are curl-free, (81) is Because at the DB boundary automatically valid. The DB condition (56) implies (82) at the boundary. This combined with (79) and the reasoning for a closed surface given in the Appendix leads to (83) at the DB boundary. This equals the PEMC condition (84)

(74) ,

As an example, for the spherical coordinates the metric coefficients are

,

Following a similar path of reasoning, one can see that at the boundary (79) is automatically valid for a field while (81) and

(75) -boundary conditions (59) at the surface and the have the form

(85) which follows from the PEMC condition

conditions (59), correspond to the

(76) ,

instead of D.

and

as could be suggested by (14).

Boundaries

The mixed conditions (57) and (58) can be handled in the same way. In the case (57), (60) implies on while (69) implies . From the reasoning given in the on . Thus, (57) corresponds to Appendix we obtain the PMC conditions for any fields. Similarly, we can show that (58) corresponds to the PEC conditions for any fields. E. The

Field

The DB and conditions were tested above for the and fields. Let us finally consider their generalization in terms of a combined field (77) where is a parameter. Its component 3 is assumed to satisfy everywhere the condition (78) Any field satisfying (78) is called a everywhere, from (94) the

field. Since field satisfies (79)

Inserting the Maxwell equations we can expand (80) whence the

field satisfies (81)

Both (79) and (81) are valid in source-free regions.

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(86) As a conclusion, for a field, which is a generalization of the and fields, both DB and boundaries appear and as PEMC boundaries with the respective admittances . This includes as special cases the results for the and fields given above. V. DISCUSSION The four boundary conditions (56)–(59) involving only field components normal to the boundary surface and their normal derivatives, form an interesting set. Two of these, (57) and (58) are alternatives for the respective PMC and PEC conditions which in terms of tangential fields are normally taken in the form (3) and (2). The other two conditions, dubbed as DB and conditions, (56), (59), appear new as kind of mixtures of PEC and PMC conditions or variants of the PEMC condition. The DB conditions have been introduced already in 1959 [15] condibut have not been applied for half a century; the tions have not been discussed earlier to the knowledge of these authors. Since their representations are so basic, DB and conditions seem to have their right of existence along with the PEC and PMC conditions. In introducing new boundary conditions the first question is to find their properties for the electromagnetic fields. This has been done here and some of the previous publications. The next step would be to ponder about their possible applications. In case there are some of enough interest, the problem of practical realization of such boundary surfaces as an interface of some material comes up. As explained in the introduction, the planar DB boundary may have some application as an isotropic soft surface [30]. The DB boundary can be realized, e.g., by an interface of a uniaxial anisotropic medium with zero axial paramboundary acts as an eters (ZAP medium) [13], [23]. The isotropic hard surface [33]. Finding a realization for the corresponding medium is still an open problem.

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In [35] it has been shown that both DB and boundaries roare self dual which implies that scatterers symmetric in tation have no backscattering. This is an interesting property, shown to be valid by numerical computations, and it may have potential applications. As a continuation of the present study, it has been further shown that it is possible to define generalized conditions DB boundary conditions of which the DB and are special cases. Such boundaries are also self dual and have the same zero-backscattering property [33]. In addition to the interesting physical properties of the novel boundary conditions, there may be some advantage in numerical electromagnetics by considering normal field components on the surface instead of the tangential components in the cases and boundaries which correspond to the respective of PMC and PEC conditions for the tangential fields.

(94) (95) (96) If a scalar function the Laplace equation

satisfies on the surface

(97) and if the surface is closed, we can expand the surface integral as (98)

APPENDIX A ORTHOGONAL COORDINATES The following review of differential operators for orthogonal , and coordinates defined by three functions has been taken from [36], [37]. The boundary surface is de. In this case, and define orfined by thogonal coordinates on . The coordinate unit vectors can be represented as (87)

Both integrals on the right-hand side vanish. The last one because (97) and the middle one because the surface is closed. on . This elementary Thus, we can conclude that proof is, however, valid for simply connected surfaces only, and requires a more involved analysis for more complicated surfaces like the torus, see [19], [20]. The condition remains valid for an open surface extending to infinity when the integral over the boundary contour of in infinity,

where the are the metric-coefficient functions. These follow from the definition of the gradient, (88) The divergence and curl of a vector function (89) are defined by (90)

(99)

is the unit vector normal to the contour and vanishes. Here we parallel to the surface in infinity. For a planar surface , the radial unit vector. As an example, if is have the tangential component of the electric field from a localized corresponds to the rasource, on the contour integral dial component of the far field which is known to decay as along the plane. Since , the integral vanishes and on the plane which corresponds to the PEC condition. However, this is not necessarily the case when the source extends to infinity. For example, for the normally incident TEM we have , whence the plane wave with constant integral (99) becomes infinite. A more complete analysis of the open surface case remains still to be done. ACKNOWLEDGMENT

(91) and the Laplacian by (92) The differential operations can be split in parts along the speand transverse to it: cial coordinate (93)

Discussions on the anomaly appearing for normally incident plane waves with Prof. P.-S. Kildal are gratefully acknowledged. REFERENCES [1] G. Pelosi and P. Y. Ufimtsev, “The impedance boundary condition,” IEEE Antennas Propag. Mag., vol. 38, pp. 31–35, 1996. [2] I. V. Lindell, Methods for Electromagnetic Field Analysis, 2nd ed. New York: IEEE Press, 1995. [3] A. Monorchio, G. Manara, and L. Lanuzza, “Synthesis of artificial magnetic conductors by using multilayered frequency selective surfaces,” IEEE Antennas Wireless Propag. Lett., vol. 1, no. 11, pp. 196–199, 2002.

LINDELL AND SIHVOLA: ELECTROMAGNETIC BOUNDARY CONDITIONS DEFINED IN TERMS OF NORMAL FIELD COMPONENTS

[4] A. P. Feresidis, G. Goussetis, S. Wang, and J. C. Vardaxoglou, “Artificial magnetic conductor surfaces and their application to low-profile high-gain planar antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 209–215, 2005. [5] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2059–2074, 1999. [6] S. Clavijo, R. E. Diaz, and W. E. McKinzie, III, “Design methodology for Sievenpiper high-impedance surfaces: An artificial magnetic conductor for a positive gain electrically small antennas,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2678–2690, 2003. [7] O. Luukkonen, C. Simovski, G. Granet, G. Goussetis, D. Lioubtchenko, A. V. Räisänen, and S. A. Tretyakov, “Simple and accurate analytical model of planar grids and high-impedance surfaces comprising metal strips or patches,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1624–1632, Jun. 2008. [8] I. V. Lindell and A. H. Sihvola, “Transformation method for problems involving perfect electromagnetic (PEMC) structures,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 3005–3011, Sep. 2005. [9] B. Zhang, H. Chen, B.-I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett., vol. 100, p. 063904, Feb. 15, 2008, (4 pages). [10] A. D. Yaghjian and S. Maci, “Alternative derivation of electromagnetic cloaks and concentrators,” New J. Phys., vol. 10, p. 115022, 2008, (29 pages). [11] I. V. Lindell and A. H. Sihvola, “Electromagnetic DB boundary,” in Proc. XXXI Finnish URSI Convention, Espoo, Oct. 2008, pp. 81–82 [Online]. Available: http://www.URSI.fi [12] I. V. Lindell and A. H. Sihvola, “DB boundary as isotropic soft surface,” in Proc. Asian Pacific Microw. Conf., Hong Kong, Dec. 2008, IEEE Catalog number CFP08APM-USB. [13] I. V. Lindell and A. H. Sihvola, “Uniaxial IB-medium interface and novel boundary conditions,” IEEE Trans. Antennas Propag., vol. 57, no. 3, pp. 694–700, Mar. 2009. [14] I. V. Lindell and A. Sihvola, “Electromagnetic boundary condition and its realization with anisotropic metamaterial,” Phys. Rev. E, vol. 79, no. 2, p. 026604, 2009, (7 pages). [15] V. H. Rumsey, “Some new forms of Huygens’ principle,” IRE Trans. Antennas Propag, Special Supplement, vol. 7, pp. S103–S116, 1959. [16] K. S. Yee, “Uniqueness theorems for an exterior electromagnetic field,” SIAM J. Appl. Math., vol. 18, no. 1, pp. 77–83, 1970. [17] R. Picard, “Zur Lösungstheorie der Zeitunabhängigen Maxwellschen in anisotropen, Gleichungen mit der Randbedingung n 1B n1D inhomogenen Medien,” Manuscr. Math., vol. 13, pp. 37–52, 1974. [18] R. Picard, “Ein Randwertproblem für die zeitunabhängigen Maxwellschen Gleichungen mit der Randbedingungen n 1 E n 1 H in beschränkten Gebieten beliebigen Zusammenhangs,” Appl. Anal., vol. 6, pp. 207–221, 1977. [19] R. Kress, “On an exterior boundary-value problem for the time-harmonic Maxwell equations with boundary conditions for the normal components of the electric and magnetic field,” Math. Methods Appl. Sci., vol. 8, pp. 77–92, 1986. [20] V. Guelzow, “An integral equation method for the time-harmonic Maxwell equations with boundary conditions for the normal components,” J. Integral Equations, vol. 1, no. 3, pp. 365–384, 1988. [21] I. V. Lindell, “The class of bi-anisotropic IB media,” Prog. Electromag. Res., vol. 57, pp. 1–18, 2006. [22] F. W. Hehl and Y. N. Obukhov, Foundations of Classical Electrodynamics. Boston: Birkhäuser, 2003. [23] I. V. Lindell and A. H. Sihvola, “Zero axial parameter (ZAP) sheet,” Prog. Electromag. Res., vol. 89, pp. 213–224, 2009. [24] N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: Nano-inductors, nano-capacitors and nano-resistors,” Phys. Rev. Lett., vol. 95, p. 095504, Aug. 26, 2005, (4 pages). [25] M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B, vol. 75, p. 075119, 2007.

=

=0 =

=0

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[26] A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilonnear-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B, vol. 75, p. 155410, 2007. [27] O. Luukkonen, C. R. Simovski, and S. A. Tretyakov, All-angle magnetic conductors realized as grounded uniaxial material slabs ArXiv: 0811.3493v1, Nov. 21, 2008. [28] P.-S. Kildal, “Definition of artificially soft and hard surfaces for electromagnetic waves,” Electron. Lett., vol. 24, pp. 168–170, 1988. [29] P.-S. Kildal and A. Kishk, “EM modeling of surfaces with stop or go characteristics—Artificial magnetic conductors and soft and hard surfaces,” ACES J., vol. 18, no. 1, pp. 32–40, 2003. [30] P.-S. Kildal, “Fundamental properties of canonical soft and hard surfaces, perfect magnetic conductors and the newly introduced DB surface and their relation to different practical applications included cloaking,” in Proc. ICEAA’09, Torino, Italy, Aug. 2009, pp. 607–610. [31] I. V. Lindell and A. H. Sihvola, “Spherical resonator with DB-boundary conditions,” Prog. Electromag. Res. Lett., vol. 6, pp. 131–137, 2009. [32] I. V. Lindell and A. H. Sihvola, “Circular waveguide with DB-boundary conditions,” IEEE Trans. Microw. Theory Tech., to appear. [33] I. V. Lindell, H. Wallén, and A. H. Sihvola, “General electromagnetic boundary conditions involving normal field components,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 877–880, 2009. [34] A. H. Sihvola, H. Wallén, P. Ylä-Oijala, M. Taskinen, H. Kettunen, and I. V. Lindell, “Scattering by DB spheres,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 542–545. [35] I. V. Lindell, A. H. Sihvola, P. Ylä-Oijala, and H. Wallén, “Zero backscattering from self-dual objects of finite size,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2725–2731, 2009. [36] J. Van Bladel, Electromagnetic Fields, 2nd ed. Piscataway, N.J.: IEEE Press, 2007, pp. 293-300–1025-1030. [37] A. Jeffrey, Handbook of Mathematical Formulas and Integrals. San Diego: Academic Press, 1995, ch. 24.

Ismo V. Lindell (S’68–M’69–SM’83–F’90–LF’05) was born in 1939 in Viipuri, Finland. He received the Ph.D. degree (1971) from Helsinki University of Technology (TKK), Espoo, Finland. Currently, he is Professor Emeritus of Electromagnetic Theory at the Department of Radio Science and Engineering, TKK. He has authored or coauthored 260 refereed scientific papers and 11 books, for example, Methods for Electromagnetic Field Analysis (IEEE Press, 2002), Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994), and Differential Forms in Electromagnetics (IEEE Press, 2004). Dr. Lindell received the IEEE S. A. Schelkunoff Award (1987), the IEE Maxwell Premium (1997 and 1998) and the URSI van der Pol gold medal (2005).

Ari H. Sihvola (F’06) is Academy Professor at the Helsinki University of Technology (TKK), Espoo, Finland. His visiting positions include the Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge (1985–1986), Pennsylvania State University, State College (1990–1991), Lund University, Sweden (1996), Electromagnetics and Acoustics Laboratory, Swiss Federal Institute of Technology, Lausanne (2000–2001), and University of Paris 11, Orsay (2008). His research interests include waves and fields in electromagnetics, modeling of complex materials, remote sensing and radar applications.

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A Cloaking Metamaterial Based on an Inhomogeneous Linear Field Transformation Stefano Maci, Fellow, IEEE

Abstract—A new type of bianisotropic metamaterial is theoretically investigated on the basis of a linear inhomogeneous field transformation applied to an arbitrary free-space Maxwellian field. This transformation does not include any space compression as predicted by transformation optics, and consists of a linear combination with space-dependent coefficients of the electric and magnetic incident fields. Duality conditions are applied to select an appropriate shape of the constituent dyads, thus resulting in a metamaterial completely defined by two real differentiable and . When these functions satisfy the functions of space 2 on the medium contour, the condition 2 and medium becomes globally lossless, and when imposing at the same boundary, the medium does not scatter for any arbitrary incident field, that is, it becomes invisible. When an additional internal boundary is introduced with boundary , the medium becomes a perfect and conditions cloak. Explicit analytical results are given for an invisible sphere and for a spherical cloak to provide additional physical insight.

=1

+

=0

= constant

=0

=0

Index Terms—Cloaking, EM theory, invisibility, metamaterials.

I. INTRODUCTION

I

N 2006, Pendry et al. [1] presented a general method, based on form-invariant transformations of Maxwell’s equations, that reveals the material parameters necessary for different structures to exhibit unusual electromagnetic scattering characteristics. The methodology, called “transformation optics”, was applied to find electromagnetic cloaks, shells of anisotropic material capable of rendering any object within their interior cavities invisible to detection from outside the cloaks. The perfect cloak ensures that for any incident field the electromagnetic scattered field vanishes in the free space external to the cloaking shell, and the total field vanishes inside the free-space cavity of the shell. Thus, any object placed in the cavity does not perturb the electromagnetic field outside the cloak and to an external observer it appears as if the object and cloak were absent. Such invisibility cloaks require materials with inhomogeneous, anisotropic permittivities and permeabilities that cannot be found in nature. However, an approximation to the ideal constituent parameters for a circular cylinder was realized using metamaterials and characterized experimentally by Schurig et al. [2] with results that have generated considerable interest and discussions in the scientific communities.

Manuscript received March 07, 2009; revised August 14, 2009; accepted October 05, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. The author is with the University of Siena, Department of Information Engineering, Via Roma 56, 53100, Siena, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2041272

The formulation given in [1] is based on spatial coordinate transformations and corresponding transformations of Maxwell’s equations that provide expressions for the required inhomogeneity and anisotropy of the permittivity and permeability of the cloaking material. A similar approximate method (in the geometrical optics limit) for cloaking was presented by Leonhardt [3] where the Helmholtz equation is transformed by conformal mapping to produce cloaking effects. Subsequently, Leonhardt and Philbin discussed the conformal mapping theory in the context of general relativity by analogy with the deviation of optical rays close to a gravitational mass [4]. In [5], cloaking is reformulated as a boundary value problem with a single first-order Maxwell differential equation for linear anisotropic media. This alternative formulation avoids the extensive reliance on coordinate transformations [4] and reveals the boundary values of the fields at the inner and outer surfaces of a cloak that yield zero scattered fields outside the cloak and zero total fields inside the free-space cavity of the cloak. In particular, it was shown that the boundary conditions that properly describe the behavior of the inner surface of both two-dimensional and three-dimensional cloaks are the normal components of and equal to zero. Although the transformation in [1] is physically appealing, different types of field-transformations can be adopted that do not resort to any space compressions. In [6], a space-dependent linear combination of electric and magnetic free-space fields is used to obtain a new field that satisfies Maxwell’s equations in a bianisotropic medium. Depending on the linear coefficients, the above transformation defines an artificial media that can produce the fields in a prescribed fashion in the volume occupied by the medium. In [6], the linear transformation is introduced to categorize different types of bianisotropic metamaterials. In this paper, the additional use of a duality condition allows one to define a new medium that possesses interesting invisibility and cloaking properties. The paper is structured as follows. In Section II, the linear EM transformation is applied to an arbitrary Maxwellian free-space field to obtain a general form of a bianisotropic medium. In Section III, application of duality conditions leads to a medium defined in terms of the two real differentiable functions of space and . The same section presents a derivation of the Poynting vector, the volumetric density of losses, and the global losses inside the metamaterial, the latter expressed in terms of a surface integration. In Section IV, the invisibility and cloaking conditions are identified as subspecies of local and global lossless conditions; Section V illustrates the general ideas with specific examples of the invisible sphere and spherical cloak. After some general physical considerations presented in Section VI, summary and conclusions are drawn in Section VII.

0018-926X/$26.00 © 2010 IEEE

MACI: A CLOAKING METAMATERIAL BASED ON AN INHOMOGENEOUS LINEAR FIELD TRANSFORMATION

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, followed by the imposition of (3); thus, we obtain

(5) Fig. 1. Geometry of the problem, (a) bianisotropic medium and definition of the internal field, (b) reference problem for the incident field.

(6)

II. LINEAR EM TRANSFORMATION Consider a general bianisotropic metamaterial medium placed in a finite region of space delimited by a surface and immersed in free space (Fig. 1(a)). The metamaterial is characterized by the constituent relations

Equations (5) and (6) are identically satisfied for any field if the dyadic coefficients vanish. This leads to the following solution:

(1a) (1b) (We use bold characters for defining vectors and bold characters with double bars to denote dyads; we suppress the time dependence ). Assume that the object is illuminated by placed outside . an EM field with arbitrary sources inside satisfy The total electric and magnetic fields the homogeneous Maxwell’s equations

(7a)

(7b) (2a) (2b) Note that the above equations remains invariant for the duality substitutions ; , . Let us denote by the incident (unperturbed) field, namely the field generated in free-space by the same enforced sources of the original problem (Fig. 1(b)). This field obeys the homogeneous Maxwell’s equations in free space inside (3a) (3b) We are interested in defining the constituent dyads , , , of the metamaterial in such a way that the EM field inside the , namely medium is a linear combination of (4a) (4b) where is the free-space impedance and the non-di, , , are conmensional parameters tinuous and differentiable arbitrary functions of the space observation vector . In order to provide the constituent dyads, we insert (4) in (2) and we use the identity

(7c)

(7d)

where is the free-space speed of light. The above solution coincides with the one presented by Tretyakov et al. in [6], who categorized through them various typologies of metamaterials. III. DUALITY CONDITIONS It can be seen by inspection of (5) and (6) that the application (i.e., of the duality transformation to both the fields ; , ), and (i.e., ; , ) leads to (5) (6), provided that , and . Therefore, we assume

(8)

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that will be referred to as “duality conditions.” These conditions simplify the general expressions (7) to (9a)

C. Power Lost in the Metamaterial The volumetric density of power lost in the medium is given by [7, Eq. (54)]

(9b) (14)

where the magneto-dielectric dyads may be also rewritten as

(9c) Under (8), for real values of and , the medium obtained is not reciprocal and punctually lossy or gainy. In fact, although the tensors and are Hermitian, (property satisfied by lossless media), the further condition for the absence of losses [7, Eq. 57] (where denote conjugate and superscript “ ” denotes transpose) does not hold for real values of and . Thus, the medium in a specific point is active or passive depending on the spatial variation of and and to the incident field.

Under duality conditions (8), and for and real functions, and are Hermitian; therefore the first two terms on the rhs of ; moreover, if and are (14) does not contribute to , thus obtaining real, one has

(15) After using (9c) in (13) and reversing the outcoming order of dot and cross product, one gets

(16)

A. Fields and Inductions Inside the Metamaterial The internal fields are obtained by using (8) in (4), yielding

that by means of (13) becomes

(10a) (10b) The “induced fields” or inductions are derived from (9), (10), and (1); by straightforward algebraic manipulations, one gets (11a)

(17) where . Equation (17) shows that the medium may locally transfer or subtract energy to the incident field depending with respect on the orientation of the local gradient of to the incident Poynting vector. When the additional condition

(11b) can The expression of the inductions fully in terms of be obtained by substituting (10) in (11). It can be easily proved by applying (3) to (11) that

(18) is imposed at any point of , the medium becomes lossless and the constituent dyads simplifies to

(12) independently on the choice of and , as expected. It is worth noting that, as evidenced in [6], there are not yet proofs to demonstrate that the general form in (7) obey (12) for any . This is values of differentiable functional parameters indeed proved under duality conditions. B. Poynting Vector in the Medium The Poynting vector of the medium expressed by the constituent dyads in (9) is given by

(19) Thus, the metamaterial loses the bianisotropic structure and becomes anisotropic and lossless at any point. The condition in (18), however, limits the degrees of freedom in designing the metamaterial. For the succeeding reasoning about the cloaking properties, it is more interesting to look for the vanishing of the total power lost in more than that of the volumetric density power. Toward this purpose, we use to (17) the identity and the property , to obtain

(20a)

(13) where is the Poynting vector of the unperturbed field. In particular, the real part of the Poynting vector always has the same sign and direction as one of the incident fields.

or, from (13)

(20b)

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given by (10)) respects the continuity of the tanfield gential fields across the boundary . The final solution thus obtained is indeed consistent with the Maxwell’s equations both inside and outside , with boundary conditions (b.c.) at the interface, and with b.c. at infinity. Denoted by the normal to at a point , the continuity conditions can be obtained through (10) as (23a) (23b)

Fig. 2. Geometry, field distributions, and boundary conditions for (a) non-scattering (invisibility) problem and (b) cloaking problem.

where . The latter equation confirms the general result from the Poynting theorem applied to bianisotopic media. Let us concentrate now on the first equality of (20); through it, the total power lost in the volume can be expressed in terms of a surface integration

(21) where is the external normal to . It is important to note that assume constant value on the surface if the quantity , the total power losses in the metamaterial is zero, due to ; i.e.,

identifies a limit from outside (upper sign) where the symbol or inside (lower sign) the volume . The b.c. in (23) are simultaneously respected for and . Under this condition the volume does not scatter any field, i.e., it is “inas the “invisivisible.” We will refer to bility” b.c. The invisibility condition actually ensures also the conservation of the normal component of the fields. Indeed, the conservation of the normal component of the fields across the boundary implies for the present medium the conservation of the normal components of the inductions. In fact, uniform values of and on implies so that from (11) it follows that the normal component of and is equal to the and , respectively. normal component of B. Cloaking Conditions

(22) denotes an arbitrary point of . The condition (22), where that is weaker than (19), allows one to keep the freedom to chose and independently inside . IV. INVISIBILITY AND CLOAKING CONDITIONS Consider now the two different problems shown in Fig. 2(a) and (b). The first problem (Fig. 2(a)), is equal to the original one in Fig. 1(a), except that the medium satisfies the duality conditions. In the second problem (Fig. 2(b)), we inside , and we assume define a continuous closed surface that the bianisotropic medium occupies the region of space between the two closed surfaces and , thus forming a cavity inside . We may assume for simplicity that this cavity is filled by free space but it can be filled by an arbitrary medium without changing the final result. Both the problems produced by the are subjected to the incident field sources . In the following paragraphs, we look for the boundary conditions on and for which the medium in in Fig. 2(a) does not scatter any field (“invisibility condition”) and the medium in Fig. 2(b) cloaks the region inside (“cloaking condition”). A. Invisibility Conditions In order to render the medium in Fig. 2(a) invisible (i.e., with zero scattered field), it is sufficient that the total external field is equal to the incident field , and that the total internal

With reference to Fig. 2(b), the material region forms an invisibility cloak if in addition to the non-scattering conditions , the total field in the cavity implied by vanishes. This ensures that any object placed in the inside cavity does not perturb the pre-existing situation; that is, it is masked by the cloak without causing any scattering. We use here normal components of both the inductions in place of tangential components of one type of field according with what done in vanishes, the continuity [5]. Since the total induction inside implies of the normal components of the inductions across

(24a)

(24b) By assuming uniform values of one has

and on

and using (10),

(24c) (24d) and . It is noted that are satisfied if that the vanishing of the normal components of the inductions at the inner surface of the cloak, boundary conditions that were boundary,” first introduced for cloaking in [5], defines a “ whose properties are studied in [8], [9].

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C. Summary

The total internal field is obtained using (28) and (31) in (10)

In summary, with reference to Fig. 2(a)

(32a) (32b) (25)

and with reference to Fig. 2(b)

(26) We emphasize the fact that since both (25) and (26) imply (22), both the invisibility and the cloaking b.c. imply absence of global losses in the medium, as expected.

. The plane wave penand from (30) etrates with perfect matching in the sphere and its polarization rotates in the plane orthogonal to at the angle in each point of the medium while preserving invariant the amplitudes of the fields and of the Poynting vector. Approaching the boundary, the polarization gradually approaches that of the external unperturbed wave. This consideration is not related to the specific case, but is generally applicable to any shape and any ray-type incident field under invisibility conditions (24) and lossless conditions (18). B. Spherical Cloak

V. EXAMPLES In the following, we show two examples in spherical domains, which allow us to investigate further the phenomenology. For each example, we analyze the behavior of an incident plane wave, since the relevant phenomena allows a gain of physical insight for ray-type incident fields.

Consider now a spherical cloak composed by an internal surface of radius and an internal surface of radius . Equations (9a) and (9b) become (33a) (33b)

A. Invisible Sphere Consider the case of a spherical volume of radius filled by , the bianisotropic metamaterial. Let us assume are functions of the radial distance from the center of the sphere with invisibility boundary conditions , . By introducing a polar reference system with unit vectors ( , , ), one gets , , where the prime denotes derivative with respect to . By applying the additional condition of absence of losses the metamaterial constituents parameters simplifies as

and the cloaking b.c. in (26) become , , , . These conditions are satisfied for instance by

(27)

(35a)

(34) where and

is a generic differentiable function with . This choice leads to

,

A possible choice is (35b) (28) where with

is a generic differentiable function of its argument . This choice implies

Imposing the additional condition (with finite), eliminates the singularity of and at the internal boundary. When a plane wave (31) illuminates the cloak, the field internal to the metamaterial is

(29) and from (13)

(36a) (30)

for any incident Poynting vector . No power is lost in any elementary volume of the sphere since . Let us consider now a plane-wave incident field propagating along , i.e., (31)

(36b) and the Poynting vector is obtained from (13) as (36c) In contrast with the previous case, the effect of the cloak is not only to rotate the polarization, but also to change locally the

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Fig. 4. Electric field for a spherical cloak based on transformational optics, when subjected to the incident plane wave field as that in Fig. 3. (Purpose of comparison with Fig. 3(a)).

Fig. 5. (a) Polarization of a ray-field inside a metamaterial with invisibility b.c. and lossless condition. (b) Amplitude of the Pointing vector inside a cloak.

Fig. 3. Electric field inside a spherical cloak characterized by the constituent tensors in (35) and illuminated by an x-polarized plane wave propagating along z as in (31). (a) Real part of the x-component; (b) real part of the y -component. The axes are normalized with respect to a wavelength.

shows the electric field for a “conventional” cloak based on transformational optics [5]. No cross-polarized field occurs in this case. The most evident difference between Figs. 4 and 3(a) consist on the fact that the wave-fronts in our cloak are flat, and not deformed like in conventional cloak. This implies a phase velocity equal to the speed of light, and not larger as in conventional cloak, as also evident from (35). The main consequence is that our cloak, whenever realized, does not violate causality. Further considerations are given in Section VI. VI. CONSIDERATIONS

amplitude of the Poynting vector for avoiding a penetration of energy inside the cloaked region. This implies necessarily that the Poynting vector is not solenoidal, and therefore (see (20)) a transfer of power from the incident wave to the medium or vice versa occurs at any point; however, (see (22)) the net power globally lost in the material is zero. Fig. 3 compares the field inside the cloak due to a plane wave for a spherical cloak characterized by constituent tensors as those in (35). In the cloak we see cross-polarized field components (Fig. 3(b)), according with (35a). For comparison, Fig. 4

Consider the situation in Fig. 5(a), where the illumination of a full volume of metamaterial is provided by a ray-field (let us use this term to denote a field with a real Poynting vector). Under in every point of invisibility conditions and with the volume, each ray penetrates the metamaterial at the interface without reflection and the wave polarization changes while maintaining the Poynting vector unchanged; when approaching the opposite portion of boundary, the polarization recovers gradually one of the original incident rays, and emerges from the material without internal reflections.

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The conditions in all points of the material cannot be realized for the cloaking case (Fig. 5(b)) because, to ensure the cloaking b.c., the external surface and the internal surface possess different values of . Any ray that impinges on the surface penetrates into the medium without reflection (point in Fig. 5(b) and change into the medium polarization and Poynting vector amplitude, while the direction of the ray is maintained along the original rectilinear path. When during its rectilinear path internal to the volume, the ray trajectory approaches the internal surface , the Poynting vector (point in decreases and gradually vanishes at the surface Fig. 5(b); it restarts to increase on the opposite part of (point in Fig. 5(b) till regaining at the exit point on (point in Fig. 5(b) the same amplitude that should have reached in absence of material. During its path the ray first loses and next gains power density. The global balance of power gained and lost associated with the summation of rays that travel across the cloak is exactly zero. We note that none of the rays in the metamaterial follow curved trajectories as they do in compression-type transformations; indeed, the use of a linear transformation does not modify the phase-front of the incident field. The main consequence is that there is more chance that the incident wave does not violate causality within the cloak. Let us further discussing on the physical insight of the present cloaking phenomenon, that is rather unusual. We emphasize that a bianisotropic medium (as an anisotropic medium) can be “gainy” (or “active”), lossy (or “passive”), or, let us say, “neutral” at a certain point depending on the sign of the real part of the Pointing vector divergence (see, for instance [10]). (In particular , , and denote “active”, “neutral” or “passive” media at that point, respectively). If the medium is active at a point, it means that, on time average basis, in a small volume around that point the medium gives an incremental energy to the field; the variation of energy augments the Pointing vector amplitude along its direction. On the contrary, if the medium is lossy, the field gives energy to the medium with the result that the Poynting vector amplitude decreases along its direction. The physical laws that describe the energy transfer must be embedded in the local shape of the equivalent constituent tensors. If at a point the medium is lossy, the energy can be for instance dissipated in heat in the neighborhood of that point. This dissipation (and the gradient of temperature associated with) must not change the form of the local constituent parameters; otherwise it means that the regime is not established yet. If the medium is active, an external source of energy, directly embedded in the constituent equivalent parameters, there exists to establish the active regime. Furthermore, the same medium may be active or passive at the same point depending on the field established in there due to the external source [10]. The present medium has the property so that it is active or passive that at a point depending on the direction of the gradient of with respect to the real part of the incident Poynting vector, and it is locally neutral if the same gradient vanishes. In our This condition is case this happens when compatible with the invisibility boundary condition, but not with the cloaking boundary condition. In the latter case the form of the power density is not definite positive neither negative at any

point; however, the “neutral” nature of the medium is ensured at global level by the cloaking boundary conditions. A difficulty in understanding the present medium is relevant to its memory in recovering phase and amplitude of a ray-field on opposite parts of the cavity. This conceptual difficulty can be overcome by invoking a spatial dispersivity of the medium. In our formulation, the response at a point (i.e., the inductions or ) depends on the application of a linear combination of the fields) at the same point of the medium. excitations ( and Thus, the medium is apparently not affected by spatial dispersivity. However, it is known that bianisotropy could be re-interpreted as anisotropy with spatial dispersivity. To this end, Maxwell’s equations in (1) can be rearranged in such a way to that only depends on introduce a new effective induction and its curl and a new induction that only depends on and its curl. Revising the medium as anisotropic and spatially dispersive renders more acceptable the idea of a memory in recovering field information at a distant point. Another critical issue of the present solution is concerned with the boundary condition at the internal surface. These conditions imply extreme values of the constitutive tensors, like those occurring in Transformation Optic cylindrical cloak. To overcome this impairment that may result in an additional difficulty in practical realization, a different linear combination of the incident fields can be assumed inside the cloak, where the scalar coefficients in (10) are substituted with dyadic coefficients, thus obtaining more degrees of freedom as the boundary conditions are concerned. The relevant solution is presently under study. Finally, the present cloaking medium conceptually resembles the transparent absorbing medium proposed by Maslovki and Tretyakov in [11] for implementing absorbing boundary conditions in finite difference time domain (FDTD). This suggests a numerical possible application—not constrained to the practical realizability—of the formulation presented here.

VII. SUMMARY AND CONCLUSION A new type of material is theoretically defined by using a linear inhomogeneous field transformation applied to an arbitrary free-space Maxwellian field. The linear transformation does not involve space compression as in transformation optics. Duality conditions are applied to select an appropriate shape of the constituent dyads, thus resulting in a medium completely defined by two differentiable functions of space and . The resulting relative permeability and permittivity dyads are equal and they are Hermitian and nonreciprocal. The medium is not in general lossless at any point, because the magneto-dielectric dyads do not respect the lossless conditions. The punctual lossless property is obtained by adding the condition that, however, limits the design capability. If the same condition is applied just at the medium contour, the medium becomes globally lossless. When imposing and at the boundary, the medium does not scatter for any arbitrary incident field, that is, it becomes invisible. When an additional internal boundary is introduced with boundary and , the medium becomes a perfect conditions cloak for any incident field.

MACI: A CLOAKING METAMATERIAL BASED ON AN INHOMOGENEOUS LINEAR FIELD TRANSFORMATION

A discussion has been carried out here to interpret and explain the physical phenomena associated with this solution. This discussion has not touched the main issue about the practical realizability. We do not have presently serious ideas about how this medium can be realized in practice. However, it seems that fundamental physical principles are not violated. ACKNOWLEDGMENT The author wishes to sincerely thank Dr. A. D. Yaghjian for his thoughtful comments and helpful suggestions in revising the manuscript; he also thank Prof. O. Bucci, Prof. N. Engheta, and Prof. S. Tretyakov for useful discussion about this subject. REFERENCES [1] R. F. Harrington, J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science, vol. 312, no. 5781, pp. 1780–1782, Jun. 2006. [2] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science, vol. 314, no. 5801, pp. 977–980, Nov. 2006. [3] U. Leonhardt, “Optical conformal mapping,” Science, vol. 312, no. 5781, pp. 1777–1780, Jun. 2006. [4] U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys., vol. 8, p. 247, Oct. 2006. [5] A. D. Yaghjian and S. Maci, “Alternative derivation of electromagnetic cloaks and concentrators,” New J. Phys., vol. 10, p. 115022, Nov. 2008. [6] S. A. Tretyakov, I. S. Nefedov, and P. Alitalo, “Generalized field-transforming metamaterials,” New J. Phys., vol. 10, p. 115028, Nov. 2008. [7] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 4, Apr. 2005. [8] V. H. Rumsey, “Some new forms of Huygens’ principle,” IRE Trans. Antennas Propag., vol. 7, pp. 103–116, Dec. 1959.

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[9] I. V. Lindell and A. H. Sihvola, “Electromagnetic boundary and its realization with anisotropic metamaterial,” Phys. Rev. E, vol. 79, p. 026604, 2009. [10] I. Lindel, Methods for Electromagnetic Field Analysis. New York: IEEE press, 1995, p. 90. [11] S. I. Maslovski and S. A. Tretyakov, “On the concept of the transparent absorbing boundary microwave and,” Opt. Technol. Lett., vol. 23, no. 1, Oct. 1999. Stefano Maci (S’98–F’04) received the Laurea degree (cum laude) in electronic engineering from the University of Florence, Italy. Since 1998, he is with the University of Siena, Italy, where he presently is a Full Professor. His research interests include EM theory, antennas, high-frequency methods, computational electromagnetics, and metamaterials. He was a coauthor of an Incremental Theory of Diffraction for the description of a wide class of electromagnetic scattering phenomena at high frequency, and of a diffraction theory for the high-frequency analysis of large truncated periodic structures. He was responsible of several projects funded by the European Union (EU), by the European Space Agency (ESA-ESTEC) and by various European industries, and WP leader of the Antenna Center of Excellence (ACE, FP6-EU). He was the founder and presently is the Director of the European School of Antennas (ESoA); presently, ESoA comprises 24 courses on antennas and propagation and about 150 teachers coming from 24 European research centres. He is principal author or coauthor of more than 100 papers published in international journals, (approximately 60 of which appeared in IEEE journals), 10 book chapters, and about 350 papers in proceedings of international conferences. Prof. Maci was an Associate Editor of the IEEE TRANSACTIONS ON EMC and two times Guest Editor and current Associate Editor of the IEEE TRANSACTION ON ANTENNAS AND PROPAGATION. He is a member of the Board of Directors of the European Association on Antennas and Propagation (EuRAAP), the AP-S AdCom, the Technical Advisory Board of the URSI Commission B, the Italian Society of Electromagnetism and of the Advisory Board of the Italian Ph.D. School of Electromagnetism. He was the recipient of various national and international prizes and best paper awards.

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The Design of Broadband, Volumetric NRI Media Using Multiconductor Transmission-Line Analysis Scott Michael Rudolph, Student Member, IEEE, and Anthony Grbic, Member, IEEE

Abstract—Multiconductor transmission-line (MTL) analysis is used to model a broadband, volumetric negative-refractive-index (NRI) medium. Equations for the two-dimensional dispersion characteristics and the Bloch impedance are derived using a simplified periodic MTL analysis that provides insight into the operation of the NRI structure, while still accounting for spatial dispersion. Equations for the resonant points of the NRI medium are also derived, including the magnetic and electric plasma frequencies and the low-frequency backward-wave cutoff. The nature of each resonant point is discussed as well. Finally, this paper presents the modeling of finite structures with rigorous MTL analysis. Using this method, the reflection and transmission coefficients for normal incidence on a four-cell slab of the NRI medium are calculated and compared to full-wave simulation. Index Terms—Multiconductor transmission lines, periodic structures, transmission line theory.

I. INTRODUCTION

HE first negative-refractive-index (NRI) medium was constructed and experimentally verified using an array of split-ring resonators (SRRs) and wires [1]. These initial experiments sparked great interest in the electromagnetic phenomena predicted by Veselago [2] and Pendry [3]. Unfortunately, the SRRs achieved negative permeability over a small fractional bandwidth of approximately 10%. This narrow bandwidth made the medium highly dispersive and forced the operating frequency to be close to the resonance of the ring, resulting in significant loss. These bandwidth and loss limitations continued to preclude the use of volumetric NRI structures in practical microwave systems. The term “volumetric” refers to metamaterials that are periodic in all three dimensions, forming a bulk medium capable of interfacing with waves in free space. This property differentiates volumetric metamaterials from “planar” metamaterials [4], [5], which are periodic in only one or two dimensions and are not designed to interface with free-space plane waves. Electromagnetic waves in planar metamaterials are instead confined to transmission lines. To address the problems of narrow bandwidth and high loss associated with volumetric media, a broadband NRI structure, shown in Fig. 1, was proposed in [6] and extensively analyzed in

T

Manuscript received May 18, 2009; revised September 15, 2009; accepted October 27, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. The authors are with the Radiation Laboratory, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2041314

Fig. 1. Unit cell of the broadband volumetric negative-refractive-index (NRI) medium. The central wire has an inductance of 2 1 L . The cage consists of a cube of metallic strips loaded with lumped capacitors on the horizonal surfaces.

[7]. This structure achieved a fractional backward-wave bandwidth of up to 60%, with the index of refraction ranging from 8.8 to 0 within this band. This broadband design exhibited greatly reduced dispersion when compared to earlier SRR/wire media. Similar designs were subsequently analyzed [8], achieving comparable results. Using this broadband design, a NRI lens was fabricated using printed-circuit-board technology [9]. This flat lens achieved two-dimensional sub-diffraction focusing with a resolution enhancement of 2.0 at frequencies around 2.424 GHz, demonstrating that broadband, low-loss NRI media could be designed and were suitable for use in practical applications. While these broadband, volumetric NRI media exhibited superior performance, no procedure existed to design them efficiently. Simple, accurate methods of designing planar NRI transmission-line media in one- and two-dimensions are wellestablished [10], [11]. In general, those analyses cannot be used to model the interaction between metamaterials and free-space waves. A notable exception is the structure presented in [12], where antenna elements are used to transition from free space waves to guided waves in a transmission-line network. In order to predict the behavior of metamaterials that couple directly to free-space waves, a different model must be used. In [13], it was shown that the dispersion curves of these broadband,

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RUDOLPH AND GRBIC: THE DESIGN OF BROADBAND, VOLUMETRIC NRI MEDIA USING MTL ANALYSIS

volumetric NRI structures could be predicted using a homogenized form of multiconductor transmission-line (MTL) analysis. This MTL analysis was capable of modeling the coupling between free-space waves and waves guided by the circuit network within the NRI medium. However, the homogenized analysis in [13] neglected spatial dispersion, making the predicted results valid only when the phase difference across the unit cell was small. In this paper, we present a more accurate version of the homogenized MTL analysis. The expressions reported here remain simple enough to provide insight into the operation of the structure, but also account for spatial dispersion. Considering spatial dispersion in the MTL analysis allows us to predict three important resonant points of the broadband NRI medium: the electric plasma frequency, the magnetic plasma frequency and the low-frequency backward-wave cutoff (the point where the phase difference across the unit cell is 180 ). We also present equations for the modal impedances of the infinite medium, which allow expressions for the permittivity and permeability to be found. Finally, we use MTL analysis to predict the scattering parameters of a finite structure. II. DISPERSION DIAGRAM One unit cell of the broadband NRI medium studied in this paper is shown in Fig. 1. This is the same design as the one reported in [7], which consists of two parts: the cage and the central wire. The lattice formed by the cage structures was described in [7] as an array of highly coupled SRRs. This coupling is responsible for the drastic increase in the bandwidth over which negative permeability can be achieved by this structure as compared to the traditional SRR array. The central wire is responsible for realizing negative permittivity. The NRI structure shown in Fig. 1 may not initially appear to be a good canelectric didate for MTL analysis. However, for the vertical field polarization (S-polarization), ground planes can be placed at the top and bottom of each unit cell to model infinite periodicity in the vertical direction. Image theory can be applied once more to replace the bottom half of the unit cell with a ground plane. The use of image theory in these simplifications ensures that the vertical electric field is antisymmetric with respect to each ground plane, as occurs naturally at the frequencies of interest. Finally, the unit cell can be shifted by half a cell in each of the horizontal ( and ) directions to form the two-conductor MTL system depicted in Fig. 2. To apply rigorous MTL analysis to the structure shown in Fig. 2, the structure must first be broken up into its constitutive elements: transmission lines, loading capacitors and loading inductors. Transfer matrices can be found for each of these elements and then cascaded to obtain the transfer matrix of the entire unit cell [14], [15]. From this complete transfer matrix, the propagation characteristics of the broadband NRI medium can be determined. However, this method leads to complicated expressions that do not provide intuition into the operation of the structure. A homogenized MTL analysis was used in [16] to analyze the Sievenpiper mushroom structure. In this analysis, the effects of the series and shunt loading elements were absorbed into the impedance and admittance matrices, respectively. This homogenization of the unit cell produced tractable equations

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Fig. 2. Equivalent unit cell of the broadband volumetric NRI medium that is conducive to MTL analysis.

that provided insight into the operation of the structure. However, since it neglected spatial dispersion, these equations were only valid when the phase difference across the unit cell was small. A simplified periodic MTL analysis can be derived that maintains the simplicity of the homogenized approach, while still accounting for spatial dispersion. In the homogenized analysis, once the loading elements are absorbed into the impedance and admittance matrices, the equations are further simplified by taking the limit as the cell size goes to zero. This has the benefit of producing equations that are of the same form as those in standard MTL analysis [14], [15], but the drawback of disregarding the effects caused by the finite unit cell size. In contrast, the simplified periodic method does not ignore the periodicity of the metamaterial by taking such limits. Instead, the periodicity is accounted for using the Floquet theorem, which states that the voltages and currents on each end of the unit cell only differ from each other by a complex constant, . This is enforced according to the circuit here defined as diagram in Fig. 3. As a result, the only approximation made is that of absorbing the loading elements into the impedance and admittance matrices of the transmission lines. Therefore, this method remains valid for any phase difference across the unit cell as long as the lengths of the interconnecting transmission lines are electrically short. Before beginning the MTL analysis, the values of the loading elements and the inductance and capacitance matrices must be found. This can be done in several ways. In this paper the values of the capacitance and inductance matrices are obtained in the static limit by modeling the unloaded conductors using a commercial electromagnetic simulator (Ansoft’s Maxwell). Since separation between the conductors is small compared to a wavelength, the static solution is accurate for the frequencies of inand ) are terest. The values of the loading elements ( , extracted from full-wave simulations at the frequency of operation using Ansoft’s HFSS. These values were found to deviate negligibly over the entire frequency range of interest. The values of all the variables used in this paper are tabulated in Table I. After the values of all the elements are known, the first step in the simplified periodic analysis is to solve for the homogenized impedance and admittance matrices. To find the impedance matrix, the impedance of the series loading element, , is added

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Fig. 3. Schematic of the two-conductor MTL system for on-axis propagation in terms of impedance and admittance parameters.

conductor, where and are voltages and currents on the and are the elements of the homogenized impedance and admittance matrices, respectively and is the dimension of the unit cell in the and directions. Note that reciprocity is assumed in (1)–(4). Combining (3) and (4), the current elements can be eliminated resulting in a 2 2 homogeneous system of equations (an eigenvalue problem)

TABLE I DESIGN PARAMETERS FOR THE MTL SYSTEM SHOWN IN Fig. 2

(5) where to the impedance of the self-inductance on conductor 2 , effectively absorbing the lumped element into the transmission line. Similarly, to find the admittance matrix, the admittances of and , are added to the admitthe shunt loading elements, and contances of the self-capacitances of conductor 1 , respectively. The resulting impedance and adductor 2 mittance matrices for the structure in Fig. 2 are

In order to find nontrivial solutions, the determinant of (5) must be set equal to zero. This determinant produces the dispersion equation for on-axis propagation

(1)

(6)

(2)

Applying this analysis in two dimensions gives the following 2-D dispersion equation:

The elements of the modified impedance and admittance matrices are subsequently used to form the circuit model of the unit cell, shown in Fig. 3. The Floquet theorem can then be applied to calculate the complex propagation constant, , (see Fig. 3) for the case of on-axis propagation. Application of Kirchhoff’s voltage and current laws yield the following expressions:

(3) (4)

(7) or direction. where is the propagation constant in the The dispersion equations give two unique modes, one corresponding to the choice of the plus sign in front of the radical and the other to the minus sign. The -mode is given by the plus sign and the -mode by the minus sign. As shown in Fig. 4, the -mode is in cutoff throughout the entire frequency range of interest, while the -mode supports the backward wave. The

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Fig. 6. Isofrequency contour plot (in GHz) of the backward-wave mode with x and ~y directions. respect to two-dimensional propagation in the ~ Fig. 4. On-axis dispersion diagram. The  -mode corresponds to the backwardwave mode while the c-mode is in cutoff throughout the frequency range of interest.

its extent. However, in discussing the properties of an infinite medium, a general condition is required. In [17], the representation of continuous media using lumped-element networks is considered. Representing a continuous medium as such a grid network results in spatial dispersion. However, effective medium theory asserts that these lumped-element networks can be considered an accu. The rate representation if the cell size is at most largest difference in propagation constant is between on-axis propagation and propagating 45 off-axis, as shown in Fig. 6. Expressions in [17] give the maximum percentage difference in as 0.874%. By propagation constant for a cell size of this stringent definition, the isotropic limit for this metamaterial occurs at a frequency of 2.52 GHz. This limit can also be obtained directly from (7). For con. For venience, the right hand side will be represented as on-axis propagation, (7) reduces to (6), now expressed as

Fig. 5. Two-dimensional dispersion diagram for the  -mode indicating the phase shifts 0 = (0 ; 0 ), X = (180 ; 0 ) and M = (180 ; 180 ).

two-dimensional dispersion diagram for the -mode generated using (6) is compared with full-wave analysis in Fig. 5. MTL analysis can also be used to examine how the propagation constant of the NRI medium changes with direction. In [7], full-wave analysis showed that the structure in Fig. 1 exhibited isotropic behavior (negligible spatial dispersion) when the phase difference across the unit cell was small. However, due to the amount of time required to perform the full-wave eigenand were plotted. mode simulations, only Using the two-dimensional dispersion equation given in (7), the isofrequency contours of the structure can be calculated rapidly over the entire backward-wave band, as shown in Fig. 6. Fig. 6 shows that the structure clearly exhibits isotropic propagation at high frequencies, however, at low frequencies the propagation constant changes significantly with direction due to spatial dispersion. The question then arises as to how significant these deviations in propagation constant can be while still describing the propagation as “isotropic.” Typically, this is influenced by the specific application, and the answer depends on the overall size of the material and the allowable deviation in phase over

(8) For propagation 45 off-axis, the dispersion equation can be written as

(9) Using (8) and (9), the ratio of as

to

is found in terms of

(10)

Thus, the value of determines the variation between on- and off-axis propagation. is small, (6) becomes In the limit where

(11)

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If is again taken to be the maximum unit cell size for which propagation can still be assumed isotropic and given , the value of at this limit becomes that . Using this value in (10), results in

(12) In other words, the maximum percentage difference in propagation constant is 0.874%, the same as that obtained from the equations in [17]. Fig. 7. Wave impedance of the  -mode on conductor 1 for the volumetric NRI medium.

III. IMPEDANCE AND MATERIAL PARAMETERS The impedance of a two-conductor MTL structure is typically represented by a 2 2 matrix relating the natural voltages to the natural currents on the two coupled lines [14]. Since the analysis in the previous section described the dispersion of the NRI medium in terms of the structure’s - and -modes, it may be preferable to examine the modal impedances of the infinite NRI medium. These impedances can be found in a manner similar to that presented in [18]. According to (5), the voltages on conductor 1 and conductor 2 are related by a ratio, , defined by the equation

parallel plate waveguide, which supports plane-wave propagation through the medium. Therefore, the transmission-line for the -mode should be impedance of conductor 1 proportional to the overall wave impedance of the infinite NRI medium. In order to find the exact expression for the wave impedance, the transmission-line impedance should be multiplied by the width-to-height ratio

(16)

(13) Inserting this relationship into (3) allows the currents, and , to be found in terms of the voltage . The impedances can then be found by taking the ratios of the voltage to the current on a given conductor, yielding the expressions

(14)

(15)

These equations are valid for a single mode propagating in the direction determined by the Floquet constant, . The mode is or . To reverse determined by the choice of the direction of travel, the sign of must be changed. These impedance equations are only valid for an infinite medium. If the material is finite, then multiple modes would be required to satisfy the boundary conditions at the termination of the structure. In Section V of this paper, a method for predicting the scattering parameters (and therefore, input impedance) of a finite structure is presented. Calculation of scattering parameters using modal analysis is presented in [19]. Despite not being strictly valid for finite slabs, the above impedance equations can still provide insight into the behavior of the NRI structure. Since Fig. 4 shows that the -mode is in cutoff for all frequencies of interest, this mode can be ignored in an infinite medium. Additionally, conductor 1 represents a

The wave impedance for the infinite medium composed of the unit cells shown in Fig. 1 is plotted in Fig. 7. The resonance that occurs around 3.28 GHz is due to the fact that the plasma frequencies are not exactly matched, resulting in a small stopband between the electric and magnetic plasma frequencies. From this graph, it is clear that the electric plasma frequency occurs slightly before the magnetic plasma frequency since the impedance grows rapidly (indicating close to zero) before going to zero (indicating equal to zero). It should also be noted that the impedance of the infinite medium was not designed to match that of free-space. Instead the unit cell was modeled exactly as it was in [7]. In [7], the values of the loading elements were selected to produce an impedance match at 2.45 GHz for a four-cell slab, as can be seen in Section V. In conventional materials, the wave impedance is equal to and the propagation constant is equal to . By taking either the ratio or the product of these formulae and dividing by the radial frequency, , the permeability and permittivity for the infinite medium can be defined as

(17) (18) However, to treat this metamaterial in the same way as conventional materials, it must behave like a homogeneous medium. This means that the propagation constant must not be distorted by spatial dispersion. As was discussed in Section II, homogeneous behavior is only possible when the phase difference across the unit cell is small. This restricts the frequency range

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over which effective material parameters can be defined. However, such a limitation can be advantageous as it allows the exis assumed to be small in (14), pressions to be simplified. If then the expression can be simplified to

(19) because it This equation is particularly useful in expressing depends inversely on . Therefore, by inserting (19) into (17), drops out of the expression, yielding

(20) In the equation for , having an inverse dependence on is not desirable because no cancellation would occur. This would instead result in the expression for being proportional to . To obtain a simpler expression, a different equation for must be derived. Instead of using the ratio of the voltages on conductors 1 and 2, the ratio of the currents is used. This ratio, , is defined as

(21)

Fig. 8. Material parameters for the volumetric NRI medium. The black lines correspond to the exact solutions given by equations (17) and (18). The gray lines correspond to the approximate solutions given by equations (20) and (24).

These zeros represent the magnetic and electric plasma frequencies, respectively. Using (20), the magnetic plasma frequency is found to be simply

(25) In the case of the electric plasma frequency, the numerator of (24) has two zeros. However, one occurs at a frequency well-above the range of interest. Ignoring this zero, the electric plasma frequency is given by the expression

Inserting this relationship in to (4), gives the following expresto : sion for the ratio of:

(22) (26) (23) as long as is small. In this new equation, is now directly proportional to , which will result in dropping out of (18), yielding

(24) The simplified equations (20) and (24) provide physical insight into what determines the effective permeability and permittivity. Equation (24) shows that the permittivity depends only on elements of the admittance matrix (with the exception of the current ratio, ), i.e. the shunt elements in the circuit diagram. This is expected since the unloaded admittance matrix consists of capacitive elements, whose values are directly related to the permittivity of the medium. Similarly, (20) depends only on elements of the impedance matrix (with the exception of the voltage ratio, ), which are the series elements of the circuit diagram. This is expected as well because the unloaded elements of the impedance matrix are inductances, which are dependent on the permeability of the medium. The form of (20) and (24) is also important. In both (20) and (24), the zeros of the functions are isolated in their numerators.

Both of the plasma frequencies will be discussed further in the following section. The relative permittivity and permeability of the infinite medium are shown in Fig. 8. Both the approximate ((20) and (24)) and the exact ((17) and (18)) curves are plotted, but it should be noted that neither is valid when the medium exhibits spatial dispersion. As mentioned in the previous section, the is small. medium can be considered homogeneous as long as Since being small was the only assumption made to obtain the approximate expressions, the material parameters are valid when the approximate and exact curves overlap. However, when the curves diverge, this indicates that it is no longer appropriate to define effective permittivity and permeability. IV. RESONANCES In NRI media, three important frequencies can be used to characterize the dispersion curve: the electric and magnetic plasma frequencies and the low-frequency backward-wave cutoff. The first two were introduced in the previous section, and both correspond to the propagation constant being equal to zero, while the third corresponds to the frequency at which the phase difference across the unit cell is 180 .

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Fig. 9. Schematic showing the magnetic plasma frequency resonance on the MTL system for on-axis propagation.

Fig. 10. The left half of the circuit shown in Fig. 9 after being simplified due to the shorting of the shunt elements.

In addition to deriving the plasma frequency equations (25) and (26) from the relative permeability and permittivity expressions, they can also be obtained from physical arguments combined with circuit analysis. The series inductors of an unloaded MTL system govern its magnetic response, and consequently, its permeability response. Altering the effective inductance of the transmission lines will affect the permeability of the medium. The effective inductance of a MTL system can be changed by the addition of a reactive series element. In Fig. 2, the inductance of the unloaded MTL system is changed using the series capacitor, . This results in a negative effective permeability for some frequencies, as shown in Fig. 8. At the magnetic plasma , the inductive elements of the unloaded MTL frequency, system will resonate with the loading capacitor such that the effective permeability is zero. When this occurs, the middle and both ends of the unit cell appear to be shorted to ground, as depicted in Fig. 9. This short-circuits the shunt elements, leaving only the series elements, as shown in Fig. 10. The effective permeability goes to zero when the series-only impedance of conductor 1 goes to zero. The series-only impedance of conductor 1, which accounts for the elements of conductor 2 through the , is given by mutual inductance,

(27) Setting (27) equal to zero and solving for as was given in (25). expression for

yields the same

A similar analysis can be applied to find the electric plasma . The shunt capacitors of the unloaded MTL frequency, system are responsible for determining the electric response, and therefore, the permittivity of the medium. Again, reactive and , are placed in shunt to loading elements, such as change the effective capacitance and, consequently, the effective permittivity. Fig. 8 shows that the introduction of these loading elements creates a negative permittivity over certain frequencies. At the electric plasma frequency, , the capacitive elements of the unloaded MTL system will resonate with the loading shunt inductors such that the effective permittivity is zero. In this case, the middle and both ends of the unit cell appear as open circuits (shown in Fig. 11), thereby eliminating the effects of the series elements. Without the series elements, the circuit appears as shown in Fig. 12. The effective permittivity goes to zero when the shunt-only impedance of conductor 1 to ground goes to infinity. The shunt-only impedance of conductor 1 to ground is given by the expression

(28) By setting the denominator equal to zero and solving for , two solutions are found, one of which is exactly the same as the exgiven by (26). The second solution corresponds pression for to a resonance for the -mode, which occurs at a frequency well above those of interest. The equation for the low-frequency backward-wave cutoff for in (6), which on-axis propagation is found by setting is the “X” point on the Brillouin diagram in Fig. 5. For this resonant condition, each end of the unit cell is shorted to ground, as shown in Fig. 13. Unfortunately, this results in a complicated expression with multiple solutions. Noting that the desired condition describes the lowest frequency in the backward-wave

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Fig. 11. Schematic showing the electric plasma frequency resonance on the MTL system for on-axis propagation.

Fig. 12. The left half of the circuit shown in Fig. 11 after being simplified due to the elimination of the series elements.

band, higher order frequency terms can be ignored, giving the following approximation

(29) It is interesting to note that this is the exact expression for the low-frequency cutoff of the backward-wave transmission line (conductor 2) in isolation. This result indicates that the coupling between conductor 1 and conductor 2 is insignificant near this cutoff frequency. Off-axis propagation can support propagation at even lower frequencies. The two-dimensional low-frequency backward-wave cutoff can be found by simply setting and in (7), which is the “M” point of the Brillouin diagram in Fig. 5. After applying the same approximations as were used to obtain (29) for the case of on-axis propagation, the equation for the two-dimensional low-frequency cutoff is given by

(30)

V. FINITE STRUCTURES The discussion in the previous sections pertains to the modal analysis of the infinite medium, however, in practice, NRI structures are finite. The simplified periodic analysis does not capture the electromagnetic effects that occur at the edges of a

finite structure because the analysis is derived using periodic boundary conditions. Such conditions do not accurately represent the unit cells at the edges of a finite slab. Unless the slab contains sufficient unit cells such that the edge effects are negligible, a different method of analysis is required to accurately model finite NRI structures. It was shown in [20], [21] that rigorous MTL analysis can be used to obtain accurate reflection and transmission data for a finite structure. As described in Section II, rigorous analysis requires that the individual transfer matrices of each constitutive element of the unit cell are found and then cascaded to produce the transfer matrix for the entire unit cell. Figs. 14 and 15 diagram this process for the structure studied in this paper. Fig. 14 breaks up the unit cell into five elements: a shunt inductor on conductor 2, a transmission line section, a shunt inductor on conductor 1 together with a series capacitor on conductor 2, a second transmission line section, and a second shunt inductor on conductor 2. The complete 4 4 matrices for these elements are depicted in Fig. 15, with the exception of the transmission line matrices, whose elements are given explicitly in [14, p. 24]. After these matrices are multiplied together, they form the transfer matrix for the entire unit cell. For a slab that is unit cells thick, the transfer matrix of a single unit cell should power to find the transfer matrix for the enbe raised to the tire slab. This final matrix is a 4 4 matrix of the form:

(31)

The reflection and transmission coefficients for a slab of the NRI medium are typically found for a free-space wave normally-incident on the structure. In our MTL system, this scenario can be modeled by connecting ports to the terminals of conductor 1 and terminating conductor 2 with an impedance that appropriately represents the physical situation. For example, if conductor 2 ends in an open circuit, the impedance applied to that terminal would be infinity, whereas if the conductor ends in a short circuit, the impedance would be zero. These terminations provide additional equations that allow the four-port MTL structure shown in Fig. 16(a) to be transformed into a simple

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Fig. 13. Schematic showing the low-frequency backward-wave cutoff resonance on the MTL system for one-dimensional propagation.

Fig. 14. Diagram showing how the MTL transfer matrices of the constitutive elements were cascaded to form a unit cell. 0 unloaded impedance and admittance matrices, respectively.

p

= ZY, where Z and Y are the

Fig. 15. Diagram depicting the transfer matrices of each constitutive element, which, when multiplied together, form the transfer matrix of the entire unit cell.

two-port network, as shown in Fig. 16(b). The equations that make this possible are (32) (33) is the impedance used to terminate conductor 2 at where each end of the NRI slab. By using a variable for the termination impedance, this method remains valid for any symmetric 4-port structure. Further, this method can model the fringe capacitance in the case of an open circuit or the via inductance in the case of a short circuit. Equations (32) and (33) are used as boundary conditions to reduce the 4 4 transfer matrix of (31) to the 2 2 matrix (34)

where

(35) (36) (37) (38) The finite structure consists of four of the unit cells shown in Fig. 14, with the second conductor being terminated with an additional inductor on either end of the slab. To account for

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Fig. 16. Transformation of the four-port MTL system to a two-port system that can be excited by a plane wave in free space. (a) Block diagram of the unterminated, finite MTL structure. (b) Block diagram of the terminated, finite MTL structure.

the added inductance, . To provide a direct comparison, the finite structure was also simulated in Ansoft’s HFSS. The full-wave model consisted of four of the unit cells shown in Fig. 1 in the direction of propagation. Perfect electric conductor boundary conditions were enforced on the top and bottom of each unit cell to achieve infinite periodicity in the vertical direction for the polarization of interest. Periodic boundary conditions with zero phase delay were applied to the sides of the four-cell slab to enforce infinite periodicity in the transverse direction. The reflection and transmission coefficients (equivalently S-parameters) of the four-cell slab were calculated for normal incidence and are compared with full-wave simulation results in Fig. 17(a) and (b). Good agreement is shown, except for the low-frequencies where there is a slight frequency shift. To provide further evidence that (34) is correct, the multiconductor circuit used in the MTL analysis was simulated using lumped-element and transmission-line components in Agilent’s Advanced Design System (a commercial microwave circuit simulator). The S-parameters from this simulation agree exactly with those obtained through MTL analysis. With this verification, the shift in the low-frequency values can be attributed to inaccuracies in the extraction of the loading elements or the capacitance and inductance matrices of the unloaded transmission line. VI. CONCLUSION This paper demonstrated that MTL analysis can be used to model volumetric, broadband NRI media. The two-dimensional dispersion equation, the modal impedances and the material parameters were all predicted using a simplified periodic analysis that provides simple expressions and intuition into the operation of the NRI structure, while still accounting for spatial dispersion. These parameters allow for the complete characterization

Fig. 17. Scattering parameters of a four-cell NRI slab. (a) Reflection coefficients calculated by full-wave analysis (HFSS), MTL analysis and a commercial circuit simulator (ADS). (b) Transmission coefficients calculated by full-wave analysis (HFSS), MTL analysis and a commercial circuit simulator (ADS).

of the electrical properties of the medium. Equations for the resonant points of the NRI medium were also derived, including the magnetic and electric plasma frequencies and the low-frequency backward-wave cutoff. The equations provided in this paper allow for NRI structures to be designed analytically and rapidly, rather than by full-wave simulation alone. This paper also detailed how finite structures can be modeled using a rigorous MTL analysis. Using this method, the effects of truncating the medium can be observed and the reflection and transmission coefficients calculated. This type of analysis is not specific to the topology presented in this paper. All of the techniques described in this paper can easily be extended to other volumetric metamaterials (such as the NRI medium in [21] or the electromagnetic bandgap structure in [16]) as well as four-port transmission-line components (such as directional couplers). In [19], coupled mode theory is used to analyze a microstrip coupler by predicting its dispersion equation and S-parameters. The homogenized MTL analysis presented in this paper allows for more accurate analysis since it accounts for the periodicity (spatial dispersion) of the structure. Lastly, the generality of derived expressions allows the analysis to accurately model any two-conductor system, providing insight into different geometries. REFERENCES [1] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001.

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[2] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of  and ,” Sov. Phys. Usp., vol. 10, pp. 509–514, Jan.–Feb. 1968. [3] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2075–2084, Nov. 1999. [4] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [5] C. Caloz and T. Itoh, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip ‘LH line’”,” in Proc. IEEE Antennas and Propagation Society Int. Symp., San Antonio, TX, June 16–21, 2002, vol. 2, pp. 412–415. [6] A. Grbic, “A 2-D composite medium exhibiting broadband negative permittivity and permeability,” in Proc. IEEE AP-S Int. Symp., Albuquerque, NM, Jul. 9–14, 2006, pp. 4133–4136. [7] S. M. Rudolph and A. Grbic, “A volumetric negative-refractive-index medium exhibiting broadband negative permeability,” J. Appl. Phys., vol. 102, p. 013904, Jul. 2007. [8] M. Stickel, F. Elek, J. Zhu, and G. V. Eleftheriades, “Volumetric negative-refractive-index metamaterials based upon the shunt-node transmission-line configuration,” J. Appl. Phys., vol. 102, p. 094903, Nov. 2007. [9] S. M. Rudolph and A. Grbic, “Super-resolution focusing using volumetric, broadband NRI media,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2963–2969, Sep. 2007. [10] G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials. New York: Wiley, 2005. [11] C. Caloz and T. Itoh, Electromagnetic Metamaterials. New York: Wiley, 2005. [12] P. Alitalo, O. Luukkonenand, and S. Tretyakov, “A three-dimensional backward-wave network matched with free space,” Phys. Lett. A, vol. 372, no. 15, pp. 2720–2723, April 2008. [13] S. M. Rudolph and A. Grbic, “A printed circuit implementation of a broadband volumetric negative-refractive-index medium,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Honolulu, HI, Jun. 10–15, 2007, pp. 1067–1070. [14] J. A. Brandão Faria, Multiconductor Transmission-Line Structures: Modal Analysis Techniques. New York: Wiley, 1993. [15] C. R. Paul, Analysis of Multiconductor Transmission Lines. New York: Wiley, 1994, 605 Third Avenue, 10158. [16] F. Elek and G. V. Eleftheriades, “Simple analytical dispersion equations for the shielded sievenpiper structure,” in Proc. IEEE MTT-S Int. Microwave Symp., San Fransisco, CA, Jun. 11–16, 2006, pp. 1651–1654. [17] C. R. Brewitt-Taylor and P. B. Johns, “On the construction and numerical solution of transmission-line and lumped network models of maxwell’s equations,” Int. J. Numer. Methods Eng., vol. 15, no. 1, pp. 13–30, 1980.

[18] K. C. Gupta, R. Garg, and I. J. Bahl, Microstrip Lines and Slotlines. Dedham, MA: Artech House, 1979. [19] R. Islam and G. V. Eleftheriades, “Printed high-directivity metamaterial MS/NRI coupled-line coupler for signal monitoring applications,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 4, pp. 164–166, Apr. 2006. [20] S. M. Rudolph and A. Grbic, “Modeling of volumetric negative-refractive-index media using multiconductor transmission-line analysis,” presented at the Applied Computational Electromagnetics Society Conf., Niagara Falls, CA, Mar. 30–Apr. 4 2008. [21] J. Zhu and G. Eleftheriades, “Fully printed volumetric negative-refractive-index transmission-line slabs using a stacked shunt-node topology,” in Proc. IEEE MTT-S Int. Microwave Symp., Atlanta, GA, Jun. 15–20, 2008, vol. 1, pp. 173–176.

Scott Michael Rudolph (S’06) received the BSE and MSE degrees in electrical engineering from the University of Michigan, Ann Arbor, in 2006 and 2008, respectively, where he is currently working toward the Ph.D. degree. His research interests include the design of volumetric metamaterials, frequency-selective surfaces and antennas. Mr. Rudolph received the IEEE Microwave Theory and Techniques Society Graduate Fellowship in 2009.

Anthony Grbic (S’00–M’06) received the B.A.Sc., M.A.Sc., and Ph.D. degrees in electrical engineering from the University of Toronto, ON, Canada, in 1998, 2000, and 2005, respectively. In January 2006, he joined the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, where he is currently an Assistant Professor. His research interests include engineered electromagnetic structures (metamaterials, electromagnetic bandgap materials, frequency selective surfaces), printed antennas, microwave circuits and analytical electromagnetics. Dr. Grbic received the Best Student Paper Award at the 2000 Antenna Technology and Applied Electromagnetics Symposium and an IEEE Microwave Theory and Techniques Society Graduate Fellowship in 2001. In 2008, he was the recipient of an AFOSR Young Investigator Award as well as an NSF Faculty Early Career Development Award. In January 2010, he was awarded a Presidential Early Career Award for Scientists and Engineers.

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Uniform Asymptotic Evaluation of Surface Integrals With Polygonal Integration Domains in Terms of UTD Transition Functions Giorgio Carluccio, Matteo Albani, Senior Member, IEEE, and Prabhakar H. Pathak, Fellow, IEEE

Abstract—The field scattered by a scattering body or by an aperture in the free space (or in an unbounded homogenous medium) can be described in terms of a double integral. In this paper we show how a canonical integral on a polygonal domain, with a constant amplitude function and a quadratic phase variation, can be exactly expressed in terms of special functions, namely Fresnel integrals and generalized Fresnel integrals. This exact reduction represents a paradigm for deriving a new asymptotic evaluation for a more general integral. This new asymptotic uniform integral evaluation is expressed in the format of the uniform geometrical theory of diffraction which is convenient for numerical computations. Index Terms—Asymptotic diffraction theory, geometrical theory of diffraction, scattering, uniform theory of diffraction (UTD).

I. INTRODUCTION S is well known, the field scattered by an object or by an aperture, which is surrounded by free space (or an unbounded homogenous medium) can be described in terms of a double integral of the form

A

(1) where is a slowly varying function and is a phase defining function, both depending on the 2D vector the pair of variables which parameterize the scattering surface over the integration domain ; also, is the wavenumber of the external medium. Typically, in these scattering problems, comprises the amplitude of the (equivalent) current over the scattering object, or aperture, and the amplitude of the Green’s comprises the local function (spreading factor), while phase of the (equivalent) current at any point on the radiating surface, and a measure of the optical path between that point and the observation point.

Manuscript received June 16, 2009; revised September 29, 2009; accepted October 11, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. G. Carluccio and M. Albani are with the Department of Information Engineering, University of Siena, 53100 Siena, Italy (e-mail: giorgio.carluccio@dii. unisi.it; [email protected]). P. H. Pathak is with the ElectroScience Laboratory, Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43212 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.2010.2041171

In [1] and [2], it is demonstrated that, when is large the dominant contributions to come from the neighborhood of some “critical” points located in the interior of or on its boundary . Various types of critical points are: a) stationary points in the interior of where the gradient of the phase function vanishes; b) partial stationary points on , i.e., points on the boundary of the integration domain where the tangential derivative of the phase vanishes; c) corner points, i.e., discontinuities ; d) stationary points of higher order of of the curvature of the phase function in the interior of or on its boundary (parabolic points or critical points associated with caustic phenomena); and e) integrable singularities associated with the amplitude function. Furthermore, for large, it is shown in [1], [2] that one obtains a complete asymptotic expansion of the double integral in (1), in terms of the critical point contributions, that can be expressed as series of inverse powers (both integer and fractional) in the wavenumber . Also, it is seen from [1], [2] that different types of critical points give rise to different contributions in inverse powers of . In particular, the contributions a) arising from a stationary point in the interior have an asymptotic order ; partial stationary points of b) on the boundary lead terms of asymptotic order ; ; while contributions given by corner points c) are of order etc. However, this asymptotic evaluation is valid for isolated critical points. Such an asymptotic evaluation becomes invalid when two or more critical points become close. For example, the latter occurs when an interior stationary point a) approaches , thus merging with a partial stationary point the boundary b). In those situations, the expansions given in [1] and [2] are said to be non-uniform. In [3] an integral of the form (1) is considered which contains a unique stationary point a) in the integration domain. They found an asymptotic expansion in which is the value of the amplitude function the first term of order at the stationary point multiplied by a double Fresnel integral, and the second term is a line integral whose asymptotic order to the order, according to the stamay vary from the tionary phase point location. Thus, the asymptotic order of the if the stationary phase point is isolated in second term is if it belongs to the boundary the integration domain, and is . More recently, the work in [4] also considers a unique stationary point a) in the domain, and provides a uniform asymptotic evaluation of in terms of the standard Fresnel integral and the Generalized Fresnel Integral [5]–[7]. Namely, various contributions arise from the stationary point a), the partial stationary points b) and the corner points c) of the integration domain ; and each contribution comprises the sum of two terms. The first

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is the respective term of the non-uniform asymptotic expansion of and the second contains an appropriate corrective transition function involving Fresnel or generalized Fresnel integrals that guarantees the uniformity of the solution. When the critical points are isolated, the contribution of the transition function terms is negligible with respect to the non-uniform terms. In contrast, when two or more critical points are close, the relevant transition functions provide the necessary correction to the non-uniform solution. Physically, when represents a radiation integral, as previously described, the critical point contributions represent specific ray field contributions; namely, the stationary phase point contribution a) corresponds to the geometrical optics (GO) ray field, the partial stationary point contributions b) represent the boundary (or edge) diffracted ray fields, while corner point contributions correspond to vertex diffracted ray fields. When critical points are isolated, the different ray field contributions lie outside their transition regions, and a non-uniform description can be given as that provided by the Geometrical Theory of Diffraction (GTD) [8]. When one or more of these critical points are close, the transition terms represent the contribution that arise from the influence of these nearby critical points. These additive contributions in [4] become significant within the transition regions associated with the relevant ray shadow boundaries, and smoothly vanish outside those transition regions. Thus a uniform solution is obtained in [4] that recovers the GTD solution outside the transition regions. That uniform formulation in [4] has a format analogous to the Uniform Asymptotic Theory of Diffraction (UAT) [9], because of the “additive” type transition terms that are present besides the non-uniform or GTD part. In this paper, we also consider an integral whose phase function contains only one stationary point a) as in [4]; moreover, we suppose that the integration domain has a polygonal form. For this type of integral we propose a new asymptotic solution that is expressed in the format of the Uniform Theory of Diffraction (UTD) [10]. Thus, in contrast to [4], we introduce multiplicative or factor type transition region correction terms to the non-uniform solution of the integral by using standard UTD transition functions involving the Fresnel integral and the Generalized Fresnel Integral. Indeed, to each ray field of the non-uniform GTD solution, arising from its relevant critical point, a multiplicative transition function correction term is included. Outside the transition regions, the value of these functions is unity and the GTD ray solution is recovered. In the shadow boundary transition regions, these functions serve to preserve the uniformity or continuity of the solution across the ray shadow boundaries; i.e., for example, at the GO shadow boundary, the boundary edge diffracted field and/or the corner diffracted field contributions compensate for the relevant GO discontinuities. Our formulation gives a compact UTD solution as compared to the previous one in [4] in the sense that no other terms need to be added to the non-uniform GTD solution to make it continuous; i.e., we simply introduce a multiplicative correction term to each ray optical contribution thereby retaining the simpler format of the UTD as opposed to the UAT format. This paper is organized as follows. In the second section, we consider a canonical integral on a polygonal domain with a constant amplitude function and a quadratic phase variation. We show that it can be exactly ex-

Fig. 1. Integration domain A, associated reference system and geometrical quantities. Equiphase lines (dotted) '(p) = const are also shown.

pressed in terms of special functions, namely Fresnel Integrals and Generalized Fresnel Integrals, and the final expression is cast in the UTD format. This exact reduction represents a paradigm for casting the asymptotic evaluation in a UTD format for a more general integral which presents a rapidly varying phase function and a slowly varying amplitude function at the critical points, as shown in the third section. In section four, we present some numerical results in order to show the effectiveness of our uniformly asymptotic (UTD) type solution. II. EXACT REDUCTION OF A CANONICAL DOUBLE INTEGRAL INTO UTD FORMAT TRANSITION FUNCTIONS A. Reduction of the Double Integral Into a Line Integral We start by considering a double canonical integral with an integrand that exhibits a quadratic phase on a polygonal domain , with boundary (Fig. 1), i.e. (2) where

(3) and are the values is a quadratic phase function. of the function and of its gradient, respectively, at a generic point in the integration domain , and is its constant symmetric Hessian matrix, defined as

(4) In this paper a bold letter denotes a vector, a hat denotes a unit vector and an underlined bold letter denotes a dyad. In what follows, the canonical integral (2) is reduced to a line integral on its boundary and then calculated in terms of special functions. This procedure can be considered an extension of Gordon’s formula [11], where the phase function is allowed only to be linear. The gradient of the phase function in (3) is (5)

CARLUCCIO et al.: UNIFORM ASYMPTOTIC EVALUATION OF SURFACE INTEGRALS WITH POLYGONAL INTEGRATION DOMAINS

hence, the integrand of (2) presents a unique stationary phase , that may be a minimum, point at a maximum, or a saddle point for the phase function . The stationary phase point may lie inside or outside . To accomplish the above mentioned line integral reduction, we introduce the vector potentials

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. The expression of for asymptotic contribution of order two cases can be combined by introducing a unit step function defined by if , or otherwise. Next, referring to Fig. 1, when the integration domain is a side , the line intepolygonal domain with vertices can be decomposed into the sum of gral on its boundary the integrals along each th side. Accordingly, (7) is rearranged as

(6) for , with . Note that , whereas , everywhere except at . One notes and are singular at , and their divergence is not that defined. Nevertheless, it is easy to verify that everywhere, including where is a regular vanishing function, because the two singularities cancel each other in the difference. Hence, by applying the divergence (Gauss) theorem for the bi-dimensional case, the integral in (2) is rewritten as (7) where contour sense.

is the outgoing unit vector normal to integration which is covered in the positive counterclockwise

B. Stationary Phase Point Contribution and Boundary Line Integral By integrating separately the two vector potentials in (7), is split into the sum of two analogous integrals involving and , respectively; namely, . Since everywhere except at , it follows that, if , then . , the integration contour of can be Conversely, if arbitrarily reduced to a small circle around with radius (see Fig. 1). Then, by mapping the circular contour as , and its normal unit vector as , in the limit as one obtains

(10) with

(11) where is the outgoing unit vector normal to the th side. is re-tagged For compactness of notation, the first vertex as when considered as final end point in the integration along the th side (see Fig. 1). By using (6) in (11), any is found to be a line integral where a quadratic polynomial appears both in the phase and in the denominator, corresponding observed on the straight line through and . to , Namely, we introduce the parametrization ; i.e., with and denoting the with th side length and tangent unit vector, respectively. Then, (11) is rewritten as (12) where (13) in which the derivatives of the phase function along the direction of the th side are given by and . The quadratic , which function (13) exhibits a partial stationary point at and may lie on the th side (i.e., between the two vertices ) or outside it, and it is given by

(8) (14) Introducing the change of variable lated in closed form as

, (8) can be calcu-

Then, the change of variable one to rewrite (11) as

leads

(9) in which if the eigenvalues of the Hessian matrix are both positive, if they are both negative, and if they origihave opposite sign, as shown in [4]. The contribution nates at the stationary point and, when present, is the leading

(15) in which the branch of the square root is chosen so that . In the next subsection, (15) will be calculated in terms of special functions whose

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arguments are square root of the phase differences between critical points

where

denotes the UTD Fresnel transition function [10]

(23) (16) and

(17) with

. To simplify (16), we used .

C. Representation of the Line Integral in Terms of Fresnel Integrals As mentioned above, the line integral along the th side (15) is exactly decomposed into its various asymptotic constituents and expressed as

which is the special function associated to partial stationary (edge) points and which uniformly describes the coalescence of the partially stationary point with a stationary point. Note that (21) contributes to (18) only when the partial stationary lies on the th side between the two vertices and ; inin (18) is defined as deed in such a case the unit step function . On the other hand, when there is no partial stationary point on the th side of the polygon . When ; and the the partial i.e., outside the transition region, exhibits an asymptotic order stationary point contribution . However, when ; i.e. in the transition region occurring when approaches , and (21) to provide continuity. properly grows to the order Finally, by using (18) in (10) the closed form expression for the canonical integral (2) is achieved

(24) (18) In (18)

(19) is the end-point contribution at the vertex and denotes the generalized Fresnel integral [7] defined as

where the contribution associated to the th vertex comprises the sum of the two line integral endpoints and arising from the integration on the th and th side side, respectively. For compactness of notation, the of the polygon coincides with the th side. Note that the arguments of the two generalized Fresnel integrals involved in and are not independent because it holds

(25) (20)

when , and otherwise. The generalized Fresnel integral is the special function asymptotically associated with a vertex type critical point which permits a uniform description of the coalescence between such or even a a vertex point and a partial stationary point . stationary point Furthermore, in (18) (21) is the partial stationary point contribution at using

Therefore it is convenient to introduce the parameter

(26)

by which one can express the parameters and as of

and

in terms

(27)

obtained by and define the two dimensional UTD transition function [6]

(22)

(28)

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So far, the vertex contributions in (24) are given by

there. Also note that therefore verify that

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has exactly the same form of and everywhere except at . It is easy to

(29) Outside the transition regions, i.e., when , and the vertex contributions are of asymptotic order . In simple transition, when either or ; i.e., when either or approaches , then the terms in the small argument bounded and continuous have expansion of which keep or , rethe form . In the double spectively, so that (29) grows to the order and vanish transition region, when both simultaneously, i.e., when , and in turn all , then approach the th vertex and the vertex . contribution (29) grows up to the order Summarizing, the canonical integral (2) is reduced to the closed form expression (24) with (9), (21) and (29), where each term is cast in a UTD format by resorting to suitable UTD transition functions which properly describes the transitional behavior of the contributions inside the associated transitions, while tend to unity out of the transitions. This result is used in the following section as a paradigm for the uniform asymptotic approximation of a class surface integrals to arrive at a UTD type format for the resulting solution. III. GENERALIZATION TO THE ASYMPTOTIC EVALUATION OF DOUBLE INTEGRALS ON POLYGONAL DOMAINS Let us now consider a general integral of the type (1), where and are analytical functions of the we suppose that on the entire polygonal domain. We also variables exhibits a unique stationary phase point suppose that where . Therefore the phase function exhibits a shape similar to that of the quadratic function in (3) and the integral (1) is asymptotically dominated by the same kind of critical points as those in (2). Namely, a unique stationary phase point that may or may not belong to , partial stationary points on the boundary and corner points at the vertices of the polygonal domain . The uniform asymptotic reduction of (1) to the sum of its asymptotic constituents relevant to the various critical points follows the same procedure as that outlined in the previous section for the exact reduction of (2) to (24). Specifically, we introduce the vector potentials

(32) is a smooth, slowly varying function everywhere. where Indeed, on the right hand side of (32), the leading term is equal to the integrand of (1), and arises from the differentiation of the exponential function in (30), whereas the differentiation of the other factors leads to higher order terms containing the exponential function as a common factor. The regularity everywhere and ensures the regularity of also at of . Hence, by using (32) in (1) yields

(33) is analogous to defined in (1) except that it inin which instead of . Since and share the same phase volves function they also have the same critical points which asymptotically dominate both the integrals, and provide analogous contributions of the same asymptotic order. However, because of factor in front of in (33), the leading asymptotic conthe tribution of at each critical point can be asymptotically calcuand by considering just the first integral lated by neglecting term in (33). By invoking the divergence theorem, such a surface integral is converted into a line integral along the rim of the integration domain. Following the same outline as in the on results in the previous section, the line integral of contribution at the stationary point , whereas the line integral is decomposed into the sum of line integrals along of each edge of . Each line integral is then evaluated asymptotically as the sum of the contributions at its three critical points: a partial stationary point along the edge (if present) and the two end points at the vertices. The asymptotic evaluations are conducted in a uniform fashion by introducing the same transition functions appearing in the exact reduction in the previous section, and are based on a local quadratic expansion of the phase function at each critical point. The final result is cast in the typical UTD format as follows

(34) where the stationary phase point contribution is

(30) (35) and the partial stationary point contribution at the

th edge is

(31) and are Analogously to the previous exact case, while singular at , is both regular and differentiable

(36)

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and the contribution at the

th vertex is

(37) In (34), the stationary phase point contribution (35) may or may lies inside or outside not be present depending on whether , as expressed by or 0, respectively. Analogously, each th stationary phase point contribution (36) may or may lies on the th edge of or not, and not be present if or 0, respectively. Also, in (36) the argument of the UTD transition function is defined as

(38) Finally, in (34) vertex contributions (37) are always present and the arguments of their transition function are defined as

Fig. 2. Integration domain for the integral I . The dash-dotted line represents the trajectory of the stationary phase point of the quadratic phase ' from the initial value p to the final point p .

(39) for

; and

(40)

Outside the transition regions, i.e., when critical points are well separated, all transition functions tend to unity and the asymptotic order of each contribution is shown explicitly in (35)–(37). When two or more critical points merge and some of them appear/disappear, the associated transition functions uniformly describe the proper contributions within their shadow boundary transition regions, in the same way as described in the previous section for the exact case. Note that, unlike the exact closed form reduction (24) which is always uniform, the asymptotic formula (34) may suffer from singularities, when second order derivatives of the phase vanish. In a ray picture, such singularities corresponds to caustics of GO or diffracted rays. The uniform description of such caustics, which are not very frequent in standard applications because they occur at specific observation points at a precise distance from the scattering surface, is beyond the purpose of this paper. Here we provide a uniform description across various shadow boundaries that are observed at any distance and therefore commonly encountered in the observation scans. IV. NUMERICAL RESULTS In this section we present some numerical results to show the effectiveness of the proposed formulation. First we consider an integral for which the quadratic phase function is equal to , where the Hessian matrix , , , and coefficients are

. Hence, is convex; accordingly, its stationary phase is a minimum and the equiphase lines, , point domain is a triangle whose are ellipses. The integration , , and , corners are as shown in Fig. 2. We move the stationary phase point along up to a oriented line (dash-dotted line) from . In Fig. 3, the contributions relevant to the different kind of critical points are separately plotted: stationary point contribution (9) (dashed line), sum of the partial stationary point contributions (21) (dash-dotted line), sum of the corner , only point contributions (29) (dotted line). When corner point contributions are present in the evaluation of the the corner point contributions comintegral . Next, at pensate for the discontinuity of a partial stationary point contriside which abruptly appears. When bution on the approaches , , a double transition is present. Indeed, the stationary phase point and other two partial staat , 3 appear, and the associated contritionary phase points butions exhibit jump discontinuities. The corner point contribucompensates for these discontinuities. Finally, tion from leaves the integration domain across the triangle side; consequently, the stationary phase point contribution abruptly disappears but its discontinuity is smoothed by the contribution associated to the partial stationary phase point . Across all these transitions and discontinuities of individual contributions, the total sum (24) of all the contributions (continuous line) is always smooth. To check the exactness of the representation (24) in terms of Fresnel and generalized Fresnel functions, we also report the numerical evaluation of the integral (circles) that perfectly coincides, within the numerical precision. In the second example, we consider an integral with a quadratic phase function, whose stationary point is a saddle point; thereby equiphase line are hyperbolas. Specifically, , we consider an hessian matrix with coefficients

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Fig. 3. Contributions of different critical points and comparison between the exact closed form reduction and the numerical evaluation of the integral I , for the example in Fig. 2 when the phase function ' exhibits a minimum. Stationary phase point contribution (dashed line), sum of partial stationary phase point contributions (dash-dotted line), sum of corners contributions (dotted line), sum of all different contributions (continuous line), numerical evaluation (circles).

Fig. 4. Contributions of different critical points and comparison between the exact closed form reduction and the numerical evaluation of the integral I , for the example in Fig. 2 when the phase function ' exhibits a saddle point. Stationary phase point contribution (dashed line), sum of partial stationary phase point contributions (dash-dotted line), sum of corners contributions (dotted line), sum of all different contributions (continuous line), numerical evaluation (circles).

, . The integration domain and the is the same of the trajectory of the stationary phase point previous example (Fig. 2). In this case, the second order partial derivatives of the phase function vanish in the directions and . Along these directions, reduces to a linear phasing, and the partial stationary phase point along these directions does not exist. side of the integration domain Since the direction of the , then , and only the partial stais relevant to the , 3 sides of the tionary phase points integration domain can contribute to the integral , when they belong to the domain boundary. Like in the previous example, the contributions relevant to the different kind of critical points are separately plotted in Fig. 4, with the same respective line , both corner and partial stationary phase styles. When point contributions are present in the evaluation of the integral . When approaches , , again a double transition is present but different to that of the previous example. Indeed, at the stationary phase point contribution is discontinuous, enters into the integration domain, and the partial since is also stationary phase point contribution associated to side no partial stationary discontinuous, but on the phase point occurs because of the linear phasing along this side. Consequently, beside the contribution associated to experiences a transition corners , also that associated to , to restore the continuity of the total solution. Next, at the partial stationary point merges the corner, and moves out of the integration domain , so that the sum of the partial stationary phase point contributions presents a discontinuity which is compensated by the contribution at that corner. leaves the inteFinally, as in the previous example, when passing across the side; the gration domain contribution of the stationary phase point abruptly disappears and the partial stationary phase point contribution at

compensates for this discontinuity. Also in this example, the various transitional discontinuity compensation mechanisms are verified by the continuity of the total sum (24) of all the contributions (continuous line), which again exactly fits with the numerical evaluation of (circles). The asymptotic method presented in this paper is applicable to any generic integral that satisfies the hypotheses mentioned previously. In order to show a possible practical application we use the present formulation to uniformly evaluate a physical optics (PO) integral. Referring to Fig. 5, let us consider a smooth convex parabolic surface illuminated by a oriented electric , with a unit momentum. The dipole located at parabolic surface is parameterized directly by the and coorand ) while the surface height coordidinates (i.e., nate is given by the parabola equation . The surface is trimmed on a rectangular domain with corplane) , , ners (in the , and ; hence, the scattering curved surface has a curvilinear quadrilateral boundary with curved edges and corners. Fig. 6 shows the PO scattered field at an obradius circle, with center at the servation point that scans a , origin of the reference system and an azimuth angle while the elevation angle ranges from to 180 . In the scattered field is the region above the surface mainly due by the reflected GO contribution, whereas below by the negative of the direct GO the surface field, which cancels the incident field in the shadow region. The interference of the edge diffracted contributions (dash-dotted line) with the dominating GO field (dashed line) creates ripples. and reflection When observing at the Incidence shadow boundaries, the stationary phase point merges the partial stationary phase point on the side, and in turn the transmission/reflection point on the scatedge. tering surface merges the diffraction point at its

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verified by the smoothness of the total field (continuous line) across various transitions. Its accuracy is highlighted by the comparison against the numerical evaluation of the same PO integral (circles). The two lines overlaps except at the minimum where the difference is about field grazing aspects 1 dB. However, when the surface is electrically large, the numerical integration becomes very time consuming, conversely the UTD approach remains really efficient, since size independent and calculated by closed form expressions. V. CONCLUSION

Fig. 5. Geometry relevant to the example of the scattering of a smooth convex parabolic surface illuminated by an electric dipole.

In this paper an exact closed form reduction was presented for a canonical surface integral with a quadratic phase function. The final expression involves Fresnel and generalized Fresnel functions and it is cast in the typical UTD format. This result is then used to provide a general algorithm for the uniform asymptotic evaluation of surface integrals in the framework of the UTD. In the next future the authors will apply the present result to the rigorous calculation of UTD edge and vertex diffraction coefficients for curved surfaces. Another future work will deal with the use of the same canonical integral to develop a quadrature rule for the efficient numerical calculation of radiation-type integrals. REFERENCES

Fig. 6. Contributions of different critical points and comparison between the asymptotic uniform evaluation of the PO integral J and its numerical evaluation. Stationary phase point contribution (dashed line), sum of partial stationary phase point contributions (dash-dotted line), sum of corners contributions (dotted line), sum of all different contributions (continuous line), numerical evaluation (circles).

Here a standard UTD edge transition and discontinuity compenthe diffraction sation mechanism occurs. In addition, at merges the corner and disappears up point at edge where it appears again at the same corner. At these to aspects, the vertex transition occurs and the vertex contribution (dotted line) compensates for the edge contribution discontinuity. On the opposite side of the scan, at the Incidence and reflections shadow boundaries, the GO and edge diffracted rays disappear simultaneously, while the transmission/reflection point and the two edge diffraction and edges all merges the points on the corner. Accordingly, the contribution associated to the corner experiences the double transition , thus providing the total field continuity. The uniformity of our closed form asymptotic representation (34) of the PO integral is

[1] D. S. Jones and M. Kline, “Asymptotic expansion of multiple integrals and the method of stationary phase,” J. Math. Phys., vol. 37, pp. 1–28, 1958. [2] N. Chako, “Asymptotic expansions of double and multiple integral,” J. Inst. Math. Applicat., vol. 1, no. 4, pp. 372–422, 1965. [3] N. Bleinstein and R. Handelsman, “Uniform asymptotic expansions of double integrals,” J. Math. Anal. Appl., vol. 27, pp. 434–453, 1969. [4] V. A. Borovikov, “Uniform stationary phase method,” IEE Electromagnetic Waves, 1994, Inst. Elect. Eng.. [5] D. S. Jones, “A uniform asymptotic expansion for a certain double integral,” Proc. R. Soc. Edinburgh Ser., vol. 69, no. 15, pp. 205–226, 1971. [6] K. C. Hill, “A UTD solution to the EM scattering by the vertex of a perfectly conducting plane angular sector,” Ph.D. dissertation, Dept. Elect. Eng., Ohio State Univ., Columbus, 1990. [7] F. Capolino and S. Maci, “Simplified closed-form expressions for computing the generalized Fresnel integral and their application to vertex diffraction,” Microw. Opt. Tech. Lett., vol. 9, no. 1, pp. 32–37, May 1995. [8] J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am., vol. 52, no. 2, pp. 116–130, Feb. 1962. [9] S. W. Lee and G. A. Deschamps, “A uniform asymptotic theory of EM diffraction by a curved wedge,” IEEE Trans. Antennas Propag., vol. AP-24, no. 1, pp. 25–34, Jan. 1976. [10] R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE, vol. 62, no. 11, pp. 1448–1461, Nov. 1974. [11] W. B. Gordon, “Far-field approximations to the Kirchhoff-Helmholtz representations of scattered fields,” IEEE Trans. Antennas Propag., vol. AP-23, no. 4, pp. 590–592, July 1975. Giorgio Carluccio was born in 1979 and grew up in Ortelle, Lecce, Italy. He received the Laurea degree in telecommunications engineering from the University of Siena, Siena, Italy, in 2006, where he is currently working toward the Ph.D. degree. From October 2008 to March 2009, he was an invited Visiting Scholar within the ElectroScience Laboratory, Department of Electrical and Computer Engineering, The Ohio State University, Columbus. His research interests are focused on asymptotic high-frequency methods for electromagnetic scattering and propagation, complex source and Gaussian beam electromagnetic field diffraction.

CARLUCCIO et al.: UNIFORM ASYMPTOTIC EVALUATION OF SURFACE INTEGRALS WITH POLYGONAL INTEGRATION DOMAINS

Matteo Albani (M’93–SM’98) received the Laurea degree in electronic engineering (1994) and the Ph.D. degree in telecommunications engineering (1999) from the University of Florence, Italy. He is an Adjunct Professor in the Information engineering Department, University of Siena, Italy, where he is also Director of the Applied Electromagnetics Lab. His research interests are in the areas of high-frequency methods for electromagnetic scattering and propagation, numerical methods for array antennas, antenna analysis and design. Dr. Albani was awarded the “Giorgio Barzilai” prize for the Best Young Scientist paper at the Italian National Conference on Electromagnetics in 2002 (XIV RiNEm).

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Prabhakar H. Pathak (F’86) received the Ph.D. degree from The Ohio State University, Columbus, in 1973. Currently, he is a Professor Emeritus at The Ohio State University where his main area of research is in the development of uniform asymptotic theories (frequency and time domain) and hybrid methods for the analysis of electrically large electromagnetic (EM) antenna and scattering problems of engineering interest. He is regarded as a co-contributor to the development of the uniform geometrical theory of diffraction (UTD). Presently, he is developing new UTD ray solutions, for predicting the performance of antennas near, on, or embedded in, thin material/metamaterial coated metallic surfaces. Recently his work has also been involved with the development of new and fast hybrid asymptotic/numerical methods for the analysis/design of very large conformal phased array antennas for airborne/spaceborne and other applications. In addition, he is working on the investigation and development of Gaussian beam summation methods for a novel and efficient analysis of a class of large modern radiation and scattering problems including the analysis/synthesis of very large spaceborne reflector antenna systems. He has published over a 100 journal and conference papers, as well as authored/coauthored chapters for seven books. Prof. Pathak has presented several short courses and invited lectures both in the U.S. and abroad. He has often chaired and organized technical sessions at national and international conferences. He was invited to serve as an IEEE Distinguished Lecturer from 1991 through 1993. He also served as the chair of the IEEE Antennas and Propagation Distinguished Lecturer Program during 1999–2005. Prior to 1993, he served as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION for two consecutive terms. He received the 1996 Schelkunoff best paper award from the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He received the George Sinclair award in 1996 for his research contributions to the O.S.U. ElectroScience Laboratory, and the Lumley Research Award in 1990, 1994 and 1998 from the O.S.U. College of Engineering. In July 2000, he received the IEEE Third Millennium Medal from the Antennas and Propagation Society. He was elected an IEEE Fellow in 1986, and is an elected member of US Commission B of the International Union of Radio Science (URSI). Currently he is serving as an elected member of the IEEE Administrative Committee (AdCom) for the Antennas and Propagation Society.

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Non-Orthogonal Domain Parabolic Equation and Its Tilted Gaussian Beam Solutions Yakir Hadad and Timor Melamed

Abstract—A non-orthogonal coordinate system which is a priori matched to localized initial field distributions for time-harmonic wave propagation is presented. Applying, in addition, a rigorous paraxial-asymptotic approximation, results in a novel parabolic wave equation for beam-type field propagation in 3D homogeneous media. Localized solutions to this equation that exactly match linearly-phased Gaussian aperture distributions are termed tilted Gaussian beams. These beams serve as the building blocks for various beam-type expansion schemes. Application of the scalar waveobjects to electromagnetic field beam-type expansion, as well as reflection and transmission of these waveobjects by planar velocity (dielectric) discontinuity are presented. A numerical example which demonstrates the enhanced accuracy of the tilted Gaussian beams over the conventional ones concludes the paper. Index Terms—Electromagnetic propagation, electromagnetic theory, Gaussian beams, propagation.

I. INTRODUCTION AND STATEMENT OF THE PROBLEM HE parabolic wave equation (PWE) models propagation of linear waves which is predominant in one direction. In this paper we refer to this direction as the “paraxial direction.” Under this assumption, as well as other asymptotic considerations, the Helmholtz equation is reduced to a much simpler form of a PWE [1]–[3]. Solutions of the PWE are subject to boundary conditions which are obtained by matching the field initial distribution on a given surface to the PWE near-field model. For problems in which the aperture distribution is given over a surface that is transverse to the initial paraxial direction, it may be conveniently matched to the PWE. Thus the field can be propagated in a step-by-step manner, by evaluating the field in each step from its distribution in the previous one. The concept of PWE was first introduced with relation to radio propagation over realistic earth surfaces [4], followed by its generalization which combined it with the well-known ray coordinates of geometric optics that resulted in a new diffraction theory [5]. Mathematical refinements and further developments of the PWE method have been achieved in [6]–[11], along with some more recent work [12]–[17]. Important solutions of the PWE include its different beam-type waveobjects [18]–[23]. PWE methods may also be utilized for solving beam-type waveobjects propagation

T

Manuscript received April 05, 2009; revised September 29, 2009; accepted October 22, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. The authors are with the Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2041161

in generic media profiles such as inhomogeneous [24]–[26], anisotropic [27]–[31], and time-dependent pulsed beams in dispersive media [32]–[36]. The need for these solutions arises from beam-type expansions such as Gabor-based expansions [37], [38] or the frame-based field expansion [39], [40]. The latter utilizes the key feature of the beam’s continuous spectrum [38], [41], [42], and discretized the spectral representation with no loss of essential data for reconstruction. A theoretical overview of frame-based representation of scalar time-harmonic fields was presented in [39], with an extension to electromagnetic fields in [43], and for time-dependent scalar fields in [40]. Exact beam-type expansions require beam solutions that match localized aperture planar distributions. In these solutions, as well as in other different propagation scenarios, the boundary plane over which the initial field distribution is given is generally not perpendicular to the paraxial direction of propagation. Therefore, in order to use conventional (orthogonal coordinates) Gaussian beams, apart from asymptotic approximations, an additional approximation is carried out to project the initial field complex curvature matrix on a plane normal to the beam axis direction. This additional approximation reduces the accuracy of the resulting beam solutions especially for large angle departures and, moreover, it becomes inconsistent with respect to asymptotic orders. The need for the additional approximation may be avoided by applying a non-orthogonal coordinates system. This work is concerned with obtaining a novel form of PWE in non-orthogonal coordinates such that its beam-type solutions are matched exactly to linearly-phased Gaussian distributions over a tilted plane with respect to the beam-axis. We seek for asymptotically-exact Gaussian beam (GB) solutions , to the 3D scalar Helmholtz equation

(1) in the half-space, where is the homogeneous being the medium wave-speed medium wavenumber with and a time-harmonic scalar field with suppressed time-de. In (1), is the convenpendence of denoting tional Cartesian coordinate frame with the transverse coordinates (see Fig. 1). Another objective is to obtain GB which can serve as the building blocks for beam-type expansion schemes of scalar [39], as well as electromagnetic [43] fields. These expansion schemes decompose the field over a spatial-directional (spectral) lattice according to spatial and directional spectral variables (see (37)). The propagating elements

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termed tilted GBs). In Section V, we apply the scalar tilted GBs to beam-type expansion of electromagnetic waves. The reflection and transfer of these novel waveobjects through planar velocity discontinuity is discussed in Section VI. Finally, in Section VII a numerical example which demonstrates the enhanced accuracy of the tilted GBs over the conventional (orthogonal coordinate) ones, is given. II. THE COORDINATE SYSTEM

=

Fig. 1. Non-orthogonal local beam coordinate system. Observation point r x ; x ; z where z is the origin location along the paraxial propagation direction, z , and the transverse coordinates, x and x , lie on a plane parallel to the x ; x plane. Thus, the transverse coordinates are tilted with respect to the propagation direction z . The phase term in (13) is accumulated according to the (perpendicular) optic length (Eikonal) s.

(

) ^ ) (

A. System Definition We define a non-orthogonal local coordinate system as a system in which the transverse beam coordinates, and , , lie on a plane parallel to the initial distribution plane at whereas the longitudinal coordinate, , is directed along the and to be orthogonal and tilted beam-axis. We chose moreover, parallel to and axes, respectively (see Fig. 1). and form angles of and with the Coordinates coordinate, respectively. It is assumed that angles and remain constant for all observation points. In view of (2), they are related to the beam-expansion directional variables in (2) via (3) as the angle between the -axis and the paraxial Denoting propagation direction , the unit-vector in the direction of (in terms of the coordinates) is given by

 0

Fig. 2. The fields in z are evaluated by a superposition of shifted tilted EM GB propagators which emanate from the initial distribution plane over the discrete spatial-directional lattice in (37). Each beam propagator emanates from a lattice point x ; x m x ; m x , in a direction of #  with respect to the corresponding x -axis.

= cos (  )

(  ) = ( 1

1 )

are GBs which are identified by their aperture planar field displane of the form [39], [42] tributions over the

(4) where, here and henceforth, hat over a vector denotes a unit-vector. Using these definitions, the -axis is identified as the paraxial propagation direction, and the transverse coordiand , lie on a plane parallel to the plane nates, plane with and are centered at the intersection of the the -axis. Therefore, the transformation from to is given by

(2) (5) where are the expansion (directional) spectral vari. Additional details regarding electroables and magnetic beam-type expansion and its spectral variables are given in Section V. Throughout this work, all vectors are column denotes the matrix (or the vector) vectors and superscript . transpose, so that the linear phase-term is a 2 2 complex symmetric matrix with a negative In (2), definite imaginary part. Here and henceforth, bold minuscule letters are used to denote vectors, whereas bold capital letters are used to denote matrices. Note that (2) consists of a Gaussian distribution with a linear phase term that causes the beam to tilt plane (see (27) as well as Fig. 2). with respect to the initial The paper layout is as follows. In the next section, we introduce a non-orthogonal coordinate system which is a priori matched to the field aperture distribution. In Section III, this coordinate system is utilized to obtain a novel (non-orthogonal) PWE whose localized solutions are obtained in Section IV (and

and the observation vector

is given by (6)

, and are the unit-vectors in the direction of the where , and axes, respectively, i.e., , , whereas is given in (4). B. Metric Coefficients The three so-called unitary vectors of the non-orthogonal system in (5), , are given by [44]

(7) elements of the 3 3 metric coefficients tensor and the are given by . By inserting (6) with (4) into (7),

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we obtain the metric tensor of the non-orthogonal coordinate system

(12)

(8) (15) its inverse (9)

and so-forth for all partial derivatives in (12). Next, in the following derivation, following conventional paraxial ray-theory [3], [24], [25], we assume that the transverse coordinates, and , are on the order of (this assumption is justified by (27)). Under this assumption

and its determinant

(16) (10)

III. THE NON-ORTHOGONAL DOMAIN PARABOLIC EQUATION

By inserting (15) with (16) into (12), we find that the and order elements cancel out, so that the parabolic equation is obtained by setting the next higher order, the -coefficient, to zero. This procedure yields

The Laplacian operator in a general non-orthogonal coordiis given by [44] nate system (17) (11) are the coordinate where is given in (10) and system metric coefficients in (9). By substituting (11) with (9) and (10) into (1), we obtain the Helmholtz equation in the nonorthogonal coordinates

Equation (17) is termed the Non-orthogonal Domain parabolic equation (NoDope). Note, that by setting (namely ) in (17), the latter reduces to the wellknown PWE in orthogonal coordinates [3], [24], [25]. IV. TILTED GAUSSIAN BEAMS A. Field Solutions

(12) where coordinate subscripts denote partial derivatives with re, spect to the coordinates, i.e., , etc. We are concerned with asymptotically evaluating the field that satisfies the 3D Helmholtz equation (12) with boundary condition (2). High-frequency wave-fields propagate along predominant directions (ray trajectories). Thus, solutions of the Helmholtz equation at an arbitrary observation point close to a ray trajectory, can be evaluated asymptotically by solving the PWE along the trajectory. By referring to boundary condition (2), we identify in (4) as the initial direction of the on the ray path which emanates from point plane. Therefore, we assume here that the wave field has the following high-frequency form

In view of the initial field distribution in (2), we are seeking localized beam solutions of the NoDope which carry Gaussian decay away from their axes. Therefore, we assume a beamtype field of the form (18) where and the so-called complex curvature denoting its matrix is a complex symmetrical matrix with th element so that the exponent in (18) is of the quadratic form . The matrix has a negative definite imaginary part, hence beam-field (18) exhibits a Gaussian decay over a plane that is tilted by with respect to the beam-axis direction . Beam-fields of form (18) (with (13)) which carry initial plane are termed Gaussian distributions over the tilted here tilted GBs. Using beam-field (18), we may now evaluate

(13) where the Eikonal (14) is the projection of the observation vector on the direction of the beam-axis , and denotes the ray-field amplitude. Using the ray-field (13), (14), we can evaluate the derivatives in

(19)

HADAD AND MELAMED: NON-ORTHOGONAL DOMAIN PARABOLIC EQUATION AND ITS TILTED GAUSSIAN BEAM SOLUTIONS

where the prime denotes a derivative with respect to the arpartial derivatives in (19) into gument. Next, we insert all and NoDope (17) and collect elements of the same order in , which results in

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Using a straightforward separation of variables, we obtain (26) The tilted GB field may now be written explicitly by using (24) and (26) in (18), and inserting into (13), which yields

(27)

(20) Relation (20) holds for all near the beam-axis, and therefore, consists of four equations which are obtained by setting all four coefficients to zero. These equations may be written in a compact way by defining

and is given where is given in (14), , we verify that GB (27) satisin (24). By setting fies the aperture field distribution in (2) exactly. This type of GB waveobjects exhibits frequency-independent collimation (Rayleigh) distance and therefore have been termed iso-diffracting [45] (see also (34)). The iso-diffracting feature makes these waveobjects highly suitable for UWB radiation representations [31], [33], [34], [39]–[41], [46]. B. Field Parametrization

(21) The first three equations yield a vector Riccati-type equation for (22) whereas setting the last (free) term in (20) to zero yields (23) Equation (23) serves as the amplitude equation once the Ricatti equation (22) is solved for . The procedure of solving the Ricatti equation is straightforward in that by setting , relation (22) transforms into . Thus

Applying radially-symmetric Gaussian windows to some aperture field distribution results in a beam-type expansion in which the propagating waveobjects are iso-axial tilted GBs. Such GBs are characterized by a diagonal initial complex curvature matrix of the form [39], [42], [43] (28) where denotes the 2 2 unity matrix. The complex curvature matrix is obtained by using (28) in (24), which yields (29) at bottom of page. The real and imaginary parts of the complex iso-axial curvature matrix may be diagonalized simultaneously by rotating the -axes over constant -planes, by a -independent angle , where (30)

(24) is the complex curvature matrix of the initial field where plane in (2). The beam amplidistribution over the , may now be found by inserting tude, , as well as (24), into (23), which results in the linear ODE (25)

and with . i.e., is identified as the angle between the -axis and the Angle plane. The resulting projection of the -axis on the rotation transformation of to the transverse coordinates in which is a diagonal matrix is given by (31)

(29)

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Thus, the quadratic phase in (27) in the by

coordinates, is given

and denoting the unit-cell dimenwith and coordinates, respectively. The sions in the is used to tag the lattice points (see index Fig. 2). These unit-cell dimensions satisfy

(32) (38) and, by using (29) with (31) in (32), we obtain (33) where and are given in (28). Using (33) with (32) in (27), we find that the iso-axial tilted directions GB exhibits (pure) quadratic decay in the beam-widths of with corresponding

are the overcompleteness parameters in the where axes. The initial field distribution is expanded using a spatially-lo, whereas the expansion coefficalized synthesis window, cients are obtained by evaluating the inner product of the ini. tial distribution with the corresponding analysis window Several methods for evaluating the analysis window from the synthesis window are provided in [39]. Following [43], the coefficients vector of the initial electric field is given by

(34) (39) where (35) are the principal beam-widths at the and and as the beam waists. By using (34), we identify collimation-lengths and waist-locations on the prinplanes, the beam cipal planes, respectively. On the field remains collimated near the waists where , whereas away from the waists, it opens up along constant diffraction angles of in axes. V. APPLICATIONS TO ELECTROMAGNETIC WAVES Expansion of electromagnetic (EM) fields using GBs has been applied in [47]–[49] for analyzing large reflector antennas in which the expansion coefficients were obtained by numerically matching GBs to the far zone field of the feed antenna. However these methods do not employ exact field expansion schemes and therefore, cannot be applied for near-field analysis, or in exact field calculations. Exact EM field beam-type expansion was investigated in [43], but the asymptotic EM beam propagators presented there were limited to validity of the conventional (orthogonal coordinates) paraxial approximaangles. Here, we briefly describe the EM tion, i.e., for small expansion scheme presented in [43] and then derive accurate asymptotics in term of scalar tilted GBs in (27). The discrete exact expansion scheme synthesizes the timedue to sources in harmonic EM field propagating in , given the transverse electric field over plane (36) and deThe propagation medium is homogeneous with noting the free space permittivity and permeability, respectively. The expansion scheme is constructed on a discrete spatial-direcand , where tional lattice

where the function set via

is related to the analysis window

(40) Using the coefficient vector in (39), the electric field expansion is given by (41)

where the electric fields of the EM beam propagators, and , are obtained from the scalar beam propagator via

(42) Here we denote, , , etc. Note that these operators may be evaluated asymptotically in closed form for Gaussian windows (see (48) and (49)). The scalar beam , satisfies the Helmholtz equation (1) subpropagator, plane ject to the aperture distribution over the (43) where is the synthesis window. Equation (41) together , as a discrete superwith (42) represent the electric field, position of EM beam waveobjects, and , which are the and electric field propagators due to the initial electric field components present over the plane, respectively. is is obThe corresponding magnetic field in tained by inserting (41) with (42) into Faraday’s law, . This yields (44)

(37)

HADAD AND MELAMED: NON-ORTHOGONAL DOMAIN PARABOLIC EQUATION AND ITS TILTED GAUSSIAN BEAM SOLUTIONS

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where the expansion coefficients are given in (39) and the magand , are given netic field of the EM beam propagators, by

(45) is the free space wave and where impedance. Next, the general spectral representation in (41) is applied for the special case of Gaussian windows. By choosing a Gaussian synthesis window (46) and inserting it into (43), we find that the scalar beam propa, satisfy the Helmholtz equation subject to the gators, initial distribution (2). Hence, these waveobjects are recognized as the tilted GBs in (27) subject to a transverse shift by to the in (5)). spectral lattice (i.e., by replacing with The asymptotic paraxial tilted GB can be used for evaluating the EM Gaussian propagators by inserting (27) into (42), and evaluating the partial derivatives in closed form. Thus, by using (5) and maintaining only terms of the highest asymptotic order, we may replace (47) such that the resulting asymptotic electric field propagators are given by (48) In a similar manner, the magnetic field, using (47) in (45), giving

, is obtained by

Fig. 3. Scattering of the tilted GB from a planar discontinuity in the medium’s velocity profile. The reflected and transmitted fields are tilted GBs which are characterized by the corresponding ' (beam-axis direction) and # (non-orthogonal system tilt) parameters. d is a length coordinate along the interface.

the plane of incidence, . Subscripts 1,2 denote quantities on either side of the interface (see Fig. 3). We assume that both reflected and transmitted fields are all of the same canonical form and differ only by amplitude , Eikonal and initial complex curvature . Therefore, we may write

(50) , , and denote incident, reflected where subscript and transmitted wave constituents, respectively. Note that whereas where and are the wave velocities on either side of the interface. Each tilted GB is idenwhich denotes the angle between and in tified by the non-orthogonal coordinate systems, and which denotes the incidence, reflection or transmission conventional (snell) angles. In order to compare the beam-fields along the interface, we express them as a function of the length coordinate along this interface, which is denoted by . Without loss of generality, and at the incident point, i.e., we assume that at the intersection of the incident beam-axis and the scattering interface. Using the 2D analog of the tilted beam complex curvature in (24), we may write (51)

(49)

where denotes the tilted beam-field complex curvature at . Using simple trigonometry, we may the incident point, express the beam-field coordinates along the interface in terms of as

VI. REFLECTION AND TRANSMISSION FROM PLANAR DISCONTINUITIES In this section, we explore a 2D tilted GB incidence on planar wave-speed discontinuity as well as the resulting reflected and transmitted fields. The incident scalar field propagating in medium is assumed to have the 2D tilted beam canonical form, in GB (27). which is obtained by setting The scalar field satisfies continuity of the total field at the inter. These boundary conditions corface, as well as of correspond to an electromagnetic field incidence where or for a TE or TM field incidence, respond to either respectively, and the scalar field being either the electric (TE) or magnetic (TM) field component in the direction normal to

(52) By applying a continuity condition to the total field, it can be concluded that the three phases of should all coincide at the interface. By using (52) in (50) and comparing the linear phase-term, we obtain the conventional Snell’s law (53) while the quadratic phase-term in (51) yields (54)

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The transmitted complex curvature is given by

. By inserting (62) into the 2D analogue of where (27), we obtain the 2D asymptotically-exact tilted GB (55)

Note that relation (54) may be further simplified in the form (56) where (57) One may readily observe that the righthand side of (56) is smaller than one. Hence, a real solution for is assured for is incident angles smaller than the critical angle, where real. Once is found from either (54) or (56), the transmitted , is obtained from (55). Following complex curvature, essentially the same procedure, we obtain for the reflected field (58) Finally, we apply the continuity condition to the total field over the interface, as well as to its weighted normal derivative, , and obtain the well-known reflection and transmission amplitude coefficient relations

(63) , the local non-orthogFor a given observation point, and onal coordinates are given by where is a coordinate normal to the beam-axis and is the distance between the exit point to the intersection point of the normal coordinate and the beam-axis. Thus, the local orthogonal coordinates are given by (64) B. Conventional GB Simulation We compare the tilted GB (63) to the conventional one which over is obtained by projecting the initial complex curvature a plane perpendicular to the beam axis direction [3], [26]. Hence, the projected complex curvature matrix is given by and the conventional (orthogonal coordinates) is given by GB,

(65)

(59) where and are the conventional fresnel reflection and transmission coefficients (60) Note that for the TE case component yields

, whereas, the TM

where

and

are given in (64).

C. Reference Field Solution The reference field solution is obtained by evaluating numerically the plane-wave spectrum integral corresponding to the aperture distribution (2), i.e.,

.

VII. NUMERICAL EXAMPLE The general formulation in Section IV is demonstrated in this section by a numerical simulation of a 2D tilted GB which is in (27). Hence, the obtained by setting is transformed into the scalar , the vectors , , matrix and , are replaced by the scalars , , and , respectively, and by . The initial field distribution is the initial tilt angles given by (2), with (61) The simulation compares the error of the tilted GB to the error of the conventional (orthogonal coordinate) GB solution, all with respect to a reference solution according to the following details.

(66) with

, , and is given in (61). The integration in (66) is replaced with a truncated integration over an effective contribution interval around the phase , i.e., with on-axis stationary point, where is the required threshold for spectrum attenuation. The sampling rate, , is chosen according to the condition (which is valid in the collimation zone), implying that the integrand oscillation period is much larger than the sampling rate and that the integrand attenuation in the sampled interval is sufficient for convergence. It was found that in order to an achieve accuracy range , it is sufficient to chose and . of

A. Tilted GB Simulation The 2D complex curvature, in (24), giving

, is obtained by using (61)

(62)

D. Error Comparison In order to verify the accuracy of the tilted GB solution, we compare the error norm of the tilted GB in (63) with respect to the reference field in (66), to the corresponding error norm

HADAD AND MELAMED: NON-ORTHOGONAL DOMAIN PARABOLIC EQUATION AND ITS TILTED GAUSSIAN BEAM SOLUTIONS

Fig. 4. error with respect to the reference solution as a function of #. Continuous and dashed lines are corresponding to the tilted GB in (63) and the con= ; (b) s =F = . ventional GB in (65), respectively. (a) s =F

=1 2

=1 4

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Fig. 5 plots the error vs. different observation line locations , for with respect to the collimation length, , and for the three values 40 , 60 and 80 . The figure demonstrates that the tilted GB exhibits better . Beaccuracy within the localized beam domain, yond the collimation-length, where the beam opens up, the tilted beams have no advantage over the conventional ones. Since beam-type expansions are generally used to take advantage of analytic simplicity which is introduced by the spatial and spectral localization of these waveobjects, these expansion schemes are tuned such that the GBs remains well-collimated within the domain of interest. The results in Fig. 5 clearly show that in the well-collimated regime, tilted GBs exhibit enhanced accuracy over the conventional ones. Full parameterization of different types of tilted GB waveobjects can be found in [50]. REFERENCES

Fig. 5. error with respect to the reference solution as a function of the on-axis location, s =F , for different # values. Continuous and dashed lines correspond to the tilted GB (63) and conventional GB (65) errors, respectively. The results clearly show that the tilted GBs exhibit enhanced accuracy over the conventional ones in the well-collimated zone.

of the conventional GB solution in (65). The error norm be, is evaluated along line tween GB and the reference GB, according to (67) in (65) or in (63). where stands for error norm is evaluated along an observation In Fig. 4, the line which is normal to the beam-axis and located at an on-axis from the initial plane. The figure plots distance of the error as a function of the beam-axis angle for different , setting the initial complex curvatures , all with . The curves are arranged in couGB waists location to ples with the continuous and dashed lines corresponding to the tilted GB error and the conventional one, respectively. Each of the curve couples in Fig. 4 share a gray shade which correspond (recall from (35) to different beam parameter that the collimation-length ). The observation line is located at 1/2 or 1/4 of the corresponding collimation-length in Figs. 4(a) and 4(b), respectively. The plots are . These figures clearly demonstrate evaluated for the lower error of the tilted GBs with respect to the conventional ones for a wide range of collimation lengths and angles.

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[21] S. Patil, M. Takale, V. Fulari, and M. Dongare, “Propagation of Hermite-cosh-Gaussian laser beams in non-degenerate Germanium having space charge neutrality,” J. Mod. Opt., vol. 55, pp. 3527–3533, 2008. [22] M. V. Takale, S. T. Navare, S. D. Patil, V. J. Fulari, and M. B. Dongare, “Self-focusing and defocusing of TEM0p Hermite-Gaussian laser beams in collisionless plasma,” Opt. Commun., vol. 282, pp. 3157–3162, 2009. [23] S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, “Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption,” Opt. Lasers Eng., vol. 47, pp. 604–606, 2009. ˆ erveny, M. M. Popov, and I. Psˆencˆik, “Computation of wave [24] V. C fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. Roy. Asro. Soc, vol. 70, pp. 109–128, 1982. [25] E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag., vol. 42, pp. 311–319, 1994. [26] T. Melamed, “Phase-space Green’s functions for modeling time-harmonic scattering from smooth inhomogeneous objects,” J. Math. Phys., vol. 46, pp. 2232–2246, 2004. [27] S. Y. Shin and L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys., vol. 5, pp. 239–250, 1974. [28] R. Simon, “Anisotropic Gaussian beams,” Opt. Commun., vol. 46, pp. 265–269, 1983. [29] I. Tinkelman and T. Melamed, “Gaussian beam propagation in generic anisotropic wavenumber profiles,” Opt. Lett., vol. 28, pp. 1081–1083, 2003. [30] I. Tinkelman and T. Melamed, “Local spectrum analysis of field propagation in anisotropic media, part I—Time-harmonic fields,” J. Opt. Soc. Am. A, vol. 22, pp. 1200–1207, 2005. [31] I. Tinkelman and T. Melamed, “Local spectrum analysis of field propagation in anisotropic media, part II—Time-dependent fields,” J. Opt. Soc. Am. A, vol. 22, pp. 1208–1215, 2005. [32] A. G. Khatkevich, “Propagation of pulses and wave packets in dispersive gyrotropic crystals,” J. Appl. Spectrosc., vol. 46, pp. 203–207, 1987. [33] T. Melamed and L. B. Felsen, “Pulsed beam propagation in lossless dispersive media, part I: Theory,” J. Opt. Soc. Am. A, vol. 15, pp. 1268–1276, 1998. [34] T. Melamed and L. B. Felsen, “Pulsed beam propagation in lossless dispersive media, part II: A numerical example,” J. Opt. Soc. Am. A, vol. 15, pp. 1277–1284, 1998. [35] T. Melamed and L. B. Felsen, “Pulsed beam propagation in dispersive media via pulsed plane wave spectral decomposition,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 901–908, 2000. [36] A. P. Kiselev, “Localized light waves: Paraxial and exact solutions of the wave equation (a review),” Opt. Spectrosc., vol. 102, pp. 603–622, 2007. [37] J. J. Maciel and L. B. Felsen, “Systematic study of fields due to extended apertures by Gaussian beam discretization,” IEEE Trans. Antennas Propag., vol. 37, pp. 884–892, 1989. [38] B. Z. Steinberg, E. Heyman, and L. B. Felsen, “Phase space beam summation for time-harmonic radiation from large apertures,” J. Opt. Soc. Am. A, vol. 8, pp. 41–59, 1991. [39] A. Shlivinski, E. Heyman, A. Boag, and C. Letrou, “A phase-space beam summation formulation for ultra wideband radiation,” IEEE Trans. Antennas Propag., vol. 52, pp. 2042–2056, 2004. [40] A. Shlivinski, E. Heyman, and A. Boag, “A pulsed beam summation formulation for short pulse radiation based on windowed radon transform (WRT) frames,” IEEE Trans. Antennas Propag., vol. 53, pp. 3030–3048, 2005.

[41] B. Z. Steinberg, E. Heyman, and L. B. Felsen, “Phase space beam summation for time dependent radiation from large apertures: Continuous parametrization,” J. Opt. Soc. Am. A, vol. 8, pp. 943–958, 1991. [42] T. Melamed, “Phase-space beam summation: A local spectrum analysis for time-dependent radiation,” J. Electromag. Waves Appl., vol. 11, pp. 739–773, 1997. [43] T. Melamed, “Exact beam decomposition of time-harmonic electromagnetic waves,” J. Electromag. Waves Appl., vol. 23, pp. 975–986, 2009. [44] L. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. [45] E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag., vol. 42, pp. 518–525, 1994. [46] E. Heyman and T. Melamed, Space-Time Representation of Ultra Wideband Signals. The Netherlands: Elsevier, 1998, vol. 103, Advances in Imaging and Electron Physics, pp. 1–63. [47] H.-T. Chou, P. H. Pathak, and R. J. Burkholder, “Application of Gaussian-ray basis functions for the rapid analysis of electromagnetic radiation from reflector antennas,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 150, pp. 177–183, 2003. [48] H.-T. Chou, P. H. Pathak, and R. J. Burkholder, “Novel Gaussian beam method for the rapid analysis of large reflector antennas,” IEEE Trans. Antennas Propag., vol. 49, pp. 880–893, 2001. [49] H.-T. Chou and P. H. Pathak, “Fast Gaussian beam based synthesis of shaped reflector antennas for contoured beam applications,” IEE Proc., Microw. Antennas Propag., vol. 151, pp. 13–20, 2004. [50] Y. Hadad and T. Melamed, “Parameterization of the tilted Gaussian beam waveobjects,” Progr. Electromagn. Res., PIER, vol. 102, pp. 65–80, 2010.

Yakir Hadad was born in Beer Sheva, Israel, in 1984. He received the B.Sc. (summa cum laude) and M.Sc. (summa cum laude) degrees in electrical and computer engineering from Ben-Gurion University of the Negev, Israel, in 2005 and 2008, respectively. Currently, he is a Ph.D. student in the Department of Physical-Electronics, School of Electrical Engineering, Tel-Aviv University, Israel. His main fields of interest are asymptotic methods, artificial materials and analytic modeling in electromagnetics.

Timor Melamed was born is Tel-Aviv, Israel, in January 1964. He received the B.Sc. degree (magna cum laude) in electrical engineering in 1989 and the Ph.D. degree in 1997, both from Tel-Aviv University, Israel. From 1996 to 1998, he held a postdoctoral position at the Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA. From 1999 to 2000, he was with Odin Medical Technologies. Currently he is with the Department of Electrical and Computer Engineering, Ben Gurion University of the Negev, Israel. His main fields of interest are analytic techniques in wave theory, transient wave phenomena, inverse scattering and relativistic electrodynamics.

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Emissivity Calculation for a Finite Circular Array of Pyramidal Absorbers Based on Kirchhoff’s Law of Thermal Radiation Junhong Wang, Senior Member, IEEE, Yujie Yang, Jungang Miao, and Yunmei Chen

Abstract—Internal calibration source provides reference data for remote sensing system. Realistic calibration source is constructed by finite periodic structure, and hence, the emissivity calculation is different from the infinite periodic array case. In this paper, conventional indirect method based on the scattering field is modified first, so that it can be used to calculate the emissivities of finite homogeneous objects. Subsequently, a new method based on antenna concept is proposed for calculating the differential scattering coefficients and emissivities of the inhomogeneous finite objects. The methods are then applied to finite circular pyramid arrays covered with an absorbing material, and the results are analyzed. Furthermore, the effect of the illuminating scope of incident wave on the emissivity and differential scattering coefficient of circular finite pyramid array is studied. Index Terms—Calibration source, emissivity, finite object, pyramid array, scattering coefficient.

I. INTRODUCTION

I

N microwave remote sensing system, calibration source is a very important component, which provides reference data for system calibration. Before fabrication, the scattering property and emissivity of the calibration source must be predicted and calculated. In our previous work [1], the scattering property and emissivity of infinite periodic pyramid array covered with an absorbing material were calculated and analyzed, which is helpful in choosing the structure type and absorbing material of the calibration source; however, it cannot provide detailed information, such as edge effect, of the realistic calibration source with finite size. Although some works on optical calibration sources can be found in the literature [2]–[5], only few works concerning microwave calibration sources can be found. Conventional indirect method for calculating emissivity of the object in microwave band is based on Kirchhoff’s law [6], in which scattering field from the object must be found first. Manuscript received February 04, 2009; revised October 15, 2009. Date of manuscript acceptance October 24, 2009; date of publication January 22, 2010; date of current version April 07, 2010. This work was supported by the NSFC Project under grant nos. 40525015, 60825101 J. Wang is with the Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China (e-mail: [email protected]) Y. Yang is with the Beijing Institute of Radio Metrology & Measurement, Beijing, China (e-mail: [email protected]). J. Miao is with the Electromagnetics Laboratory, Beihang University, Beijing, China (e-mail: [email protected]). Y. Chen is with the National Key Laboratory of Metrology and Calibration Technology, Beijing, China 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.2010.2041148

Most of the available works concerning emissivity calculation are conducted from the angle of remote sensing, where problems with infinite surfaces or with surfaces that are very large when compared with the illuminating area of incident wave are considered, such as the land, ocean, [7], [8] or even artificial foam [9]. In these cases, the incident wave is illuminated only on one part of the surface, and only the scattered field in the upper half space needs to be considered. However, when we solve the scattering problem with a finite-size object, the object is considered to be immersed in the incident wave, and the whole object, including those in the shadow region are assumed to be illuminated by the incident wave, so that the scattered power is transformed from the incident wave by the whole object, and not just by the physically illuminated boundary. Therefore, the intercepted power of the object must be changed from the physical illumination power to a theoretical interception power, if we want to use the method based on scattering field to calculate the emissivity of the finite object. The theoretical interception power, as will be discussed in Section II.A, varies with the property of the medium of the object. The modified method can be used to solve the problem with finite homogeneous objects; however, for inhomogeneous cases, such as finite objects with stratified medium, it cannot provide correct results. Hence, in Section II.B, a new method based on antenna concept is proposed for calculating the emissivity of finite object constructed by inhomogeneous medium. The method is then applied to finite pyramid array covered with absorbing material, which is presented in Section III. Conclusions are given in Section IV. II. THEORY AND FORMULAS A. Emissivity Calculation Based on Scattering Field By conventional indirect method, emissivity of an object, as shown in Fig. 1, can be expressed as

(1) is the differential scattering coefficient of where the object and is expressed by

(2) and are the power where densities of the incident wave and scattered wave, respectively.

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can be determined by full-wave numerical methods, but cannot be easily obtained. The medium factor in (4) is . For an electrically then obtained using large perfect conducting plate illuminated by a plane wave in the and is twice that of the physical normal direction, , and hence, . For low permittivity interception power, lossless dielectric object, most of the physical intercepted power is very small, and passes through the object, and hence, approaches a zero value. However, if the boundary of the object extends infinitely and the incident wave illuminates a part of it, equals to the physical intercepted power and then . For general cases with lossy medium, is difficult to determine, and to avoid this problem, a method based on the antenna concept is proposed in the following section.

Fig. 1. Finite object with incident wave illuminated on S .

is the wave impedance of free space. in (1) and (2) are the incident angles of the incident wave in terms of the are the angles in the spherical spherical coordinates, and coordinates of a certain point at which scattering field will be in (2) is the cross-section of a tube within calculated. which all power of the incident wave is intercepted by the object. can be obtained by (3) is the unit vector of the propagation direction where is the vector differential area elof the incident wave, and in (3) is the object surface ement on . The integral path illuminated by the incident wave, as shown in Fig. 1. The above-mentioned theory and formulas are valid when applied to an object with infinite boundaries. However, if the object is finite, then these formulas based on scattering field will not provide correct results. In theory, when scattering field is mentioned, it simultaneously indicates the existence of the incident field. Incident field is the field in the same place, but without an object. This implies that the finite object is transparently immersed in the incident field, and all parts of the object are ilis also luminated by the incident field, and not just , but illuminated by the incident field from the internal side of the object. When this concept is used to establish the integral equation, the incident field is applied to all parts of the object [10], and when it is applied to the FDTD method, the incident field is set on a separation boundary enclosing the whole object. Therefore, the total power intercepted by the object is not equal to , intercepted by , it is conthe physical incident power, tributed by the whole object, and we define it as the theoretical , where is a interception power that can be expressed by medium factor that depends on the medium property of the object. Therefore, the differential scattering coefficient in (2) must be changed to

(4) , actually includes The theoretical interception power, , of the obtwo parts: one is the complete scattered power, ject, and the other is the absorbed power, , of the object.

B. Emissivity Calculated Using Antenna Concept Though (4) combined with (1) can be used to calculate the emissivity of finite objects, it is not suitable for solving problems with inhomogeneous finite objects, as will be seen later. In this case, the antenna concept can be used to calculate emissivity. As shown in Fig. 2, the object now is considered as an antenna, and a plane wave feeds the antenna on the surface . Subsequently, the differential scattering coefficient in of the antenna. The feeding (2) becomes the gain power of the antenna is the physically intercepted power of the object, and can be obtained by

(5) The gain of the antenna can be given as

(6) and represents the re-radiating where field of the antenna. The star sign in (5) denotes the conjugate of the magnetic field. By substituting (6) into (1), we get

(7) where represents the re-radiating power of the antenna and represents the antenna efficiency. Equation (7) indicates that if the efficiency of the antenna can be found, then, by Kirchhoff’s law, emissivity of the antenna can be obtained. From the above-mentioned analysis, we can conclude that the emissivity obtained by the method based on antenna concept mainly depends on the scattering property of the surface . However, if we use (1) and (4) to evaluate emissivity, both and play the same role, and the result will be the average of their contribution. For example, if the object is a perfect conis covered with an ideal absorbing mateducting plate and rial that can absorb the incident wave completely, then by the method based on the antenna concept, the emissivity is almost 1. However, if we use (4), the emissivity is only 0.5. Obviously, the conventional method based on scattering field cannot provide correct result in this case.

WANG et al.: EMISSIVITY CALCULATION FOR A FINITE CIRCULAR ARRAY OF PYRAMIDAL ABSORBERS

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Fig. 2. Finite object fed by plane wave on surface S .

Fig. 4. Configuration and parameters of the circular pyramid array considered. (a) Top view of a pyramid array, (b) cross-section of the pyramid (side view) and (c) cross-section of the pyramid array (side view). Fig. 3. Finite object fed by a feeder on surface S .

C. Realistic Problem and Error Discussion In reality, no ideal uniform plane wave exists, and hence, a feeder is used to illuminate the object, as shown in Fig. 3. Thus, the feeding power of the antenna becomes

(8) and are the fields generated by the feeder. The where efficiency of the antenna can be given as

(9) where are the spherical coordinates of the feeder is the re-radiating power density generated by center. Here, the object only, and does not include the radiating power directly from the feeder. to represent the We have already defined a parameter physical interception power of the object (the feeding power is not the input of the antenna); it should be noted that power of the feeder, but is the power generated by the feeder and intercepted by the object. If we define the input power of , then the difference between and is the feeder by , which is also generated by the feeder but the leaky power leaks away from the object. and can be found Equations (8) and (9) show that if accurately, then, using (7), the accurate value of emissivity can be obtained. Thus, the key problem is to find the fields of and accurately. Of course, the integrations in (8) and (9) also affect the accuracy of the results, and hence, they should

also be calculated with sufficient precision. However, in pracand , especially tice, it is not easy to obtain accurate for experiments in which the receiving antenna is used, and the and output of the receiving antenna is a superposition of . Thus, we cannot obtain the values of or individually. Usually, an assumption is made, i.e., the measured power is considered as the re-radiating of the receiving antenna , and the input power of the feeder, , power of the object . Thus, (7) becomes is taken as the interception power,

(10) If all the energy generated by the feeder is focused onto the will exactly be , and will exactly object, then (efficiency of the feeder is assumed to be 100%), and be accurate emissivity can be obtained by (10). However, owing to finite aperture of the feeder, some energy generated by the feeder will leak away from the object, but it can also be received by the . Thus, the error receiving antenna, so that owing to the assumption can be evaluated by

(11) Equation (11) indicates that reducing leaky power is the key to obtaining accurate result of emissivity. From the above-mentioned analysis, we can observe that there are two ways that can be used to obtain accurate results, one is to place the feeder as close as possible to the object and focus the radiating power of the feeder on the object as much as possible; the other is to try to determine the leaky power of the feeder and the re-radiated power of the object accurately, and then to find accurate emissivity using (10) and (11).

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Fig. 5. Convergence of differential scattering coefficient of a circular metallic mm, mm, pyramid array with mesh size of FDTD, mm, mm, and GHz. (a) E-plane and (b) H-plane.

w = 30

h = 120

D = 120 f = 18:7

D = 120

The scattering field or radiating field of the finite object in the above-mentioned formulas can be obtained by numerical methods. In this paper, we have used FDTD to calculate the near field, and the far field is obtained by near-to-far field transformation.

III. APPLICATION TO CIRCULAR PYRAMID ARRAY

A. Problem Description and Method Verification Fig. 4 shows the configuration of a finite pyramid array considered in this paper. The array has a circular edge and is constructed using 45 pyramids or parts of pyramids, as shown in Fig. 4(a). The metallic pyramids with height and width are covered with a layer of absorbing material with thickness of , as

2

Fig. 6. Differential scattering coefficients of a 210 210 mm square metallic pyramid array obtained from the scattering field-based method and antenna concept-based method, respectively, with mm, mm, and GHz. (a) E-plane and (b) H-plane.

18:7

w = 30

h = 120

f=

shown in Fig. 4(b). The real and imaginary parts of the permittivity and permeability of the covering material are denoted by respectively. The array is illuminated by a localized plane wave with circular aperture in the normal direction, , as shown in and the diameter of the illuminating aperture is Fig. 4(c). The amplitude of the incident field can either be uniform or in Gaussian distribution, and for the Gaussian distribu, the Gaussian parameter is determined by tion of the rule of 10 dB reduction of the electric level from the center to the edge of the illuminating aperture. In this paper, most of the results are obtained by the method based on the antenna concept, and hence, it is desired to first determine the re-radiated field of the array, then, obtain the gain (differential scattering coefficient), and finally, calculate the emissivity of the array by integrating the gain. In this paper, the FDTD method combined with perfect matched layer (PML) technique is used. The incident field is

WANG et al.: EMISSIVITY CALCULATION FOR A FINITE CIRCULAR ARRAY OF PYRAMIDAL ABSORBERS

Fig. 7. Differential scattering coefficients of an absorbing material covered 210 210 mm square metallic pyramid array obtained from scattering fieldbased method and antenna concept-based method, respectively, with w mm, h mm, and f : GHz; parameters of covering material: " : j : ; j : ; t : mm. (a) E-plane and (b) H-plane.

2

= 120 = 2 8+ 0 11

= 18 7 = 1+ 0 2 = 0 32

= 30

coupled to the FDTD iteration through separation boundary. The physical interception power of the array is

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Fig. 8. Effect of diameter of the metallic pyramid array on differential scattering coefficient, by method based on antenna concept, uniform plane wave mm, w mm, h mm, and f : GHz. illumination, D (a) E-plane and (b) H-plane.

= 120

= 30

= 120

= 18 7

24 grids per wavelength. In the following calculation, we use a mesh size corresponding to 25 grids per wavelength. B. Comparison of the Results From the Two Methods

(12) is the power density of the Gaussian incident wave in where the aperture center. The FDTD method used in this paper was verified in our previous work [1] by comparing its result with that in [11] for an absorbing wall constructed by the pyramid array. Here, only the convergence of differential scattering coefficient with the mesh size of FDTD is given, as shown in Fig. 5. It can be observed that to obtain result with adequate accuracy, the mesh size should be less than 0.681 mm at frequency of 18.7 GHz, corresponding to

The two above-mentioned methods, either based on scattering field or antenna concept, should give the same results if the object is composed of homogeneous medium. To verify it, a 7 7 metallic square pyramid array is used as an example. A uniform plane wave is set on a closed separation boundary around the array when using the method based on scattering field, and is set only on the array aperture by a square separation boundary when using the method based on antenna concept. The far field is calculated by integrating the electrical and magnetic currents on an equivalent surface that encloses the whole pyramid array and separation boundary. The emissiviand , ties obtained by the two methods are respectively. These values are actually the errors of the two methods, because the emissivity of a metallic object is almost

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Fig. 9. Effect of illumination scope of uniform incident wave on differential scattering coefficient of a circular metallic pyramid array with a diameter of 210 mm, obtained by the method based on antenna concept, w mm, h mm, and f : GHz. (a) E-plane and (b) H-plane.

Fig. 10. Effect of incident wave type on differential scattering coefficient of a circular metallic pyramid array with a diameter of 210 mm, obtained using the method based on antenna concept, w mm, h mm, and f : GHz. (a) E-plane and (b) H-plane.

zero. However, from the results, we can conclude that the two methods can produce results in agreement with each other for homogeneous objects. Fig. 6 gives the differential scattering coefficients of the square array obtained by the two methods. It can be noted that the method based on scattering field gives almost equal strong scattering lobs in both forward and backward directions, while the method based on antenna concept only gives one strong lob in the backward direction. This is because in the scattering field-based method, the bottom of the pyramid array is also illuminated by the incident field. Hence, strong scattering in forward direction is excited by the bottom. In the antenna concept-based method, the radiation from the source excited by the incident wave on the array surface is blocked by the array itself in the forward direction. Thus, the radiation in this direction becomes very small, especially for large array cases. As mentioned in Section II, if the object is composed of inhomogeneous medium, then the emissivity calculated by the method based on scattering field will differ from that of the method based on the antenna concept. To show the difference in detail, the above-mentioned 7 7 metallic pyramid array is

covered with a layer of absorbing material with parameters of , and mm. In this case, the emissivities given by the two methods are 0.3672 and 0.7492, respectively. It can be observed that the value obtained by the method based on antenna concept is almost twice that of the method based on scattering field. Fig. 7 gives the differential scattering coefficients of the two methods for the absorbing material covered square pyramid array. The properties are similar to that shown in Fig. 6, except for the reduction in the scattering level in backward direction owing to absorption.

= 18 7

= 30

= 120

= 30

= 120

= 18 7

C. Effect of Illumination Scope on the Emissivity and Differential Scattering Coefficient It is difficult to obtain the exact value of interception power, , for the circular pyramid array as shown in Fig. 4. Hence, the approximate formula (10) is used for the calculation of emissivity. As shown in (11), if we want to reduce the error, the field generated by the feeder should be focused onto the array as much as possible. This can be realized by either enlarging the array size or contracting the illuminating region of the incident wave. Fig. 8 shows the differential scattering coefficients

WANG et al.: EMISSIVITY CALCULATION FOR A FINITE CIRCULAR ARRAY OF PYRAMIDAL ABSORBERS

of the two metallic circular arrays with different diameters of mm and mm, obtained by antenna concept-based method, with the diameter of the illuminating scope fixed at 120 mm. The emissivities of the two arrays are 0.0018 , respectively. Fig. 9 gives the differential scatand tering coefficients of an array with different illuminating scopes mm and mm, obtained by the antenna of concept-based method, with the array diameter of mm. The corresponding emissivities are and 0.0018, respectively. As expected, these results show that for a fixed illuminating scope, relatively larger array gives more accurate result. Fig. 10 gives the comparison of differential scattering coefficients of a circular metallic pyramid array illuminated by uniform distribution plane wave and Gaussian distribution plane wave, respectively, and the results are obtained using the method based on antenna concept. The diameter of the array is 210 mm, and the diameter of the illuminating scope is 180 mm. For the Gaussian distribution case, the electrical level has a 10 dB reduction from the center to the edge of the circular illuminating area. The corresponding emissivities in the two cases are and , respectively. To some extent, this shows that more accurate result can be expected if the array is illuminated by the Gaussian distribution plane wave. IV. CONCLUSION Emissivity prediction of the calibration sources is an important work. It has been shown that conventional indirect method based on the scattering field must be modified for finite-size objects. Both the modified scattering field-based method and the proposed antenna concept-based method are observed to give accurate results for homogeneous objects. The antenna-based method can also be used to solve problems with finite objects constructed by stratified medium. From the results of the circular pyramid array, we can conclude that the accuracy of emissivity of the array is affected by the ratio of the array diameter and the diameter of the illuminating scope. Two ways can be used to reduce the error: one is to focus the incident power as much as possible on the object; and the other is to get accurate leaky power of the feeder and re-radiation power of the object, and then obtain the emissivity of the object using the formulas. REFERENCES [1] J. H. Wang, J. G. Miao, Y. J. Yang, and Y. M. Chen, “Scattering property and emissivity of a periodic pyramid array covered with absorbing material,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pt. II, pp. 2656–2663, Aug. 2008. [2] J. T. McLean, A. McCulloch, and E. I. Mohr, “Calibration source for remote sensors,” NASA Special Pub., no. 379, pp. 783–796, 1975. [3] D. F. Heath, “Large aperture spectral radiance calibration source for ultraviolet remote sensing instruments,” Proc. SPIE, vol. 4891, pp. 335–342, 2002. [4] E. C. Kintner, J. M. Hartley, E. S. Jacobs, and P. J. Cucchiaro, “Advanced development of internal calibration sources for remote sensing telescopes,” in Proc. SPIE, Infrared Spaceborne Remote Sensing XII, 2004, vol. 5543, pp. 313–319. [5] K. F. Carr, “Integrating sphere calibration sources for remote sensing imaging radiometers,” Proc. SPIE, vol. 1109, pp. 99–113, 1989. [6] P. Pigeat, D. Rouxel, and B. Weber, “Calculation of thermal emissivity for thin films by a direct method,” Phy. Rev. B: Cond. Matter, vol. 57, no. 15, pp. 9293–9300, 1998.

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[7] L. Zhou, L. Tsang, V. Jandhyala, Q. Li, and C. H. Chan, “Emissivity simulations in passive microwave remote sensing with 3-D numerical solutions of maxwell equations,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 8, pp. 1739–1748, Aug. 2004. [8] F. Karbou and C. Prigent, “Calculation of microwave land surface emissivity from satellite observations: validity of the specular approximation over snow-free surfaces,” IEEE Geosci. Remote Sens. Lett., vol. 2, no. 3, pp. 311–314, Jul. 2005. [9] L. A. Rose, W. E. Asher, S. C. Reising, P. W. Gaiser, K. M. St. Germain, D. J. Dowgiallo, K. A. Horgan, G. Farquharson, and E. J. Knapp, “Radiometric measurements of the microwave emissivity of foam,” IEEE Trans. Geosci. Remote Sens., vol. 40, no. 12, pp. 2619–2625, Dec. 2002. [10] R. C. Baucke, “Scattering by two-dimensional lossy, inhomogeneous dielectric and magnetic cylinders using linear pyramid basis functions and point matching,” IEEE Trans. Antennas Propag., vol. 39, no. 2, pp. 255–259, Feb. 1991. [11] C. F. Yang, W. D. Burnside, and R. C. Rudduck, “A doubly periodic moment method solution for the analysis and design of an absorber covered wall,” IEEE Trans. Antennas Propag., vol. 41, no. 5, pp. 600–609, May 1993. Junhong Wang (M’02–SM’03) was born in Jiangsu, China, in 1965. He received the B.S. and M.S. degrees in electrical engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 1988 and 1991, respectively, and the Ph.D. in electrical engineering from Southwest Jiaotong University, Chengdu, China, in 1994. In 1995, he joined the faculty of the Department of Electrical Engineering, Beijing Jiaotong University, Beijing, China, where he became a Professor in 1999. From January 1999 to June 2000, he was a Research Associate with the Department of Electric Engineering, City University of Hong Kong, Hong Kong, China. From July 2002 to July 2003, he was a Research Scientist with Temasek Laboratories, National University of Singapore, Singapore. He is currently with the Key Laboratory of all Optical Network and Advanced Telecommunication Network, Ministry of Education of China, Beijing Jiaotong Univeristy, Beijing, China, and also with the Institute of Lightwave Technology, Beijing Jiaotong University, Beijing, China. His research interests include numerical methods, antennas, scattering, and leaky wave structures.

Yujie Yang was born in Beijing, China, in 1960. He received the B.S. degree in electrical engineering from the National University of Defense Technology, Changsha, China, in 1982. His area of study was electromagnetic field theory and microwave technology. In 1982, he joined the Group of Radio Metrology, Beijing Institute of Radio Metrology and measurement, China, where he became a Professor in 1993. From 1996, he became the technical director of Beijing Institute of Metrology and Measurement (BIRMM). He is currently employed in The Beijing Institute of Radio Metrology and Measurement (BIRMM), Beijing, China.

Jungang Miao was born in Hebei, China, in July 1963. He received the B.S.E.E. degree from the National University of Defence Technology, Changsha, China, in 1982, the M.S.E.E. degree from Beijing University of Aeronautics and Astronautics (BUAA), Beijing, China, in 1987, and the Dr. rer. nat. in physics from the University of Bremen, Germany, in 1998. From 1982 to 1984, he was working in Beijing with the Institute of Remote Sensing Instrumentation, Chinese Aerospace, where he developed space-borne microwave remote sensing instruments. From 1984 to 1993, he worked at the Electromagnetic Laboratory of BUAA doing research and teaching in the field of microwave remote sensing. In 1993, he joined the Institute of Environmental Physics and Remote Sensing, University of Bremen, Germany, as a staff member conducting research on space-borne microwave

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radiometry. In October 2003, he returned to BUAA, and since then, he is the Chair Professor at the Electromagnetic Laboratory of BUAA. His research areas are electromagnetic theory, microwave engineering, and microwave remote sensing of the atmosphere, including sensor development, calibration, and data analyses.

Yunmei Chen was born in Fujian, China, in 1963. She received the B.S. and M.S. degrees in electrical engineering from the Beijing Institute of Technology, China, in 1984 and 1991, respectively. In 1991, she joined the microwave fundamental group of Beijing Institute of Radio Metrology and Measurement, China, where she became a Professor in 2003. In 2007, she became the Director of RF Electronics Lab of Beijing Institute of Metrology and Measurement (BIRMM). She is directly in charge of the microwave parameter and pulse-waveform services. The responsibilities in these services include research and development of new measurement systems and techniques, overseeing the calibration workload, and quality assurance.

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Near-Field Electromagnetic Holography in Conductive Media Earl G. Williams, Member, IEEE, and Nicolas P. Valdivia

Abstract—A new approach to the inversion of ill-posed boundary value (BV) problems is presented for an infinite conductive, homogeneous media. Our interest is to investigate the possibility of imaging underwater electromagnetic sources from remote electromagnetic sensor data when that data is measured coherently over a spatial sheet, commonly referred to as a hologram. Specifically, given independent holograms of two polarizations of the electric and/or magnetic fields on a cylindrical surface exterior to the electric and magnetic sources, we develop a frequency domain, back-projection (inverse) technique that reconstructs the complete electric and magnetic vector fields in the region between the BV (hologram) surface and the sources. Of particular interest is the Poynting vector that is constructed from the back-projected fields, providing the power per unit area radiated from the sources. We believe it may be of immense practical use in diagnosis of electromagnetic sources, such as underwater ship propulsors. Tikhonov regularization, developed here for the two component measured field, is used to stabilize the inversion. To investigate the accuracy and limitations of this new approach, we carry out a numerical experiment in which an array of either magnetic or electric dipole sources are excited in a frequency range of 1 to 1000 Hz in seawater. They are arranged 8 m apart in a line and generate coherent holograms of the axial components of the electric and magnetic fields on an imaginary cylindrical sheet of radius 30 m. Spatially random noise is added to these two holograms to simulate a relatively poor signal to noise ratio of 20 dB. Results show that we can successfully reconstruct the electric, magnetic and Poynting field vectors on the cylindrical sheet of 20 m radius (10 m closer to sources) with an accuracy of less than 30% for both magnetic dipole sources and electric dipole sources from 1 to 1000 Hz. The computations needed for this approach are easily carried out on a laptop computer and reconstructions of the complete field vectors are extremely fast, with a processing time in seconds. Index Terms—Absorbing media, array signal processing, backpropagation, boundary value problems, cylindrical arrays, electric field measurement, electromagnetic propagation in absorbing media, geophysical inverse problems, holography, inverse problems.

INTRODUCTION HE study undertaken in this paper was precipitated by a need to quantify sources of electromagnetic radiation from underwater structures such as external electric motor propulsors for surface ships or underwater vehicles by using measurements made tens of meters away from the structures. Characterization of the electromagnetic fields in the vicinity of

T

Manuscript received May 11, 2009; revised October 14, 2009; accepted October 22, 2009. Date of publication February 02, 2010; date of current version April 07, 2010. This work was supported by the Office of Naval Research. The authors are with the Naval Research Laboratory, Washington, DC 20375 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.2010.2042028

the sources including complex impedance, electric currents in the water, and the spatial distribution of power flow (normal component of the Poynting vector) at frequencies between 1 and 1000 Hz, derived from standoff measurements could offer a huge potential to the navy for the development of “quiet” structures. Having enjoyed a great deal of success in developing near-field acoustical holography (NAH) both underwater and in the air over the past 25 years, and with the increased interest in navies world-wide considering all-electric platforms, provided us motivation to apply these developments to the underwater electromagnetic source characterization problem. In this problem we assume the location and shape of a radiating source are known and we want to reconstruct the fields on and near its surface from remote measurements, an inverse problem. The NAH method is a super-resolution approach - accurate details of the source field down to a fraction of a wavelength are obtained. Since the electromagnetic wavelengths (or equivalent skin depths) underwater are 1500 m at the low end of the frequency range of interest, there is no doubt that super-resolution approaches are needed. The transition from acoustics to underwater electromagnetics also implies the transition from the wave equation to the diffusion equation, and from a real wavenumber to a complex one. Near-field acoustical holography (NAH) was developed by Williams and Maynard in 1980 [1] at The Pennsylvania State University and is now applied worldwide in solving acoustic problems of many kinds. Much of this work in acoustics is summarized in a recent book [2] by the first author of this paper. Since NAH uses a back-propagation algorithm that is ill-posed, stabilization (regularization) of the inverse solution is critical. The stabilization methods used in the acoustics arena [3] are Tikhonov regularization with parameter choice methods based on Morozov [4], L-curve [5]–[7], generalized cross-validation [7] and conjugate gradient least squares approaches [3] and Krylov subspace methods [8]. In the electromagnetism area there has been some recent work on the back-propagation of fields in non-conductive media with the inclusion of evanescent waves. It appears that the first attempt to back-propagate the evanescent field in an experiment in non-conductive media was carried out in 2001 [9] using magnetic loop probes to image the surface current on a slotted metal plate. With the inclusion of evanescent waves in the reconstruction, following developments in the acoustics area, further research in electromagnetics has appeared [10], [11]. Morgan [11] presents a detailed analysis of the inverse problem that includes evanescent waves for cylindrical geometry and presents the reconstruction formulation for two measured components of the field and discusses the resulting inverse -space propagators and filter, the latter needed to control the ill-posedness of the problem.

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When the medium is water, a different picture arises as all the waves are now evanescent, given that the wave equation turns into a diffusion equation due to the conductivity of the water. Fortunately, the transition from air to water, and for that matter from acoustics to electromagnetics is straightforward: the characterbecomes complex but the eigenfuncistic wavenumber tions that form the solution to Maxwell’s equations are the same, but now with a complex wavenumber. At DC, a region of great interest underwater due to hull magnetization problems [12], there is interesting work in similar inverse problems [13]–[15] using multipole expansions and regularization strategies. There is a large body of research in electromagnetic inverse problems in conductive media such as the earth due to the extensive applications in geophysics for mine detection and oil exploration. Most of the research centers on the air-earth interface problems for subsurface boundary imaging [16] and recently on the water-earth interface for electromagnetic mapping of prospective petroleum deposits [17], [18]. In this work, in order to avoid the ill-posed problem, generally an adjoint problem is solved, using a conjugate inverse propagator that attenuates the evanescent part. Also geophysical applications generally are coupled with a forward model who’s parameters are adjusted to produce a minimum error in the inversion, and regularization strategies are developed to include the forward model [16]. Far-field to near-field approaches are very common in air for antenna characterization studies [19] but do not include evanescent waves and thus are limited to a spatial resolution of one-half wavelength. Much of this work demonstrates the utility of reconstructing the field at the surface of an antenna [20], [21], or other source bodies [22], [23]. These antenna reconstructions are very much in the spirit of the present research, that is, providing source diagnostic techniques that can provide extensive spatial information about the source. As we have said, this paper treats the ill-posed inverse problem, backtracking from the measurement aperture towards the source. We study the inverse problem in a cylindrical geometry, motivated by the near cylindrical shapes of important naval structures of interest coupled with the desire to have a measurement surface as close as conformal to the source surface as possible. Our experience has shown that formulations based on conformal surfaces can be significantly less ill-posed than those with measurement surfaces that do not follow the contour of the source body. The layout of the paper is as follows. In the first section we discuss the theory for building the various inverse propagators for a prescribed measurement. In Section 2 we describe the regularization theory needed to back-project the field, stabilizing the ill-posed inverse problem. This work draws upon previous developments in NAH in the acoustics area. To test the theory and determine the accuracy of the back-projection we provide simulations using numerical data generated from an array of either magnetic or electric dipoles in Section III. Simulated holograms are generated to imitate an actual underwater measurement and used to back-propagate over a range of 10 meters. An error analysis is provided of the reconstructed fields as well as the prediction of the Poynting vector for both types of sources at the end of Section III. The Poynting vector results are compared with known theory reviewed in the Appendix.

I.

-SPACE FORMULATIONS AND DERIVATION OF THE PROPAGATOR MATRIX

The steady state Maxwell’s equations in “reduced form” ([24], Ch. 6) are given in a media with conductivity , permeability and permittivity by

where

, the free-space impedance and the time dependence is suppressed. The reduced form has the advantage for our work that both and have the same units of V/m. The common form of the magnetic field (units of A/m), not used here, can be determined by . If we use the symbol dividing by the impedance, i.e., to represent one of the electric or magnetic field com, we can define a twoponents in cylindrical coordinates dimensional helical wave expansion of by defining the Fourier transform set

(1) and its inverse

(2) in which is called the helical wave Fourier coefficients or alternatively -space coefficients. We introduce (1) into Maxwell’s equations and following Stratton solve the dependent differential equations that remain, ([25], p. 361), modified for the “reduced” Maxwell’s equations. The six components of the -space spectra of the electric and reduced magnetic fields defined by where represents transpose are combined by defining a 6 1 vector

(3) can be written as a function of two unknown -space funcand using the 6 2 matrix tions

(4)

(5)

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. If we insert this result into (4) we obtain the final form of our boundary value problem in which the six components of the electric and magnetic fields can be determined at any radius from the measured fields

(7)

Fig. 1. Cross-cut of geometry for the exterior domain problem solved in this paper with all sources 6 inside the cylindrical surface at  = a. The cylindrical measurement surface is shown in red at  = b located at  a. The region of validity extends down to the minimum circle represented by the thin dashed line of radius r . The insert on the right indicates the direction of the unit vectors corresponding with the directions of the electromagnetic field vectors.



The 6 2 matrix product resulting from the first two terms in (7) is called the -space propagator denoted by

(8) In this equation the Hankel function is understood in the simplified notation leaving out the superscript and

The real space electric and magnetic field components are determined by taking the inverse Fourier transform defined in (1) of (7)

(6) where we always select the root with a positive imaginary part. in a freeThis formulation is valid in the domain as field region exterior to all of the electromagnetic sources shown in Fig. 1. The cylinder of the smallest (minimum) radius . Note that (5) circumscribing the sources is located at can be arrived at using the formal expansion of the electric and magnetic fields in terms of the orthogonal vector wave functions and ) as given in Stratton [25, p. (usually denoted by 393], Leach [26, Eqs. (4) and (5)], and Yaghjian ([27, Eqs. (13a) and (13b)]. We cast (4) as a boundary value problem in which two compoand , nents (polarizations), for example of the electric/magnetic fields are specified (measured) on a ) complete surface (closed in and extending to . To be general let these compoat a constant radius nents (after transformation into -space using (2)) be given by the indices and where , marking two rows of and the two corresponding rows . Thus (4) evaluated on becomes the surface

(9)

The propagator takes the boundary value data on the surand translates it to any other surface face with a minimum radius determined by the minimum cylinder in Fig. 1) and an surrounding the surface of the source ( unrestricted maximum radius. An important distinction is made when the translated surface is inside the boundary value sur. This is the inverse problem that is ill-posed, face in which small variations in the measured boundary value data lead to large variations in the reconstructed field on the left-hand . This reside of (9). This is not the case, however, when gion defines a forward problem. We discuss later in this paper techniques for controlling these large variations, so called regularization techniques. A. Propagator Matrix for

where represents component of the matrix . This equation can be inverted to solve for the two -space unknown and functions,

System

for all posWe have derived the propagator matrices . Although a subject sible combinations of measured pairs of a future paper, we note that there are significant differences in results with different combinations. We concentrate here on what appears to be the most successful system, composed of measuring the axial components of the electric and magnetic fields. Starting with (8) we thus consider the case in which the is and , given boundary data at axial polarizations of the electric and magnetic fields with corand . We responding -space coefficients use (7) with and (the third and sixth rows of (3)).

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In this case the matrix product leads to the following result for

(10)

where

where

is the modified Bessel function, and . Thus the propa(the inverse problem) gator increases exponentially when and decreases exponentially when (the forward problem). This is true independent of the conductivity of the medium. also [28] then a similar result Since is obtained for all the terms in (10). Thus all the propagators in the propagator matrix exhibit an exponential growth for large and fixed for back-propagation and exponential . decay for forward propagation and increasing it is a slightly different story. For fixed In this case assuming we find, using the asymptotic expansion ([28, Eq. (9.3.1)]) of the Hankel function for large , that

(11)

The advantage of using the “reduced” Maxwell’s equations is evident in (10) as all the terms in the propagator matrix are unitless. The formulation in (11) is valid from the surface of the , represents an ill-posed inverse minimum cylinder to problem when and a well-posed forward propagation . problem when For the axial component we notice that it is uncoupled from other polarizations in (11)

reflecting the fact that the axial component satisfies the scalar Helmholtz equation. The corresponding propagator represented by the ratio of Hankel functions has been studied extensively for the acoustic case ([2], in chapter 5). One of the objectives of near-field electromagnetic holography (NEH) is to reconstruct the Poynting vector (average power per unit area) so that the “hot” spots on a radiator can be identified. Once the solution to (11) is obtained the Poynting vector is easily computed on any imaginary cylindrical surface using the standard expression for the Poynting vector

and the increase in follows a power law as noted in the acoustics literature [2]. This is the propagator for the acoustic case and forms the backbone for cylindrical NAH. Similar expressions can be derived for all the other propagators and it can be concluded that all the terms in the propagator matrix behave with an exponential/power-law increase for large axial/circumferential wavenumbers. Due to this exponential growth for the inverse problem, we can see that noise added to the prescribed boundary fields is amplified by this exponential growth in the propagators, thus creating what mathematicians call an ill-posed problem. This amplification is mitigated by the use of -space filters that ‘regularize’ and stabilize the reconstruction of the six field compo. We will introduce the approach to this regularnents at ization in the next section. Note that regularization of the scalar problem, presented by the third and sixth rows of (11) has been considered in detail in acoustic applications of NAH [4] and the reader can find more information there. II. SOLUTION OF THE ILL-POSED PROBLEM: REGULARIZATION at As a governing example, consider the reconstruction of , represented by the first row of (11), from a knowledge and fields given on of the

(12) The real part of is called the active intensity and provides the power radiated out of the near-field. The imaginary part is called the reactive intensity. A useful diagnostic is to vary visualizing the Poynting vector as it moves away from the source to display the often circulating active intensity in the near-field. 1) Behavior of the Propagator Matrix: Turning to (10) we consider the behavior of the propagators for large values of and . The simplest propagators and , as for fixed have [28]

(13) This equation provides without modification the solution to the forward problem when , but fails (usually catastrophinecessitating a regularization strategy. This cally) when strategy is more easily developed by writing (13) as a classic with column vector linear algebra problem representing the discretized version of the unknown and a column vector representing the “measured” and fields (each on a discrete lattice) written as . We consider only . The discretization of (13) the inverse problem given by

WILLIAMS AND VALDIVIA: NEAR-FIELD ELECTROMAGNETIC HOLOGRAPHY IN CONDUCTIVE MEDIA

is fully implemented by replacing the Fourier transform operators with the matrix provided by the discrete Fourier transform implemented in computer code with the fast Fourier transform (FFT). This FFT can be viewed as a matrix multiplication opwith the usual eration by the FFT unitary matrix ( the unit diagonal matrix) and properties that . Note that each column of spans all the values of and of -space, and each row spans all of the discretizaspace. Thus in discretized form tion grid points in real (13) becomes, as well as any of the six rows of (11) evaluated at

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(10). Given (18) for we can now consider the regularization of (17). There is a great deal of literature discussing inversion of the ill-posed problem and Hansen [3] is an excellent reference detailing most of the literature as well as Zhdanov [16] who treats ill-posed geophysical inversion problems. The standard solution to this type of problem is the use Tikhonov regularization. As noted above the scalar problem presented has been investigated in detail for in the ill-posed acoustic radiation problem [4]. Tikhonov regularization applied to (17) defines a functional that must be minimized using a fixed positive parameter by a correct choice of the dimensional solution vector . This functional is given by

(14) and represent the where the -space entries in first and second columns respectively of (10) and for clarity we to represent a diagonal matrix. As already noted using use (14) directly fails catastrophically to yield a solution for since the problem is ill-posed, that is, any small variation in due to noise produces large variations in . We write (14) in -space defining (15) where

is defined as (16)

In order to apply a regularization strategy we need to invert (15)

(19) represents the vector norm. We recognize the first where term on the right-hand-side of (19) as the normal least squares solution of (17), a solution we might use if the problem was ). However, this least not ill-posed (as is the case when squares solution is useless as the effect of measurement noise in causes very large perturbations in the solution vector (that is, is orders of magnitude too large). Tikhonov regularization mitigates this effect by including a penalty function in the funcmultiplied by an positive tional of (19) that depends upon constant (to be determined later) used to control the strength of the penalty term, suppressing the growth of and creating a solution that delicately balances the two terms, overcoming the ill-posed nature of the inverse problem. The solution vector that minimizes (19) (with dependency upon the unknown constant indicated) is well known [3], [4]

(20) (17) , and turn to results from the classic literature is not square we use the on Tikhonov regularization. Since , Moore-Penrose pseudo-inverse (represented by a ), defined in this case as

where (21) We insert (18) into (21) to determine the regularization matrix obtaining

where the -space regularization filter is given by

which yields in the limit

(22)

(18)

where represents complex conjugate and represents the sum of the squares of the corresponding diagonal terms of each matrix, i.e., the sum of the squares of one row of

All the regularization approaches when formulated in -space, for example Landweber iteration and the conjugate gradient approach result in diagonal -space filters [4]. This leads to the final result in -space from (20)

(23)

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Comparison with (15) shows that Tikhonov regularization to accomplish merely applies a filter matrix to premultiply the stabilization of the solution. Equation (22) is an extremely important expression in regu, which larization theory. First note that if the value of and corresponds to no penalty term in (19), then (23) is identical to the original ill-posed (14) or its -space form (15). For non zero values of the terms of vary between 0 and 1. At the extremities of -space we saw in Section I-A.1 that and blow up exponentially, which drives the the terms filter value to zero as (22) indicates. However, when these terms vanish the filter becomes unity. The process of regularization can be attributed solely to the application of a low-pass filter in -space. The pass and stop bands of this filter are determined by the sum of the square magnitudes of the values of the inverse propagators of (10). When one of the inverse propagator terms is zero, as we have in the third and sixth rows of (10), then the dual-field formulation of (23) and the filter (22) reduce to the scalar field problem which is identical to the formulation for the acoustics application. The value of must still be determined, and its value will depend upon the level of noise in the holograms. There are different methods to determine such as the Morozov discrepancy principle, L-curve, and the generalized cross validation procedure [3], [4]. We will not discuss them here and leave that issue to a future paper, and merely choose the best value by scanning through a set candidate values for and choosing the one that yields a minimum error (given that we know the exact answer a priori). We must compute separate regularization filters for each of the six rows in (11), for both the scalar problems represented by the third and sixth rows, and the four vector two-component problems represented by the other rows. The value of that delivers a minimum error solution also differs for each of the six components. The use of different filters for the six components of the reconstructed field does not lead to a decorrelation of these components since the -space filtering process can be interpreted as a filter for the noise of each of the components (see Chapter 4 of [3] and Section III.A.1 of [4]), thus increasing the correlation. Furthermore, since the noise level of any one of the components may be different the filter cutoff must change to reflect the difference. We proceed now to test the theory with a numerical experiment in which the magnetic and electric fields are simulated using an array of five point dipoles located within the meaand surement surface. The holograms representing at the measurement surface (at ) are synthesized and spatially random noise is added to them. The six fields using along with the Poynting vector are reconstructed at the approach just detailed. Note that is predetermined here based on the criterion that it produces the minimum error over all possible values, given the level of noise introduced. Thus the results presented in this paper provide the optimum solution to this inverse problem. Of primary interest is assessing the accuracy of imaging the Poynting vector, in particular the active (real) part, as a knowledge of the power flux should be a great asset in diagnosis of underwater electromagnetic sources.

We include separate tests of magnetic and electric dipoles sources, as they offer different challenges to our approach when imaging the Poynting vector. We will find the Poynting vector results in Sections III are in qualitatively in agreement with theory, which is summarized in the Appendix. The Appendix shows that in the near-field and at low frequencies the Poynting vector for the magnetic source is almost completely reactive, whereas for the electric source it is dominantly active. The active component is necessary to estimate the real power radiated from a source, and provides an important metric to locate radiation “hot” spots on extended sources. III. DISCUSSION OF ACCURACIES USING SIMULATIONS WITH AN ARRAY OF POINT DIPOLES We can demonstrate and quantify the accuracy of NEH using simulated hologram data with added random noise using an array point dipoles sources located in a conductive media such as seawater. This simulation generates holograms of two field that are used to reconstruct components at a given radius the six field components and the Poynting vector on the surface , the inverse problem described above. Comat and the parisons are made with the known solution at errors computed for the optimal choice of . An array of either five magnetic or five electric equally spaced dipoles located at is used to assess the viability of using and NEH approach to imaging various types of sources underwater. We are interested in the frequency band from 1 to 1000 Hz. At frequencies above 1 kHz the attenuation in water overrides the practicality of this imaging approach. The dipoles are considered to be point dipoles, so that the equations of the fields generated are very simple, and can be written in closed form [29]. The axis of each dipole initially aligned with a unit vector in the direction is then inclined at an angle of 45 towards the axis and with respect to the axis, with the purpose of generating a field rich in all three polarizations of the electric and magnetic fields. We alternate the sign of each dipole to increase the complexity of the field, i.e., the dipole amplitudes look like . We choose the following parameters for the simulations: m, reconstruction radius Measurement radius m, dipole sources centrally located on a cylinder m, each separated by 8.0 m axially and of radius 20 degrees circumferentially. (In terms of Fig. 1 the circuminstead of ). scribing minimum surface is now at The source to hologram standoff (18 m in this case) is chosen to simulate practical underwater distances for a hypothetical measurement system. The reconstruction surface is not at the source boundary, the desired location, as would be possible for more realistic sources like a ship hull. However, due to the delta function nature of our dipole sources this choice is m distance chosen represents impossible. The a compromise dictated by the need for source localization yet avoiding spatial aliasing, the later arising from the very high spatial wavenumbers contained in the spatial delta function representing the source dipoles. The high spatial wavenumber content of the source field is strongly attenuated with distance (see Section I-A.1) from the dipole consistent with the spatial spreading of the dipole, and thus a larger

WILLIAMS AND VALDIVIA: NEAR-FIELD ELECTROMAGNETIC HOLOGRAPHY IN CONDUCTIVE MEDIA

measurement lattice spacing (2.0 m in this case) can be used as the standoff increases. In contrast, since these high spatial wavenumbers provide significant spatial resolution they are needed for source localization. If a large standoff is used, the source fields spread spatially and the source locations are not revealed. The standoff distance arrived at here reflects this compromise, although no specific formula was used. In general one would expect that practical underwater electromagnetic such that their spectrum sources contain an inherent limit . Some knowledge of this limit is is limited to always necessary when setting up a measurement system so would satisfy a that, for example, the axial lattice constant spacing, i.e., . The holograms are generated on a lattice of 100 by 90 (axial by circumferential) with an axial lattice constant of 2.0 m and circumferential lattice of 4 degrees (full circle of data). Spatially random noise is added to create holograms with signal to noise ratios (SNR) of 80, 40 and 20 dB. The conductivity of the medium (seawater) is 4.0 F/m and H/m. S/m, A. Array of Electric Dipoles We begin by displaying “measurement” holograms at m generated by the five electric dipoles without added noise in Fig. 2. The total axial aperture is 200 m and the vertical axis is the circumferential direction unwrapped for the display. The real part of the phase shifted fields are plotted in an unconventional manner. What is meant by phase shifted fields follows. We prefer to display data with phase information included, as one obtains with plots of the real or imaginary parts, since from our experience physical phenomena are better illustrated by this. However, there are some subtleties in this approach that must be recognized as is demonstrated by trying to plot a standing wave field since the real or imaginary part may be identically zero for all time (never the case however for a traveling wave). Thus we preprocess the display data by first deterin the aperture where is a mining the point at this point is subtracted from maximum. The phase all the points in the hologram. That is we plot the real part of . In this way the real part of a shifted standing wave would always be maximized (with the imaginary part reduced to zero). In the plots that appear in this paper the applied to the data is indipreprocessed phase shift cated at the top of each plot, as can be seen in Fig. 2. Noise is added to these two holograms to produce a SNR of 20 dB, using the standard definition—the L2 norm of the noiseless field divided by the standard deviation of the added random noise. 20 dB corresponds to a fairly noisy experiment and we feel provides a realistic, if not pessimistic, in-situ level that one might encounter underwater. We reconstruct the field at a disof 10 m ( m) from these holograms using tance the procedure outlined in Section II above. This entails running through a set of values of the regularization parameter , constructing the Tikhonov filter of (22) and computing the resulting reconstructed field from the inverse FFT of (23); choosing the value of for which the minimum error (compared with the noiseless exact solution) is obtained. in (22)) that result for the The shape of the -space filters ( optimum value of are of interest. The following figure shows

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Fig. 2. The axial polarization of the simulated electric field E (left) and unreduced magnetic field H =Z (right) at b m generated by five electric m and a frequency of 1 Hz. The center dipole is located at dipoles at  z . The phase of the field is shifted by the amount noted m and  in the plot title, an amount that maximizes the level, with the resulting real part plotted.

=0

= 12 = 90

= 30

the -space filters generated for the reconstruction of the and polarizations, respectively, of the electric field at m and 1 Hz. The blue area represents the stop-band (level 0) and the red the pass-band (level 1). To generate this plot we note corresponds to a cothat each diagonal element of ordinate in -space, and thus the filter’s value can be mapped onto a two-dimensional plot (DC at the center) to generate the plot shown. Note that the filter for the scalar problem (for ) is circular in -space as is well known for the (mathematically identical) acoustic problem [4]. In contrast the filter for other polarizations are no longer circular, due to the increased mathematical complexity of the inverse propagators. The filters for the magnetic field reconstructions are very similar (not shown). As the frequency increases the dipole-like shape becomes circular for all components. For different values of the shapes do not change, but merely expand (smaller ) or contract (larger ) producing congruent shapes. The above is also true for the magnetic dipole source. m, determined by the The six reconstructed fields at inverse FFT of (23), for the minimum error are shown (given the electric dipole source at 1 Hz and a 20 dB SNR) in Fig. 4 below. We compute the L2 norm errors by comparison with the exact fields where the percent error is defined as

where is the exact solution ( vector) and represents the L2 norm. This error is displayed in Fig. 4 above each mosaic, and it can be seen that the errors are reasonable given the low SNR and the fact that we have solved an ill-posed problem. We do not show a plot of the exact fields for comparison as differences are nearly imperceptible. The next figure, Fig. 5 plots the Poynting vector, from (12), for this electric dipole source case with the exact field on the right for comparison. The five electric dipoles are clearly reconstructed, each separated by 8 m axially and 20 degrees (about 7.0 m circumferentially. This result is quite interesting as it shows an impressive resolution given the fact that the effective wavelength at 1 Hz is over 1400 m in seawater. Generally we would expect the resolution to be dependent upon the standoff distance

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Fig. 3. The three types of k -space regularization filters that result. From left to right filters for reconstruction of the E ; E and E polarizations.

= 20

Fig. 5. The reconstructed (a m) Poynting vector (real part) on the left versus the exact field on the right for a frequency of 1 Hz. One can clearly see the five point dipole sources (each separated by 20 ) and 8 m axially.

= 20

E

Fig. 4. The reconstructed (a m) polarizations of the electric and magnetic fields =Z are displayed for a frequency of 1 Hz. Again the shifted real parts are shown so that phase information is preserved. The mosaics display the to 90 . region near the sources from 

H

=0

[2] (10 m in this case) and the SNR, but not on the wavelength (or skin depth) in the medium. B. Array of Magnetic Dipoles With the same parameters used for the electric dipole above, we investigate the reconstruction of an array of five infinitesm are imal magnetic dipole sources. The holograms at identical to those displayed in Fig. 2 except that, due to reciand , the electric and magnetic procity, fields interchange. The results for the reconstruction of the six field components, for comparison with Fig. 4 are shown in the Fig. 6. As can be seen the results are nearly identical as well

as the errors, noting that we need to compare to and vice versa, due to reciprocity. However, the phase shifts to arrive at the maximum field amplitude between the electric and magnetic fields of Fig. 6 can be seen (phase shifts displayed in the titles) to now differ by nearly 90 degrees. This implies a highly reactive Poynting vector will arise for the magnetic dipole sources. This is unlike the electric dipole case of Fig. 4 in which the fields were almost either in phase or 180 degrees apart implying a non-reactive Poynting vector. Both these conclusions are in agreement with the discussion in the Appendix. As we now suspect, computing the real part of the Poynting vector for the magnetic source reconstructions shown in Fig. 7 produces a completely different result in comparison with the electric dipole array result in Fig. 5. First looking at the exact fields on the right we see that unlike the electric dipole array, the components of the intensity vector for the magnetic dipole array are both positive and negative indicating strong circulation of the power flux vector. However, remembering that the standoff m to the sources at m is 8 m, off-line simufrom lations show that the radial intensity field closer to the sources is positive at the five source locations confirming the fact that each source is putting energy into the medium. The reconstructed intensity field is shown on the left hand side of Fig. 7. As indicated in the titles, the errors are very large for the three reconstructed components of the Poynting vector compared with the exact result on the right. Surprisingly, the error is largely due to the enormous difference in the amplitudes between the actual and the reconstructed fields, whereas the spatial shape of the source fields are somewhat similar (at least confined to the same spatial region), an unexpected but favorable result. At this point we do not have an explanation for the fact that the spatial coherence of Poynting vector has been fairly well retained. The large errors were expected given the discussion in the Appendix

WILLIAMS AND VALDIVIA: NEAR-FIELD ELECTROMAGNETIC HOLOGRAPHY IN CONDUCTIVE MEDIA

= 20

E

Fig. 6. The reconstructed (a m) polarizations of the electric and magnetic fields =Z are displayed for the magnetic dipole array for a frequency of 1 Hz. Again the shifted (the phase shift is shown in the title of each) real parts are shown so that phase information is preserved. The mosaics display the region – . The corresponding results for the electric near the sources from  dipole array were shown in Fig. 4.

H

= 0 90

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= 20

Fig. 8. The reactive (imaginary part) Poynting vector reconstructed at a m on the left versus the exact field on the right for a frequency of 1 Hz and the magnetic sources. The radial component (top row) indicates clearly the locations of the five dipoles and thus is very useful for source localization.

Plotting the reactive Poynting vector is also very useful at source localization. The reactive energy introduced into the medium is revealed in the plot of the Poynting vector shown in Fig. 8. The top row shows the radial component of the reconstruction (left) and theory (right) and the five dipoles are clearly resolved spatially. The blue color indicates a negative imaginary part, corresponding to an inductive field (negative instead of the circuit due to the use of the time convention ). The errors are indicated on the left theory convention of column mosaics and can be seen to range from 24 to 33%. These results indicate that for the magnetic source imaging of the reactive Poynting vector is successful. C. Comparison of Electric and Magnetic Dipole Reconstruction Errors

= 20

Fig. 7. The reconstructed (a m) Poynting vector (real part) on the left versus the exact field on the right for a frequency of 1 Hz. Note that the reconstructions are nearly two orders of magnitude larger than the exact result on the right side of the plot.

that showed the components of the Poynting vector in spherical coordinates for a magnetic dipole to be highly reactive in the near-field. This reactivity arises from the 90 degree phase difference between the electric and magnetic fields, as was discussed at the beginning of this section.

We now present a more comprehensive look at the accuracy of the reconstructions over a broad frequency range (1–1000 Hz) and both possible source conditions (electric and magnetic dipoles). The errors are computed again for the best choice for each polarization. To create a concise plot we exof amine the errors summing over all polarizations by computing for the electric field reconstruction error an similarly for the magnetic and Poynting vector fields. Thus the L2 norm here is where the sum is over all points in the reconstructed aperture, except near the ends of the aperture. Since the reconstructed field is corrupted at the ends of the aperture due to “finite measurement aperture effects” (see [2, Sect. 3.9]) this region is excluded in the error computation. We see that the errors shown in Fig. 9 diminish with frequency and are all less than 30%. Given the low SNR these errors are exceptionally low for an inverse problem and are very encouraging to potential

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Fig. 9. Total error in the reconstructed fields for the electric (‘Elec Dpl’) and magnetic (‘Mag Dpl’) dipole sources with an SNR = 20 dB. The errors are presented independently for the electric (‘kE k’), magnetic (‘kM k’) and the Poynting (‘kS k’) field reconstructions. In the latter case the error is computed only for the real part of S for the electric dipole sources and the imaginary part of S for the magnetic dipole sources.

applications for underwater problems. The Poynting vector errors (imaginary part only) for the magnetic dipole (solid green curve) illustrates the fact, described above in the Appendix, (26) and (27), that the Poynting vector of the magnetic dipole becomes nearly completely reactive as the frequency drops . Due to a 90 degree phase difference between the electric and magnetic fields the real part (active part) becomes impossible to measure below 50 Hz in our example. The real part of the Poynting vector for the electric dipole source, however, is accurately reconstructed consistent with the conclusion in the Appendix, (24) and (25), that the electric dipole always has a strong active intensity even at 1 Hz. As one might expect, when the SNR is increased to 80dB the errors drop. We found that the L2 norm errors for the and fields were generally only a few percent. IV. CONCLUSION The results show that the we can successfully back-project noisy electromagnetic fields underwater over a range of 10 meters. Although not discussed above, larger ranges still provide good results. For example at a 20 meter range and 40 dB SNR the back-projection and regularization exhibit errors are almost the same as the 10 meter range and 20 dB SNR. The Poynting vector for the electric source is still resolved spatially for each of the five sources even at 1 Hz and 20 dB SNR. We have deliberately simulated a worst case scenario as regards the SNR in order to provide a realistic evaluation, and we expect we will do better than 20 dB in an actual measurement system. It is important to note, even though we have concentrated on a cylindrical system that one might implement with a movable hoop array, we expect to get similar results with a planar to planar reconstruction system (for a planar source) at similar ranges. A critical factor was not addressed in this paper, that is, the implementation of techniques to determine the cutoff when the exact solution is not known. The objective of this paper was to determine the best solution possible given the theory, without

introducing additional errors arising from parameter choice methods (for the parameter ). These approaches common in the acoustics literature use the Morozov discrepancy principle, the L-curve approach or the generalized cross-validation among others. We are concentrating on this issue in our ongoing research. It is reassuring, however, to know that we have been very successful with the parameter selection routines for the acoustics problem although extension to the electromagnetics problem is not trivial. Although we have not reported on the results here, we have made studies other cylindrical measurement systems such as and and discovered that the reconstruction errors are markedly worse. In some cases such as a system one can not get accurate results even without noise. It , discussed in this paper may be appears that the system, the best one, although a bit more difficult to implement since it relies on two different types of measurement probes. Additionally, one of the reviewers of this paper pointed out the interesting fact that the complete field (for the forward problem) can be constructed using the TE (transverse electric) and TM (transverse magnetic) separations of Maxwell’s equations from and . a knowledge of only With regard to reconstruction of the Poynting vector, one conclusion became very evident. At low frequencies we can successfully reconstruct the real part of the Poynting vector for the electric source and the imaginary part for the magnetic source, in each case the component is very useful at localizing the five dipole sources. No attempt was made in this work to introduce any boundaries into the formulation although measurements underwater can not escape their presence. Research on this issue is underway. APPENDIX In water we can use Kraichman’s formulas [29] for the electric and magnetic fields to derive the complex (real and imaginary parts) Poynting vector from an electric and a magnetic dipole. These formulas deal with an infinite homogeneous media (no boundaries exist). For the point electric dipole we have (in spherical coordinates)

which for

reduce to

(24) (25)

WILLIAMS AND VALDIVIA: NEAR-FIELD ELECTROMAGNETIC HOLOGRAPHY IN CONDUCTIVE MEDIA

where is the skin depth, is the radius and the polar angle in spherical coordinates. For the magnetic dipole

which for

reduce to

(26)

(27) Dramatic differences between magnetic and electric dipole sources in water are evidenced in these results for the Poynting vector in (24) and (25) compared with (26) and (27) seen clearly . Since at 1 Hz for the near-field governed by m in seawater, the near-field is a primary interest. in the near-field it becomes dominantly active Comparing for the electric dipole and reactive for the magnetic dipole. is purely active for the electric and purely reactive for the magnetic dipole. Past experience from work in acoustics has shown that when the reactive part is much larger than the active part then it becomes impossible to measure the Poynting vector since the phase angles between and reach close to 90 degrees. Thus as the range to the source diminishes at some point we will not be able to measure the Poynting vector for the magnetic dipole source, whereas we should always be able to reconstruct the Poynting vector for an electric dipole, even at any , as for the active and reactive components become equal, both as the equations above show. These facts depending upon manifest themselves in the simulations of electric and magnetic dipole reconstructions presented in Section III. ACKNOWLEDGMENT The authors thank G. Stimak of ONR for encouragement and fruitful discussions with regard to Navy need and potential applications. REFERENCES [1] E. G. Williams and J. D. Maynard, “Holographic imaging without the wavelength resolution limit,” Phys. Rev. Lett., vol. 45, pp. 554–557, 1980. [2] E. G. Williams, Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography. London, U.K.: Academic Press, 1999.

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[3] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems. Philadelphia, PA: Siam, 1998. [4] E. G. Williams, “Regularization methods for near-field acoustical holography,” J. Acoust. Soc. Am., vol. 110, pp. 1976–1988, 2001. [5] B.-K. Kim and J.-G. Ih, “On the reconstruction of the vibro-acoustic field over the surface enclosing an interior space using the boundary element method,” J. Acoust. Soc. Am., vol. 100, pp. 3003–3016, 1996. [6] P. A. Nelson and S. H. Yoon, “Estimation of acoustic source strength by inverse methods: Part I, conditioning of the inverse problem,” J. Sound Vib., vol. 233, pp. 643–668, 2000. [7] R. Scholte, “Fourier Based High-resolution Near-Field Sound Imaging,” Ph.D. dissertation, Technische Universiteit Eindhoven, , The Netherlands, 2008. [8] N. P. Valdivia and E. G. Williams, “Krylov subspace iterative methods for boundary element method based near-field acoustic holography,” J. Acoust. Soc. Am., vol. 117, pp. 711–724, 2005. [9] P. H. Harms, J. G. Maloney, M. P. Kesler, E. J. Kuster, and G. S. Smith, “A system for unobtrusive measurement of surface currents,” IEEE Trans. Antenna Propag., vol. 49, pp. 174–184, 2001. [10] D. Taylor and P. Loschialpo, “Imaging of helical surface wave modes in the near field,” J. Electromagn. Waves Appl., vol. 17, pp. 1593–1604, 2003. [11] M. A. Morgan, “Electromagnetic holography on cylindrical surfaces using k-space transformations,” Progr. Electromagn. Res., PIER, vol. 42, pp. 303–337, 2003. [12] J. J. Holmes, Exploitation of A Ships’s Magnetic Field Signatures. San Rafael, CA: Morgan & Claypool, 2006, vol. 9, Synthesis Lectures on Computational Electromagnetics. [13] J. J. Holmes, “Theoretical development of laboratory techniques for magnetic measurement of large objects,” IEEE Trans. Magn., vol. 37, no. 5, pp. 3790–3797, 2001. [14] A. V. Kildishev, J. A. Nyenhuis, and M. A. Morgan, “Multipole analysis of an elongated magnetic source by a cylindrical sensor array,” IEEE Trans. Magn., vol. 38, no. 5, pp. 2465–2467, 2002. [15] R. Kamondetdacha, A. V. Kildishev, and J. A. Nyenhuis, “Multipole characterization of a magnetic source using a truncated svd,” IEEE Trans. Magn., vol. 40, no. 4, pp. 2176–2178, 2004. [16] M. S. Zhdanov, Geophysical Inverse Theory and Regularization Problems. Amsterdam: Elsevier, 2002, vol. 36, Methods in Geochemistry and Geophysics. [17] S. Constable and L. J. Srnka, “An introduction to marine controlledsource electromagnetic methods for hydrocarbon exploration,” Geophysics, vol. 72, no. 2, 2007. [18] E. S. Um and D. L. Alumbaugh, “On the physics of the marine controlled-source electromagnetic method,” Geophysics, vol. 72, no. 2, 2007. [19] E. Joy, “Fundamentals of antenna measurements,” in Short Course Notes. St. Louis, MO: AMTA, Nov. 2007. [20] D. J. van Rensburg and C. Walker, “Implementation of back projection on a spherical near-field range,” in AMTA Proc., Denver, CO, Oct. 2001. [21] C. Cappellin, A. Frandsen, and O. Breinbjerg, “Application of the SWE-to-PWE antenna diagnostics technique to an offset reflector antenna,” in AMTA Proc., St. Louis, MI, Nov. 2007. [22] H. Kitayoshi and K. Sawaya, “Electromagnetic-wave visualization for emi using a new holographic method,” Electron. Commun. Jpn., vol. 82, no. 8, pp. 284–291, 1999. [23] R. C. Wittmann and M. H. Francis, “Test-chamber imaging using spherical near-field scanning,” in AMTA Proc., Denver, CO, Oct. 2001, pp. 87–91. [24] D. Colton and R. Kress, Inverse Acoustics and Electromagnetic Scattering Theory. New York: Springer-Verlag, 1992. [25] J. A. Stratton, Electromagnetic Theory.. Piscataway, NJ: IEEE Press, Wiley-Interscience, 2007. [26] W. M. Leach, “Probe compensated near-field measurements on a cylinder,” IEEE Trans. Antennas Propag., vol. 21, pp. 435–445, 1973. [27] A. D. Yaghjian, Near-Field Antenna Measurements on a Cylindrical Surface: A Source Scattering-Matrix Approach Tech. Re. Natl. Bur. Stand. Tech. Note 667, Oct. 1977. [28] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. New York: Dover publications, 1972. [29] M. B. Kraichman, Handbook of Electromagnetic Propagation in Conducting Media. Washington, D.C.: U.S. Government Printing Office, 1970.

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Earl G. Williams (M’80) received the B.S.E.E. degree from The University of Pennsylvania, in 1967, the M.S. degree in applied physics from Harvard University, Cambridge, MA, in 1968, and the Ph.D. degree in acoustics from The Pennsylvania State University, University Park, in 1979. He is a Senior Scientist for Structural Acoustics in the Acoustics Division, Naval Research Laboratory (NRL), Washington, DC, where he has worked for 27 years. He is an internationally recognized scientist whose pioneering research in the field of nearfield acoustical holography has impacted acoustics research and development throughout the world. His research was formally recognized by NRL as one of the 35 most innovative technologies there over the past 75 years. He is the author of Fourier Acoustics, Sound Radiation and Nearfield Acoustical Holography (Academic Press, 1999), which Springer, Tokyo, translated into Japanese in 2005. Dr. Williams is a member of Tau Beta Pi and is a Fellow of the Acoustical Society of America where he has been an Associate Editor over the past nine years. In 2009, he was awarded the Per Bruel Gold Medal from the ASME “in recognition of eminent achievement and extraordinary merit in the field of noise control and acoustics.”

Nicolas P. Valdivia was born in Lima, Peru, on January 20, 1974. He received the B.S. degree in computer science and M.S. degree in applied mathematics from the University of Puerto Rico, Mayagüez, in 1995 and 1997, respectively, and the Ph.D. degree in applied mathematics from Wichita State University, KS, in 2002. He is currently a Research Mathematician in code 7130, Acoustic Division,Naval Research Laboratory, Washington, DC. His research interest include inverse problems for partial differential equations, numerical inversion techniques and regularization methods. Dr. Valdivia is an associate member of the Acoustical Society of America.

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Thin Microwave Quasi-Transparent Phase-Shifting Surface (PSS) Nicolas Gagnon, Aldo Petosa, Senior Member, IEEE, and Derek A. McNamara, Senior Member, IEEE

Abstract—A thin phase-shifting surface is described that consists of three metallic and two dielectric layers. The metallic layers consist of conducting shapes, which are tuned to introduce the desired phase shift on an incident wave, while maintaining a very high amplitude of transmission coefficient. Such a quasi-transparent surface is then applied to phase-correcting devices and their operation validated experimentally. Index Terms—Gratings, lens antennas, microwave antennas.

I. INTRODUCTION RADITIONALLY, high-gain antennas designed for microwave applications have consisted of reflectors, lenses, or planar arrays of low-gain elements. Each technology has its strengths and weaknesses, and the requirements of the application will usually dictate the type of antenna selected. Conventional paraboloidal reflectors and dielectric lenses in general have a higher radiation efficiency but require a larger volume than planar arrays. Planar arrays are attractive for their low profile and capability for electronic beam scanning, but these come at the expense of complex feed network design and reduced radiation efficiency. For fixed-beam applications, conventional reflectors and lenses usually have superior electrical performance and would probably always be selected if it were not for the larger volumes they occupy. In recent years, a considerable effort has been devoted to investigating printed reflectarray technology, where the curved reflector surfaces are replaced by thin flat panels of microstrip patches [1]. The major disadvantage of reflectarrays is their limited bandwidth. Flat reflective Fresnel zone plate antennas have also been used as a replacement for paraboloidal reflectors [2], but these generally suffer from a lower aperture efficiency than such reflectors, as well as limited bandwidth [3]. Artificial impedance surfaces have also been developed to act as reflectors [4], [5]. Reflectarrays and artificial impedance surfaces (which differ essentially in terms of the electrical size of the element used) also have the capability

T

Manuscript received February 16, 2009; revised June 16, 2009. Date of manuscript acceptance October 01, 2009; date of publication January 22, 2010; date of current version April 07, 2010. N. Gagnon is with the Communications Research Centre, Ottawa, ON K2H 8S2, Canada and also with the School of Information Technology and Engineering, University of Ottawa, Ottawa, ON K1N 6N5, Canada (e-mail: nicolas. [email protected]). A. Petosa is with the Communications Research Centre Canada, Ottawa, ON K2H 8S2, Canada. D. A. McNamara is with the School of Information Technology and Engineering, University of Ottawa, Ottawa, ON K1N 6N5, Canada. 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.2010.2041150

of electronic beam scanning, similar to planar phased arrays, by the incorporation of active devices such as microelectromechanical systems (MEMS) or PIN diodes. Holographic principles have also been applied at microwave frequencies for designing low-profile scattering surfaces for high-gain antenna applications [6]–[9]. However, due to the difficulty in reproducing (“recording”) the required interference patterns at microwave frequencies, these antennas generally suffer from lower aperture efficiencies than conventional reflectors or dielectric lenses. Lenses offer certain advantages over reflectors, such as the elimination of aperture blockage by the feed antenna and reduced sensitivity to manufacturing tolerances, both of which are important for higher frequency designs. However, less work has been carried out on reducing the volume of microwave lenses. Traditionally these lenses are formed out of dielectric material with a plano-hyperbolic cross-section. These lenses are relatively thick, especially for designs with small focal length-to-diratios. The lenses can be zoned (Fresnel lenses) ameter to reduce the overall thickness, but this comes at the expense of reduced bandwidth relative to the full lens arrangement. Fresnel zone plate lenses have also been used; their thickness can be considerably less than the plano-hyperbolic lens, but they also have reduced aperture efficiency [10] relative to the full lens. Parallel plate lenses have also been designed, but they similarly suffer from thickness issues, bandwidth limitations and polarization sensitivity. The lens is an example of a device that transforms an incoming phasefront into some desired outgoing phasefront (usually from a spherical to a planar one). Shaped dielectric lenses rely on the phase delay introduced by the wave traveling through the dielectric material to carry out the phase transformation, while metal gratings and metal Fresnel zone plates rely on diffraction effects to realize the phase transformation. Factors affecting the lens performance include the impedance mismatch at the air-lens boundaries, the insertion loss in the lens material, and the bandwidth performance of the lens. An ideal lens would have minimal reflection and insertion losses, operate over a wide bandwidth, and have a thin flat profile. Recent work on a transmitarray (the transmission equivalent of a reflectarray) has shown that the lens can be realized using a series of thin dielectric layers. A prototype transmitarray consisting of four dielectric sheets was demonstrated in [11], [12]. Although thinner than a traditional dielectric lens, one drawback with the current transmitarray designs is that each layer is separated by an air gap, which increases the mechanical complexity of the design. This paper presents an electrically thin phase-shifting surface for the realization of microwave lenses and other antennas

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devices using holographic-based principles. In Section II of the present paper, we propose a configuration consisting of conducting shapes etched on dielectric layers that can be used as a planar thin quasi-transparent phase-shifting surface (PSS). Section III discusses the electromagnetic modelling of such a surface and presents some performance data. This leads to the equivalent circuit presented in Section IV, which allows us to obtain some understanding of why the specific configuration can be used as a quasi-transparent surface with controllable phase. Section V shows the use of the proposed PSS in the realization of a particular two-dimensional phase grating. In Section VI a cylindrical lens antenna is implemented using the proposed PSS configuration. In both cases the success of the PSS is demonstrated experimentally. The paper is concluded in Section VII. II. PROPOSED PHASE-SHIFTING SURFACE CONFIGURATION Consider a planar phase-shifting device, with the - and -axes oriented parallel to the plane of the device and the -axis oriented normal to it. In the ideal case each infinitesimal portion of a wavefront entering the front of the device at some would undergo a prescribed phase shift as it moves point through the device, without any change to its amplitude. In order to convert the overall incident wavefront into a desired output one, the transmission phase of the device would in , for the same reason general need to be a function of that the thickness is varied in a traditional dielectric lens. Each portion of the incident wavefront will then be differently converted in phase while passing through the device, and then reconstituted into the required output wavefront. In this paper we propose to realize a thin phase-shifting surface (PSS) using conducting discs/patches/strips (hereafter referred to simply as “conducting shapes”) etched on multiple dielectric sheets. The phase-shifting surface consists of a lattice of cells (possibly of different sizes) defined over the -plane and spanning the finite thickness of the structure, with conducting shapes of different sizes located on multiple layers in each cell. The conducting shapes may be continuous from one cell to another. The . The thickness of the individual cell size is kept to less than dielectric sheets is also purposefully electrically small (on the ) in order to obtain good transmission phase conorder of trol without significantly altering the transmission amplitude; more will be said about this in Section IV. By varying the size of the conducting shapes from cell to cell we in general obtain an approximation to the desired transmission phase through . A good design will be one the structure as a function of that simultaneously provides as little amplitude change to the incident wavefront as possible while allowing a wide range of transmission phase values to be realized through changes in the size of the conducting shapes. In order to determine the potential for realizing the proposed PSS, preliminary work [13] was conducted on single and double metal layer square elements, which revealed that no significant phase shift can be obtained with such few layers. Here, we investigate a three-layer version with strip elements whose layout is shown in Fig. 1(a) and (b). The fact that strips are used allows for reducing computation time since the problem can be treated as two-dimensional rather than three-dimensional; see

Fig. 1. The phase-shifting surface layout: (a) Side view; (b) front view (first layer only); (c) front view of a single zone (first layer only). The dielectric regions are shown in gray; the metal regions are shown in black in (a) and hatched in (b) and (c).

the Appendix for further details. The structure is composed of two thin dielectric sheets (shown in gray) with conducting strips (shown in black in Fig. 1(a) and hatched in Fig. 1(b)) etched on

GAGNON et al.: THIN MICROWAVE QUASI-TRANSPARENT PSS

two sides (inner and outer) of one sheet and on one side (outer) of the second sheet. The two dielectric sheets are then bonded together, leading to a final single piece arrangement as shown in for the Fig. 1(a). The height of the strips along the -axis is for the middle layer, and it varies along the outer layers and -axis. The overall thickness of the structure is . The dielectric , while the metal layers are assheets are of equal thickness sumed infinitely thin. The structure can be thought of as being divided into cells along the -axis, with a cell height , but not along the -axis. This particular PSS structure happens to be periodic along the -axis but not along the -axis, but a PSS need not in general be periodic along any axis. In order to minimize phase quantization error it is important to keep the cell size as small as possible. In order to generate design data for this structure we perform an electromagnetic analysis of the geometry shown in Fig. 1(c). The results obtained with the structure shown in Fig. 1(c) can then be translated locally to the corresponding region in Fig. 1(b) with the same dimensions , , and : the same phase results that apply on the infinite periodic structure also apply locally. The process is repeated for each region. This approach is validated experimentally in Sections V and VI. The transposition of the results obtained assuming infinite periodicity to a local region in a non-periodic structure is analogous to the design approach used for reflectarrays [1]. The PSS is analyzed for a plane wave propagating along the -direction, at normal incidence to the dielectric sheet and polarized in the vertical direction, as shown in Fig. 1(c). Thus the incident electric field is perpendicular to the conducting strips. In the theoretical analysis, perfect electric conductors and lossless dielectric materials were used. A portion of the incident wave is transmitted through the surface, undergoing a phase shift; the rest is reflected. The objective was to control the value of the local phase shift of the transmitted wave while simultaneously minimizing the reflected wave (in other words, this matching aspect becomes an intrinsic part of the PSS design procedure). Ideally, a phase shift range of 360 or more is desired; however, some applications may require a smaller phase shift range or could still offer excellent performances even if a phase shift range of 360 is not achieved. III. ELECTROMAGNETIC MODELLING Details of the way the commercial FDTD code [14] was used to do the electromagnetic simulation of the structure in Fig. 1(c) are provided in the Appendix. The simulations were conducted at 30 GHz with the following fixed parameters: unit cell height (which is just less than for an operating of and thickfrequency of 30 GHz), dielectric constant . The strip height parameters and are the ness variables. The unit cell height was kept as small as possible in order to reduce the quantization error, but large enough in order to achieve a significant phase shift range. Smaller values of would result in smaller quantization errors, but provide an insufficient range of phase shift values. At any rate, the values of were sub-wavelength in size and should not result in a significant quantization error. We do not wish to have an that is too large (say if the dielectric constant were too low) since

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Fig. 2. Amplitude of the transmission coefficient and normalized phase for dif,h and s obtained ferent values of a and a using " from FDTD simulations at 30 GHz: 1 1 1 1 1 1 fixed a value as a function of a ; —– fixed a value as a function of a . Values of a are shown in italic and values of a are shown in bold (units are mm).

= 22 = 1 mm

= 3 mm

this would defeat the purpose of having a “thin” device of lower weight and volume. Yet we cannot have a dielectric constant that is too high; even though this might allow a really thin device the inherent reflection coefficient of the substrate would then preclude [13] a high transmission amplitude through the resulting PSS. Of course, one is anyway constrained to use a dielectric constant that is readily available. The values of and were kept smaller than 2.85 mm since larger values resulted in etching tolerance problems due to an inability to etch very thin gaps between strips. The gaps are the difference between the cell height and the strip heights ( or ), the front layer gap being shown in Fig. 1. In other words, gaps of less than 0.15 mm were difficult to etch or had poor tolerance and repeatability with the available chemical etching or larger: smaller process. This also limited us to values of were leading to smaller phase shift ranges that would not have been applicable to a broad range of applications. An etching process that allowed smaller tolerances would have led to a bigger phase shift range whatever value of , and could more attractive than the cases have made cases with actually realized in Sections V and VI. Fig. 2 shows the phase and amplitude of the transmission and at 30 GHz obcoefficient for different values of tained from FDTD simulations assuming infinite periodicity and normal incidence. The phase was normalized assuming the bare and case, i.e., the case where there are no strips ( ) produces a phase shift of 0 . On this plot, it was desired to operate as close as possible to the top-end of the vertical axis, i.e., close to the values of total transmission (0 dB). For operating points away from this axis, the amplitude of the transmission coefficient is reduced, which translates to a lower efficiency. Fig. 3 shows the “envelope” of the curves in Fig. 2, which corresponds to the best transmission cases for different phase values. Inspection of Fig. 3 allows one to determine the transmission amplitude variation obtainable for a given range of transmission phase values, and this is summarized in Table I.

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Fig. 4. Equivalent circuit model of a unit cell of the thin microwave quasitransparent phase-shifting surface.

Fig. 3. Best case of amplitude of the transmission coefficient for different : ,h and s obtained values of normalized phase for " from FDTD simulations at 30 GHz. The regions labeled A to E are described in Table I.

= 2 2 = 1 mm

= 3 mm

TABLE I REGIONS OF PHASE SHIFT RANGE FOR GIVEN MINIMUM AMPLITUDE OF TRANSMISSION COEFFICIENT

This confirms the high potential of the proposed phase-shifting surface. IV. EQUIVALENT CIRCUIT An equivalent circuit model of the thin PSS is shown in Fig. 4. The parameter extraction capability in the commercial code [15] was used to determine the value of the equivalent circuit compoand result from the charge nents. The shunt capacitances accumulation at the edge of the strips of two adjacent unit cells, located on the same layer, but separated by a gap along the -axis as described in Section III. The series inductance is a function of the dielectric constant . It was observed in the simulation models that the electric field, which was assumed to be initially polarized in the -direction, was bent between two strips of the same unit cell, providing a -component of the field and hence the resulting series capacitance . The structure is clearly physically symmetrical, and hence so is the equivalent and will result in difcircuit. The combination of , , ferent values of both amplitude and phase of the transmission coefficient. It is important to note that the contribution of the seallows one to achieve high amplitude transries capacitance mission for a broad range of phase shift values. The equivalent circuit is apparently a topology that can exhibit the non-minimum phase properties [16] required to allow somewhat independent control of the amplitude and phase characteristics that is required here. The results using the equivalent circuit are compared to full-wave simulations in Section VI.

Fig. 5. Two-dimensional phase grating with normally-incident reference beam and desired beam propagating at angle ; the shaded phase-shift regions have the phase shifts given in Table II.

8

V. BINARY PHASE GRATING The thin PSS configuration is here applied to a two-dimensional binary phase grating shown in Fig. 5. The black and gray regions represent phase-shift regions of 0 and 180 , respectively. This phase grating is discussed in [17]. The aim of such a grating is to have an input beam (plane wave), incident normally on the grating, provide output beams that propagate at to the normal, according [17] to , angles and where is the free-space wavelength and the spatial period of the grating. The beam at angle corresponds to the spatial harmonic of the grating; they are termed “diffracted beams”. The theoretical levels of these two output beams are spatial the same. A third output beam at zero angle (the harmonic) is inevitably present; it is the “undiffracted” beam. Since there are two phase transitions per spatial period , only the three above-mentioned spatial harmonics are present at the output [17]; all other spatial harmonic beams are evanescent. The diffraction efficiency, which is related to the level of the beams relative to the output beam is desired [17]. highest when the two sub-regions are of equal width In the present realization the phase grating was designed for operation at 30 GHz, for which the free-space wavelength is 10 mm. The spatial period used was 14 mm, and consequently the theoretical propagation angle of the desired beam was 45.6 . The regions of the phase grating were realized using the PSS proposed in Section II, utilizing the data in Fig. 2. The details of the design are presented in Table II, and a photograph of the fabricated device is given in Fig. 6. In each 180 phase shift region the cell is repeated along the height of the phase grating ( -axis). This is necessary for generating the desired phase shift at each location, as described in Sections II and III. Furthermore, since the cell height is less , this structure will not act as a grating in the -dithan rection. Along the width ( -axis), the grating geometry is constant within each 180 phase shift region. In each 0 phase shift and region, the conductors are simply omitted ( ). Due to material thickness availability and the need

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TABLE II DESIGN VALUES FOR THE PHASE GRATING AT 30 GHz

Fig. 8. Normalized amplitude of transmitted power as a function of angle for Thin PSS three types of grating obtained from measurements at 30 GHz: phase grating; Dielectric phase grating; Amplitude grating.

Fig. 6. Photograph of the PSS structure that emulates the two-dimensional phase grating realized using the thin PSS concept having " : ,h and s .

= 3 mm

= 2 2 = 1 mm

Fig. 7. Photograph of the free-space quasi-optical measurement system used to characterize the gratings under test.

for a bonding film, the total thickness was not exactly 1 mm and the middle metal layer was not perfectly located in the centre, as described in Section III. The final phase grating composition was as follows: • 0.017 mm copper layer (front layer); ; • 0.508 mm low loss dielectric material with • 0.017 mm copper layer (middle layer);

2

+

TABLE III PERCENTAGE OF POWER FOR EACH BEAM AT 30 GHZ

; • 0.1016 mm low loss bonding film with ; • 0.381 mm low loss dielectric material with • 0.017 mm copper layer (back layer). This gives a total thickness of about 1.04 mm, which is close to the 1 mm used in the simulations. The PSS phase grating was inserted into a pre-calibrated free-space quasi-optical measurement system for characterization [18], [19]. The setup is shown in Fig. 7. Phase-corrected feed-horns collimate the beam to a size significantly smaller than the phase grating and ensure that quasi-plane waves are present at the input and output of the grating. The small-size beam allows one to neglect the diffraction at the edge of the grating. The output feed horn was moved at different receiving angles in order to collect data. The thin PSS phase grating was compared to two other types of grating, namely a traditional dielectric phase grating and an amplitude grating. The two-dimensional amplitude grating can be represented by the sketch in Fig. 5, except that the shaded

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TABLE IV DESIGN VALUES FOR THE CYLINDRICAL LENS ANTENNA AT 30 GHZ

Fig. 9. Cylindrical lens antenna designed using the PSS technique. The shaded regions are the constant-phase-shift regions described in Table IV.

regions represent opaque and transparent regions (that is, regions of unity and null transmission amplitude); this grating has a different response to the phase grating and will enable us to demonstrate that the PSS indeed realizes a phase grating and not an amplitude grating. The traditional dielectric phase slab grating was made of a 13-mm thick Plexiglas with grooves designed to produce a phase difference of 180 between the two different regions. The amplitude grating consisted in clear and opaque regions on a thin piece of FR4, the opaque regions being made of continuous metal. The measured results for the three types of grating are shown in Fig. 8. Note that the 0 dB level corresponds to total transmission obtained without any device inserted and the two phase-corrected feed-horns of the quasi-optical measurement system facing each other, as obtained from calibration. For clarity, only the beams resulting and spatial harmonics are shown in from the Fig. 8. It is seen that the amplitude grating had much poorer performance (given the goals stated at the start of this section) compared to the phase gratings. In fact, for the amplitude grating the fraction of power in the undiffracted beam is higher than that of the desired beam. Regarding the phase gratings, Fig. 8 clearly shows that the thin PSS phase grating has a better performance than the dielectric phase grating. This is attributed to the fact that the PSS phase grating has a higher transmission through its surface and lower reflection since the phase-shifting cases selected for the design were specifically chosen to reduce the mismatch at the air-PSS interface. Table III shows the fraction of power for each beam based on the results from Fig. 8. The diffracted beam column shows the and fraction of power in the beams resulting from the spatial harmonics. The results are quite impressive for the thin PSS phase grating, with the diffraction ratio for the desired

Fig. 10. Photograph of the cylindrical lens antenna realized using the thin PSS : ,h and s . surface concept with "

= 2 2 = 1 mm

= 3 mm

beam (that is, intensity of the desired beam to that of the incident beam) being within 4% of the idealized theoretical value (and within 2% if we consider each individual beam separately). VI. CYLINDRICAL LENS ANTENNA The thin PSS technique has also been applied to design a 90 phase correcting cylindrical Fresnel zone plate lens antenna. The cylindrical lens setup is presented in Fig. 9, with the details of each phase-shift region provided in Table IV. This cylindrical lens collimates the beam in the -plane only. Since the PSS is dependent on the polarization, it would not collimate the beam in the -plane when rotated by 90 . A different lens design would be required for such a case. The region transitions were determined using the traditional Fresnel zoning rule for flat surfaces that is based on a geometrical optics approximation [20]. Specifically,

(1) where is the size of the th Fresnel zone, is the focal length and is the total number of zones in the cylindrical lens of size . In this case, the cylindrical lens size is 152.4 mm and the ratio is 0.5. focal length is 76.2 mm. Consequently, the A photograph of the PSS cylindrical lens is presented in Fig. 10. In each phase-shift region the unit cell of height is repeated along the -axis (the -plane) but is constant along the -axis (the -plane), in order to achieve the required phase shift for each different region. The far-field realized gain patterns of the cylindrical lens were measured in an anechoic chamber. A traditional cylindrical dielectric Fresnel zone plate lens of the same exact size in the -plane but with a thickness of 15 mm and made of Plexiglas

GAGNON et al.: THIN MICROWAVE QUASI-TRANSPARENT PSS

H

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Fig. 11. -plane co-polarization far-field radiation pattern of the cylindrical lens antenna obtained from far-field measurements at 29.5 GHz: Thin PSS lens; Dielectric lens; Feed horn.

2

+

Fig. 13. Amplitude of the transmission coefficient and normalized phase for the cases presented in Table IV: FDTD simulations; Equivalent circuit model. Frequency values (in GHz) are shown in italics.

+

TABLE V COMPONENT VALUES FOR THE EQUIVALENT CIRCUIT MODEL

2

Fig. 12. Measured boresight gain of the cylindrical lens antenna: lens; Dielectric lens; Feed horn.

+ Thin PSS

was also measured. The results are presented in Figs. 11 and 12, in which the realized gain pattern of the feed horn used to feed the two cylindrical lenses is also presented for reference. With this feed, the edge taper was about 14 dB for both lenses. Fig. 11 clearly shows that the thin PSS cylindrical lens produced efficient collimation because the boresight realized gain is almost identical to that when the dielectric cylindrical lens is used. The only difference is that the main beam is slightly wider, and the sidelobe levels lower, for the PSS cylindrical lens. The minor pattern asymmetry may have been caused by slight off-axis misplacement of the feed horn with respect to the lens in the -plane. The cross-polarization levels are below dBi level for all cases over the angular range shown. the Fig. 12 shows the realized boresight gain over frequency. From 28 GHz to about 29.5 GHz the gain performance of the PSS cylindrical lens is almost identical to that of the dielectric cylindrical lens, and even higher around 29 GHz. However, from about 30 GHz, the thin PSS cylindrical lens suffers from boresight gain degradation caused by the bandwidth limitation of the PSS structure. As expected, the dielectric cylindrical lens has a

wider gain bandwidth. Nevertheless, the gain bandwidth of the PSS cylindrical lens is at least 5%, which is still more than acceptable for many different types of application. Consequently, there is a niche for the use of a thin PSS realization of a lens, especially if we recall that such a realization significantly reduces cost, weight and thickness. In order to explain the performance of the thin PSS cylindrical lens, we examine the phase shift variation over frequency for given cell dimensions. Fig. 12 shows that, for a given application (in that case a lens antenna), the bandwidth was fair but could result in a problem if a very wide band behavior is desired. Fig. 13 shows how the different cases presented in Table IV varied with frequency from 28 GHz to 32 GHz. The case corresponding to a phase shift of 0 (consisting of a dielectric layer with no strips) only presented marginal variation with frequency and so is not shown in Fig. 13. The other cases show that the variation was non-negligible. Moreover, the variation was more significant for higher phase shift ranges. The curves for the and cases were clearly sub-optimum, which also appears in Table IV. These cases presented a difference between at 30 the desired and realized phase shift values of about GHz. This, combined with possible etching and alignment errors, could provide an explanation as why the PSS cylindrical lens has a better performance at frequencies lower than 30 GHz. However, the major problem with the bandwidth of the cylinand drical lens could be the rapid amplitude decay for the cases, which is quite severe for the case. Fig. 13 also presents curves obtained from the equivalent circuit model, described in Section IV. The equivalent circuit model was optimized from DC to 45 GHz and provided an excellent agreement throughout this band. Component values

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exploit the commercial code’s periodic structure capability and the resulting computational simplicity of being able to determine the fields on a single unit cell. We emphasize that this approach is only a computational “trick” that is being used to generate design data. It has nothing to do with the actual PSS being periodic or not. In the FDTD analysis, the unit cell is simulated by placing perfect electric conductors parallel to the -axis and perfect magnetic conductors parallel to the -axis at the unit cell boundaries, as shown in Fig. 14. Since the unit cell size is electrically small, such a configuration allowed for mimicking the infinite periodic structure shown in Fig. 1(c) without the problem of propagating higher-order Floquet modes being generated. This enforces normal-incidence plane-wave propagation and an electric field polarization along the -direction. In the -direction, the length of the strip equals that of the unit cell to simulate an infinitely long strip. Consequently, the results obtained will be independent of the length of the unit cell along the -direction. This has been confirmed by simulating the same case with different values for the unit cell length along the -axis and obtaining the same results. To reduce the computation time, a very ) was used for the unit cell along the small value (roughly -axis.

Fig. 14. The unit cell of the phase-shifting surface: (a) Side view; (b) Front view (first layer only); (c) Top view. The dielectric regions are shown in gray; the metal regions are shown in black in (a) and (c) and hatched in (b).

(shown in Fig. 4) for each case of Fig. 13 are presented in Table V. Results for the dielectric layer with no strips (transmission phase of 0 ) are also included. As expected, the series is null in this case because of the absence of capacitance strips. VII. CONCLUSION A thin phase-shifting surface composed of multilayer metallic strips has been proposed. The surface offers a broad phase shift range with high transmission amplitude. Two proof-of-concept examples were presented, namely a linear phase grating and a cylindrical Fresnel zone plate lens. Experimental validation confirms the efficacy of the approach. Although the PSS concept and applications have been discussed in a “two-dimensional” device context, it can be extended to those cases where the PSS geometry is fully three-dimensional by applying a similar technique. APPENDIX USE OF THE FDTD CODE Inspection of Fig. 1(c) easily reveals that the problem is a two-dimensional one but of infinite extent; there is no variation of the incident field or the geometry along the -axis. It is also singly-periodic, since it repeats itself along the -axis. In order to analyze the structure and so generate design data for a PSS, we have used a commercial finite-difference time-domain (FDTD) simulation package [14] by treating the structure as being periodic along the -axis as well. This permits one to

REFERENCES [1] J. Huang and J. A. Encinar, Reflectarray Antennas. Hoboken, NJ: Wiley-IEEE Press, 2007. [2] M. A. Gouker and G. S. Smith, “A millimeter-wave integrated-circuit antenna based on the Fresnel zone plate,” IEEE Trans. Microw. Theory Tech, vol. 40, pp. 968–977, May 1992. [3] Y. J. Guo and S. K. Barton, Fresnel Zone Antennas. Boston, MA: Kluwer Academic, 2002. [4] D. F. Sievenpiper, J. H. Schaffner, H. Jae Song, R. Y. Loo, and G. Tangonan, “Two-Dimensional beam steering using an electrically tunable impedance surface,” IEEE Trans. Antennas Propag., vol. 51, pp. 2713–2722, Mar. 2003. [5] D. M. Pozar, “Wideband reflectarrays using artificial impedance surfaces,” Electron. Lett., vol. 43, no. 3, Feb. 2007. [6] K. Iizuka, M. Mizusawa, S. Urasaki, and H. Ushigome, “Volume-Type holographic antenna,” IEEE Trans. Antennas Propag., pp. 807–810, Nov. 1975. [7] K. Levis, A. Ittipiboon, A. Petosa, L. Roy, and P. Berini, “Ka-band dipole holographic antennas,” Inst. Elect. Eng. Proc. Microw. Antennas Propag., vol. 148, no. 2, pp. 129–132, Apr. 2001. [8] M. Elsherbiny, A. E. Fathy, A. Rosen, G. Ayers, and S. M. Perlow, “Holographic antenna concept, analysis, and parameters,” IEEE Trans. Antennas Propag., vol. 52, pp. 830–839, Mar. 2004. [9] T. Quach, D. McNamara, and A. Petosa, “Holographic antenna realised using interference patterns determined in the presence of the dielectric substrate,” Inst. Elect. Eng. Electron. Lett, vol. 41, no. 13, pp. 724–725, Jun. 2005. [10] H. D. Hristov, Fresnel Zones in Wireless Links, Zone Plate Lenses and Antennas. Norwood, MA: Artech House, 2000. [11] M. R. Chaharmir, S. Raut, A. Ittipiboon, and A. Petosa, “Cylindrical multilayer transmitarray antennas,” presented at the URSI Electromagnetic Theory Symp., EMTS-2007, Ottawa, Canada, Jul. 2007, (CDROM). [12] M. R. Chaharmir, A. Ittipiboon, and J. Shaker, “Single-Band and dual-band transmitarray,” in Proc. 12th Int. Symp. Antenna Tech. Applied Electromagn. (ANTEM 2006), Montreal, Canada, Jul. 2006, pp. 491–494. [13] N. Gagnon, A. Petosa, and S. McNamara, “Phase hologram composed of square patches on a thin dielectric sheet,” in Int. Symp. on Antennas and Propagation (ISAP 2008), Taipei, Taiwan, Oct. 2008, pp. 678–681, (CD-ROM). [14] IMST Empire XCcel™ [Online]. Available: http://www.empire.de [15] Advanced Design System (ADS) Agilent Technologies Inc, California. [16] H. W. Bode, Network Analysis and Feedback Design. New York: Van Nostrand, 1955.

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[17] W.-H. Lee, “High efficiency multiple beam gratings,” Appl. Opt., vol. 18, no. 13, pp. 2152–2158, 1979. [18] N. Gagnon, J. Shaker, P. Berini, L. Roy, and A. Petosa, “Material characterization using a quasi-optical measurement system,” IEEE Trans. Instrum. Meas., vol. 52, pp. 333–336, Apr. 2003. [19] N. Gagnon and J. Shaker, “Accurate phase measurement of passive non-reciprocal quasi-optical components,” Inst. Elect. Eng. Proc. Microw. Antennas Propag., vol. 152, no. 2, pp. 111–116, Apr. 2005. [20] J. Ojeda-Castaneda and C. Gomez-Reino, “Selected papers on zone plates,” in SPIE Milestone Series 128, 1996. Nicolas Gagnon received the B.A.Sc. (summa cum laude) and M.A.Sc. degrees both in electrical engineering from the University of Ottawa, Ontario, Canada, in 2000 and 2002, respectively, where, since 2007, he has been working toward the Ph.D. degree. Since 2001, he has been a Research Engineer in the Advanced Antenna Technology Lab, Communications Research Centre Canada, Ottawa, Ontario, Canada. His research interests include quasi-optics, permittivity measurement, microwave holography and microwave antennas. Mr. Gagnon is a licensed professional engineer in the province of Québec, Canada, and a member of the Ordre des ingénieurs du Québec.

Aldo Petosa (S’89–M’95–SM’02) received the B.Eng, M.Eng., and Ph.D. degrees in electrical engineering from Carleton University, Ottawa, Canada, in 1989, 1991, and 1995, respectively. From 1990 to 1994, he carried out research at CAL Corporation on microstrip antennas for cellular and mobile satellite communication applications. In 1995, he joined the Communications Research Centre Canada, Ottawa, Canada, where he is presently the Project Leader for Antenna Design and Development in the Advanced Antenna Technology Lab. He has published over 150 journal and conference papers and is the author of the Dielectric Resonator Antenna Handbook (Norwood, MA, Artech House, 2007). His current research interests include: microstrip antennas,

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microwave lenses, dielectric resonator antennas, reconfigurable antennas, and holographical techniques applied to antenna designs. Dr. Petosa is an Adjunct Professor with the Department of Electronics at Carleton University and is currently the Canadian National Council Chair for URSI Commission B. He also recently joined the Editorial Board of the International Journal of RF and Microwave Computer-Aided Engineering. He was a recipient of the IEEE Antennas and Propagation Society Commendation Certificate recognizing the exceptional performance of a reviewer for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION for 2008.

Derek A. McNamara (S’80–M’81–SM’90) received the B.Sc. degree (with honours) from the University of Cape Town (UCT), Cape Town, South Africa, in 1976, the M.Sc. degree from the Ohio State University, Columbus, in 1980, and the Ph.D. degree from UCT, in 1986, all in electrical engineering. During 1977 to 1979, and 1981 to 1985, he was employed as a Senior Research Engineer at the Council for Scientific & Industrial Research (CSIR), Pretoria, South Africa. He was a Professor in the Department of Electrical and Electronic Engineering, University of Pretoria, from 1985 to 1994 (Grinaker Chair in Electromagnetics from 1991–1994), and in 1992 was a Visiting Scientist at the Institut für Höchstfrequenztechnik und Elektronik (IHE), University of Karlsruhe, Germany. He was a Principal Member of Technical Staff with the COM DEV Space Group, Ontario, Canada, from 1994 through 2000, where he worked on satellite antenna design and development. Since 2000, he has been a Professor of electrical engineering in the School of Information Technology & Engineering (SITE), University of Ottawa, Canada. His research interests include antenna synthesis and design, computational electromagnetics, microwave circuits, and electromagnetic measurements. He has published 72 papers on these subjects in international journals, and 75 in conference proceedings. He is a coauthor of Introduction to the Uniform Geometrical Theory of Diffraction (Artech House, 1990). Dr. McNamara is a Licensed Professional Engineer in the Province of Ontario, Canada.

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Polarization Rotating Frequency Selective Surface Based on Substrate Integrated Waveguide Technology Simone A. Winkler, Student Member, IEEE, Wei Hong, Senior Member, IEEE, Maurizio Bozzi, Member, IEEE, and Ke Wu, Fellow, IEEE

Abstract—A novel frequency selective surface based on substrate integrated waveguide technology (SIW) is proposed and studied theoretically and experimentally. Its primary function is the selection of a linear polarization of an incident wave and its 90-degree rotation in a given frequency band. The proposed structure is based on an SIW cavity, coupled to the input and output by two orthogonal slots, and is designed using a simulation code based on the method of moments boundary integral-resonant mode expansion method, especially developed for the accurate characterization of FSS structures. Moreover, a complete design procedure for the proposed structure and a parametric study of its performance are presented. Finally, in order to verify the proposed structure, a prototype is designed at an operation frequency of 35 GHz, exhibiting a relative bandwidth of 9.1% and an impedance matching of better than 11 dB with a maximum insertion loss of 0.2 dB in the passband. Its performance is investigated in detail including the analysis of non-orthogonal incidence angles. Index Terms—Frequency selective surface, method of moments boundary integral-resonant mode expansion (MoM/BI-RME) method, polarization, substrate integrated waveguide.

I. INTRODUCTION

F

REQUENCY selective surfaces (FSS) are planar periodic arrays of metal patches or slots that function as filtering elements for free-space radiation [1]–[4]. A large variety of element shapes including rectangles, crosses, and more complex geometries have been employed as periodic elements [1]. Their shape and dimensions as well as the substrate characteristics determine the performance of an FSS structure and they are selected and optimized according to different specifications. FSS are widely used in the microwave, millimeter-wave, and sub-millimeter-wave frequency regions and find various applications, such as in the design of dichroic mirrors in deep-space Manuscript received June 08, 2009; revised October 09, 2009; accepted October 11, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. This work was supported in part by the Austrian Academy of Science, the Fonds Québécois de Recherche sur la Nature et les Technologies (FQRNT),and in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada under a Discovery grant. S. A. Winkler is with the Poly-Grames Research Center, École Polytechnique, Montréal, QC H3T 1J4, Canada and also with the Institute for Communications and Information Engineering, University of Linz, 4040 Linz, Austria (e-mail: [email protected]). W. Hong is with the State Key Laboratory of Millimeter Waves, Department of Radio Engineering, Southeast University, Nanjing 210096, China (e-mail: [email protected]). M. Bozzi is with the Department of Electronics, University of Pavia, 27100 Pavia, Italy (e-mail: [email protected]). K. Wu is with the Poly-Grames Research Center, École Polytechnique, Montréal, QC H3T 1J4, 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.2010.2041170

antennas for radio-astronomy and the band or channel separation in quasi-optical systems. Furthermore, they are used as antenna radomes and as shielding elements in microwave ovens. In this work, an FSS based on the substrate integrated waveguide (SIW) technology [5] has been developed and demonstrated. In recent years, SIW technology has experienced an enormous success as it offers a highly attractive method for the design of microwave and millimeter-wave waveguide-based integrated components and sub-systems in the form of planar circuits. SIW technology provides a low-cost and low-loss solution for system-on-substrate (SoS) designs allowing for a straightforward integration with other planar technologies such as microstrip and coplanar waveguide. Among the developed structures, SIW-based resonant cavities have been built exhibiting an excellent performance in terms of quality factor [6]. The combination of SIW resonant cavities with slotted FSS designs offers a combination of two different resonances and therefore provides a low-loss and low-cost architecture with a high quality factor and thus sharp roll-off at the passband edges [7]. In this paper, a novel concept has been introduced in order to implement a polarization-selective filtering element. In previous works, a polarization-selective surface with two dipoles connected by a via hole as the unit cell has been proposed [8]. The presented paper makes use of an SIW-based FSS architecture and extends its functionality further by the selection of a linear polarization of an incident wave and its subsequent 90-degree polarization rotation. The proposed structure can be used in a variety of applications such as so-called filtennas combining an antenna with a filtering shield in a single unit [8]. Moreover, it can be employed in quasi-optical systems requiring a separation of polarization, such as it is the case e.g., in polarimetric imaging radars and/or radiometers. Such systems are taking advantage of the concept inherent in the proposed structure where an incident wave is transmitted with a rotated polarization, while the reflected wave maintains the same polarization, thus allowing for a separation of the two polarizations and filtering of an incident wave in a single component. Moreover, the proposed structure provides a number of advantages in terms of performance: the design is based on a dualmode configuration, which offers a second-order resonance and therefore provides a low-loss and low-cost architecture with a broader bandwidth, as well as a high quality factor and thus a sharp roll-off at the passband edges. In addition, a very good decoupling of incident and outgoing waves is achieved using this dual-mode configuration. For the design of the proposed structure, a code based on the method of moments boundary integral-resonant mode expansion (MoM/BI-RME) [10], [11] has been adopted: this code pro-

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vides a fast and accurate modeling of single-layer and multi-layered FSS with arbitrarily shaped elements, allowing for a considerable reduction in computation time over commercial fullwave electromagnetic software. In addition, the paper includes a detailed parametric performance study based on this method, providing a profound and computationally efficient insight into the performance optimization of SIW-based FSS. Finally, a prototype operating at 35 GHz has been fabricated in order to verify the proposed concept. Its performance is investigated in detail, and measurements of different incidence angles are included in this paper in order to manifest the applicability of the proposed structure in practical high-end millimeter-wave applications. The presented paper is organized as follows: Section II describes the FSS architecture and design concept, Section III outlines the simulation with particular emphasis on the analysis based on the MoM/BI-RME method. Section IV introduces a performance model with a parametric analysis of the FSS, and in Section V the experimental implementation and results are outlined and compared to the simulation results. In Section VI the work is summarized in a brief conclusion. II. FSS ARCHITECTURE AND CONCEPT A. Concept The primary function of the FSS proposed in this work is the selection of a linear polarization of an incident wave and its subsequent rotation of 90 degrees while, at the same time, filtering the signal in a given bandwidth around the center frequency. The proposed structure is shown in Fig. 1(a). It consists of a 2-D array of SIW cavities with two orthogonal slots in each cavity on the front and back sides of the FSS, respectively. The SIW cavity is designed providing a resonance at the desired center frequency. Then, a vertical slot on the input (front) plane of the FSS is used for selecting the horizontal polarization from the incident radiation. The energy coupled from the slot is exciting a field in the cavity, and the outgoing wave is subsequently coupled through an orthogonal (horizontal) slot at the back side of the structure. In this way, it is possible to introduce the desired 90-degree rotation of the polarization of the incident wave. B. SIW Cavity Design The main element in the proposed design is the SIW cavity. Its resonance frequency for TM modes can be determined as follows [8]:

Fig. 1. (a) Structure of the proposed FSS architecture using a 2-D array of SIW cavities; (b) a unit cell of the FSS structure with the coupling slots including the notation for the dimensions used in this paper.

are effective measures relating the SIW cavity to a stanrepresents the dard metallic rectangular cavity. Moreover, wavelength at the operation frequency, and is the relative dielectric permittivity of the substrate. Finally, and represent the modal indices with respect to the cavity resonances. In the case of the proposed structure, it is favorable to use a square cavity in order to maintain symmetry between the two polariza. tion planes, i.e.,

(1)

C. Slot Geometry

(2)

The second element required in the presented structure are the slots, which couple the incident energy into and out of the is related to the length of cavity. Their resonance frequency the slots, which should be selected approximately equal to half a wavelength, thus giving the following approximate equation : for

with

which is valid for (3)

(4) stands for the diameter of the SIW posts, for their distance, and for the width and length of the cavity, respectively, and is the speed of light in vacuum. The dimensions and

with as the slot length, and as the effective dielectric permittivity. This frequency needs to be selected close to the reso-

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nance frequency of the SIW cavity, so that a coupled incident wave can determine the excitation of the cavity mode. As a starting point, the length of the slot is calculated according to (4) using the desired center operation frequency.

D. Selection of Cavity Mode For the SIW resonator an appropriate cavity mode needs to be selected. In general, SIW cavity structures only support modes with and (considering the axis definitions used as a convention in resonators as shown in Fig. 1(b)): first of all, SIW structures are usually very thin and therefore permit no field variation in -direction; secondly, TE modes cannot exist in the structure, as the electric current density on the lateral side walls (for the case of resonators these - and -planes) can flow only in -direction are the along the via holes. Fig. 2 shows an illustration of theoretically possible cavity modes for the use in our structure. In order to realize the desired polarization rotating behavior, we need to find a slot arrangement, in which, first of all, only horizontally polarized waves are coupled into the cavity, as well as only vertically polarized waves are emitted from the cavity. To this aim, the -field-vectors need to be aligned orthogonally with respect to these slots in order to realize a proper coupling between the incident wave and the modal field in the cavity. mode offers a field As shown in Fig. 2(a), e.g., the distribution suitable for such a configuration: the input coupling slot can be aligned vertically near the cavity edge, whereas the output slot is aligned horizontally so that for both input and output slots the -field-vectors are aligned orthogonally with respect to the slot. Another suitable configuration is given by a dual-mode and modes as shown in configuration using Fig. 2(c) and (d), respectively. In this concept, the two modes coexist in the cavity, and the incident wave is coupled into mode. the cavity through the vertical slot exciting a input mode is coupled with the output Then, the mode in the cavity. Finally, the outgoing wave is coupled from mode. The combination or a horizontal slot using the coupling of these two modes in the cavity needs to be induced by a small perturbation, which, in the proposed structure, is given by the presence of the SIW via holes and the slots. Additional solutions for realizing the desired functionality are given by any other dual-mode configuration combining and modes with (e.g., and modes, see Fig. 2(e) and (f).). Furthermore, in principle, modes with and , such as the mode, also offer a proper mode configuration for the proposed structure when two separate slots are used on either side of the cavity. However, as the slots need to be located at a distance corresponding to a value in the range of a wavelength, they bear the problem of creating transmission zeros under certain conditions due to field interference problems. Thus, the possible implementations include the mode and the dual-mode configurations combining and

Fig. 2. A unit cell of the FSS structure with the coupling slots and an overlaid vector plot of the respective mode (arrows: E -field vectors, solid lines: input and (b) TM mode suitable for slot, dashed lines: output slot): (a) TM both input and output coupling, (c)–(e) TM =TM and TM =TM dual-mode configurations suitable for input/output coupling (the bold red lines indicate the respective slot coupling with the mode shown in the figure).

modes . For the final selection of the mode, also geometrical restrictions and fabrication limitations posed by the structure need to be considered. In the following, a and a Rogers 5880 substrate with a dielectric constant thickness mil (1.57 mm) is used. These substrate values are selected for their advantages in terms of bandwidth and of reasonably big geometric dimensions allowing for the use of a standard printed circuit board (PCB) fabrication process at an operation frequency of 35 GHz in this case study. The for this substrate lower limit for the SIW hole diameter thickness is 0.5 mm. Table I shows the size of the SIW cavity mm, and a for a resonance frequency of 35 GHz using spacing mm. The slot length determined from (4) for the above denoted substrate using a first-order approximation

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TABLE I CAVITY DIMENSIONS FOR DIFFERENT MODES FOR A RESONANCE FREQUENCY OF 35 GHz

for the effective permittivity yields a preliminary value mm. Comparing the slot length to the dimensions in Table I, it is mode, the slot length is almost equal noted that, for the to the cavity width. Considering the diameter of the SIW posts, the required slot would be too long to fit into a cavity using the mode. One possibility consists in forming the slot in a -shape or -shape, but in that case, the suppression of the undesired polarization is limited. and Thus, the selection of one of the dual-mode configuration appears viable. When using a higher order mode configuration, the cavity size increases (see Table I) and therefore the slot length is smaller than dual-mode the cavity width. Among them, the configuration appears the most suitable solution, as the cavity size is still comparably small and easily allows forming a periodic structure with several cells. For higher modes, such dual-mode configuration and higher, the as the coupling with the incident field is weaker, as the direction of the field vectors vary more and more, and thus, performance is degraded. In addition, compared to single-mode configurations -based solution, the dual-mode configurasuch as the tion offers a number of additional advantages, as the two modes are inherently separated, and therefore a better decoupling of the incident and outgoing waves can be achieved. In addition to the advantage of decoupling, with the dualmode configuration we also achieve a broader bandwidth, as the two dual modes form a second-order resonance and thus two distinct poles in the frequency spectrum. E. Slot Position configuration, two different possiUsing the bilities can be considered for the position of the slots: the two orthogonal slots can be located either near the edge of the SIW cavity (Fig. 3(a)), or in the center of the cavity (Fig. 3(b)). Since this latter configuration exhibits higher transmission leakage of the undesired polarization, the solution shown in Fig. 3(a) was finally adopted in the design of the FSS. F. Operation Concept For a better understanding of the operation principle of the proposed structure, all the different possible signal paths are illustrated in Fig. 4. Four different scenarios are of interest using the structure in a realistic environment (note that in the following description an arrangement with vertical input slots and horizontal output slots as depicted in Fig. 1 is considered).

Fig. 3. Different possible slot positions for the TM =TM dual-mode configuration: (a) near the cavity edges; (b) in the center of the unit cell.

Fig. 4. Measurement of different incident angles: Transmission and reflection paths for (a) orthogonal and (b) non-orthogonal incidence angles.

1. The desired effect, i.e., transmission in cross-polarization, is given by an incident wave of horizontal polarization (orthogonal to the input slot), which is coupled to the output with a vertical polarization (orthogonal to the output slot); 2. Impedance matching, i.e., reflection in co-polarization, is described by an incoming wave of horizontal polarization with a reflected wave of the same polarization; 3. Transmission leakage (in our case corresponding to transmission in co-polarization) is defined by an incident wave of horizontal polarization, and its non-rotated outgoing wave of the same polarization; 4. Reflection leakage (here reflection in cross-polarization) describes the reflection with a polarization rotation, i.e., a wave incident with horizontal polarization and a reflected wave with vertical polarization. Certainly, in an analogous manner, the same characteristics (with reversed polarization considerations) can be seen from the output side of the structure as the proposed structure is passive in nature.

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TABLE II FINAL DIMENSIONS FOR SIW CAVITY FSS

III. SIMULATION The analysis and optimization of the proposed FSS structure has been performed using the MoM/BI-RME method, a very efficient numerical technique based on the method of moments (MoM) with entire-domain basis functions, calculated by the boundary integral-resonant mode expansion (BI-RME) method. This method was originally developed for the modeling of inductive and capacitive FSS [10], [11], and subsequently applied to the modeling and design of FSS [12], [13], of boxed microstrip circuits [14] and of electromagnetic band-gap structures [15]. Thanks to the use of entire-domain basis functions, the code based on the MoM/BI-RME method is generally much faster than commercial general purpose electromagnetic simulators. For a further reduction in simulation complexity and computing time, the SIW cavity was modeled as a standard waveguide cavity using dimensions according to the basic equivalent waveguide concept for SIW structures. As a starting point, (1)–(4) are evaluated for a dual-mode configuration at the desired operation frequency of mm and a 35 GHz, using a selected hole diameter mm. This yields an SIW cavity width spacing mm (square cavity) and a slot length mm. A preliminary simulation with these values is performed setting the iniand positioning the slot close to the cavity tial slot width to mm). edge ( A simulation of this preliminary structure gives a frequency shift compared to the originally calculated SIW cavity resonance in (1). This is an expected behavior, as the presence of the slots perturbs the field distribution in the cavity given the fact that the slot dimensions are comparable to the size of the cavity. Furthermore, for the final design, impedance matching needs to be adjusted and bandwidth maximized. Therefore, an opti, mization iteration is performed including the parameters and as shown in Fig. 1(b) yielding the final design dimensions given in Table II. Fig. 5(a) shows a simulation of the wideband performance from 20 GHz to 40 GHz for the optimized design obtained with the MoM/BI-RME method. The desired pass-band is obtained at the center frequency of 35 GHz, with an impedance dB over a relative bandwidth of 9.1%. matching better than The simulation also shows another resonance occurring at 23.4 cavity mode. Comparing GHz, which is attributed to the the two transmission peaks in Fig. 5(a), the performance advantages in terms of bandwidth and roll-off introduced by the dual-mode resonances in our structure are clearly visible and correspond to the design considerations that have been taken in

Fig. 5. Simulation results for the proposed FSS structure with optimized dimensions: (a) transmission and reflection coefficients, (b) simulation of leakage with the MoM/BI-RME method.

Section II.B. Fig. 5(b) also shows the results for transmission/reflection leakage (refer to Fig. 4 for details in path definition). dB. More detailed simulation results All values are below can be found in the parametric analysis in Section IV and in the comparison with measured results in Section V. IV. PERFORMANCE STUDY In order to establish a generally applicable design procedure for structures such as the FSS proposed in this paper, a performance study based on a parametric analysis is developed. The analysis is carried out simulating different variations of the proposed structure with the help of the previously outlined MoM/BI-RME method, which has been proven to yield very accurate analysis results. As explained earlier, in the proposed FSS design, we combine two resonances originating from the selected dual-mode configuration in order to achieve the desired frequency response. In order to understand its working principle, it is important to note the following details: in the proposed structure, the cavity provides a resonance that occurs exactly at the mode (input coupling) and same frequency for both the

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W : (a) transmission

Fig. 7. Simulation results for different slot lengths : (a) transmission coefficients, (b) reflection coefficients.

Fig. 6. Simulation results for different SIW cavity widths coefficients, (b) reflection coefficients.

the mode (output coupling), respectively. When a slot is inserted into the cavity, the field in the cavity is perturbed by the presence of the slot. Considering, for example, the input slot, its mode (input) is not the same as its effect on effect on the mode (output), as the field distributions for the two the modes are orthogonal to each other. Thus, for the two modes, the presence of the slot results in a different field perturbation. In an analogous manner, also the output slot gives a different effect on the two modes. In combination, this effect leads to the development of two different distinct poles with a certain frequency spacing as shown in the simulation result in Fig. 5(a). A. Resonance Frequency For the determination of the desired resonance frequency, the and two main design parameters are the SIW cavity width the slot length . The cavity width directly defines the frequency modes. The slot length defines of the resonant the frequency, at which an incident wave is coupled into the cavity, and therefore needs to be selected close to the cavity resonance. In both cases, smaller values lead to a higher resonance

l

frequency. A plot for a parametric sweep of these two parameters is shown in Figs. 6 and 7, respectively. When increasing , the center frequency of the passband moves to higher values. Varying the slot length leads to a similar effect. Optimizing the two parameters together in the proposed demm and mm are sesign, the dimensions lected, as they provide a center operation frequency of 35 GHz as desired. B. Bandwidth For bandwidth considerations, the design aim is to increase the frequency spacing between the two resonances. In principle, any variation, which changes the perturbation of the cavity field caused by the slot, leads to a change in frequency spacing and thus bandwidth. First of all, the selection of the substrate thickness represents an important design step: if this value is changed, the effective permittivity of the substrate varies, which plays a significant role for the field caused by the slot. With a thicker substrate, the effective permittivity increases, and thus the perturbation becomes more distinct, because the field is more confined within the substrate. This leads to a higher bandwidth as shown in Fig. 8. In

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Fig. 8. Simulation results for different substrate heights h: (a) transmission coefficients, (b) reflection coefficients.

the presented structure, we select a substrate with a thickness mil (1.57 mm). Moreover, the relative values of and , respectively, largely influence bandwidth, as varying them indirectly also influences the field perturbation created by the slot. For instance, for higher values of , the slot becomes shorter in comparison, and therefore perturbs the cavity field less resulting in a smaller bandwidth. On the other hand, when increasing the slot length , the field perturbation is stronger, and so bandwidth increases. C. Impedance Matching The above described approach for maximizing bandwidth performance always has to be carried out in accordance with impedance matching specifications. If the two resonances move away from each other, the level of the return loss becomes higher. Thus, all the parameters need to be optimized in order to find a suitable trade-off between bandwidth and matching specifications. In addition to these considerations, the slot position is an important parameter: it has only a marginal influence on the slot

Fig. 9. Simulation results for different slot positions t: (a) transmission coefficients, (b) reflection coefficients.

resonance, but impedance matching is largely affected. Theoretically, the best performance in terms of impedance matching is achieved if the slot is positioned at the edges of the cavity, as the field distribution at that point is orthogonal to the slot for a metallic waveguide resonator. The effect is shown in Fig. 9, where matching becomes better and better for higher values of . However, in reality, the via holes of the SIW cavity perturb the ideal field distribution given by a metallic rectangular cavity, and so it is necessary to position the slot at a certain distance from the edge where the field is only negligibly influenced by the via posts. Note that in Fig. 9, which shows an analysis using the MoM/BI-RME method, this effect is not visible, as the simulation is carried out with the equivalent cavity model as discussed in Section III. For this reason the position of the slot from the SIW cavity was set to an approximate distance of edges. Moreover, the variation of the slot width enhances matching while maintaining a constant bandwidth as shown in Fig. 10. For a wider slot, the coupling is higher while maintaining an almost constant bandwidth, and so impedance matching is increased. Here, once more it becomes clear, why the selected mode is a good choice for the proposed structure: selecting a

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Fig. 11. A photograph of the proposed 35 GHz polarization rotating FSS including a detailed view of front and back side unit cells.

has to be selected in order to minimize the amount of radiation dB). This distance illuminating the edges of the FSS ( can be determined from the radiation pattern of the horn antenna used in the measurement and yields a minimum distance of 50 cm between each antenna and the FSS in our case. Moreover, the size of the FSS has to be chosen sufficiently large in order to let this distance fall in the far field of the antennas. In addition, an arrangement of anechoic material in the form of absorbing cones is placed around the FSS as shown in Fig. 12(a)–(c) in order to minimize the edge effects in the measurement. A. Transmission and Reflection Measurement Setup

Fig. 10. Simulation results for different slot widths s: (a) transmission coefficients, (b) reflection coefficients.

higher mode such as the mode, the available space for the slot decreases (see Fig. 2), and so the maximum slot width that can be achieved is limited, which reduces coupling and thus impedance matching. In our design, the slot position is selected to 1.55 mm with a slot width of 0.57 mm. V. EXPERIMENTAL RESULTS To verify the functionality of the proposed architecture, the FSS has been fabricated with the optimized dimensions in Table II and measured. A photograph of the prototype is shown in Fig. 11. It consists of a 30 30 array of SIW cavities with vertical and horizontal slots on the front and back sides, respectively. The experimental setup used in the measurement of the proposed FSS is shown in Fig. 12 using two standard gain K-band horn antennas and a vector network analyzer for measuring the -parameters in order to determine the transmission/reflection characteristics. A number of important considerations need to be taken into account for achieving good measurement accuracy: first of all, the distance of the FSS with respect to the antennas

All the important signal paths that have to be measured are shown in Fig. 4 for both orthogonal and non-orthogonal incidence angles. The measurement of transmission characteristics is carried out with two antennas on either side of the FSS as shown in Fig. 12(a). One antenna is facing the front side of the FSS and the second antenna its back side. A reference measurement is taken without the FSS for normalization in order to exclude the effect of the free-space transmission and any possible interferences with surrounding objects. Both cross-polarization transmission (the desired function of the FSS) and co-polarization transmission (leakage) are measured. The former is obtained by orienting the polarizations of both antennas orthogonal to the input and output slots of the FSS, respectively, while the latter is measured with both antennas in the same orientation. Fig. 12(b) shows the measurement arrangement for a reflection measurement with two adjacent antennas. In theory, the paof the transmission measurement is sufficient for rameter characterizing reflection behavior. However, in practice, in most cases, the level of the reflected signal is lower than the return loss of the antenna, as it also includes losses introduced by the free-space transmission path between the antenna and the FSS, and thus a separate configuration for the reflection measurement with two adjacent antennas needs to be used. Once again, in this setup, a reference measurement is carried out for eliminating the effects of free-space radiation from the measurement. This measurement step is carried out using a reflecting metal plane of the

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Fig. 12. A photograph of the measurement setup for the proposed 35 GHz polarization rotating FSS: (a) overall setup with network analyzer in transmission configuration; (b) reflection measurement with two adjacent antennas and crank handle setup for non-orthogonal incidence angle measurements; (c) alignment procedure with reflected laser beam.

size of the FSS. Moreover, the crosstalk between the two adjacent antennas has to be taken into account and is measured separately for reference, as the antennas are positioned very close to each other, especially in the case of a measurement of orthogonal incidence (Fig. 12(b)). In addition, the antennas need to be aligned correctly with each other and with the FSS ensuring that the maximum of the antenna beam illumination lies at the center of the FSS, which is realized in practice using a laser beam together with a small mirror in order to align the signal paths as shown in Fig. 12(c). Finally, the reflection in both coand cross-polarizations is measured with the two antennas in parallel and orthogonal polarization orientations, respectively. B. Orthogonal Incidence Fig. 13(a) and (b) show the results for the case of an orthogonal incidence for transmission and reflection, respectively. We dB for a 3-dB-bandobtain a very good matching below width of 3.2 GHz or 9.1% with a maximum insertion loss of 0.2 dB in the passband. The ripple on the measurement data can be attributed to the diffraction at the edges of the FSS: the maximum size of the FSS is limited by the PCB fabrication process, and therefore on one hand the illumination cannot be considered as locally planar, whereas on the other hand the antenna beam

Fig. 13. Transmission and reflection in simulation and measurement for incidence angles of a) 0 degrees, b) 10 degrees, c) 20 degrees.

is scattered at the edge of the FSS surface. The measured data agree very well with the simulation using the MoM/BI-RME method, which proves once more its high accuracy and practical applicability. Considering also the immense reduction in terms of computational time, the proposed simulation method represents a highly interesting solution in the design of high-end passive microwave- and millimeter-wave structures.

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Fig. 14. Leakage in simulation and measurement for the proposed FSS polarization rotating architecture: (a) leakage due to transmission in co-polarization; (b) leakage due to reflection in cross-polarization.

Fig. 14(b) shows the results for transmission and reflection leakage in comparison with the simulation results. All values dB for the entire frequency band. The slight are below deviation between simulations and measurements is due to the low level of the leakage measurement signals, as they partly fall below the return loss of the two antennas. C. Non-Orthogonal Incidence A very important criterion for the performance of FSS structures is its behavior for the case of non-orthogonal incidence angles. Usually, inclining an FSS structure, the dimensions such as e.g., slot lengths an incident wave sees, appear shorter, and therefore frequency shifts may occur. In the proposed structure, for the case of the desired transmission in cross-polarization, a horizontally polarized wave is coupled through a vertical input slot into the cavity. Varying the incidence angle by rotating the FSS around its vertical axis, the transmission antenna always sees the same slot length as for an orthogonal incidence. However, the horizontal slot at the output plane seen from the receiving antenna appears shorter and thus its resonance appears shifted to a higher frequency. In addition, the slot width of the

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input slot appears narrower, and so input coupling is decreased. The combination of these two effects leads to a decrease in bandwidth on one hand, and to a worsened matching performance on the other hand. Considering this problem, it becomes clear why it is important to describe the FSS behavior for non-orthogonal incidence angles. The measurement of non-orthogonal incidence angles is carried out in the same way as the above described reflection and transmission measurements. The rotation of the FSS is achieved using a crank handle as shown in Fig. 12(b), on which the FSS support including its anechoic embedding is mounted. This setup allows for a precise rotation of the structure while maintaining the remaining setup unchanged. The measurement is carried out for incidence angles of 0 degrees, 10 degrees, and 20 degrees, respectively. Attention needs to be paid in the case of reflection measurements as the two antennas need to be positioned at the correct angles and positions relative to the FSS structure as shown in Fig. 4. Their alignment is again achieved using a laser beam that is redirected with a small mirror positioned in the plane of the FSS. Again, for every angle, a reference measurement is taken as described for the measurement of orthogonal incidence. Fig. 13 shows the results for transmission and reflection. It is clearly visible that the FSS shows good performance for all the measured incidence angles. As previously described, the measurement setup includes an arrangement of anechoic cones around the FSS in order to reduce the edge effects. At 30 degrees and higher, the measurement result is strongly influenced by the presence of these cones, as the shadow they drop on the FSS strongly absorbs the incident energy. For an accurate measurement, a different measurement setup using a thin layer of plane anechoic material should be used, which was not available at the time of writing. Moreover, once again, the excellent prediction of the FSS performance by the MoM/BI-RME method is visible: simulations and measurements agree very well and prove the high quality of the proposed method.

VI. CONCLUSION A novel FSS architecture based on SIW technology is proposed and demonstrated providing a rotation of the polarization of an incoming plane wave by 90 degrees. An SIW cavity is used as a resonant cavity, and the energy is coupled via two orthogonal slots on front and back sides of the FSS, respectively. A MoM/BI-RME simulation code especially developed for the characterization of FSS is used to predict performance and to compare and validate the results obtained from a commercial simulation code. Moreover, a complete design procedure for the proposed structure and a parametric study of its performance, based on the MoM/BI-RME method, are presented. The proposed structure is designed at a frequency of 35 GHz with a relative bandwidth of 9.1%, a matching of less than 11 dB, and an insertion loss of 0.2 dB and thus provides a low-loss high-Q filtering screen for various polarimetric applications. Its performance is investigated in detail including the analysis of non-orthogonal incidence angles.

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ACKNOWLEDGMENT The authors would like to express their gratitude to L. Liu, H. Zhang, and Y. Zhang from the Southeast University, Nanjing, China, as well as L. Han and J. Gauthier from the École Polytechnique Montréal, Canada, for their valuable help in the fabrication and measurement process of this work. REFERENCES [1] T. K. Wu, Frequency Selective Surface and Grid Array. New York: Wiley, 1995. [2] J. C. Vardaxoglou, Frequency Selective Surfaces. New York: Wiley, 1997. [3] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley Interscience, 2000. [4] R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces—A review,” Proc. IEEE, vol. 76, pp. 1593–1615, Dec. 1988. [5] D. Deslandes and K. Wu, “Single-substrate integration technique of planar circuits and waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 593–596, Feb. 2003. [6] Y. Cassivi, L. Perregrini, K. Wu, and G. Conciauro, “Low-cost high-Q millimeter-wave resonator using substrate integrated waveguide technique,” in Proc. European Microw. Conf., Oct. 2002, pp. 1–4. [7] G. Q. Luo, W. Hong, Z.-C. Hao, B. Liu, W. D. Li, J. X. Chen, H. X. Zhou, and K. Wu, “Theory and experiment of novel frequency selective surface based on substrate integrated waveguide technology,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 12, pp. 4035–4043, Dec. 2005. [8] J. E. Roy and L. Shafai, “Reciprocal circular-polarization-selective surface,” IEEE Antennas Propag. Mag., vol. 38, no. 6, pp. 18–33, Dec. 1996. [9] G. Q. Luo, W. Hong, J. X. Chen, X. X. Yin, Z. Q. Kuai, and K. Wu, “Filtenna consisting of horn antenna and substrate integrated waveguide cavity FSS,” IEEE Trans. Antennas Propag, vol. 55, no. 1, pp. 92–98, Jan. 2007. [10] M. Bozzi, L. Perregrini, J. Weinzierl, and C. Winnewisser, “Efficient analysis of quasi-optical filters by a hybrid MoM/BI-RME method,” IEEE Trans. Antennas Propag., vol. 49, no. 7, pp. 1054–1064, Jul. 2001. [11] M. Bozzi and L. Perregrini, “Analysis of multilayered printed frequency selective surfaces by the MoM/BI-RME method,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2830–2836, Oct. 2003. [12] P. Besso, M. Bozzi, L. Perregrini, L. Salghetti Drioli, and W. Nickerson, “Deep space antenna for rosetta mission: Design and testing of the S/X-Band dichroic mirror,” IEEE Trans. Antennas Propag., vol. 51, no. 3, pp. 388–394, Mar. 2003. [13] S. Biber, M. Bozzi, O. Guenther, L. Perregrini, and L.-P. Schmidt, “Design and testing of frequency selective surfaces on silicon substrates for sub-mm wave applications,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2638–2645, Sep. 2006. [14] M. Bozzi, L. Perregrini, A. Alvarez Melcon, M. Guglielmi, and G. Conciauro, “MoM/BI-RME analysis of boxed MMICs with arbitrarily shaped metallizations,” IEEE Trans. Microwave Theory Tech., vol. 49, no. 12, pp. 2227–2234, Dec. 2001. [15] M. Bozzi, S. Germani, L. Minelli, L. Perregrini, and P. de Maagt, “Efficient calculation of the dispersion diagram of planar electromagnetic band-gap structures by the MoM/BI-RME method,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 29–35, Jan. 2005. Simone A. Winkler (S’05) was born in Mittersill, Austria, on May 4, 1982. She received the Dipl.-Ing. (M.Sc.) degree in mechatronics (with distinction) from the Johannes Kepler Universität Linz, Linz, Austria, in 2005, and is currently working toward the Ph.D. degree at the École Polytechnique Montréal, QC, Canada. Her dissertation project concerns a self-oscillating mixing technique for passive millimeter-wave imaging receiver applications. From July to November 2004, she spent a research term with the University of Waikato, Hamilton, New Zealand. Also, in 2008, she was an Invited Researcher at the State Key Lab of Millimeter-Waves, Southeast University, Nanjing, China. Her main research

interests are active circuit design, receiver design, and passive millimeter-wave imaging. Ms. Winkler is a student member of the European Microwave Association (EuMA) and the Austrian Electrotechnical Association (ÖVE). She received the IEEE MTT-S Graduate Student Fellowship 2009 and was selected to the first rank throughout Québec for the FQRNT 2007–2008 Merit Scholarship Program. Moreover, she was the recipient of the 2006 Best Paper Award presented at the IEEE Canadian Conference on Electrical and Computer Engineering (CCECE) and the 2006 Hedy-Lamarr Award for her ongoing doctoral research. She was also the recipient of the 2005 Erwin–Wenzl–Preis, the 2005 GIT–Förderpreis, and the 2005 tech2b Award for her diploma thesis. She was also selected for a long-term doctoral fellowship from the Austrian Academy of Science and serves as the vice-president for the student committee of the Centre de Recherche en Électronique Radiofréquence (CRÉER) Québec, Canada. Wei Hong (M’92–SM’07) received the B.S. degree from the University of Information Engineering, Zhengzhou, China, in 1982, and the M.S. and Ph.D. degrees from Southeast University, Nanjing, China, in 1985 and 1988, respectively, all in radio engineering. Since 1988, he has been with the State Key Laboratory of Millimeter Waves, Southeast University, and is currently a professor of the School of Information Science and Engineering. In 1993, 1995, 1996, 1997 and 1998, he was a short-term Visiting Scholar with the University of California at Berkeley and at Santa Cruz, respectively. He has been engaged in numerical methods for electromagnetic problems, millimeter wave theory and technology, antennas, electromagnetic scattering, inverse scattering and propagation, RF front-end for mobile communications and the parameters extraction of interconnects in VLSI circuits, etc. He has authored and coauthored over 200 technical publications, and authored two books of Principle and Application of the Method of Lines (in Chinese, Southeast University Press, 1993) and Domain Decomposition Methods for Electromagnetic Problems (in Chinese, Science Press, 2005). He twice awarded the first-class Science and Technology Progress Prizes issued by the Ministry of Education of China in 1992 and 1994 respectively, awarded the fourth-class National Natural Science Prize in 1991, and the first-class, second-class and third-class Science and Technology Progress Prize of Jiangsu Province. Besides, he also received the Foundations for China Distinguished Young Investigators and for “Innovation Group” issued by NSF of China. Dr. Hong is a senior member of CIE, Vice-Presidents of Microwave Society and Antenna Society of CIE, and served as the reviewer for many technical journals such as the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IET Proc.-H, Electron. Lett. etc., and now serves as an Associate Editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. Maurizio Bozzi (S’98-M’01) was born in Voghera, Italy, on June 1, 1971. He received the “Laurea” degree in electronic engineering and Ph.D. degree in electronics and computer science from the University of Pavia, Pavia, Italy, in 1996 and 2000, respectively. Since March 2002, he has been an Assistant Professor of electromagnetics with the Department of Electronics, University of Pavia, where he currently teaches the courses of Numerical Techniques for Electromagnetics and of Computational Electromagnetics and Photonics. He has held research positions with various European and American universities, including the Technical University of Darmstadt, Darmstadt, Germany, the University of Valencia, Valencia, Spain, and the Polytechnical University of Montreal, Montreal, QC, Canada. His main research activities concern the development of numerical methods for the electromagnetic modeling of microwave and millimeter-wave components (frequency-selective surfaces, reflectarrays, electromagnetic bandgap structures, metallic and integrated waveguide components, and printed microwave circuits). He authored the chapter “Periodic Structures” in the Wiley Encyclopedia of RF and Microwave Engineering (2005) and co-edited the book Periodic Structures (2006). Moreover, he authored more than 45 journal papers and more than 130 conference papers. Prof. Bozzi was the recipient of the Best Young Scientist Paper Award presented at the XXVII General Assembly of the International Union of Radio Science (URSI) in 2002, and the MECSA Prize for the best paper presented by a young researcher at the Italian Conference on Electromagnetics (XIII RINEM) in 2000.

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Ke Wu (M’87–SM’92–F’01) is a Professor of electrical engineering, and Tier-I Canada Research Chair in RF and millimeter-wave engineering at the Ecole Polytechnique (University of Montreal). He also holds the first Cheung Kong Endowed Chair Professorship (visiting) at the Southeast University, the first Sir Yue-Kong Pao Chair Professorship (visiting) at Ningbo University, and an Honorary Professorship at the Nanjing University of Science and Technology, and the City University of Hong Kong, China. He has been the Director of the Poly-Grames Research Center and the founding Director of the Center for Radiofrequency Electronics Research of Quebec (Regroupement stratégique of FRQNT). He has authored or coauthored over 700 referred papers, and a number of books/book chapters and patents. His current research interests involve substrate integrated circuits (SICs), antenna arrays, advanced CAD and modeling techniques, and development of low-cost RF and millimeter-wave transceivers and sensors for wireless systems and biomedical applications. He is also interested in the modeling and design of microwave photonic circuits and systems.

Dr. Wu is a member of Electromagnetics Academy, the Sigma Xi Honorary Society, and the URSI. He has held key positions in and has served on various panels and international committees including the chair of technical program committees, international steering committees and international conferences/symposia. In particular, he will be the general chair of the 2012 IEEE MTT-S International Microwave Symposium. He has served on the editorial/review boards of many technical journals, transactions and letters as well as scientific encyclopedia including editors and guest editors. He is currently the chair of the joint IEEE chapters of MTTS/APS/LEOS in Montreal. He is an elected IEEE MTT-S AdCom member for 2006–2012 and serves as the chair of the IEEE MTT-S Member and Geographic Activities (MGA) Committee. He was the recipient of many awards and prizes including the first IEEE MTT-S Outstanding Young Engineer Award, the 2004 Fessenden Medal of the IEEE Canada and the 2009 Thomas W. Eadie Medal of the Royal Society of Canada. He is a Fellow of the Canadian Academy of Engineering (CAE) and a Fellow of the Royal Society of Canada (The Canadian Academy of the Sciences and Humanities). He is an IEEE MTT-S Distinguished Microwave Lecturer from January 2009 to December 2011.

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Design and Analysis of a Tunable Miniaturized-Element Frequency-Selective Surface Without Bias Network Farhad Bayatpur, Student Member, IEEE, and Kamal Sarabandi, Fellow, IEEE

Abstract—A novel, tunable miniaturized-element frequency-selective surface that does not require additional bias networks is presented. This spatial filter is composed of two wire-grids printed on opposite sides of a substrate and connected to each other with an array of varactors using plated via holes. Varactor diodes are positioned between the grids. Via sections and metallic pads are fabricated and create a dc path for biasing the varactors with the grids themselves. This configuration eliminates the need for any additional network, and therefore resolves the design difficulties associated with the spurious response of the bias network. An equivalent circuit model is developed to facilitate the design procedure. Full-wave numerical simulations are used to validate the results based on the circuit model. Simulations show that by altering the capacitance of the varactors from 0.1 to 1 pF, a frequency tunability from 8 to 10 GHz with an almost constant bandwidth can be achieved. Index Terms—Electronic tuning, frequency-selective surface (FSS), varactor.

I. INTRODUCTION

F

REQUENCY-SELECTIVE surface (FSS) structures are the free-space counterparts of microwave filters in a transmission line. Once exposed to electromagnetic radiation, an FSS is expected to act like a filter independent of the angle of incidence and polarization. Frequency-selective surfaces have been used traditionally in stealth technology for reducing the radar cross-section (RCS) of communication and radar antennas as well as protecting receivers of such systems from interfering signals and jamming. Since the early 1960’s, because of potential military applications, FSS structures have been the subject of intensive study [1]–[3]. FSS structures are commonly made up of planar, periodic metallic arrays printed on dielectric substrates. Frequency behavior of an FSS is entirely determined by the geometry of the surface in one period (unit cell) provided that the surface size is Manuscript received July 08, 2009; revised September 10, 2009; accepted October 02, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. This work was supported in part by the Alexander von Humboldt Foundation. F. Bayatpur was with the Radiation Laboratory, Department of Electrical Engineering and Computer Science, The University of Michigan, Ann Arbor, MI 48109-2122 USA. He is now with the Center of Excellence for Advanced Materials, Department of Mechanical and Aerospace Engineering, University of California, La Jolla CA 92093-0416 (e-mail: [email protected]). K. Sarabandi is with the Radiation Laboratory, Department of Electrical Engineering and Computer Science, The University of Michigan, Ann Arbor, MI 48109-2122 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2041173

infinite. Over the years, a variety of FSS elements were introduced for bandpass, [4]–[6], and band-stop, [7], applications. Independent of their different geometries, the principle of operation of conventional FSS structures is based on the electromagnetic resonance of the unit cell geometry. The induced current on metallic traces and geometries are significantly enhanced when geometries’ dimensions or perimeters are an inor , respectively [1]–[3]. teger multiple of A new approach for designing frequency-selective surfaces recently has been proposed in [8] and further developed in [9]. In this method, instead of using resonant structures as the building blocks of the FSS, special sub-wavelength unit cells are used. These unit cells interact with the incident wave in the fundamental TEM mode and act as lumped inductive and capacitive elements. By arranging the unit cells properly, coupling among different layers or structures within a single layer can be utilized to achieve a desired frequency response. The creation of such lumped elements is possible in the sub-wavelength regime or smaller in diwhere the unit cells are on the order of mension. The most unique feature of the proposed approach is its ability to create a localized frequency-selective behavior [9]. This allows for the creation of FSSs with high-order frequency responses that have very low sensitivities with respect to the incidence angle of the wave. The localized characteristic also implies that the new surfaces, unlike traditional designs, can operate even with small dimensions (compared to the wavelength) and still maintain their frequency-selective properties. Moreover, given the small dimensions of their elements, the frequency responses of the surfaces are harmonic-free up to a frequency where their elements’ dimensions become comparable with the wavelength. The ability to electrically tune or alter the frequency response is also a practical feature in FSS design. Generating dynamic frequency behavior requires the reactive characteristics of the FSS to change with a tuning voltage or current. The literature concerning tunable FSS includes studies for the application of ferrite substrates [10]–[16], some type of liquid materials as the substrate [17]; and use of plasma instead of metallic traces [18]. Also, the theory of tunability by using the GA algorithm to control an array of switches is demonstrated in [19]. A more recent method uses microelectromechanical systems (MEMS) technology to build capacitors or switches to vary the shape of the unit cell [20], [21]. A well-established method for tuning microwave filters uses solid-state diodes like varactors diodes, PIN diodes, and

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BAYATPUR AND SARABANDI: DESIGN AND ANALYSIS OF A TUNABLE MINIATURIZED-ELEMENT FSS WITHOUT BIAS NETWORK

Schottky diodes [22]–[25]. Previously, this approach has been applied experimentally in the design of grid arrays [26]–[29]. The grid arrays are FSSs with series resonance characteristics resonator) which were employed for active beam control ( applications, i.e., beam shaping (by changing the phase of the incident plane-wave) and wave amplitude control. For biasing the active components, these grid arrays require a separate circuitry. This complementary circuit basically shorts all the grid elements that are in the same row and connects them to a common dc voltage. The grid, consequently, works only for the polarization perpendicular to its rows, and therefore, the array is polarization dependent. Moreover, because of being of the resonators with reflective characteristics, such type of grids are typically less efficient in transmit mode (amplitude control). The reason for this low level of performance is that the bandpass spatial filtering has not been the goal of the past work. Nevertheless, biasing the varactors properly is still an area of on-going research because of the number of varactors needed to build a tunable array. Also, a major difficulty in this research is the effect of bias network on the FSSs’ frequency response. Switchable grid arrays loaded with PIN and Schottky diodes in larger scales were later appeared in [30] and [31], respectively. These arrays yet suffered from the issues of biasing and poor selectivity. A varactor-tuned, dipole-array FSS using a resistive grid for individually biasing the varactors was proposed in [32]; however, the fabrication cost/complexity and the loss associated with the bias grid remained a problem. Finally, in [33], a reconfigurable FSS was provided with a simple method for individual biasing of the varactors. Although being practical, this method required additional components (resistors) besides varactors, thus making its design less cost-effective. This article proposes a tunable, bandpass frequency-selective surface with an embedded bias network. In this design, the varactors are biased in parallel and thus are controlled individually.

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Fig. 1. Unit cell drawing for the tunable FSS for operation in the free-space environment- The FSS is comprised of two wire-grids translated with respect to one another by half of the dimension of the unit cell. It also includes a varactor which is biased using the two grids along with a pad and a via-hole that make a dc circuit for individual biasing of the varactor.

Fig. 2. Circuit model of the tunable FSS consisting of two parallel inductive branches connected to one another through a capacitor (varactor) in the Wheatstone bridge fashion. The FSS substrate is modeled as a small piece of transmission line (Z , l ) with an effective electrical length of l 5 .



TABLE I DESIGN PARAMETERS OF THE TUNABLE FSS AT X-BAND

II. FSS’S CONFIGURATION AND OPERATION MECHANISM The new FSS is a double-sided circuit-board comprised of two wire-grids built on opposite sides of a very thin substrate. The unit cell drawing of this FSS is shown in Fig. 1. As illustrated, the grids are laterally shifted with respect to one another by half of the unit cell size in both and directions. A small pad is fabricated inside the grid that lies on front side of the substrate. The pad is connected at a specific point to the bottom grid through a metalized, vertical post (via). Finally, each unit cell is loaded with a varactor placed between the pad and the grid surrounding the pad. The bandpass characteristic of the proposed FSS structure can be described using circuit theory: As shown in [34], wire-grids are inductive which together with the varactor create a circuit topology similar to that of the Wheatstone bridge. Equivalent circuit model of the FSS is shown in Fig. 2. In this model, inand represent the metallic traces of the top grid, ductors and model the traces of the bottom grid. The varand actor is shown by in the circuit. The pad behaves like a small and piece of transmission line which is modeled by inductor [35]. Other elements of the circuit model are the capacitor representing the via and the mutual inductance . inductor This mutual coupling is created as the pad and the grid in bottom

are overlaid on top of one another [36]. Given this arrangement of inductors and capacitors, a bandpass response is produced if the bridge is unbalanced. This happens when the ratio of the two inductors connected to the left terminal ( and ) differs from that of the two inductors connected to the right terminal ( and ). As described above, on one side all the varactors are connected to the top grid and on the other side are attached to the bottom grid. Hence, by applying a dc voltage between the two grids, all the varactors are biased at the same voltage (parallel biasing). Obviously, the wire-grids are functioning simultaneously as the elements of the FSS and the bias network. III. FSS DESIGN AND OPTIMIZATION This section outlines the FSS design procedure deploying full-wave and circuit simulators. The design goals include: 1) ) in order to achieving a small unit cell dimension ( obtain a better uniformity and thus less sensitivity to the incidence angle [9]; 2) achieving a reasonably large tuning range while keeping the capacitance variations within a practical range (0.1–1 pF).

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TABLE II CIRCUIT MODEL VALUES FOR THE TUNABLE FSS AT X-BAND

Fig. 3. Free-space simulations of the tunable FSS using the periodic boundary condition setup in HFSS at normal incidence compared to ADS circuit simulations– The HFSS and ADS parameters values are provided in Tables I and II, respectively.

As mentioned earlier, ADS analysis of the circuit model reveals that a bandpass response can be produced by the FSS provided that the bridge is unbalanced. This requirement can be accomplished in practice by positioning the via so that the unit cell is asymmetrical (unbalanced). The proposed unit cell is shown in Fig. 1. After finding an appropriate place for the via, other design parameters are optimized to further improve the bandpass characteristics of the FSS. The simulations use the periodic boundary conditions (PBC) in Ansoft HFSS. This setup simulates a large array of unit cells on an FSS that are exposed to a plane-wave incident at an arbitrary angle. The following simulations assume a plane-wave polarized parallel to the pad. The effects associated with the dielectric/copper loss and a finite -factor of the varactor are included in the simulation model. The model, however, does not account for the parasitic effects in the varactors. A. Optimized FSS For operation at X-band, a unit cell size ( , ) of 4.8 mm is chosen according to the insights gained from our previous work, [9], and to attain the afore mentioned design goals. With such periodicity, the initial values assigned to the width of the wires ( for top grid and for bottom grid) become 0.1 mm ) that which is well above the minimum feature size ( can be fabricated using the standard copper etching process. The optimization is then focused on other parameters of the design including representing the width of the pad, as the length of the pad, and the substrate thickness, . The gap between the pad and the top grid is kept constant and equal to 0.354 mm

Fig. 4. Free-space simulations of the tunable FSS using the periodic boundary conditions setup in HFSS– Frequency tuning from 8 to 10 GHz with an almost , 0.3, and fixed bandwidth is achieved by changing the capacitance, C 0.1 pF; Q .

= 25

=1

for the following simulations (see Fig. 1). The substrate used is Rogers RT/duroid 5880 material with a 1/2 Oz. copper cladding. Table I provides the optimized values for the parameters. Given these values, the frequency response of the FSS was calculated using HFSS. The simulated results compared to those obtained by ADS are given in Fig. 3, showing a good agreement between the FSS and its circuit model. The circuit model values are shown in Table II. These values are the result of a curve-fitting process using ADS. First, each inductive wire layer is analyzed in HFSS individually. In the frequency range considered here, the transmission response of each wire layer has a linear characteristic in decibel scale. From the slope of the transmission graph, the inductance value of each and ). Next, the pad, which wire is calculated ( acts like a small piece of transmission line, is analyzed and its parameters are calculated ( and ). Finally, the position of the via post on the bottom grid and the point where the lumped capacitor is connected to the top grid determine the ratios and (see Fig. 1). These analyses determine the initial values for the parameters of the circuit model which are fine tuned using an optimization algorithm in ADS to get the best fit between the circuit response and the HFSS response. The tunability of the FSS is shown in Fig. 4. The frequencyband 8–10 GHz is swept by varying the capacitance from 1 to 0.1 pF. Fig. 5 shows the scan performance of the FSS. The FSS preserves its frequency-selective characteristic; however a lower selectivity is observed while scanning at a 45 angle. In the next section, we present the results of sensitivity analyses performed over other design parameters. These results are then interpreted using the circuit model, demonstrating the accuracy of the model.

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Fig. 5. Free-space simulations of the tunable FSS using the periodic boundary conditions setup in HFSS–Scan performance of the surface at angles 30 and 45 are compared to the case of a normal incidence (the thicker line) with a polarization along the pad. Fig. 7. Free-space simulations of the tunable FSS using the periodic boundary conditions setup in HFSS– response sensitivity to the pad width (w ) is shown for : ; w : , 0.3, 0.5, 0.7, and 0.9 mm. Other parameters are set to:   : ;d : ; D and D : ;C ; Q .

=01 = 0 12 mm = 25

= 3 73 mm

= 4 8 mm

= 0 24mm = 1 pF

Fig. 6. Free-space simulations of the tunable FSS using the periodic boundary conditions setup in HFSS– esponse sensitivity to the pad length (d) is shown for d : , 3.13, 3.33, 3.53, and 3.73 mm. Other parameters are set to:  : ; : ;w : ; D and D : ;C ; Q .

= 2 93 0 24 mm = 0 12 mm = 0 5 mm = 25

= = 4 8 mm = 1 pF

B. Sensitivity Analysis and Circuit Analogy The FSS design approach is investigated in more depth in this section through a number of sensitivity analyses. In addition to designing an FSS with optimum characteristics, such analyses give insight into the behavior of the FSS which can also be explained qualitatively using the circuit model. Given the knowledge obtained from the sensitivity analyses and the circuit model, the behavior of the FSS can be predicted readily with some level of accuracy. The optimization criteria, as discussed previously, consist of staying within the fabrication limits while decreasing the unit cell size and minimizing the insertion loss. As for the filtering characteristics, a roll-off factor of more than 10 dB/GHz around the center frequency of the passband is sought. The sensitivity of the FSS’s response with respect to the pad length is shown in Fig. 6. In these simulations, the length is varied slightly from

Fig. 8. Free-space simulations of the tunable FSS using the periodic boundary conditions setup in HFSS– response sensitivity to top grid width ( ) is shown : , 0.24, 0.34, 0.44, and 0.54 mm. Other parameters are set to: for  : ;w : : ; ; D and D d : ; C ;Q .

= 0 14 = 3 73 mm = 0 12 mm = 1 pF = 25

= 0 5 mm

= 4 8 mm

the optimum value of 3.73 mm. Increasing the length decreases the center frequency, a result of a larger bridge inductance, . Fig. 7 shows the effect of the pad width which is also related to the inductor . A wider pad produces a smaller inductance [35], thus pushing the passband to higher frequencies. Changing , to the pad dimensions also affects the shunt capacitance, some extent. However, for this structure, the inductive effect ( ) dominates the change in capacitance ( ). As mentioned above, a wire-grid produces an inductive response which depends on the width of the traces constructing the grid. Wider traces result in lower inductance [34]. According

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in turn pushes the notch to lower frequencies according to the circuit simulations. For this specific circuit, is dominant over ; therefore, the center frequency goes down. Nevertheless, the effect of smaller inductance, i.e., shift to higher frequencies, is observed again at the lower band frequencies (see Fig. 9). Finally, the effect of the substrate thickness is provided in Fig. 10 which is also validated by the circuit model. A thicker substrate corresponds to a longer transmission line, ( , ), in the model. According to the circuit model, increasing the length of the transmission line decreases the center frequency of the passband slightly. The full-wave simulations (Fig. 10), however, predict a much faster change in the center frequency. This is because increasing the thickness reduces the coupling between the pad and the top wire-grid. Circuit model well verifies that decreasing the coupling coefficient, , changes the center frequency rapidly to smaller values. IV. CONCLUSION Fig. 9. Free-space simulations of the tunable FSS using the periodic boundary conditions setup in HFSS– response sensitivity to the bottom grid width ( ) is : , 0.22, 0.32, 0.42, and 0.52 mm. Other parameters set to: shown for   : ;d : ;w : ; D and D : ; C ;Q .

= 0 12 = 0 24 mm = 3 73 mm = 1 pF = 25

= 0 5 mm

= 4 8 mm

In this article, design of a tunable frequency-selective surface with sub-wavelength periodicity is presented. The tunability is achieved by using a solid-state varactor diode per period. This research demonstrates a new architecture that enables biasing the varactors in parallel without any external biasing circuitry. The concept for the new biasing architecture consists of two wire layers along with connecting via sections built on a very thin substrate. This simple, practical method allows for implementation of large-scale tunable surfaces with high performance. The sensitivity analyses of different parameters in the numerical simulations of the FSS structure verify the application of the new method. The lack of experimental data is purely a financial restriction as testing in a free-space environment requires a few thousands of varactors. An alternative, less expensive approach is the waveguide test, but in this particular case, the waveguide flanges short-circuit the varactors. As a result, the varactors cannot be biased. REFERENCES

Fig. 10. Simulations of the tunable FSS using the periodic boundary conditions setup in HFSS– Response sensitivity to the substrate thickness (t) is shown for : ; t : , 0.2, 0.4, 0.6, and 0.8 mm. Other parameters set to:  : :  ;d : ;w : ; ; D and D . C ;Q

= 01 = 0 24 mm = 3 73 mm = 1 pF = 25

= 0 5 mm

= 0 12 mm = 4 8 mm

to circuit model, decreasing the inductance ( ) increases the center frequency of the response. Sensitivity of FSS to the width of the top and bottom grids is provided in Figs. 8 and 9. As shown, the FSS demonstrates a little dependence on the changes in the top grid (Fig. 8). The notch and the passband frequencies remain almost fixed. However, the effect of smaller inductance predicted by circuit model is clearer at lower band frequencies. Fig. 9, however, shows a different, unexpected behavior. By increasing the width of the bottom grid, the center frequency of the passband decreases. This behavior happens because the bottom grid ( ) is coupled with the pad inductance ( ). Decreasing also decreases the coupling coefficient ( ) which

[1] B. A. Munk, Frequency-Selective Surfaces: Theory and Design. New York: Wiley, 2000. [2] T. K. Wu, Frequency-Selective Surface and Grid Array. New York: Wiley, 1995. [3] J. C. Vardaxoglou, Frequency-Selective Surfaces: Analysis and Design. Taunton, U.K.: Research Studies Press, 1997. [4] T.-K. Wu, “Bandpass Frequency Selective Surface,” U.S. patent 5384575, Jan. 24, 1995. [5] T.-K. Wu, “High Q bandpass structure for the selective transmission and reflection of high frequency radio signals,” U.S. patent 5103241, Apr. 7, 1992. [6] E. L. Pelton and B. A. Munk, “Periodic antenna surface of tripole slot elements,” U.S. patent 3975738, Aug. 17, 1976. [7] T.-K. Wu, “Double-loop frequency selective surfaces for multi frequency division multiplexing in a dual reflector antenna,” U.S. patent 5373302, Dec. 13, 1994. [8] K. Sarabandi and N. Behdad, “A frequency selective surface with miniaturized elements,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1239–1245, May 2007. [9] F. Bayatpur and K. Sarabandi, “Single-layer, high-order, miniaturizedelement frequency-selective surfaces,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 4, pp. 774–781, Apr. 2008. [10] T. K. Chang, R. J. Langley, and E. A. Parker, “Frequency selective surfaces on biased ferrite substrates,” IEE Electron. Lett., vol. 30, no. 15, pp. 1193–1194, Jul. 1994. [11] G. Y. Li, Y. C. Chan, T. S. Mok, and J. C. Vardaxoglou, “Analysis of frequency selective surfaces on biased ferrite substrate,” in IEEE AP-S Dig., Jun. 1995, vol. 3, pp. 1636–1639.

BAYATPUR AND SARABANDI: DESIGN AND ANALYSIS OF A TUNABLE MINIATURIZED-ELEMENT FSS WITHOUT BIAS NETWORK

[12] Y. Liu, C. G. Christodoulou, P. F. Wahid, and N. E. Buris, “Analysis of frequency selective surfaces with ferrite substrates,” in IEEE AP-S Dig., Jun. 1995, vol. 3, pp. 18–23. [13] 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. [14] Y. Liu, C. G. Christodoulou, and N. E. Buris, “Fullwave analysis method for frequency selective surfaces on ferrite substrates,” J. Electromagn. Waves Applicat., vol. 11, no. 5, pp. 593–607, 1997. [15] E. A. Parker and S. B. Savia, “Active frequency selective surfaces with ferroelectric substrates,” in Proc. Inst. Elect. Eng. Microwaves, Antennas, Propag., Apr. 2001, vol. 148, pp. 103–108. [16] J. C. Vardaxoglou, “Optical switching of frequency selective surface bandpass response,” IEE Electron. Lett., vol. 32, no. 25, pp. 2345–2346, Dec. 1996. [17] A. C. de C. Lima, E. A. Parker, and R. J. Langley, “Tunable frequency selective surface using liquid substrates,” IEE Electron. Lett., vol. 30, no. 4, pp. 281–282, Feb. 1994. [18] T. Anderson, I. Alexeff, and J. Raynolds, “Plasma frequency selective surfaces,” in Proc. IEEE Int. Conf. Plasma Science, Jun. 2003, p. 237. [19] J. A. Bossard, D. H. Werner, T. S. Mayer, and R. P. Drupp, “A novel design methodology for reconfigurable frequency selective surfaces using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1390–1400, Apr. 2005. [20] B. Schoenlinner, A. Abbaspour-Tamijani, L. C. Kempel, and G. M. Rebeiz, “Switchable low-loss RF MEMS Ka-band frequency-selective surface,” IEEE Trans. Microw. Theory Tech., vol. 52, pp. 2474–2481, Nov. 2004. [21] J. P. Gianvittorio, J. Zendejas, Y. Rahmat-Samii, and J. Judy, “Reconfigurable MEMS-enabled frequency selective surfaces,” IEE Electron.Lett., vol. 38, no. 25, pp. 1627–1628, Dec. 2002. [22] I. C. Hunter and J. D. Rhodes, “Electronically tunable microwave bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. 30, no. 9, pp. 1354–1360, Sep. 1982. [23] M. Makimoto and M. Sagawa, “Varactor tuned bandpass filters using microstrip-line ring resonators,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1986, pp. 411–414. [24] S. R. Chandler, I. C. Hunter, and J. G. Gardiner, “Active varactor tunable bandpass filter,” IEEE Microw. Guided Wave Lett., vol. 3, no. 3, pp. 70–71, Mar. 1993. [25] A. R. Brown and G. M. Rebeiz, “A varactor-tuned RF filter,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 7, pp. 1157–1160, Jul. 2000. [26] W. W. Lam, H. Z. Chen, K. S. Stolt, C. F. Jou, N. C. Luhmann Jr., and D. B. Rutledge, “Millimeter-wave diode-grid phase shifters,” IEEE Trans. Microw. Theory Tech., MTT, vol. 36, no. 5, pp. 902–907, 1988. [27] R. J. Langley and E. A. Parker, An equivalent circuit study of a PIN diode switched active FSS Rep. Br. Aerosp. plc, 1990. [28] T. K. Chang, R. J. Langley, and E. A. Parker, “Active frequency-selective surfaces,” in Proc. IEEE Microwaves, Antennas Propag., Feb. 1996, vol. 143, pp. 62–66. [29] H. X. King, N. C. Luhmann Jr., X. H. Qin, L. B. Sjogren, W. Wu, D. B. Rutledge, J. Maserjian, U. Lieneweg, C. Zah, and R. Bhat, “Millimeterwave quasi-optical active arrays,” in Proc. Int. Conf. Space Terahertz Technol., 2nd, 1991, pp. 293–305. [30] K. D. Stephan, F. H. Spooner, and P. F. Glodsmith, “Quasioptical millimeter-wave hybrid and monolithic PIN diode switches,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 10, pp. 1791–1798, 1993. [31] L. B. Sjogren, H. N. Liantz Liu, F. Wang, T. Liu, W. Wu, X. H. Qin, E. Chung, C. W. Domier, and N. C. Luhmann Jr., “A monolithic millimeter-wave diode array beam transmittance controller,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 10, pp. 1782–1790, 1993.

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[32] C. Mias, “Varactor-tunable frequency selective surface with resistivelumped-element biasing grids,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 9, Sep. 2005. [33] F. Bayatpur and K. Sarabandi, “A tunable metamaterial frequency-selective surface with variable modes of operation,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 6, pp. 1433–1438, Jun. 2009. [34] W. R. Smythe, Static and Dynamic Electricity. New York: McGraw Hill, 1968, pp. 63–120. [35] J. A. Kong, Electromagnetic Wave Theory. Cambridge, MA: EMW Publishing, 2000, pp. 180–274. [36] F. Bayatpur and K. Sarabandi, “Multipole spatial filters using metamaterial-based miniaturized-element frequency-selective surfaces,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 2742–2747, Dec. 2008.

Farhad Bayatpur (S’06) received the B.Sc. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 2005 and the M.S. and the Ph.D. degrees in electrical engineering from the University of Michigan at Ann Arbor, in 2007 and 2009, respectively. Currently, he is a Postdoctoral Researcher at the Center of Excellence for Advanced Materials (CEAM), Department of Mechanical and Aerospace Engineering, University of California, San Diego.

Kamal Sarabandi (S’87–M’90–SM’92–F’00) received the B.S. degree in EE from Sharif University of Technology, Tehran, Iran, in 1980, the M.S. degree in EE (1986), the M.S. degree in mathematics, and the Ph.D. degree in electrical engineering from The University of Michigan, Ann Arbor, in 1989. He is the Rufus S. Teesdale Professor of Engineering and Director of the Radiation Laboratory in the Department of Electrical Engineering and Computer Science, University of Michigan. His research areas of interest include microwave and millimeter-wave radar remote sensing, metamaterials, electromagnetic wave propagation, and antenna miniaturization. He has published many book chapters and more than 170 refereed journals and 420 conference papers 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. Dr. Sarabandi is a member of NASA Advisory Council appointed by the NASA Administrator. He is an Adcom member and served as a Vice President of the IEEE Geoscience and Remote Sensing Society (GRSS). He is the founding editor of the IEEE Book Series on Geoscience and Remote Sensing and is serving on the Editorial Board of the IEEE PROCEEDINGS. He was the recipient of the Henry Russel Award from the Regent of The University of Michigan. In 2005 he received two prestigious awards, namely, 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 for Senior U.S. Scientist from The Alexander von Humboldt Foundation of Germany.

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A Novel Band-Reject Frequency Selective Surface With Pseudo-Elliptic Response Amir Khurrum Rashid and Zhongxiang Shen, Senior Member, IEEE

Abstract—A novel frequency selective surface (FSS) exhibiting pseudo-elliptic band-reject response is presented. The proposed FSS consists of a two-dimensional periodic array of microstrip lines. For an incident wave of linear polarization perpendicular to the printed lines, virtual magnetic walls are formed between the microstrip lines. Since a microstrip line shielded with magnetic side walls supports two quasi-TEM modes, the two-dimensional array effectively forms a dual-mode resonator. The elliptic response is thus realized by coupling the incident wave to both modes of the resonator. A wideband pseudo-elliptic response is achieved by including higher-order modes of the array. To analyze this FSS, we treat it as a cascaded junction of air-to-microstrip line discontinuities, and investigate the problem using the full-wave mode-matching method. We provide useful design guidelines for both narrow-band and wide-band FSSs, while explaining the effect of various geometrical parameters on the performance of this FSS. Narrow-band and wideband FSS prototypes are fabricated and measured results validate our design concept. This new class of FSS is easy-to-assemble, exhibits a superior performance, and is thus potentially useful for many practical applications. Index Terms—Band-reject response, dual-mode resonator, elliptic response, frequency selective surface, microstrip line shielded with magnetic side walls.

I. INTRODUCTION REQUENCY selective surfaces (FSS) have traditionally been widely used in antenna radomes, reflectors, sub-reflectors, and spatial filters [1], [2]. Classical FSS consists of a two-dimensional periodic array of certain shapes that are either printed on a dielectric substrate or etched out of a conductive layer. These are classified as inductive or capacitive surfaces. Filtering response of a single layer FSS is known to suffer from poor selectivity and narrow bandwidth. Cascading single layers with inclusion of a dielectric spacer is the most common technique to increase the bandwidth, and significant improvement of the filtering response has been reported. Most of the conventional multi-layer designs following Butterworth or Chebyshev function exhibit a monotonic increase in their out-of-band rejection. It is usually difficult to obtain elliptic response from the cascaded single-layer FSS structures. The level of cross-coupling required for producing elliptic filtering response is generally not achievable in this case, and multi-

F

Manuscript received May 10, 2009; revised September 14, 2009; accepted October 01, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (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.2010.2041167

layer designs are usually comprised of only the direct-coupled resonators. Recently, a quasi-elliptic band-pass FSS has been proposed [3] that uses substrate integrated waveguide (SIW)based dual-mode resonator. Introducing cross coupling through dual-mode resonators has conventionally been a very popular technique in the design of waveguide elliptical filters [4]–[6]. Dual-mode resonators realized in SIW and microstrip line are, however, very limited to the respective structures and it is difficult to extend them to applications in spatial FSSs, as it will result in bulky structures that may not be feasible for many practical applications. To the best of our knowledge, no significant research work has been reported on the elliptical band-reject FSSs. We present a novel band-reject FSS that exhibits pseudo-elliptic filtering response. This has been realized with the help of a novel dual-mode resonator based on shielded microstrip line with magnetic side walls. The proposed FSS structure is illustrated in Fig. 1, which shows a two-dimensional periodic array of microstrip lines. It can be constructed by stacking up a number of identical printed circuit boards (PCB) on a common plane. Each PCB is printed with a one-dimensional periodic array of microstrip lines of width . Thickness of the printed and strip line is kept negligibly small. Periods along the - axes are denoted by and , respectively. Thickness of the substrate is denoted by . Under a linear polarization perpendicular to the strips as indicated in Fig. 1, virtual magnetic walls are formed between the microstrip lines assuming that the adjacent strips are excited in phase. Effectively, a unit cell of this periodic array consists of a microstrip line shielded with magnetic side walls. Since a microstrip line shielded with perfect magnetic conductor (PMC) side walls supports two quasi-TEM modes, the operating principle of the proposed FSS turns out to be similar to that of elliptic waveguide filters employing a dual-mode resonator. The structure of our proposed FSS is easy to fabricate and has not been previously studied in the past. In this work we use the full-wave mode-matching method to analyze the proposed FSS. The formulation of this problem is given in Section II. Based on the calculated S-parameters of the air-to-microstrip line discontinuities, we provide an equivalent circuit model of this FSS. The same section also includes a discussion that relates this FSS to a well established topology of elliptical microwave filters whereby the same design and synthesis procedures become applicable to the proposed FSS. To validate this novel concept, we design and fabricate two structures. Measured results are in a good agreement with those calculated using our full-wave analysis. Some useful extensions of this work have also been suggested in the conclusion section.

0018-926X/$26.00 © 2010 IEEE

RASHID AND SHEN: A NOVEL BAND-REJECT FREQUENCY SELECTIVE SURFACE WITH PSEUDO-ELLIPTIC RESPONSE

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Fig. 2. A cascaded junction of air-to-microstrip line discontinuities.

It is known that Region I (parallel-plate waveguide) supports TEM, TE and TM modes. Modes in Region II are, however, hybrid in nature. Expressions for the transverse field components of Region I can be written as (1) (2) where are the forward and backward amplitudes of the -th mode in Region I. Expressions for the transverse modal function and the admittance can be found in [7]. Similarly, the transverse field components of Region II can be expressed as (3) (4)

Fig. 1. Geometry of the proposed FSS. (a) Perspective view, (b) top view, (c) front view, (d) enlarged view of the cross-section of a unit-cell.

II. ANALYSIS As explained in the previous section, the proposed structure can be divided into the identical unit-cells shown in Fig. 1. Each cell consists of a microstrip line of length that is shielded with magnetic side walls. Under the plane wave incidence of a polarization perpendicular to the strips, air region can be effectively represented as a parallel-plate waveguide. A side view of the unit cell, sandwiched between air at two sides, is shown in Fig. 2. The air region is denoted as Region I, and the air-to-microstrip line discontinuities are shown as vertical broken lines. Region II consists of a microstrip line shielded with magnetic side walls. The problem of the proposed FSS can thus be formulated as a cascaded junction of the air-to-microstrip line discontinuities. In the following analysis, we primarily solve the air-to-microstrip line discontinuity using the full-wave modematching method, and the S-parameters of a cascaded junction are then straightforwardly calculated from the S-parameters of the single discontinuity.

where are the forward and backward amplitudes of the -th mode in Region II. Region II is further divided into subregions 1 and 2, as indicated in Fig. 1(d). Fields in each subregion are written as a superposition of TE and TM modes, and they are derived from the following Hertzian potentials: (5) (6) (7) (8) where

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and . Applying the boundary conditions at the leads to a set of equations that are solved for interface using the singular integral equation the propagation constant (SIE) method. We can then substitute back into these equa, tions and evaluate the expansion coefficients . The hybrid transverse mode functions and are and then calculated from these modal coefficients using the expressions given in the Appendix. The air-to-microstrip line discontinuity is represented by the in Fig. 2. Enforcing the condition that tangential field surface components must be continuous along the regional interface , we obtain (9) (10)

Fig. 3. Comparison of S-parameters of a cascaded junction of air-to-microstrip line discontinuities calculated by our full-wave method, and those obtained from mm, b mm, d mm, h mm, " ;L CST MWS. (t mm).

(11)

III. EQUIVALENT CIRCUIT MODEL

3

=2

= 10

=3

=5

=5 =

where

(12) (13) Writing (9) and (10) in the matrix form and solving them for S-parameters leads to the following relations [8] (14) (15) (16) (17) and is an identity where matrix. is the admittance matrix used in the field expressions and are diagonal matrices given in (12) and of Region I. (13), respectively. Finally, S-parameters of the cascaded junction of air-to-microstrip line discontinuities are calculated using the expressions given in [9]. The overall approach is primarily a combination of the SIE and the mode-matching methods. Efficiency and accuracy of the SIE method for two-dimensional transmission line problems have already been demonstrated [10], [11]. Similarly, the mode-matching method is also well known to be a highly efficient and accurate technique for waveguide discontinuity problems [12]. In order to verify the accuracy of the above formulation, we compare our results for the S-parameters of a cascaded junction of air-to-microstrip line discontinuities, with those obtained from CST Microwave Studio (MWS). As shown in Fig. 3, an excellent agreement between them has been observed. It should be mentioned that, due to the superiority of the SIE and mode-matching methods, our approach is highly efficient. It is pointed out that the mode-matching method exhibits rapid convergence because only 10 modes of Region II need to be considered for the results shown in Fig. 3.

Unlike a microstrip line shielded with perfect electric conductor (PEC) side walls, the microstrip line shielded with PMC side walls supports two quasi-TEM modes at any frequency because the top and bottom conductors can have different electric potentials in this case. It is observed that one of these two modes is primarily concentrated in the substrate region and thus may be termed as the “substrate mode.” The second quasi-TEM mode is concentrated mainly in the air region and can be called as the “air mode.” A discontinuity of air-to-microstrip line shielded with magnetic side walls thus results in a three-port network as the incident plane wave is coupled to two modes of the microstrip line. Parameters of the equivalent circuit can be extracted from the S-parameters of the discontinuity obtained through the full-wave mode-matching method. The cascaded junction of two such discontinuities then leads to the circuit model shown in Fig. 4 where the first two quasi-TEM modes of the shielded microstrip line have been represented by two resonators. In order to relate this FSS with an already established microwave filter theory, the representation has been retained in a general form where the air region on the two sides of the FSS is termed as source/load. In our actual case of the proposed FSS, source and load are identical i.e.,

where , and denote the couplings between the air region and the two fundamental modes of the microstrip line shielded with magnetic side walls. The source-load direct coupling is given with a dotted line because it is observed only at very low frequency. For the polarization under consideration, the air-region can be modeled as a parallel-plate waveguide, and the FSS provides a direct link between the air-regions at two sides. At higher frequencies, however, this direct source-load coupling becomes negligible and instead the incident wave is coupled to two modes of the microstrip line shielded with magnetic side walls. The direct source-load coupling can hence be safely ignored at higher frequencies.

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Fig. 4. Equivalent circuit model of the proposed FSS.

Based on the theory of cross-coupled resonators [13], [14], the topology given in Fig. 4 is known to generate two reflection zeros and one transmission zero at finite frequencies. Each resonator contributes to one reflection zero at a frequency where its electrical length becomes equal to [15], [16]. The topology is applicable for both band-pass and the band-reject filtering functions [17]. The coupling matrix elements needed for the band-reject response are, however, easier to realize with this structure. Also due to the presence of a strong direct source-load coupling, the FSS is inherently suitable for band-reject applications, and we thus focus on the band-reject behavior although other filtering response may also be possible. Fig. 5 gives an example to show the two reflection zeros and one transmission zero associated with the block diagram in Fig. 4. Also illustrated in Fig. 5 is the effect of oblique incidence of a plane wave upon the FSS. It is seen that the reflection zeros move towards a higher frequency with an increase of the angle of incidence. However, the position of the transmission zero is not affected significantly for a large variation of the angle of incidence, and the resulting overall frequency response is stable, as shown in Fig. 5. The reflection zero is known to occur at a frequency where the electrical length of a resonator becomes a multiple of [15], and its shift towards higher frequencies is due to a reduction in the propagation constants of the two modes with an increase in the angle of incidence. Furthermore, since the transmission zero is related to the difference between the electrical lengths of the two resonators, it is less affected when the propagation constants of the two modes change by a similar variation. It should also be mentioned that CST MWS has been and in Fig. 5. used to obtain the results for For a wideband FSS, more transmission zeros may be needed and they can be realized if higher-order modes of the two resonators are considered. Reflection zeros corresponding to these modes occur at frequencies where the electrical length of the resonators becomes a multiple of . In that case, the proposed FSS can be generalized as the block diagram shown in Fig. 6. is an maThe resulting coupling matrix trix, where refers to the number of resonances. This configuration can generate reflection zeros at finite frequencies [13], [14], [18], [19]. IV. DESIGN GUIDELINES AND EXAMPLES The frequency response of the proposed FSS is mainly controlled by three design parameters, dielectric constant , strip

Fig. 5. Effect of different angles of incidence on the S-parameters of the proposed FSS for a polarization perpendicular to the strips. (t : mm, b mm, h : mm, d : : ;L : mm). (a) j j, mm, " j. (b) j

10

S

= 2 524

= 1 524

= 01 = 3 38 = 9 5

= S11

Fig. 6. A generalized equivalent circuit model of the proposed FSS with resonators.

N

width to unit cell width ratio , and substrate height to unit . Effect of these parameters on the S-pacell height ratio rameters of the FSS is related to the propagation characteristics of a microstrip line shielded with magnetic side walls. Based on that, we deduce useful guidelines for designing such FSS. These guidelines are intuitive and lead to a good estimate of the required FSS dimensions. We then verify and fine-tune those dimensions using our efficient full-wave mode-matching method. Alternatively, this refining process can also be performed using a suitable commercial software package if available.

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For a given center frequency , bandwidth , and min, the following steps are imum out-of-band return loss considered. 1. Based on the given bandwidth requirement, one has to determine the number of transmission zeros needed in the reject-band. For narrow-band cases, single transmission zero may serve the purpose. Two or more transmission zeros are, however, desirable for wideband designs. The number of transmission zeros is directly related to the number of resonances required for the FSS. If a single transmission zero is required, a lower value of may be selected. For two or more transmission zeros, a higher is more suitreduces the thickness of the able. A higher value of structure and yet the pass-band (out of band response) may not be very large as further higher-order modes may deteriorate the upper end of the out-of-band response. 2. As the first estimate, length of the strips may be chosen where and is the free as space wavelength at the center frequency . The final value of may slightly be different from the first estimate due to the parasitic effects of the air-to-microstrip line discontinuity. is obtained by setting a proper ratio 3. The required between the substrate width and the unit-cell height. varies beFor a given , the minimum return loss as tween . The parameter also affects the bandwidth of the transmission zero. For an extremely narrowis a relatively indeband or a notch-band response, pendent design parameter, and a higher value of it may be preferred for an improved out-of-band rejection. For wideband designs, it is required to choose a highest possible that can be allowed by the given specvalue of increases coupling to the ification. Higher value of higher-order modes of the resonators. The principle here is similar to that of the known ultra wideband filters based on microstrip line multi-mode resonators [15], [16]. With inclusion of the higher-order modes, more transmission zeros are obtained at finite frequencies and the response curve is flattened through an increased coupling to the higher-order modes. 4. Since the reflection zeros corresponding to the two modes of a microstrip line are placed close to each other for is selected. For narrow-band design, a lower value of ratio to increase wideband FSS, we choose a higher the gap between the reflections zeros so that the second reflection zero does not deteriorate the in-band response. Table I gives a summary of the above discussions of the design parameters, and compares them for the two cases of narrow-band and wideband FSSs. In the following subsections, we provide two designed examples. These FSS structures are fabricated based on the following steps. (i) A large double-layer PCB is used to fabricate the one-dimensional periodic array of long microstrip lines on its top layer. (ii) The board is then cut into a number of small parts containing one-dimensional periodic array of short microstrip lines. (iii) The individual small PCBs of equal length are then stacked vertically up to form the proposed FSS through two plastic rods at two ends.

TABLE I COMPARISON OF DESIGN PARAMETERS FOR NARROW-BAND AND WIDEBAND FSSs

A. Narrow-Band FSS The equivalent network of the basic configuration shown in Fig. 4 is inherently narrow-band, and it is thus easy to obtain a notch like response. The aim is to place the two reflection zeros close to each other with a transmission zero in between. A lower can effectively achieve this goal. The separation between the ratio. two reflection zeros can be controlled by changing the also affects the separation between the Though the ratio reflection zeros, that effect is negligibly small in comparison with its effect on the out-of-band return loss performance of the structure. Fig. 7 presents an example that is designed for the center frequency at 10 GHz. Since we are interested in only one transvalue of 3.38 is selected. The FSS is mission zero, a lower fabricated using RO-4003 substrate material. Reflection coefficient measurements are performed in an anechoic chamber. Due to the small size of the anechoic chamber available to us, its transmission coefficient is measured in an open lab hall. Fig. 7 shows the simulated and measured reflection and transmission coefficients of this narrow-band FSS. It is seen that the measured results are in a good agreement with those obtained from our full-wave mode-matching method. Fig. 8 shows the photo of the designed narrow-band FSS. The overall size of the fabricated structure is 250 mm 220 mm, which contains 41 62 unit-cells. B. Wideband FSS As discussed earlier, wideband response can be obtained with an inclusion of more transmission zeros through higher-order resonances. We thus prefer a higher of 10.2 in this example, and we include one higher order resonance in the reject-band. is chosen to be high so that the reflection zero corAlso responding to the second quasi-TEM mode of the microstrip line may not occur in the frequency band of interest. Within that stop-band, one more transmission zero is generated due to the first higher-order resonance, and the overall stop-band is then flattened through increased coupling to the substrate mode by ratio. This FSS is fabricated using Rogers decreasing the RT Duroid 6010 substrate material. Similar to the previous case, the reflection and transmission coefficients are measured in an anechoic chamber and an open lab hall, respectively. As shown in Fig. 9, measured results are in good agreement with those calculated by our full-wave mode-matching method. There are small differences in the location of the reflection zeros and the insertion loss, which may be attributed to the fabrication tolerances and the measurement errors. The measurement error at the upper end of frequencies is expected as it is not possible to fulfill the far-field criterion at higher frequencies, due to the small size of the anechoic chamber we have.

RASHID AND SHEN: A NOVEL BAND-REJECT FREQUENCY SELECTIVE SURFACE WITH PSEUDO-ELLIPTIC RESPONSE

= 02 = 3 38 = 9 5

=6

Fig. 7. S-parameters of a narrow-band FSS. (t : mm, b mm, h : mm, d : mm, " : ;L : mm). (a) j j, (b) j j.

3 524

= 1 524

S

S

=

Fig. 8. Photo of the narrow-band FSS with pseudo-elliptic response.

V. CONCLUSION A new class of band-reject FSS with pseudo-elliptic response has been presented. It consists of a simple geometry based on a two-dimensional periodic array of microstrip lines. The operating principle of the proposed FSS has been explained using

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=4 S

= 4:2 mm, h = 2 mm, S .

Fig. 9. S-parameters of a wideband FSS. (t mm, b : ;L : mm). (a) j j, (b) j mm, " d :

= 1 27

= 10 2 = 6 8

j

its circuit model that is similar to that of known dual-mode resonator based microwave filters. A full-wave mode- matching approach has been used for the accurate and efficient analysis of the FSS structure, and based on that some useful design guidelines have been presented. Since the new FSS offers flexibility in terms of bandwidth control, two designed examples have been demonstrated, and the measured results validate our design concept. The proposed FSS works for a linear polarization that is received perpendicular to the printed strips. However, we believe that the concept can be further extended to a combination of two such arrays where one of them is rotated by 90 degrees, and with a proper selection of the dimensions this may result in an FSS working for any combination of two linear polarizations. The proposed FSS has been assembled by periodic placement of identical PCBs. If two or more different PCBs are periodically arranged, it is also possible to obtain multi-band FSS with pseudo-elliptic response. The thickness of the proposed FSS can also be reduced with inclusion of lumped elements in the printed strip lines. Due to the presence of microstrip lines, this structure supports an easy loading of lumped elements, which can provide great flexibility in designing multi-band and even tunable FSSs.

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APPENDIX Expressions of the Mode Functions for Region II

REFERENCES [1] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [2] T. K. Wu, Frequency Selective Surface and Grid Array. New York: Wiley, 1995. [3] G. Q. Luo, W. Hong, Q. H. Lai, K. Wu, and L. L. Sun, “Design and experimental verification of compact frequency-selective surface with quasi-elliptic bandpass response,” IEEE Trans. Microw. Theory Tech., vol. 55, pp. 2481–2487, 2007. [4] I. C. Hunter, J. D. Rhodes, and V. Dassonville, “Dual-mode filters with conductor-loaded dielectric resonators,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2304–2311, 1999. [5] X.-P. Liang, K. A. Zaki, and A. E. Atia, “Dual mode coupling by square corner cut in resonators and filters,” IEEE Trans. Microw. Theory Tech., vol. 40, pp. 2294–2302, 1992. [6] M. Ohira, H. Deguchi, M. Tsuji, and H. Shigesawa, “Novel waveguide filters with multiple attenuation poles using dual-behavior resonance of frequency-selective surfaces,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 3320–3326, 2005. [7] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005. [8] A. S. Omar and K. Schunemann, “Transmission matrix representation of finline discontinuities,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, pp. 765–770, 1985. [9] J. Uher, J. Bornemann, and U. Rosenberg, Waveguide Components for Antenna Feed Systems: Theory and CAD. Norwood, MA: Artech House, 1993. [10] Y. F. Huang and S. L. Lai, “Regular solution of shielded planar transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 42, pp. 84–91, 1994.

[11] R. Mittra and T. Itoh, “A new technique for the analysis of the dispersion characteristics of microstrip lines,” IEEE Trans. Microw. Theory Tech., vol. 19, pp. 47–56, 1971. [12] 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, pp. 2371–2380, 1994. [13] S. Amari, “On the maximum number of finite transmission zeros of coupled resonator filters with a given topology,” IEEE Microw. Guided Wave Lett., vol. 9, pp. 354–356, 1999. [14] S. Amari and J. Bornemann, “Maximum number of finite transmission zeros of coupled resonator filters with source/load-multiresonator coupling and a given topology,” in Proc. Asia-Pacific Microwave Conf., 2000, pp. 1175–1177. [15] L. Zhu, H. Bu, and K. Wu, “Aperture compensation technique for innovative design of ultra-broadband microstrip bandpass filter,” in IEEE MTT-S Int. Microwave Symp. Digest, 2000, vol. 1, pp. 315–318. [16] W. Menzel, L. Zhu, K. Wu, and F. Bogelsack, “On the design of novel compact broad-band planar filters,” IEEE Trans. Microw. Theory Tech., vol. 51, pp. 364–370, 2003. [17] S. Amari and U. Rosenberg, “Direct synthesis of a new class of bandstop filters,” IEEE Trans. Microw. Theory Tech., vol. 52, pp. 607–616, 2004. [18] S. Amari, “Direct synthesis of folded symmetric resonator filters with source-load coupling,” IEEE Microwa. Wireless Compon. Lett., vol. 11, pp. 264–266, 2001. [19] R. J. Cameron, “Advanced coupling matrix synthesis techniques for microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 51, pp. 1–10, 2003.

Amir Khurrum Rashid 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 toward 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.

Zhongxiang Shen (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, 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 coauthored over 80 journal articles and more than 90 conference papers.

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Broadening of Operating Frequency Band of Magnetic-Type Radio Absorbers by FSS Incorporation Yuri N. Kazantsev, Alexander V. Lopatin, Natalia E. Kazantseva, Alexander D. Shatrov, Valeri P. Mal’tsev, Jarmila Vilˇcáková, and Petr Sáha

Abstract—Problems of the theory of radio absorbers (RAs) involving frequency selective surfaces (FSSs) are considered. A design procedure for these RAs is described that takes into account the multiparameter character of the problem and allows one to determine the optimal characteristics of an FSS that provide the maximal operating bandwidth. RAs for a range of 1–17 GHz are obtained on the basis of polymer composites filled with carbonyl iron and Co2 Z ferrite, into which an FSS in the form of a biperiodic array of thin metal rings is embedded. It is shown that the application of such an FSS allows one to increase the operating frequency bandwidth of a magnetic-type RA by a factor of more than 1.5 virtually almost without increasing the thickness of the absorber. Substantially, several types of FSS-based layered magnetodielectric RAs have been produced by compressing molding technique, including a 3-mm-thick RA with an operating bandwidth of 1.05–2.7 GHz, a 2-mm-thick RA with an operating bandwidth of 1.7–4.4 GHz, and a 1.6-mm-thick RA with an operating bandwidth of 6.7–16.1 GHz. Index Terms—Electromagnetic interference, electromagnetic shielding, ferromagnetic materials, frequency selective surfaces (FSSs).

I. INTRODUCTION ADIO ABSORBING MATERIALS (RAMs) have been the subject of intensive study since the World War II. The first studies were focused on military applications and reached their peak in the 1970–1980s due to the development of the Stealth technology. Recently, there has been a renewal of interest of the scientific community in RAMs, which is associated with an increasing penetration of electronics into all areas of

R

Manuscript received June 25, 2009; revised September 24, 2009; accepted October 01, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. This work was supported in part by the Ministry of Education, Youth and Sports of the Czech Republic (MSM 708 8352 101 and ME 883 KONTAKT) and in part by the Russian Foundation for Basic Research (projects no. 08-02-90418, 08-08-90006). Y. N. Kazantsev is with the Laboratory of Diffraction Problems, Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Fryazino, Moscow region, Russia (e-mail: [email protected]). A. V. Lopatin, J. Vilˇcáková, and P. Sáha are with the Tomas Bata University, Zlin 762 72, Czech Republic N. E. Kazantseva is with the Laboratory of Diffraction Problems, Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Fryazino, Moscow region, Russia and also with the Tomas Bata University, Zlin 762 72, Czech Republic. A. D. Shatrov, V. P. Mal’tsev, with the Laboratory of Diffraction Problems, Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Fryazino, Moscow region, Russia. Digital Object Identifier 10.1109/TAP.2010.2041316

modern life. There are a variety of electron devices that radiate electromagnetic energy into the environment, thus causing many serious problems such as electromagnetic interference (EMI), electromagnetic compatibility (EMC), and a hazardous effect of electromagnetic waves on living organisms. One of the main means for solving these problems is RAMs. The frequency interval of 0.8–10 GHz, in which many communication and information-transmission systems operate, is of particular importance. There are a large number of publications [1]–[16] devoted to the design of RAMs in this and other frequency intervals. An RA, which reduces the reflection of incident electromagnetic wave (EMW), represents a layer (layers) of a RAM placed on a metal surface. Any RA operates in a limited operating frequency band. The usual method for expanding the operating frequency band of an RA consists in using multilayer structures instead of singlelayer ones. However, this increases the thickness and the weight of RAs. Today there exist several leading manufacturers of a wide assortment of RAs, such as the FDK Corporation (Japan) and Laird Technology Company (USA). Thin RAs offered by these manufacturers operate in narrow frequency bands, while broadband RAs have thickness of about 30–60 mm. Thus, the design of thin and simultaneously broadband RAs is a topical problem. Any single-layer RA is characterized by a certain matching frequency and matching thickness for which the reand can flection coefficient is minimal. The values of be determined either by graphical [16] or analytic [5] methods from the complex permittivity and permeability data. In [5], the and were determined for polymer composites values of filled with various types of carbonyl iron (CI). It was shown that the matching frequency can be tuned within a wide range by varying the type and concentration of CI. However, although belong to the relevant microwave band, the the frequencies absorption bands of the RA are relatively narrow. For instance, the ratio of the extreme frequencies of the operating bands (meadB level of the reflection coefficient) lies in the sured at interval 1.5–1.8. One of efficient methods for expanding the range of operating frequencies of RAs is the application of FSSs to the design of RAs [6]–[14]. The effect of FSSs on the reflection characteristics of an RA based on lossy dielectrics is demonstrated in [6] and [7]. In [8]–[10], the authors considered an RA based on dielectrics and FSSs made of resistive elements. In [11] and [12], the authors proposed a microgenetic algorithm (MGA)

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Fig. 2. (a) FSS in an infinite magnetodielectric medium, (b) an equivalent circuit.

Fig. 3. Multilayered structure of RA with FSS.

Fig. 1. Radio absorbers with an FSS: (a) FSS embedded into a RAM, (b) FSS is separated from a RAM by dielectric layers, and (c) the FSS structure.

for the design of broadband RAs. In [13], the authors investigated the frequency and angular characteristics of the reflection coefficient of the so-called Circuit Analog Absorbers, which are made of resistive films with frequency selective properties. These authors showed that broadband RAs with small reflection coefficient can be designed for both normal and oblique incidence of TE- and TM-polarized waves. However, since the RAs considered in the above-cited papers were based on FSSs embedded in nonmagnetic materials, they had relatively large thicknesses. In [14], the control of the absorption characteristic of an RA with the use of an FSS is illustrated by an example of polymer composites filled with different types of CI. The theoretical methods for studying FSSs (including cascaded FSSs in a dielectric medium) developed earlier in [17]–[23] allow one to calculate the characteristics of RAs with FSSs. In the present paper, we describe a design procedure for RAs based on polymer magnetic composites that include FSSs. For different regions of the frequency band 1–17 GHz, we use composites filled with different types of CI and a ferrite of type Co Z. The measured reflection characteristics are compared with the results of computation by the method of integral equations. II. FUNDAMENTAL CONSIDERATION AND ANALYSIS

When choosing a preliminary structure of an RA, namely, the type of the FSS and the thicknesses and of layers, it is expedient to apply an approximate approach that takes into account the generalized characteristics of an FSS in free space: and the quality factor . In this the resonance frequency approach, one first determines the equivalent impedance of the FSS embedded into an infinite magnetodielectric with electric and magnetic losses (Fig. 2(a) and (b)) and then applies the transmission line technique to calculate the reflection coefficient of the structure shown in Fig. 1(a). The formula for the equivalent impedance [14] is expressed as (1) and where and permeability, respectively;

are complex permittivity

(2) (3) The reflection coefficient of the RA with an FSS can easily be obtained by well-known formulas from transmission line theory. When the FSS is brought close either to the metal plane or to the external surface of the RA, formulas (1)–(3) cease to be valid. For example, when the FSS is placed on the surface, the permittivity and the permeability in these formulas should be and replaced by their averaged values .

A. Approximate Theory for the Design of RA With FSS Schemes of RAs with FSSs are shown in Fig. 1(a) and (b). In the scheme of Fig. 1(a), an FSS (1) is embedded into the layer of a RAM (2) placed on a metal plane (3). In the scheme of Fig. 1(b), an FSS is separated from the RAM by thin layers (4) of a low-loss dielectric. An FSS in the form of a two-dimensional array of thin metal rings is shown in Fig. 1(c).

B. Rigorous Theory Consider the problem of reflection of a plane EMW from a multilayer structure shown schematically in Fig. 3. The structure consists of four magnetodielectric layers placed on a metal plane and a planar patch biperiodic metal array enclosed between the second and third layers.

KAZANTSEV et al.: BROADENING OF OPERATING FREQUENCY BAND OF MAGNETIC-TYPE RAs BY FSS INCORPORATION

The method of analysis is based on the numerical solution of an integral equation of the first kind for the surface current in the metal elements of the array. The integral equation is derived by a procedure similar to that used in [17], [22], and [23]. The multilayer structure considered in the present study is different from the structures considered in [22] and [23] both in the position of the FSS in the material and in the presence of a metal plane on the backside of the structure. Since the integral equation for multilayer structures is too bulky, we will use a recurrence expression for this equation. Suppose that a plane EMW is incident on the structure. The Cartesian coordinates of the electromagnetic field in this wave , where is the wave vector are proportional to and the dot denotes the scalar product of vectors. Below, we will denote by a bar below a character two-dimensional transverse vectors that have only - and -components. For example, we onto the plane by . It is denote the projection of obvious that (4)

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modes propagating in the positive and negative directions of are propagation constants of the the axis . The quantities Floquet modes in a magnetodielectric medium with permittivity and permeability (9) The quantities are the characteristic admittances of the traveling Floquet modes:

(10) In the absence of a metal array, the problem of transmisthrough a multilayer sion of a Floquet mode with indices magnetodielectric medium is solved easily, because there are no mode conversion at the boundaries of the media and on the metal plane. Therefore, the fields in the th layer are expressed as

where and are the polar and azimuth angles that determine are the unit the propagation direction of the plane wave and vectors along the coordinate axes. The kernel of the integral equation for current in a conducting element of the array is expressed in terms of Floquet functions, which are defined as follows: (5) (11) where (6) and are the dimensions of the unit cell (periods) of the biperiodic array. Let us introduce two mutually orthogonal unit vectors and by the formulas

(7) The symbol denotes vector multiplication. The transverse coordinates of the electromagnetic field of the Floquet modes propagating in a homogeneous medium with number are expressed in [17] as

The unknown amplitudes and of counterpropagating Floquet modes can be determined from a system of algebraic equations that are obtained from the continuity conditions for and on the boundaries between media. the vectors The kernel of the integral equation for current contains terms expressed through the solutions of two auxiliary electrodynamic problems on the propagation of Floquet modes in a multilayer medium without a metal array. 1. Consider a Floquet mode satisfying the condition that the vanishes on the metal plane situated on the back vector side of the multilayer magnetodielectric structure. Denote by the ratio of the field amplitudes and on the boundary with number between layers and . It is obvious that (12) The quantities with are determined by successive application of the descending recurrence formula

(8) and correspond to TM- and The subscripts TE-modes. The upper and lower signs in (8) correspond to the

(13)

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We also introduce

(14) This function possesses the following property: if we denote the amplitude of the electric vector of a standing by mode on the boundary with number , then (15) 2. Consider another field pattern of the Floquet mode that has a form of a wave propagating in the negative direction of in the left half-space ( in expression (11)). Denote by the ratio of amplitudes of the vectors and on boundary . On boundary 1 we have (16)

Fig. 4. Electromagnetic characteristics of RAMs; (1) Material I, (2) Material II, (3) Material III, and (4) Material IV.

dices . The amplitudes of these modes are represented in terms of the solution of integral equation (20) by the formula

Here with are determined by the following ascending recurrence formula:

(21) (17) Let us pass on to the problem of diffraction of a Floquet mode by the structure shown in Fig. 3. We will assume that a mode incident on the structure has unit amplitude of the electric field . Denote by and is characterized by indices and the tangential components of the magnetic field on the left and right of the metal array situated on boundary 3. In the unit cell of the array, we introduce a vector function by the formula (18) is proportional to the surface current and The function is different from zero only in the domain occupied by metal. Introduce (19) where the asterisk denotes complex conjugation. The required integral equation is expressed as

(20) When a plane wave is incident on the structure, the reflected field is expressed as a superposition of Floquet modes with in-

is the Kronecker delta. where The components of the matrix of reflection coefficients of a as plane incident wave are expressed in terms of follows:

(22) Here and are the copolarization reflection coeffiis the cross-polarization reflection coefficient for a cients, is the cross-polarization reflection coeffiTM-wave, and cient for a TE-wave. Integral equation (20) is solved numerically by the Galerkin method. C. An Example of Calculating an RA With an FSS The aim of the analysis is to determine the effect of the FSS characteristics and the position of the FSS in the maon the operating bandwidth of the RA. To this terial end, we calculated the frequency dependence of the reflection coefficient from the RA for the scheme shown in Fig. 1(a) and for an FSS in the form of a biperiodic array of thin metal rings embedded in a radioabsorbing material representing a polymer composite filled with carbonyl iron—Material II described in Section III.A. The electromagnetic characteristics of Material II are shown in Fig. 4. , and for several variants of FSS The dimensions and the values of and used in the calculations are given in Table I. This table also presents the resonance frequencies

KAZANTSEV et al.: BROADENING OF OPERATING FREQUENCY BAND OF MAGNETIC-TYPE RAs BY FSS INCORPORATION

TABLE I COMPUTED PARAMETERS OF RA BASED ON MATERIAL II FOR d

Fig. 5. Reflection coefficient of RA calculated for normal and oblique incidence of TE and TM polarized waves.

and the Q-factors of these FSSs in free space calculated and according to [22], as well as the extreme frequencies at a level of dB of the reflection coefficient. The results of rigorous calculation of the reflection coefficient from the RA for normal incidence of EMW are shown in Fig. 5(a)–(d) by solid lines. The numbers of curves in the figures correspond to the numbers of variants in Table I. The number “0” denotes the case when there is no FSS in the composite. This curve attains its minimum at the matching frequency. The dashed curves correspond to variant 3 calculated by the approximate method described in Section II.A. Fig. 5(a) shows the frequency dependence of the reflection and the same coefficient from the RA for several values of and . An increase in the resonance values of frequency of the FSS leads to a gradual change in the frequency dependence of the reflection coefficient. First, an additional minimum arises in the left part of curve 1, which then moves to the right and simultaneously becomes deeper (see curve 2). In this case, the main minimum slowly moves to the right and becomes shallower. Then both minima become equal in amplitude (see curve 3), in which case the absorption bandwidth at a dB attains its maximum. The resonance frequency level of of the FSS embedded in the material is close to the matching GHz. frequency of the RA,

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=d

Fig. 5(b)–(d) represent the frequency dependence of the rewhen the values of flection coefficient for three values of and guarantee the maximal operating bandwidth. Calcudepend on the ratio lations show that these values of and as follows: the greater this ratio, the larger the value of , and the smaller the . For example, when GHz and , and when, GHz and . Note that Q-factors can be obtained with an FSS with circular metal rings, while Q-factors smaller than 1.2 can be obtained with an FSS with square rings. The value of is the minimum possible value of the Q-factor for an FSS with square rings. According to variants 3, 5, 6 and 7 in Table I, the maximal relative operating bandwidth of the RA weakly depends on the position of the FSS in the material in a rather wide range of – , although any position of the FSS corvalues of responds to specific values of the generalized characteristics of the FSS ( and ). This situation also occurs with other types of RA made of materials I, III, and IV, whose compositions and characteristics are presented in Section III.A. Fig. 5(b)–(d) also shows the frequency dependence of the reflection coefficient for an oblique incidence of TE- and TM-polarized waves by dotted line and dash-and-dot lines, respectively. As the angle of incidence increases, the reflection coefficient for TM-polarized waves decreases, whereas that for TE-polarized waves increases. However, the value of the reflection coefficient averaged over both polarizations varies little, up . However, at angles of incidence above 60 , the to value of the reflection coefficient for TM-polarized waves starts . to increase and approaches unity as To illustrate the role of FSS in expanding the operating frequency bandwidth of a magnetic-type RA, we consider an absorber structure in which FSS is situated on the surface of the RA (variant 7). Fig. 6 shows the frequency dependence of the of a layer without an FSS (Fig. 6(a)), equivalent admittance of the FSS on the surface of the layer equivalent admittance (Fig. 6(c)). These functions (Fig. 6(b)), and the sum are also shown in the Smith chart (Fig. 6(d)), where curve 1 . At corresponds to , curve 2, to , and curve 3, to , the imaginary part of vanishes, the matching frequency while its real part is close to unity, which guarantees low redB) at this frequency. The deviation of flection coefficient ( leads to a small variation of frequency to either side from

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TABLE II COMPUTED PARAMETERS OF RA BASED ON MATERIAL II FOR d d

=

(VARIANT 8) AND d

>d

(VARIANTS 9 AND 10)

ficient as a function of frequency has a shape of a double-hump curve. Note that one can reduce the reflection coefficient of an RA dB in a relatively wide frequency band (Table II). down to Calculations have shown that this can be attained at high of (variant 8 the FSS in the following two cases: when (variants 9 and 10 in Table II). in Table II) and when The frequency dependence of the reflection coefficient for these variants is shown in Fig. 7. D. Design Procedure of an RA With an FSS

Fig. 6. Equivalent admittance as a function of frequency: (a) admittance of a layer without an FSS ( ), (b) admittance of an FSS on the surface of a layer ( ), (c) the sum of equivalent admittances of the layer and the FSS ( + ), and (d) the Smith chart for (1), (2), and + (3).

Y

Y

Y

Y

Y

Y Y

Y

Fig. 7. Reflection coefficient of RA calculated with parameters that guarantee a maximal bandwidth at a reflection coefficient of 15 and 20 dB.

0

0

the real part of admittance; however, the absolute value of the rapidly increases, thus leading to the inimaginary part crease of the reflection coefficient. When an FSS is placed on an , RA layer, the admittance of the RA is given by the sum where the imaginary parts of and have opposite signs and compensate each other. As a result, the imaginary part of the approaches zero in a wide frequency range, the opsum erating bandwidth of the RA increases, and the reflection coef-

Now, let us describe the design procedure of an RA with an FSS, which allows considering the multivariable character of the task. First, we choose an RAM for which the matching frequency lies in the required operating frequency band of the RA. Naturally, the complex permittivity and permeability of this material are already known. The matching frequencies correspond , which are equal to . Next, to matching thicknesses within the interval 0–2. Applying the we choose the ratio and that guartransmission-line method, we determine antee the maximal operating bandwidth. The next step consists in choosing the type of a stopband FSS, i.e., choosing the period, as well as the shape and size of individual elements of the two-dimensional array. A mandatory requirement imposed on and under different polarizations FSS is the stability of and angles of incidence of an electromagnetic wave. According to [18], an FSS with elements in the form of circular and square rings or Jerusalem crosses exhibits high angular stability. In the case of circular or square rings, the period and the size of elecan be obtained ments of the FSS for given values of and by the methods described in [17]–[22]. The shape of the rings is determined by the required value of . For example, a value of is obtained in FSSs made , one should use FSSs with of circular rings; to obtain square rings. After evaluating all the basic parameters of the FSS, we should determine them accurately by rigorously calculating the reflection coefficient; in this way we can take into account the effect of gaps between metal elements of the FSS and the RAM. Therewith, to compensate the effect of these gaps, it suffices to reduce the resonance frequency of the FSS by proportionally , and increasing the dimensions of the FSS structure ( ). The procedure described allows one to determine the paramdB-bandwidth of a eters of the FSS and thus expand the matched to free space at thin single-layer RA of thickness

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TABLE III THE MAIN CHARACTERISTICS OF RA SAMPLES

frequency . If necessary, one can reduce the reflection coeffidB in a wide frequency cient of this FSS-based RA down to , and ) rather than two ( band by optimizing three ( and at ) RA parameters. III. EXPERIMENTAL SETUP

The main characteristics of RA samples, as well as the di, and ) and its position in the RA mensions of the FSS ( structure ( and ) are shown in Table III. The RA samples were designed from polymer composite layers and an FSS. The use of several layers of composites of different thickness allowed one to vary both the thickness of a sample and the position of the FSS in the sample.

A. Materials and Samples The basic components for the fabrication of soft magnetic polymer composites are the following fillers: CI powders of types HQ and ES (BASF, Germany), ferrite powders of type Co Z (Ba Co Ti Fe O ) (Ferrite Domen Co, Russia), glass microspheres of type MSVP A9 (NPO Stekloplastik, Russia), and a polymer binder—a silicon elastomer SYLGARD 184 (Dow Corning, USA). The glass microspheres were used to improve the homogeneity of the CI particle distribution in the composite, as well as to control its complex permittivity. Individual layers of polymer composites were fabricated as follows. First, the basic components were mixed in the required proportion to give a homogeneous mixture. Then, the mixture was placed in a press form and cured for four hours at temperature of 80 C. After curing, the final product was extracted from the press form. In this work, we used four types of polymer composites, which had the following compositions and density Material I: 50 vol. % ES and 50 vol. % SYLGARD 184, g/cm ; Material II: 40 vol. %. HQ, 15 vol. % MSVP A9, and 45 vol. g/cm ; % SYLGARD 184, Material III: 30 vol. %. HQ, 20 vol. % MSVP A9, and 50 vol. g/cm ; % SYLGARD 184, Material IV: 50 vol. % Co Z and 50 vol. % SYLGARD 184, g/cm . These polymer composites have low dielectric losses irrespective of the filler type. However, magnetic losses of CI filled composites are higher when compared with composites based on Co Z (Fig. 4). The FSS represents a two-dimensional grating of ring-shaped conducting elements (Fig. 1(c)).

B. Test Apparatus The complex permittivity and permeability of polymer composites were measured by an Agilent E49991A RF Impedance/ Material Analyzer in the frequency range of 0.01–2 GHz and by a resonator method involving R2 scalar network analyzers, in the frequency range of 2–17 GHz. The reflection coefficient of the samples was measured by a reflectometer [24] in combination with an HP 8720C Vector Network Analyzer in a laboratory anechoic chamber. To increase the accuracy of these measurements, we applied the technique of subtracting the background level and the time-domain separation of the reflected pulse. IV. RESULTS AND DISCUSSION The measured reflection coefficients as a function of frequency for ten samples are shown by solid lines in Figs. 8–12. The numbers of the curves in these figures correspond to the numbers of samples in Table III, and the number “0” indicates that a sample does not contain an FSS. The sample thicknesses are close to the matching thickness. Fig. 10 clearly illustrates the variation of the frequency dependence of the reflection coefficient as the size of the rings decreases (which corresponds to the increase in the resonance frequency of the FSS). This variation qualitatively reproduces the variations of computed curves in Fig. 5(a). The main difference is that the size of the rings for calculated curves is approximately half that for the measured ones. This result is attributed to the fact that the computations did not take into account the gaps between metal elements of the FSS and the RAM. The results of rigorous calculations, which take into account the above gaps, are shown

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Fig. 8. Measured (solid curves) and computed (dotted curve) reflection coefficient versus frequency for samples 1 and 2.

Fig. 11. Measured (solid curves) and computed (dotted curve) reflection coefficient versus frequency for sample 9.

Fig. 9. Reflection coefficient versus frequency for samples 3 and 4.

Fig. 12. Measured (solid curves) and computed (dash curve) reflection coefficient versus frequency for sample 10.

Samples 9 and 10, which are made of Materials III and IV, respectively, have higher operating frequency bands. For example, the operating frequency band of sample 9 is 6.7–16.1 GHz with the ratio of extreme frequencies 2.4. For samples 1–4, 7, and 9, this ratio lies within the range 2.3–2.6. These results do not contradict the bandwidth estimates by using the data in Table I. V. CONCLUSION

Fig. 10. Measured (solid curves) and computed (dotted curve) reflection coefficient versus frequency for samples 5–8.

by dotted lines in Figs. 8, 10, 11, and 12 for samples 2, 7, 9, and 10, respectively. and of the operating freThe extreme frequencies quency bands at a level of dB of the reflection coefficient are shown in Table III. One can see that the type of a material allows one to vary the operating frequency bands of RAs over the region from 1 to 17 GHz. For instance, operating frequency bands for samples made of Material I, which is characterized , and , lie in a lower frequency by the maximal values of domain than those for samples made of other materials. For example, for sample 1, the operating frequency band is 1.05–2.7 . GHz with the ratio of extreme frequencies

We have demonstrated the efficiency of application of FSSs to controlling the characteristics of thin RAs based on polymer magnetic composites in the range of frequencies from 1 to 17 GHz. Two types of CI that differ in electromagnetic characteristics and a microwave ferrite with Co Z-type structure have been used as fillers of RAMs. Approximate analytic expressions for calculating the reflection coefficient from an RA with an FSS and a rigorous theory of multilayer magnetodielectric structures with FSS allow one to carry out a full design of microwave absorbers. As FSSs, it is recommended to take biperiodic arrays of metal elements in the form of circular or square rings, because the characteristics of these arrays are stable with respect to the angle of incidence of EMW. In the approximate calculations, we have used generalized and the characteristics of FSSs (the resonance frequency ), and the electromagnetic characteristics of the Q-factor material into which an FSS is embedded.

KAZANTSEV et al.: BROADENING OF OPERATING FREQUENCY BAND OF MAGNETIC-TYPE RAs BY FSS INCORPORATION

The results of rigorous calculation of the reflection coefficient, based on the solution of an integral equation by the Galerkin method, do not contradict the results of calculations obtained by the approximate method. We have shown that, for any position of the FSS in the RA layer, there exist optimal values of generalized characteristics of the FSS that provide the maximal operating bandwidth at a dB or lower. reflection coefficient of We have proposed a design procedure for thin RA FSS structures that allows one to take into account a large number of parameters of the problem (the characteristics of the material and the parameters and position of the FSS). First, one determines the thickness of the RA, which is equal to the matching thickness of the given RAM. Then, one chooses the position of the FSS in the RAM layer and determines the generalized characteristics of the FSS, which are optimal for the given position. Finally, one determines the size of the FSS structure by rigorous calculation and makes corrections to take into account the gaps between metal elements of the FSS and the RAM. Substantially, several types of FSS-based layered magnetodielectric RAs have been produced by compressing molding technique, including a 3-mm-thick RA, a 2-mm-thick RA, and a dB bandwidths of 1.05–2.7 GHz, 1.6-mm-thick RA with 1.7–4.4 GHz, and 6.7–16.1 GHz, respectively. Thus, we have shown both theoretically and experimentally that the application of an FSS allows one to increase the operating frequency bandwidth of a magnetic-type RA by a factor of more than 1.5 virtually without increasing the thickness of the absorber.

REFERENCES [1] J. L. Wallace, “Broadband magnetic microwave absorbers: Fundamental limitation,” IEEE Trans. Magn., vol. 29, no. 6, pp. 4209–4214, Nov. 1993. [2] K. N. Rozanov, “Ultimate thickness to bandwidth ratio of radar absorbers,” IEEE Trans. Antennas Propag., vol. 48, no. 8, pp. 1230–1234, Aug. 2000. [3] M. Amano and Y. Katsuka, “A method of effective use of ferrite for microwave absorber,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 238–245, Jan. 2003. [4] M.-J. Park, J. Choi, and S.-S. Kim, “Wide bandwidth pyramidal absorbers of granular ferrite and carbonyl iron powders,” IEEE Trans. Magn., vol. 36, no. 5, pp. 3272–3274, Sep. 2000. [5] A. V. Lopatin, N. E. Kazantseva, Yu. N. Kazantsev, O. A. D’yakonova, J. Vilˇcáková, and P. Sáha, “The efficiency of application of magnetic polymer composites as radio-absorbing materials,” J. Comm. Tech. Electron., vol. 53, no. 5, pp. 487–496, May 2008. [6] F. Terracher and G. Bergnic, “Thin electromagnetic absorber using frequency selective surface,” in Proc. IEEE Antennas Propag. Society Int. Symp., 2000, vol. 2, pp. 846–849. [7] Y. Sha, K. A. Jose, C. P. Neo, and V. K. Varadan, “Experimental investigation of microwave absorber with FSS embedded in carbon fiber composite,” Microw. Opt. Technol. Lett., vol. 32, no. 4, pp. 245–249, Jan. 2002. [8] L. Hai-Tao, C. Hai-Feng, C. Zeng-Yong, and Z. De-Yong, “Absorbing properties of frequency selective surfaces absorbers with cross-shaped patches,” Mater. Design, vol. 28, no. 6, pp. 2166–2171, Jun. 2007. [9] H. Rahman, J. Dowling, and P. K. Saha, “Application of frequency sensitive surfaces in electromagnetic shielding,” J. Mat. Proc. Techn., vol. 54, no. 1, pp. 21–28, Oct. 1995. [10] J. Yang and Z. Shen, “A thin and broadband absorber using double square loops,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 388–391, Dec. 2007.

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[11] S. Chakravarty, R. Mittra, and N. R. Williams, “On the application of the microgenetic algorithm to the design of broadband microwave absorbers comprising frequency-selective surfaces embedded in multilayered media,” IEEE Trans. Microw. Tech., vol. 49, no. 6, pp. 1050–1059, Jun. 2001. [12] S. Chakravarty, R. Mittra, and N. R. Williams, “Application of a microgenetic algorithm (MGA) to the design of broadband microwave absorbers using multiple frequency selective surface screens buried in dielectrics,” IEEE Trans. Antennas Propag., vol. 50, no. 3, pp. 284–296, Mar. 2002. [13] B. A. Munk, P. Munk, and J. Prior, “On designing Jaumann and circuit analog of absorbers (CA absorbers) for oblique angle of incidence,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 186–193, Jan. 2007. [14] A. V. Lopatin, Yu. N. Kazantsev, N. E. Kazantseva, V. N. Apletalin, V. P. Mal’tsev, A. D. Shatrov, and P. Sáha, “Radio-absorbing materials on the base of polymer magnetic composites and frequency selective surfaces,” J. Comm. Tech. Electron., vol. 53, no. 9, pp. 1114–1122, Sep. 2008. [15] K. N. Rozanov, Z. W. Li, L. F. Chen, and M. Y. Koledintseva, “Microwave permeability of Co Z composites,” J. Appl. Phys., vol. 97, no. 1, pp. 013905-1–013905-7, Jan. 2005. [16] Z. W. Li, L. Chen, Y. Wu, and C. K. Ong, “Microwave attenuation Co Fe O composproperties of W-type barium ferrite BaZn ites,” J. Appl. Phys., vol. 96, no. 1, pp. 534–539, Jun. 2004. [17] J. C. Vardaxoglou, Frequency Selective Surfaces: Analysis and Design. London, U.K.: Research Studies Press, 1997, p. 284. [18] B. A. Munk, Frequency Selective Surface: Theory and Design.. New York: Wiley, 2000, p. 410. [19] 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, Dec. 1988. [20] S.-W. Lee, G. Zarrillo, and C.-L. Law, “Simple formulas for transmission through periodic metal grids or plates,” IEEE Trans. Antennas Propag., vol. 30, no. 5, pp. 904–909, Sep. 1982. [21] R. J. Langley and E. A. Parker, “Equivalent circuit model for arrays of square loops,” Electron. Lett., vol. 18, no. 7, pp. 294–296, Apr. 1982. [22] V. N. Apletalin, Yu. N. Kazantsev, V. P. Mal’tsev, V. S. Solosin, and A. D. Shatrov, “Frequency-selective ring-element gratings,” J. Comm. Tech. Electron., vol. 48, no. 5, pp. 469–479, May 2003. [23] V. N. Apletalin, Yu. N. Kazantsev, V. P. Mal’tsev, and A. D. Shatrov, “A cascade of two island gratings separated by a multilayer magnetodielectric structure,” J. Comm. Tech. Electron., vol. 53, no. 6, pp. 631–635, Jun. 2008. [24] V. N. Apletalin, Yu. N. Kazantsev, and V. S. Solosin, “Microwave and millimeter wave reflectometers based on hollow metal-dielectric waveguides,” Radiotehnika, no. 8, p. 44, Aug. 2005. Yuri N. Kazantsev, photograph and biography not available at the time of publication.

Alexander V. Lopatin, photograph and biography not available at the time of publication.

Natalia E. Kazantseva, photograph and biography not available at the time of publication.

Alexander D. Shatrov, photograph and biography not available at the time of publication.

Valeri P. Mal’tsev, photograph and biography not available at the time of publication.

Jarmila Vilˇcáková, photograph and biography not available at the time of publication.

Petr Sáha, photograph and biography not available at the time of publication.

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Embedding Calderón Multiplicative Preconditioners in Multilevel Fast Multipole Algorithms Joris Peeters, Kristof Cools, Ignace Bogaert, Femke Olyslager, Fellow, IEEE, and Daniël De Zutter, Fellow, IEEE

Abstract—Calderón preconditioners have recently been demonstrated to be very successful in stabilizing the electric field integral equation (EFIE) for perfect electric conductors at lower frequencies. Previous authors have shown that, by using a dense matrix preconditioner based on the Calderón identities, the low frequency instability is removed while still maintaining the inherent accuracy of the EFIE. It was also demonstrated that the spectral properties of the Calderón preconditioner are conserved during discretization if the EFIE operator is discretized with Rao-Wilton-Glisson expansion functions and the preconditioner with Buffa-Christiansen expansion functions. In this article we will show how the Calderón multiplicative preconditioner (CMP) can be combined with fast multipole methods to accelerate the numerical solution, leading to an overall complexity of ( log ) for the entire iterative solution. At low frequencies, where the CMP is most useful, the traditional multilevel fast multipole algorithm (MLFMA) is unstable and we apply the nondirectional stable plane wave MLFMA (NSPWMLFMA) that resolves the low frequency breakdown of the MLFMA. The combined algorithm will be called the CMP-NSPWMLFMA. Applying the CMP-NSPWMLFMA at open surfaces or very low frequencies leads to certain problems, which will be discussed in this article. Index Terms—Electromagnetic scattering, fast solvers, numerical stability.

I. INTRODUCTION

NTEGRAL equations discretized by the method of moments (MoM) are very popular for handling scattering problems in the frequency domain as these result in fully error controllable solutions. In this paper we focus on scattering at perfectly electrically conducting (PEC) objects. There exist two independent boundary integral equations (BIE) for this scattering problem, namely the electric field integral equation (EFIE) and the magnetic field integral equation (MFIE) [1], which can be linearly combined to form the combined field integral equation (CFIE). The MFIE is inherently well-posed and as such results in a limited number of iterations when solved iteratively. However, for the same level of discretization it is significantly less accurate than the EFIE [2], [3]. Regrettably, the EFIE is ill-

I

Manuscript received January 10, 2009; revised May 12, 2009; accepted September 15, 2009. Date of publication January 22, 2010; date of current version April 07, 2010. J. Peeters, K. Cools, I. Bogaert, and D. De Zutter are with the Department of Information Technology (INTEC), Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium (e-mail: [email protected]). F. Olyslager (deceased) was with the Department of Information Technology (INTEC), Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2041145

posed, with the situation becoming worse as the frequency drops (or, a related problem, as the discretization becomes finer). We also need to make the distinction between closed and open objects. For the latter ones, the MFIE is not valid and only the EFIE remains viable. In high frequency (HF) simulations of closed objects, both the EFIE and MFIE show spurious solutions at certain resonance frequencies, leading to strongly increased condition numbers (only discretization error prevents it from becoming infinite). The CFIE is resonance-free but still suffers from a breakdown due to its EFIE contribution, when the mesh-density becomes high. The above outline shows that there are a few situations for which no satisfactory approach is available without the use of efficient preconditioners. Recently [4], [5], a new type of preconditioners has displayed impressive results in stabilizing the EFIE at all frequencies, making it an ideal candidate to solve the low frequency (LF) stability problems described above. The preconditioners are based on the Calderón identities, exploiting the fact that the square of the EFIE operator is a second kind operator. Initial formulations [5] of the preconditioner suffered from problems when it was discretized using Rao-Wilton-Glisson (RWG) [6] expansion functions, because the second kind behavior was lost during the discretization. This was recently remedied by using Buffa-Christiansen (BC) [7] functions for the preconditioner. With every RWG, a BC is associated on a refined so-called barycentric mesh. Using the RWG-BC combination allows the formulation of the Calderón preconditioner as a matrix multiplication of the MoM matrix with the so-called Calderón multiplicative preconditioner (CMP). Among others, the multilevel fast multipole algorithm (MLFMA) [8] has been shown to reduce the computational and memory cost in the iterative solution of the MoM system from to in the high frequency case. Recently, seamless extensions of the MLFMA to low frequencies have been proposed, such as the nondirective stable plane wave MLFMA (NSPWMLFMA) [9]. In this paper we will develop a new combined CMP-NSPWMLFMA. We want to emphasize that the discussion in the remainder of this article is equally applicable to a combination of the CMP with the traditional MLFMA. However, since the MLFMA itself breaks down at low frequencies [8], [9], it is less suitable to combine with Calderón preconditioning that focuses on the low frequency regime. It is also possible to replace the NSPWMLFMA by alternative approaches that are stable at low frequencies, most notably the spectral methods [10] and traditional LF-FMM [8]. This paper is organized as follows. First, in Section II we will briefly revisit the CMP. Section III introduces the NSPWMLFMA and some concepts of fast multipole algorithms that

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PEETERS et al.: EMBEDDING CALDERÓN MULTIPLICATIVE PRECONDITIONERS IN MFMAs

are applied further in the paper. In Section IV, the CMP-NSPWMLFMA is applied to closed surfaces, in both the low and high frequency regime, the latter requiring the linear addition of the MFIE to form the CFIE. Section V demonstrates how to incorporate open PEC objects in the CMP-NSPWMLFMA, while Section VI explains and solves the problems that occur when using the CMP-NSPWMLFMA at extremely low frequencies. Finally, Section VII contains a number of numerical examples that demonstrate the validity and capability of the CMP-NSPWMLFMA combination. In addition, the different sections contain numerical experiments that assert the staked claims therein.

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dealing with the EFIE at fine mesh-densities although they can be very effective when considering large scatterers requiring not too fine meshes (see also the numerical experiment at the end of this Section). Recently, a new type of preconditioner was proposed, that approaches the problem at the level of the integral equation. It is based on the Calderón identities [15], [12] and preconditions the EFIE by operating on the EFIE, resulting in a combined operator that is second kind. One Calderón identity is

(5) II. CALDERÓN MULTIPLICATIVE PRECONDITIONER

with

time dependence is assumed In frequency domain (an and suppressed) the EFIE on a PEC scatterer is defined as (6) (1) with the incident electric field, the unknown induced surface current density on the scatterer, the permittivity, the the characteristic impedance and permeability, the unit normal on the scatterer surface. The electric-electric operator is given by

the magnetic-electric operator. The operator is second kind and has a bounded spectrum, which means that the integral equation (7) is well-posed. When decomposing the operator in contributions from vector and scalar potentials, we arrive at

(2) (8) (3) (4) with the wavenumber, the homogeneous space scalar Green’s function and the homogeneous space Green’s dyadic with the unit dyadic [11]. The first term represents the contribution from the vector potential, while the second term describes the influence from the scalar potential. The singular value spectrum of this operator has two branches, one going to infinity and one going to zero [12]. As the mesh-density increases, the singular functions associated with these very large and very small singular values can be better resolved and therefore the condition number of the resulting impedance matrix in the MoM discretization will increase. More precisely, with the wavenumber and the typical discretization size. This means that it becomes increasingly difficult to solve the MoM EFIE system iteratively without preconditioning if the mesh-density (unknowns per square wavelength) increases. There exists a wide variety of preconditioners, most directly based on forming approximate inverses of the impedance matrix. The most popular of this kind are block-Jacobi and Incomplete LU (ILU) factorization [13], which are essentially applicable to any system of linear equations, although their effectiveness may vary significantly. Recent developments include the approximate MLFMA [14]. Experience shows that this type of preconditioners is generally not performant when

The range of is also its kernel, such that [15], [5]. However, when discretizing the operators, one must be careful regarding the choice of the expansion functions. When using and RWG functions to discretize both the preconditioning is not the EFIE , it appears that the property the discretized version of . A proposed preserved, with solution is to use the decomposed form (8) and manually omit term. This approach [5] introduces a discretization error, the limiting the accuracy. Recent research [15], [16] has revealed that the use of BC expansion functions for the preconditioning operator and RWG functions for the EFIE operator maintains the second kind behavior also after discretization. This method yields a true Calderón multiplicative preconditioner (CMP). While more accurate than the approach using solely RWG functions, it is also faster and more memory efficient because less matrices need to be stored, added and multiplied. The BC functions are constructed on a refined, so-called barycentric mesh and are div-conforming and quasi curl-conforming. The former property makes them suitable for use as basis functions, the latter assures that the Gram matrix between RWG functions and BC functions is well-conditioned. With every RWG function, a BC function is associated, defined in [7], [15], [16] and illustrated in Fig. 1. Each BC function can be constructed by a weighted sum of ‘small’ RWG functions on the barycentric mesh. In order to make the BC functions scale identically as their associated RWG function on the original mesh, the BC functions are multiplied by (see Fig. 1), the sum of the lengths of the two central edges in the barycentric

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Fig. 1. An example BC function. The filling colors (magnitude) and arrows (orientation) illustrate the vectorial behavior. Compared to the original definition [7], [15], [16], the BC function is multiplied with a constant factor equal to the length L, i.e., the sum of the lengths of the two central edges of the barycentric mesh. The two large empty triangles (named a and b) indicate the support of the RWG function that is associated with this BC function.

mesh. This will simplify the handling of non-uniform meshes and make the formulation more symmetric. The following scheme is used to discretize the preconditioned EFIE (similar to [15], but in the frequency domain)

(9)

Fig. 2. Condition number of the discretized operator for EFIE and Calderón preconditioned EFIE as a function of the inverse mesh size d (in wavelengths).

from (5) and the definition of RWG and BC functions it can be deduced that they scale proportionally to the area of the RWG or its associated BC function. The most general formulation for diagonally preconditioning the entire system is

with

(10) and

with the Gram ma. The inner product trix is defined as the matrix with , with the a a set of basis functions. The set of test functions and the inner product is defined as , with the identity operator. Finally, the ‘right hand side’ inner product is defined as the vector with . and represent the curl-conforming Here, functions obtained by rotating the RWG and BC functions. The Gram matrix is sparse and well-conditioned and its evaluation and multiplication causes barely any additional computational resources. An example demonstrates the stability properties of the CMP of is comas shown in Fig. 2. The condition number of over pared to the condition number a wide range of mesh sizes. The scattering object is a 1 m 1 m 1 m PEC cube, discretized in triangles of approximate m, leading to 450 RWG basis functions. edge size By increasing the wavelength, the mesh-density increases. The non-preconditioned EFIE has a rapidly rising condition number with increasing mesh-density, while the Calderón preconditioned matrix remains stable. Some care must be taken when the mesh is non-uniform (to efficiently discretize a large object with important sub-wavelength detail). To guarantee a low condition number of the Gram matrix, a diagonal preconditioner can be applied that makes all elements on the diagonal equal [16]. A second problem is the condition number of the entire . The diagonal elements must be of the same order of magnitude, which can again be achieved by applying a diagonal preconditioner. Unlike for the diagonal elements can not be calculated efficiently, but of

which is solved for , after which follows immeand are a left and right prediately. In these expressions, conditioner, respectively. The most balanced system is formed and are diagonal matrices containing the inverse when square root of the areas of the RWG and BC functions, respectively. However, the condition number and iteration count are containing the inverse areas almost equally low when only a containing the inverse areas of the BC functions or only a of the RWG functions is used. Diagonal preconditioners of this kind are very effective when the only source of ill-conditioning is a scaling mismatch, as is the case for both the Gram matrix . and If the object is closed (i.e., it has a finite and nonzero volume) and if it has dimensions about half a wavelength or larger, then may become singular due to internal resonances. the matrix This is typically alleviated by forming the CFIE as a linear combination of the EFIE and the MFIE. The latter one is deassures that it is fined as (the additional cross product with test functions) well-tested by

(11) Because we have

(12) contains all resonances of the MFIE operator and more. Therefore, forming the Calderón preconditioned CFIE (in the remainder of this article denoted as the operator

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the modified CFIE (MCFIE), in accordance with [12]) in the traditional way as

(13) does not lead to a resonance-free integral equation because both terms contain the MFIE resonances. However, the stabilizing properties of the Calderón preconditioner are local [12], which of the preconditioner. allows the use of a localized version does not contain the MFIE resonances [12]. Unlike We will discuss in Section IV how we construct a localized version of the preconditioner, which removes the resonances in the MCFIE. In [17] another localization technique for the MCFIE is proposed. Let us now consider the choice of . In the unpreconditioned CFIE, the number of iterations is minimized around – [3]. However, the EFIE is more accurate and for obtains a certain tolerance on the RCS it appears that the result most efficiently, as a balance between the number of unknowns and the amount of iterations required [18]. In the MCFIE, on the other hand, both the MFIE and locally preconditioned EFIE contributions are well-conditioned and the number of iterations is almost independent of . Fig. 3 displays the backscattered RCS as a function of for scattering at a 2 m 2 m cube in a high frequency and low frequency case. This illustrates that the typical accuracy considerations still apply. In the low frequency case the number of iterations relative accuracy), while was always limited to 6 or 7 (for in the high frequency simulation the number of iterations varied from 16 to 8 as was increased from 0 to 1, regardless of the number of unknowns. The conclusion can be drawn that should be chosen fairly high, in order to profit from the high accuracy of the EFIE, but not so high as to have a negligible MFIE contribution, which could lead to a strong increase in as iterations in case of a resonance. We propose an approximation, but a detailed study (beyond the scope of this article) might reveal an optimal choice. In Section IV we will briefly revisit this topic when an object is simulated at a frequency which displays spurious solutions. A further improvement could include a different expansion scheme for the MFIE (for instance [3]), which enhances its accuracy, to make the choice of almost completely irrelevant. Of course, the MFIE impedance matrix leads to additional calculation time per iteration and increased memory-usage for storage and, as displayed in Fig. 3, leads to a loss of accuracy. For these reasons, we will only use the MCFIE when the situation requires so, namely for closed objects larger than half a wavelength. To conclude this section, we compare the Calderón preconditioner with one of the most popular algebraic precondioners, the so-called incomplete LU (ILU) decomposition. ILU preconditioning has been extensively discussed in [13] (and it is beyond the scope of this article to give a detailed overview and comparison). However, as Calderón preconditioning is aimed in particular at stabilizing the low frequency (or dense grid) breakdown

Fig. 3. The influence of  in the MCFIE on the backscattered RCS of a 2 m 2m 2 m. The high frequency (left, f : e Hz) and low frequency : e Hz) simulations show the value of the backscattered RCS (right, f for three different mesh densities, indicated by the number of unknowns N .

2

2 = 4 77 5

= 4 77 7

of EFIE, it is useful to compare both preconditioners in a broadband sense, rather than just the high frequency regime which is most often studied when evaluating algebraic preconditioners. We simulate scattering at a 16 m 40 m (open) PEC plate (7568 unknowns), for a range of frequencies, and compare the number of iterations for solution of the EFIE integral equation. The drop tolerance [13] of the ILU preconditioner was chosen for each % frequency such that the sparsity of the preconditioner is approximately equal to that of the near interaction matrix, from which it was built. The Calderón preconditioner has 0% sparsity. In this example neither the impedance matrix nor the CMP were accelerated using fast multipole methods, due to its limited size. The results of the comparison (using TF-QMR iterative algorithm [19]) are shown in Table I. It is clear that the ILU preconditioner performs well at high ), as was alfrequencies (discretization approximately ready shown by previous authors [13], but quickly breaks down when the frequency drops (or the grid density increases), while the Calderón preconditioner is less effective at high frequencies

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TABLE I COMPARISON OF ILU AND CALDERÓN PRECONDITIONER AS A FUNCTION OF THE FREQUENCY (f ) AND THE ASSOCIATED ELECTRIC DISCRETIZATION SIZE (d=). THE ITERATION COUNT IS SHOWN FOR A SOLUTION TO RELATIVE ACCURACY 10 FOR THE UNPRECONDITIONED CASE (‘NONE’) AND USAGE OF THE ILU AND CALDERÓN (‘CAL’) PRECONDITIONER, FOR SCATTERING AT A PEC PLATE

Fig. 4. The vectors required for aggregation, translation and disaggregation.

but is very stable at low frequencies. The low frequency breakdown of ILU is in accordance with an observation made later in this article, namely that the sparsity of the preconditioner must decrease when the frequency goes down, in order to remain effective. The Calderón approach leads to a fully dense matrix and as such does not suffer from this effect. Because of this, it is however more computationally expensive (although, as will be shown, this is not prohibitive to the complexity after applying fast multipole methods). As such, at high frequencies the algebraic preconditioners are at least competitive with and probably preferable to Calderón techniques and the focus in the remainder of the article will be predominantly on the performance in a broadband frequency range. III. NON-DIRECTIONAL STABLE PLANE WAVE MULTI-LEVEL FAST MULTIPOLE ALGORITHM The MLFMA [8] is based on a propagating plane wave decomposition of the Green’s function

(14) with

(15) with the second kind spherical Hankel function of order the Legendre polynomial of degree and the vectors and illustrated on Fig. 4. All basis and test functions are divided into localized boxshaped groups. Using (14), a matrix-vector product can be executed as follows: all basis functions (multiplied by their expansion coefficient) in a group can be aggregated into an outgoing radiation pattern, which is then translated to an incoming radiation pattern at the receiving group. The incoming radiation pattern is then disaggregated to the test functions, completing the interaction between those two groups. The addition theorem is

only valid when the largest possible sum of the aggregation and disaggregation distance is smaller than the distance of translation, requiring at least one box separation between groups that interact through MLFMA. However, in practice, the MLFMA scheme can only be used for boxes that are separated by at least boxes, with and defined by the desired accuracy and the box size at the lowest level [8], [9]. In practical simulations, typically lies between 1 and 3. Interactions between expansion functions of boxes that are closer to each other are not accelerated with the MLFMA scheme and need to be calculated by direct evaluation of the Green’s function. The radiation patterns are sampled in the quadrature points used for the integration in (14). The number of plane waves required for an accurate integration depends on the group size. When using a multilevel scheme, where nearby groups are hierarchically assembled into larger groups, some form of interpolation and anterpolation is required between the levels. In [8], uniform sampling in the coordinate and Gauss-Legendre sampling in the coordinate is used, while aggregating to a higher level relies on local Langrange interpolation. An alternative approach [20] suggests the use of uniform sampling for both coordinates, allowing for fast Fourier transform (FFT) interpolation and anterpolation, which is global and more accurate. In the next sections, we will opt for the latter method. The MLFMA suffers from the so-called low frequency (LF) breakdown [9], which is caused by a numerical instability rooted in the supra-exponential behavior of the Hankel function when the order becomes larger than its argument. The higher order terms, even though they should largely cancel out after integration, swamp the dominant lower order terms which are lost then in the process. When the electric translation distance drops, the argument of the Hankel function decreases and the instability becomes more prominent. In practice, the MLFMA cannot obtain a reasonable accuracy for boxes below a quarter of a wavelength. This is especially relevant to this paper because we are looking at Calderón preconditioning techniques for fine meshes. Recently, research for alternatives that are stable over a wide frequency range has resulted into a number of efficient algorithms. A first one is based on a spectral decomposition [10] of the Green’s function and includes evanescent waves in addition to the propagating waves. Because the formulation is only valid in one half-space, it requires multiple radiation patterns. Nevertheless, very efficient interpolation routines have been developed, making this method suitable for broadband simulations. Another approach, the nondirective stable plane wave multilevel fast multipole algorithm (NSPWMLFMA) [9], is closely related to the MLFMA and stabilizes the plane wave decomposition for all frequencies. By shifting the coordinate into the complex

PEETERS et al.: EMBEDDING CALDERÓN MULTIPLICATIVE PRECONDITIONERS IN MFMAs

plane over a certain distance , the terms in the translation operator sum are normalized. However, this scheme only works for -directed translations. Translations in other directions can be treated by a rotation, aligning them with the z-axis. To avoid a different radiation pattern for every direction of translation, a basis of plane waves is constructed through a QR-procedure that contains enough information for all the different orientations. The rotation and transformation to uniform points can be included in the translation operator, which keeps its diagonal property. The result is a scheme that is very similar to the MLFMA, the only drawback being that the interpolations can no longer be efficiently treated with the FFT algorithm but now require a dense matrix. This drawback limits the scheme to the low frequency region, where is approximately independent of the electric box size. However, the entire method still becomes broadband by switching to the MLFMA as soon as the boxes become large enough. This can be done seamlessly by setting and using the uniform sample points and FFT interpolation from a certain level. Compared to the spectral methods, the NSPWMLFMA requires aggregation of only one radiation pattern, but has less efficient interpolations between the levels. In all simulations and tests, we will opt for the NSPWMLFMA, but it needs to be stressed that all techniques discussed further are essentially independent of the specific algorithm used. The remainder of this section is devoted to introducing and clarifying a few subtle aspects of the MLFMA that are important to Calderón preconditioning. First of all, we will generally use the vectorial formulation, based on the dyadic representation in (2). The Green’s dyadic can be decomposed as

(16) Because and because it is a projection operator, the radiation patterns are fully determined with only two independent components, namely a and component (as one could expect from a far field pattern). The scalar formulation of the MLFMA relies on (3) and uses four components (three Carthesian components for the vector potential and one for the scalar potential). The number of components can be reduced to three by exploiting the Lorenz gauge, but it is still less efficient than the vectorial formulation which we will use in the next section. An important aspect of vectorial formulations is the choice of the truncation limit . A number of implicit and explicit expressions for exist for the scalar case [21], [8]. In the HF regime it can be shown [8] that the vectorial case requires two extra terms to be included for the EFIE, to compensate for operation in the Green’s dyadic. However, an expresthe sion valid over the entire frequency range is significantly more analytically involved. The MFIE contains only one operation and as such it is expected to require a lower than the EFIE. A detailed study is conducted in [21] and we will limit ourselves here to Fig. 5, displaying the required for the scalar case, the MFIE and the EFIE, to approximate respectively the Green’s , the electric-electric Green’s dyadic and the function with accuracy electric-magnetic Green’s dyadic and .

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Fig. 5. Truncation L required for approximating within a tolerance of 10 the Green’s function (denoted ‘scalar’), the electric-electric Green’s dyadic (denoted ‘EFIE’) and the magnetic-electric Green’s dyadic (denoted ‘MFIE’), as a function of box size for n = 3.

As expected, in the HF-regime (i.e., for boxes that are of the order of a wavelength or larger) the differences are small and the Green’s dyadics require at most one or two extra terms, compared to the scalar Green’s function. However, in the LF regime the differences are much more pronounced and are rooted in the differential operators that occur in the Green’s dyadics. For the purpose of this article, we observe that at lower frequencies the MFIE requires significantly less terms in the series than the EFIE for the same accuracy. The previously mentioned denotes the number of terms in the translation operator. The required sampling rate for the Ewald sphere integration (in the case of NSPWMLFMA) can be . However, this is not the number of sample shown to be points required to store the aggregation or disaggregation patterns with sufficient accuracy. One can use a smaller amount of , expressed by means of a pasample points, namely ). is defined as the smallest number rameter (with for which sample points at the aggregation and disaggregation stages still lead to the desired accuracy after interposample points at the lation to and anterpolation from translation stage. This is particularly useful at the lowest level, where aggregation and disaggregation are done through dense matrices that need to be stored. In the high frequency limit one , which would mean a fourfold expects to approximate reduction in memory. A detailed analysis is given in [21], in this article we will restrict ourselves to Fig. 6, showing the required for accurate aggregation and disaggregation. significantly saves memory, it will inWhile using crease computational cost due to the extra interpolations and anterpolations required. If the number of iterations is limited, e.g., by using a Calderón preconditioner, such that the overall run-time is dominated by setup, then the reduced memory-usage can be more important than the increased computational iteration cost. In the next three sections, we will demonstrate how the NSPWMLFMA (or, for high frequencies, the MLFMA alone) can be efficiently used to accelerate the Calderón preconditioned EFIE.

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TABLE II STATISTICS FOR SCATTERING SIMULATIONS OF A 2 M 2 M 2 M CUBE AT 4:77 1 10 HZ AS A FUNCTION OF THE NUMBER OF UNKNOWNS (N ) AND MESH-DENSITY ( AS NUMBER OF UNKNOWNS PER SQUARE WAVELENGTH)

2

2

Fig. 6. Aggregation L required such that after interpolation to L the Green’s function (‘scalar’), the electric-electric Green’s dyadic (‘EFIE’) and the magnetic-electric Green’s dyadic (‘MFIE’) can still be approximated within a tolerance of 10 , as a function of box size for n = 3.

We will start with the most straightforward case: that of closed objects. IV. CLOSED OBJECTS complexity due Storage and calculation of (9) is of and . Both these complexities to the dense matrices by applying the NSPWMLFMA can be reduced to (including the transition to MLFMA when the boxes become large), which makes the scheme useful for objects requiring many unknowns. Application of fast methods to the impedance has been studied extensively before and has been matrix shown to be highly effective. The oct-tree used for the RWG functions can be re-used for the BC functions. Since every BC function is associated with one RWG function and has approximately the same centroid and spatial extent, the BC function can be assigned to the same box as the corresponding RWG function. This makes extension of existing codes significantly easier. Let us first look at the simulation of (closed) objects below their first resonance frequency, in which case the EFIE alone is sufficient to obtain an accurate solution. As an example, a 2 m 2 m 2 m cube is considered, illuminated by a plane wave with freHz, which is well below the first resonance frequency , the time quency. Table II shows the number of iterations

and the memory for aggregation/disaggregaper iteration tion matrices , near interactions and translation as a function of the number of unknowns operators and mesh-density ( as number of unknowns per square wavelength). Additionally, the memory is indicated that would be required for a classical MoM simulation without use of fast . The relative accuracy of the itermultipole techniques . The transpose free quasi minimal residual ative solution is (TFQMR) iterative algorithm [19] was used in this and all later examples. The table shows that, even for very high mesh-densities, the scheme including NSPWMLFMA maintains the LF stability of the integral equation. Also note that the electrical size of the cube is approximately 0.3 , which coincides more or less with the lowest level box size of the traditional (high frequency) MLFMA. For stability reasons (see Section III), this box size can not be reduced in the MLFMA and therefore it can not be applied in this example. This illustrates the importance of having an LF stable fast multipole technique (like the NSPWMLFMA) in combination with the CMP. When the frequency is high such that resonance solutions cannot be excluded, the MCFIE (13) is needed. When not using a Calderón preconditioner, the impedance matrices resulting from the EFIE and MFIE can simply be added at setup-time, resulting in no additional memory or CPU time during the iterative process. However, in our scheme this is no longer must be applied to possible, because alone. Therefore, the memory required for the near interactions of the MFIE must be stored separately. However, the distant interactions can be stored efficiently since the aggregations and are identical, with and translations of . The sharing of aggregations and translations can also be exploited during the iterative process, by treating MFIE and EFIE together for aggregation and translation and only separating them for disaggregation. The sampling rate for the radiation patterns of the MFIE is significantly smaller than that of the EFIE, such that the disaggregation matrices of the MFIE can be stored more compactly than those of the EFIE. We have previously indicated that contains all the resonances of and more, such that we must use a localized version of (denoted as in order to obtain a truly resonance-free equation. A number of methods exist to create a , but here we will opt to omit interactions over a distance longer than 1 . Omitting distant interactions beyond a certain translation distance has the

PEETERS et al.: EMBEDDING CALDERÓN MULTIPLICATIVE PRECONDITIONERS IN MFMAs

secondary advantage of saving some memory and accelerating the matrix-vector product, even though these gains are generally marginal because most computational effort is done on the lowest levels, i.e., at short distances. This implies that when the frequency goes down (and the mesh remains identical), becomes less sparse, eventually being fully dense when the scatterer is smaller than the wavelength. An alternative method for instead of in [17]) shows eslocalization (using sentially the same behavior. Although a detailed study is beyond the scope of this article, this behavior could indicate why the (sparse) algebraic preconditioners break down when the frequency drops. Although in general high frequency simulations can be treated efficiently with algebraic preconditioners like ILU, using a Calderón preconditioner for electrically large objects can be necessary when the mesh-density is high (for instance, due to fine geometrical features), leading to a high condition number due to the EFIE contribution in the unpreconditioned CFIE. In the following example we consider a resonating cube with a locally refined mesh near the edges, to more accurately catch the singular behavior of the induced currents. The object has dimensions larger than half a wavelength, requiring CFIE or MCFIE. For a fair comparison between the two solution methods it is necessary to solve the solution vectors for the same accuracy. During the , the iterative process, when solving the system stopping criteria is such that . When using a preconditioner , the stopping criterion used . However, these do is not lead to solutions that are equally accurate, because of the . We will use the difference in condition number of and , with a posteriori stopping criterion of an approximation of the exact solution that we obtained by first solving for a few additional orders of magnitude accuracy. While this has no practical purposes, because the exact solution is not known at run-time, it allows for a more accurate and fair assessment of preconditioners. A significant increase in (or just if iteration count is observed for those where no preconditioner is applied) is ill-conditioned. . The nuThe results are displayed in Table III, for merical value after CFIE and MCFIE indicates the coefficient . Note that both the EFIE and the MFIE have spurious solutions. However, only those of the MFIE will effectively radiate, leading to incorrect radar cross sections. The EFIE will lead to incorrect current densities but results in the correct field values. is finite, Hence, even though the MFIE (its exact solution due to the discretization, so the iteration count can be determined) is included in Table III for the sake of completeness, it cannot be considered a dependable solution method in the high frequency region. This is illustrated in Fig. 7, displaying the radar cross section of the cube at large distance, obtained using most of the integral equations from Table III. The accuracy of the CFIE and MCFIE are good and, as expected, depend on the choice of (a high means the accurate EFIE dominates the slightly more inaccurate MFIE, as previously discussed). Table III explains our earlier recommendation in the MCFIE: the accuracy is high and the number of

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Fig. 7. Comparison of the radar cross section, using various integral equations (see also Table III). The absolute difference of the results using MFIE, CFIE and MCFIE with the EFIE (used as a reference) are plotted along a large circle around the cubic scattering object. For convenience, the entries in the legend are ordered by descending value along the 180 direction.

TABLE III COMPARISON OF THE VARIOUS INTEGRAL EQUATIONS FOR THE SOLUTION OF A RESONATING CUBE ( = 150 390)

N

of iterations is low. The MFIE alone leads to a completely incorrect far field, as a result of the radiating spurious mode. It is obvious that NSPWMLFMA-accelerated Calderón preconditioners have significant value in situations where the meshdensity is too high for the EFIE to converge fast. The additional cost per iteration is more than compensated for by the highly reduced number of iterations. The improved spectral properties guarantee robustness. V. OPEN SURFACES One of the most powerful features of BC functions is that effectively preconditions even in the case of open structures [16], something hitherto impossible using only RWG functions. As shown in [12], on open surfaces we solve for

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TABLE IV STATISTICS FOR SCATTERING SIMULATIONS OF A 2 M 2 M PLATE AT 4:77 1 10 HZ AS A FUNCTION OF THE NUMBER OF UNKNOWNS (N ) AND MESH-DENSITY ( AS NUMBER OF UNKNOWNS PER SQUARE WAVELENGTH)

2

as the sum of the current densities on both sides of the surface, using the EFIE (17) cannot be linked to However, contrary to closed surfaces, , such that cannot be shown to be a well-posed opover erator. Some arguments indicating the superiority of are given in [12]. Still, through a Helmholtz-decomposiis the kernel tion it can be shown that the range of [15]. In the remainder of this section, will of simply be denoted as . Due to the particular construction of the BC functions at the edges, the scheme described by (9) applies to open structures as well. However, application of the vectorial faces some difficulties due to the fact NSPWMLFMA to that BC functions, contrary to RWG functions, have a component normal to the edge. In order to be able to calculate the near interactions for the contribution due to the scalar potential, the integral

(18) must be calculated. When does not have a component normal to the edge, this can be reduced to the less singular form

(19) of which the singular part of the inner integration can be integrated analytically for linear functions [22]. However, in the case of BC functions, the normal component gives rise to equivalent line charges. These line charges lead to non-integrable integrals over the edge. While this may seem an unsurmountable issue, in previous publications [15] these line charges were simply omitted. Practice shows that it leads to the desired results. In the NSPWMLFMA the omission of the line charges requires a different formulation for the distant interactions. The vectorial formulation, based on the decomposito the basis tion of the Green’s dyadic, does not move a and test functions and thus includes the contribution of the line charges. This would lead to a mixed matrix, where the near interactions omit the line charges and the far interactions include them, jeopardizing the preconditioning effect. For consistency, the scalar formulation is required, using four scalar radiation patterns. The extra computational cost is relatively limited. , nor to the calThere is no modification to the storage of . The near interactions of are treated culation of

identically as for closed objects. Only the distant interactions rerequire twice as much memory and computational lated to time, due to a doubling of the number of the radiation pattern components. As an illustration, the simulation of a 2 m 2 m , for varying plate is considered, at a frequency of mesh-density. The results are displayed in Table IV and show , the time per iteration and the number of iterations , the memory for aggregation/disaggregation matrices and translation operators as near interactions a function of the number of unknowns and mesh-density as number of unknowns per square wavelength). Addition( ally, the memory is indicated that would be required for a classical MoM simulation without use of fast multipole techniques . The relative accuracy of the iterative solution is . The results indicate that the preconditioner has the same effect on open surfaces as it has on closed ones. While in general for open surfaces the condition number of is slightly higher than that for closed surfaces, the same independence of mesh-density is observed. The previous example demonstrates the LF behavior. The Calderón preconditioner can also be applied to HF simulations (characterized by mesh-sizes . In Table V the number of iterations is comof the order , the full pared when using the unpreconditioned matrix matrix and a local preconditioner as in . The dimensions of the plate are systemHz. atically increased, while the frequency remains at . The relative accuracy of the iterative solution is As expected, the number of iterations for the unpreconditioned EFIE increases rapidly. More surprising, however, is the rises relatively fact that the iteration count for fast as well (even though it remains considerably lower than the seems to keep things EFIE alone), while under control. In order to get a better understanding of this phenomenon, the eigenvalue spectra for both are displayed in Fig. 8 for the 20 m 20 m plate. can not be Unlike for closed objects, the global operator analytically related to a second-kind operator [12] and displays eigenvalues in all quadrants of the complex plane. In addition to a slightly increased condition number, this scattering of the eigenvalues causes difficulties for iterative solvers, which perform optimally when the eigenvalues are clustered and restricted to certain areas of the complex plane [23]. When using the local preconditioner, the scattering is limited and as a consequence the solution is found in less iterations. In general, it is better to use a local preconditioner, not just as a way to save resources but also to obtain faster convergence. A detailed study of why the local preconditioner reduces the scattering is beyond the scope of this article. It is, however, related to the fact that, using a local

PEETERS et al.: EMBEDDING CALDERÓN MULTIPLICATIVE PRECONDITIONERS IN MFMAs

N T

COMPARISON OF THE ITERATION COUNT UNKNOWNS) FOR MATRICES 

M ( MB)

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TABLE V

AND MEMORY (DENOTED

USAGE  ), 

T T

IN

1

G

1

FOR A HF SCATTERING SIMULATION AT A SQUARE PLATE OF INCREASING (DENOTED  ) AND  1  1  (DENOTED   )

T

T

G

T

T

TT

SIZE (

N

T

Fig. 9. Condition number of  as a function of inverse wavelength for various truncation errors.

tical applications of electrodynamics. The next Section investigates what happens at extremely low frequencies. VI. VERY LOW FREQUENCIES  Fig. 8. Eigenvalue spectra (in the complex plane) of the matrices T T (left) and T 1 G 1 T (right) for a square plate.

 1G

1

instead of global preconditioner, the influence from the edges on the rest of the surface is reduced. This explains why open structures with a high edge to surface ratio (for instance Split Ring Resonators) still cause compactness issues, even when using a local preconditioner. The memory comparison in Table V shows that for open surfaces the scheme with a preconditioner requires about 2.5 times the memory of the unpreconditioned scheme. This is due to the previously discussed fact that the scalar formulation of (NSPW)MLFMA with four radiation patterns must . be employed for Both this Section on open surfaces and the previous one on closed surfaces deal with frequency ranges that cover most prac-

When using the NSPWMLFMA to accelerate the Calderón preconditioner and the impedance matrix at even lower frequencies, the Calderón preconditioner, as described previously, will eventually break down. In Fig. 9 the condition number of is shown as a function of the frequency, for scattering at a configuration of two small cubes positioned such that they interact through the NSPWMLFMA. It appears that the condition number increases very rapidly from a certain point. A comparison of the three curves for various truncation errors indicates that it is essentially caused by the error introduced by the NSPWMLFMA. Indeed, the explosion of the condition number can be suppressed by increasing , but obviously this comes with the cost of additional memory and CPU time. The instability is caused by cancellation errors, prohibiting to vanish. For reasons of compactness, in the remainder of this section we will omit the notations BC

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TT

TT

T T

Fig. 10. j   j; j   j and j  for truncation error 10 .

k

j

as a function of inverse wavelength Fig. 11. The lines represent the convergence behavior of the iterative process in the simulation of a small cube, at three different frequencies. The circle indicates the estimated convergence limit, predicted by (21).

and RWG, as well as the Gram matrix , unless the context requires their presence. As can be derived from (3), with varying behaves like and like . In Fig. 10 frequency, are displayed the norm of various constituents of the matrix as a function of the frequency, for the same two-cube example truncation error. The norm is ten orders with a , of magnitude smaller than the product of the norms demonstrating that the cancelling of the hypersingular contribution is very effective. However, no matter how accurate the NSPWMLFMA calculates the distant interactions, there will always , destroying the be a frequency range where well-behaved properties of the operator. and in (3) shows that Analysis of the expressions for

(20) with a certain function of the accuracy of the distant interactions, the wave number and the typical size of the mesh. is difficult to derive, because it depends An expression for and on many factors, including box size at the lowest level, whether or not (see Section III). As also illustrated by Fig. 9, this means that increasing the accuracy with an order of magnitude delays the instability by the same amount for this small configuration. In general, also for larger objects, Fig. 9 is a good indicator to verify when the instability occurs. Another problem, related to the previous one but independent of the use of the NSPWMLFMA, limits the achievable accuracy must be negligible, of the iterative process. The term in comparison with . Because increases in magnitude with decreasing frequency and the solution of is of limited precision in double precision, with the condition number of the Gram matrix), round-off effects in the last significant digit cause the process to eventually stagthat can be solved for can easily nate. The number of digits be estimated as

(21)

and the accuracy used to solve . Fig. 11 displays the convergence process (lines) and the estimated limit (21) (circle) for three different frequencies, demonstrating that (21) provides an accurate estimate. It is obvious that a purely multiplicative scheme for the preconditioner can never obtain more accurate digits than a direct inversion, . hindered by the condition number of Even though usually neither of the above two problems occurs in electrodynamic simulations of practical interest, it is useful to provide a solution. Both can be avoided with the same technique, albeit at a cost of memory and CPU time. Both issues , hence the only soluare caused by the failure to cancel tion is to manually eliminate the term from (8) and give up the strictly multiplicative scheme. A similar approach was used in [5], before the invention of the CMP. In order to save time and memory, the decomposition can be partially recombined as

with

(22) Note that eliminating this term does not introduce a discretization error (unlike the case of RWG-only schemes [5]), because only numerical errors cause it to be different from zero. There are now two matrix-vector products instead of one. The calcuand requires a formulation lation of of the NSPWMLFMA with respectively one and three radiation must also be treated in a scalar pattern components. requires eiway, with three components, while ther four scalar radiation patterns (for open surfaces) or can rely on the vectorial formulation with two components (when the object is closed). The recombination (22) also saves significant memory and time for the near interactions. Considering the two cube problem at different frequencies treated with the decomposed operators now yields results that are fully stable. When the frequency goes down (experiments Hz) the condition number and iterawent as low as tion count quickly converge to 2.475 and 11, respectively, when

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Fig. 12. The mesh for one spiral object.

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Fig. 14. The geometry for the simulation (not to scale), displaying the cylindrical object, the two incoming plane waves and the circle along which the far field is calculated. The first plane wave has k ; ; and E = ; ; and ; ; , while the second one is defined by k E ; ; .

(0 01 0) = (0 01 0)

= (1 0 0) = = (1 p2)(1 0 1)

Fig. 15. The bistatic RCS (for the Y-component) of the two incoming fields displayed in Fig. 14.

Fig. 13. A cylinder-shaped constellation of 222 spiral objects.

solved for a relative iterative accuracy of . The omission makes the formulation unconditionally stable. of VII. NUMERICAL EXAMPLES In this section we will illustrate the capabilities of our approach by means of three examples of increasing electrical size. The first one features metamaterials, which require very fine discretization. Metamaterials have very fine electric details and are usually only effective at certain resonance frequencies, complicating accurate simulation. In this example we study a material built from small identical spirals (which can be considered as closed PEC objects), leading to macroscopic chirality. To accurately describe the geometrical behavior, a single spiral requires 4584 unknowns. The discretization is shown in Fig. 12. A time domain analysis revealed the resonance frequencies of Hz) will be used to one spiral [24], one of which (

execute the simulations for a large constellation of spirals as well. A cylinder-shaped metamaterial is constructed using 222 spirals that are oriented identically, shown in Fig. 13, leading to a total of 1 017 648 unknowns. Note that despite the large number of unknowns, the structure itself is small compared to the wavelength, allowing us to use the EFIE alone. For an efficient handling, a parallelized version of the NSPWMLFMA was employed, relying on an asynchronous algorithm [25], which is highly suited for complicated geometries. As explained in [24], the presence of the chiral effect depends on the direction and polarization of the incoming wave. Two simulations are executed, using the incoming fields schematically represented in Fig. 14. The bistatic radar cross section (RCS) for both is displayed in Fig. 15. In agreement with the predictions, the RCS resulting from the field with oblique incidence displays some asymmetry, while the other one is perfectly symmetrical. In the case of a homogenous and isotropic scatterer, both results would be symmetrical. Due to the resonance of the structure, the number of iteraaccuracy is about 100. Note tions required to get below that these are lossy resonances, which are not related to spurious solutions. If the simulation is repeated at a non-resonant frequency, the number of iterations required becomes as low as

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Fig. 17. Left: the geometry of the PC case. A slot is shown, through which the field penetrates. The areas of local refinement model small holes. Right: a detail from the backside of the PC case, showing the local refinement around the ventilation holes.

Fig. 16. Left: the mesh of the ISS. The habitable parts and the laboratories form one closed object. The solar panels and the two radiators (the zigzag structures) are modeled as open surfaces. Each radiator consists of 24 small plates placed closely together. Right: the induced surface current densities as a result from an incoming left circularly polarized plane wave.

4, but this triggers a much weaker chiral response and is not interesting from an engineering point of view. Due to the very fine mesh, in comparison with the wavelength, these accurate simulations would be nearly impossible without a powerful preconditioner. In a second example we will demonstrate the performance of the CMP-NSPWMLFMA through the simulation of a geometry featuring many open surfaces, thin objects and sharp corners. In addition, there is a significant amount of non-uniformity present (the ratio of the areas of the largest and smallest triangles present is approximately 50). Fig. 16 shows a mesh representing a simplified form of the International Space Station (ISS) as it would look after completion in 2010 (at the time of writing it is approximately 81% finished). As was shown in Section II, the CMP-NSPWMLFMA is particularly useful at low frequencies, and we will study the effect of an incoming left circular polarized plane wave with a wavelength about the size of the ISS. The mesh contains 164 768 unknowns. We again employ the EFIE formulation (the closed object is sufficiently small to be below the first resonance). The CMP-NSPWMLFMA algorithm , each iterarequired 121 iterations (for a residual error of tion requiring 13 seconds. Without preconditioning, a single iteration takes 5 seconds. However, even after 1000 iterations the relative residual error was still close to one. This example, due to its non-uniformity, also demonstrates the need for the diagonal scalers (see Section II). Without the scalers, no significant improvement in the solution is obtained after 1000 iterations. In both this example and the previous one, successful usage of algebraic preconditioners would be extremely difficult due to the low frequencies (see also Section II). Both examples also required the use of the NSPWMLFMA for an efficient solution, because the lowest level box sizes are much smaller than

Fig. 18. The amplitude of the z-component of the total electric field (V/m) in the YZ-plane, with x = 0:1 m (see also Fig. 17). The black line indicates the contours of the PC case.

(which would be about the smallest box size in an MLFMA scheme, see Section III). In a final example we will look at a structure of multiple wavelengths in size. Fig. 17 shows the realistic (open PEC) model of a PC case (0.21 0.405 0.425 m ), as can be used for the calculation of shielding efficiencies. This structure is illuminated m, and by a plane wave ( . For a normal high frequency problem (discretization , algebraic preconditioners are arguably approximately more performant than the Calderón preconditioner. However, in this case, a fine discretization was used for very accurate modm) with local refinement near the ventilation eling ( holes (see Fig. 17), leading to 200 127 unknowns. This relatively makes the application of sparse dense mesh algebraic preconditioners undesirable, because the near interaction matrix alone is not sufficient to create an effective preconditioner (see also Section II). In addition, the cavity-like nature of this structure makes preconditioning a necessity. When

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using a fully dense Calderón preconditioner, the number of itrelative accuracy on the iterative solution) is erations (for 541. When employing a local preconditioner (cutting interactions beyond ) as described in Section V, the number of iterations is reduced to 283, illustrating its effectiveness. Without a preconditioner, the same amount of accuracy is only reached after 6270 iterations. The amplitude of the z-component of the total field is shown in Fig. 18.

VIII. CONCLUSION The combination of the NSPWMLFMA and the CMP leads to a very stable and efficient scheme for scattering simulations at PECs. The application, however, is not trivial and the various problems that occur have been explained and solved in the previous sections. Further research will focus on the use of CMP’s for dielectric objects as well, as these preconditioners have proven to be very promising in removing the Achilles’ heel of surface integral equations in practical applications, namely the lack of guaranteed fast convergence when solved iteratively.

REFERENCES [1] P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Analysis of surface integral equations in electromagnetic scattering and radiation problem,” Engineering Analysis With Boundary Elements, vol. 32, no. 3, pp. 196–209, Mar. 2008. [2] Ö. Ergül and L. Gürel, “The use of curl-conforming basis functions for the magnetic-field integral equation,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 1917–1926, Jul. 2006. [3] Ö. Ergül and L. Gürel, “Improving the accuracy of the magnetic field integral equation with the linear-linear basis functions,” Radio Sci., vol. 41, Jul. 2006. [4] S. Borel, D. P. Levadoux, and F. Alouges, “A new well-conditioned integral formulation for Maxwell equations in three dimensions,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2995–3004, Sep. 2005. [5] R. Adams and N. Champagne, “A numerical implementation of a modified form of the electric field integral equation,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2262–2266, Sep. 2004. [6] 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. [7] A. Buffa and S. H. Christiansen, “A dual finite element complex on the barycentric refinement,” Tech. Rep. PV-18 IMATI-CNR, 2005. [8] W. C. Chew, J. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics. Bostonm, MA: Artech House, 2001. [9] I. Bogaert, J. Peeters, and F. Olyslager, “A nondirective plane wave MLFMA stable at low frequencies,” IEEE Trans. Antennas Propag., to be published. [10] H. Wallén and J. Sarvas, “Translation procedures for broadband MLFMA,” Progr. Electromagn. Res., no. 55, pp. 47–78, 2005. [11] I. V. Lindell, Methods for Electromagnetic Field Analysis, ser. Oxford Engineering Science Series. Oxford: Clarendon Press, 1992. [12] R. J. Adams, “Physical and analytical properties of a stabilized electric field integral equation,” IEEE Trans. Antennas Propag., vol. 55, pp. 362–372, Feb. 2004. [13] 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. [14] T. Malas, Ö. Ergül, and L. Gürel, “Approximate MLFMA as an efficient preconditioner,” in Proc. Antennas and Propag. Society Int. Symp., Jun. 2007, pp. 1289–1292. [15] K. Cools, F. Andriulli, F. Olyslager, and E. Michielssen, “Time-domain Calderón identities and their application to the transient analysis of scattering by 3D PEC objects—Part I: Preconditioning,” IEEE Trans. Antennas Propag., submitted for publication.

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[16] F. Andriulli, K. Cools, H. Bagˆci, F. Olyslager, A. Buffa, S. Christiaensen, 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. [17] H. Ba˘gci, F. Andriulli, K. Cools, F. Olyslager, and E. Michielssen, “A Calderon multiplicative preconditioner for the combined field integral equation,” IEEE Trans. Antennas Propag., submitted for publication. [18] P. Ylä-Oijala and M. Taskinen, “Electromagnetic scattering analysis with combined field integral equations,” in Proc. Antennas and Propag. Int. Symp., Jun. 2007, pp. 4869–4872. [19] R. R. Freund, “A transpose-free quasi-minimal residual algorithm for non-hermitian linear systems,” SIAM J. Sci. Comput., vol. 14, pp. 137–158, 1993. [20] J. Sarvas, “Performing interpolation and anterpolation entirely by fast Fourier transform in the 3-D multilevel fast multipole algorithm,” SIAM J. Numer. Analy., vol. 41, no. 6, pp. 2180–2196, 2003. [21] I. Bogaert, J. Peeters, and F. Olyslager, “Error control of the vectorial NSPWMLFMA,” IEEE Trans. Antennas Propag., submitted for publication. [22] P. Ylä-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n RWG functions,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1837–1846, Aug. 2003. [23] W. Schönauer and R. Weiss, “An engineering approach to generalized conjugate gradient methods and beyond,” Appl. Numer. Math., vol. 19, pp. 175–206, 1995. [24] I. Bogaert, J. Peeters, and F. Olyslager, “Homogenization of metamaterials using full-wave simulations,” Metamaterials, vol. 2, no. 2–3, pp. 101–112, Sep. 2008. [25] J. Fostier and F. Olyslager, “An asynchronous parallel MLFMA for scattering at multiple dielectric objects,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2346–2355, Aug. 2008.

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Joris Peeters was born on April 16th, 1983. He received the M.S. degree in science and engineering/ applied physics from Ghent University, Ghent, Belgium, in 2006, where he is currently working toward the Ph.D. degree. Both his interests and research are focused on computational electromagnetism, more in particular the fast and accurate simulation of complicated geometries in broadband frequency ranges.

Kristof Cools received the B.S. degree in mathematics in 2002, the M.S. degree in physical engineering in 2004, and the Ph.D. degree in electrical engineering in 2008 from Ghent University, Ghent, Belgium. His advisors were Prof. Femke Olyslager (Ghent University) and Prof. Eric Michielssen (University of Michigan). Since 2008, he has been a Postdoctoral Researcher at Ghent University. His research interests are in computational electromagnetics with focus on the spectral properties and discretization schemes of the boundary integral equations encountered in computational electromagnetics. Dr. Cools was the recipient of the Young Scientist Best Paper Award at the 2009 ICEAA Symposium in Torino.

Ignace Bogaert was born in Ghent, Belgium, in 1981. He received the M.S. degree in physical engineering and the Ph.D. in applied physics from Ghent University, Ghent, Belgium, in 2004 and 2008, respectively. After graduating, he joined the Electromagnetics Group, Department of Information Technology (INTEC), Ghent University. His research interests include optimization problems and the modeling of various physical systems, with the emphasis on robustness, efficiency and accuracy. Prof. Bogaert’s research is supported by a postdoctoral grant from the Research Foundation-Flanders (FWO-Vlaanderen).

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Femke Olyslager (deceased) was born in 1966 and died in 2009. She received the Master degree in electrical engineering in 1989 and the Ph.D. degree in electrical engineering in 1993, both from Ghent University, Ghent, Belgium. She was a Full Professor in electromagnetics at Ghent University. Her research concerned different aspects of theoretical and numerical electromagnetics. She authored or coauthored close to 300 papers in journals and proceedings. She coauthored Electromagnetic and Circuit Modeling of Multiconductor Transmission Lines (Oxford, U.K.: Oxford University Press, 1993) and authored Electromagnetic Waveguides and Transmission Lines (Oxford, U.K.: Oxford University Press, 1999). Prof. Olyslager was Assistant Secretary General of the International Union of Radio Science (URSI), an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and was an Associate Editor of Radio Science. In 1994, she became Laureate of the Royal Academy of Sciences, Literature and Fine Arts of Belgium. She received the 1995 IEEE Microwave Prize for the best paper published in the 1993 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and the 2000 Best Transactions Paper Award for the

best paper published in the 1999 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. In 2002 she received the Issac Koga Gold Medal at the URSI General Assembly and in 2004 she became Laureate of the Royal Flemish Academy of Belgium. She was a Fellow of the IEEE.

Daniël De Zutter (F’00) received the M.Sc. degree in 1976, the Ph.D. degree in 1981, and the Habilitation in 1984 from Ghent University, Ghent, Belgium. He is now a Full Professor of Electromagnetics at the Engineering Faculty of Ghent University, where he is the Head of the Department of Information Technology. He published more than 160 journal publications. In the period 2004–2008 he served as Dean of the faculty. Prof. De Zutter was elected to the grade of Fellow of the IEEE in 2000. He serves as an Associate Editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES.

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3D Isotropic Dispersion (ID)-FDTD Algorithm: Update Equation and Characteristics Analysis Woo-Tae Kim, Il-Suek Koh, Member, IEEE, and Jong-Gwan Yook, Member, IEEE

Abstract—Three 3D isotropic dispersion-finite-difference timedomain (ID-FDTD) algorithms are formulated based on two new spatial difference equations. The difference equations approximate the spatial derivatives in Maxwell’s equation using more spatial sampling points distributed in an isotropic manner. The final spatial difference equation is a weighted summation of the two new difference and the conventional central difference equations. Therefore, based on the proposed spatial difference equation and choices of the weighting factors, seven different FDTD schemes can be formulated, which include the Yee scheme. Among the seven schemes, three methods can show isotropy of the dispersion superior to that of the Yee scheme. The weighting factors for the three schemes are numerically determined to minimize the anisotropy of the dispersion by using an optimization technique. In this paper, the dispersion & stability characteristics and the numerical complexity of the three ID-FDTD schemes are addressed. Also, the upper bound of the Courant number of the three ID-FDTD schemes is heuristically proposed and numerically verified. One scattering problem is considered to show the improved accuracy of the proposed ID-FDTD scheme. Index Terms—Finite-difference time-domain (FDTD) methods, isotropic dispersion, stability analysis.

I. INTRODUCTION

T

HE standard finite-difference time-domain (FDTD) method (Yee scheme) [1] has been widely used to solve Maxwell’s equations in numerous electromagnetic applications [2]. However, the anisotropic numerical dispersion inherent in the Yee FDTD method prohibits its application to electrically large or phase sensitive problems such as long-range wave propagation and radar cross section (RCS) computation of large objects due to the accumulation of the dispersion error [3]. Many researchers have proposed various FDTD schemes to reduce the dispersion error over the past decades. The research trend for the low-dispersion FDTD schemes can be mostly categorized into two groups. The first approach is to use a higher-order approximation for time and/or space derivatives Manuscript received January 22, 2009; revised September 10, 2009; accepted October 22, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. This work was supported in part by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-003-D00398) and the Ministry of Knowledge Economy, Korea, under the Information Technology Research Center support program supervised by the Institute for Information Technology Advancement (IITA-2009-C1090-0904-0002). W.-T. Kim and J.-G. Yook are with the Department of Electrical and Electronics Engineering, Yonsei University, Seoul, Korea. I.-S. Koh is with the Graduate School of Information Technology and Telecommunication, Inha University, Incheon, Korea. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2041311

such as Fang’s (2, 4) method [4]. The second approach approximates the spatial derivative in second-order accuracy using more spatial sampling points (larger computational stencil), and includes such methods as Forgy’s scheme [5], non-standard (NS) FDTD scheme [6]–[8] and ID-FDTD scheme [9]–[11]. The properties, advantages and disadvantages of these low-dispersion FDTD schemes are well addressed and compared in [12], [13]. The ID-FDTD scheme was originally proposed for a 2D problem, and it uses more spatial sampling points on the Yee grid than the Yee scheme to approximate the spatial derivative in Maxwell’s equation [9]. The final spatial difference equation known as ID-finite difference (FD) equation, is a linear combination of the traditional central difference equation and a new difference equation based on the extra sampling points. The weighting factor for the linear combination is analytically determined based on the dispersion relation, which can generate a negligible anisotropy of the dispersion. However, the numerical phase velocity is slower than the exact one in a dielectric or conducting medium. Thus, scaling factors are introduced to scale the material properties such as the dielectric constant and the conductivity of the given medium, which can produce the exact phase velocity and the attenuation rate [10]. The final ID-FDTD update equation is the same as that of the Yee method by replacing the spatial central FD equation with the ID-FD equation. This is the major difference between the ID-FDTD and the NS-FDTD schemes: the ID-FDTD update equation satisfies the duality principle, but the NS-FDTD does not. In this paper, a 3D ID-FDTD scheme is formulated by following a procedure identical to that of the 2D case. In Section II, a 3D ID-FDTD update equation is proposed based on the standard Yee grid and two new 3D FD equations. In Section III, the dispersion relation of the 3D ID-FDTD scheme is analytically formulated and the weighting factors for the ID-FDTD scheme are calculated in order to minimize the anisotropic dispersion. Additionally, the fitting functions of the two weighting factors are proposed as a function of cell-per-wavelength (CPW) for practical convenience, and the effect of the significant figure of the weighting factors on the resulting isotropy is examined. In Section IV, the stability condition is numerically analyzed since the dispersion relation for the 3D ID-FDTD method is very complicated, which makes analytical formulation of the stability condition very hard. However, a heuristic equation for the maximum time step is proposed. In addition, the computational complexity and the computational time of the Yee method and the proposed ID-FDTD method are compared. In Section V, a dielectric sphere scattering problem is considered to show the improved accuracy of the proposed ID-FDTD scheme.

0018-926X/$26.00 © 2010 IEEE

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Fig. 1. Graphical illustration of three spatial difference equations using : (a) surface points (Yee algorithm), (b) vertex points for the vertex scheme, and (c) edge points for the edge scheme.

II. 3D ID-FDTD UPDATE EQUATION In this section, three 3D ID-FDTD update equations are proposed based on the standard Yee grid and two new 3D finite difference (FD) equations that are extended versions of the known 2D ID-FD equation [9]. As for the 2D ID-FDTD scheme, to achieve an isotropic phase velocity, the field values at sampling points isotropically distributed on the Yee grid are used to ap, where proximate the spatial derivative operator and . Fig. 1 graphically shows the proposed FD equation for the . The additional field values can be used approximation of at the vertexes or edges of the Yee grid as seen in the Figs. 1(b) and (c). Unfortunately, the values at the new sampling points are not calculated in the standard FDTD scheme, so that the values must be estimated by using a linear interpolation of the values at the adjacent points as seen in the Fig. 1. To formulate the 3D ID-FDTD schemes based on the new sampling points, first, shift and , are introduced for a simple expression of operators, the ID-FD equations, which are defined as

(1) Then, the central difference operator terms of the shift operator as

and weighted summation of where and are arbitrary constants and . For , the proposed ID-FD operator can be reduced to the central difference operator. The resulting and and and FDTD schemes with are denoted as the vertex scheme and the edge scheme, respectively. Thus, the final ID-FDTD schemes can be expressed as a linear combination of the Yee scheme, the vertex scheme and and the edge scheme. The FDTD scheme of the case in this paper is conceptually similar to Bi et al.’s collocated FDTD scheme [14]. The major difference between two schemes is that Bi’s scheme calculates all magnetic field components at the vertex point of the Yee grid, while the proposed ID-FDTD scheme computes the magnetic field at the original Yee grid points. Therefore, the resulting update equations and their dispersion relation are different. To obtain the ID-FDTD update equation, first, the new differand are required to be explicitly exence equations of pressed in terms of the shift operators previously defined. Using the field values at the eight vertex points of the Yee grid, the difcan be given as ference equation for the vertex scheme for

can be represented in

(2) uses the field values on the surface of the Yee grid. For is approximated by the central difference example, equation [2] given as

(3) and are the spatial indexes corresponding to the where x-, y- and z-axes in the Yee grid, respectively. and are the size of the grid. To formulate the ID-FDTD schemes, the two new difference operators and , are required, which use the values at the vertex and the edges of the Yee grid, respectively. Therefore, the 3D ID-FD operator is defined by a

(4) where is the field value at the vertex point. Since is not calculated on the Yee grid, can be computed by using a linear interpolation with the field values at the four adjacent points, as seen in Fig. 1(b). For example, at a vertex point, can be interpolated and simply written as

(5)

KIM et al.: 3D ID-FDTD ALGORITHM: UPDATE EQUATION AND CHARACTERISTICS ANALYSIS

where . Therefore, the comcan be represented in terms of the new plete expression of , and the central difference equation, shift operator, as shown in (6) at the bottom of the page. Thus, can be simplified to . Similarly, and can be written as and , respectively. By following a similar procedure for , the spatial difference operator for the edge scheme, , can be represented with and , which is given by a new shift operator of

is replaced with the new spatial difference scheme, update equation can be explicitly written as

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. The

(7)

where . For example,

and and , and , is explicitly expressed as

(8)

As mentioned above, the ID-FD equation is defined as

(9) The 3D ID-FDTD scheme has the same update equation as that of the Yee scheme, in which the central difference equation,

(10) where , the central difference operator with respect to time, and are the permittivity, permeability and conductivity of and are a time-step and the time the medium, respectively. index, respectively. Based on the ID-FD equation (9), seven different FDTD algorithms can be established including the Yee algorithm: (Yee scheme), and (similar to Bi’ scheme), and and and and and (similar to Forgy’s scheme), and

(6)

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TABLE I SEVEN CATEGORY OF ID-FDTD SCHEMES ARE SUMMARIZED

and and . Combining the two equations in (11), the following eigenvalue equation can be obtained as (12)

and , and and and , summarized in Table I. It can be numerically observed that the anisotropy of the numerical dispersion of the FDTD algorithms is comparable to that of the Yee scheme. For exwith and , the dispersion has the comparable ample, for anisotropy over the propagation directions, but the anisotropy behavior is opposite to that of the Yee scheme [5], [14]. Hence, case is not interesting, since the anisotropy of the the dispersion cannot be reduced. Therefore, in this paper, the properties of the other three FDTD schemes are investigated and and and (ID-FDTD scheme compared: I), and and (ID-FDTD scheme II) and and and (ID-FDTD scheme III). The ID-FDTD scheme II is similar to Forgy’s scheme [5] since two schemes conceptually use the same sampling points, but adopt different interpolation method to compute the fields at the new sampling points. Hence, the update equation, dispersion relation and numerical complexity of the scheme II look very similar but not exactly identical to those of Forgy’s scheme. III. DISPERSION RELATION AND WEIGHTING FACTORS To formulate the numerical dispersion relation of the proposed scheme, a plane wave propagation is assumed in and where and the free space ( are the free-space permittivity and permeability, respectively). The plane wave can be expressed in the discretized domain as and where is the angular frequency and and are constant vectors. The numerical wavenumber vector is given by in spherical coordinates, and is a numerical wavenumber. After and into (10), (10) can be modified to an inserting eigen-matrix equation as

is a non-zero vector, the where is the identity matrix. Since numerical dispersion relation (eigenvalue) can be formulated by where is a determinant, solving which can be given by (13) where . The two weighting factors of and can be obtained to achieve an isotropic dispersion over the propagation directions such as and . Unlike for the 2D case [9], and cannot be uniquely defined since there is only one (13) for two unknowns of and . Hence, in this paper, two optimization techniques are used to find the values of and for given parameters such as cell size, frequency and so on, which will be explained in detail. To begin with, the obtained dispersion relation is slightly modified to equalize the numerical phase ve, the dispersion locity to the physical phase velocity. At relation can be exactly solved for , which has the same value as for the 2D case [9]. Therefore, the identical scaling factor , for the 2D case [9], of the permittivity and permeability can be used. Here, is the Courant number. Using the scaling factor, can become at , and due to the isotropic dispersion beover havior, can be highly accurately approximated by all values of and . Therefore, replacing and with and , respectively, in (13) and assuming the uniform grid , the dispersion relation, (13) can be modified as (14) It is worth noting that (14) is used to estimate and , but cannot be solved for . From (14), it can be easily observed that the optimal weighting factors, and are not functions of . As was mentioned above, the optimal weighting factors can be obtained using an optimization method. A cost function is defined as

(11) where

(15) The optimal factors should minimize the fluctuation of over the propagation directions such as and , whose computation is very difficult. However, if the cell size is very small (the usual assumption for a FDTD simulation), the minimal fluctuation of

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TABLE II THE OPTIMAL VALUES OF  AND , THE MAXIMUM ERROR AND THE MAXIMUM VARIATION OF THE NORMALIZED PHASE VELOCITY ARE SUMMARIZED AS A FUNCTION OF N COMPUTED BY THE OPTIMIZATION METHOD I

TABLE III THE OPTIMAL VALUES OF  AND , THE MAXIMUM ERROR AND THE MAXIMUM VARIATION OF THE NORMALIZED PHASE VELOCITY ARE SUMMARIZED AS A FUNCTION OF N COMPUTED BY THE OPTIMIZATION METHOD II

the cost function can guarantee the minimal fluctuation of . In this paper, two optimization methods are considered

(16a)

(16b) denotes “minimization” of the cost function where and are minimum over the and spaces. and maximum values for the ranges of and , respectively. Due to the symmetry of about and , it is sufficient to confor the optimization process. sider the range of Tables II and III show the estimated optimal weighting factors , the factors for three as a function of CPW, . For ID-FDTD schemes are compared. Tables II and III also provide the maximum fluctuation of for the given optimal weighting factors, which are defined as

(17) For smaller cells, only the error of ID-FDTD scheme III is shown in the tables, and it decreases with decreasing cell size as expected. Fig. 2 shows the maximum phase velocity error, , of the ID-FDTD scheme III as a function of

= 1 p3

Fig. 2. The maximum errors of the normalized phase velocity of the 3D = are compared as a function of N for the ID-FDTD scheme III with S two optimal weighting factors calculated by the optimization method I and II.

for the two optimized methods. As expected from Tables II and III and Fig. 2, the maximum error and the values of the optimal weighting factors are insensitive to the optimization method. Hence, in this paper, optimization method I is used for the remaining computations. It is interesting to compare the isotropy of the three ID-FDTD schemes. Fig. 3 shows the maximum phase velocity errors versus CPW for the six schemes; Yee scheme, Forgy’s scheme [5], NS-FDTD scheme [6] with parameters [8], and ID-FDTD scheme I, II, and III. From the

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Fig. 3. Comparison of the maximum error of the normalized phase velocity of the ID-FDTD scheme I, II and III and Yee scheme and Forgy’s scheme and is assumed. NSFDTD as a function of .

N S = 1p3

Fig. 3, it can be observed that ID-FDTD scheme I and II, and Forgy’s scheme show nearly identical error characteristics, and ID-FDTD scheme III and NS-FDTD scheme show the most isotropic one among the six schemes. Unfortunately, the optimal weighting factor cannot be analytically formulated. Hence, in this paper, empirical expressions of the two factors are given based on the numerical values for a wide range of . It is sufficient to formulate and as a function of since the factors are independent on . The considered fitting function for and is given by

Fig. 4. The optimal weighting factors for the ID-FDTD scheme III are plotted as a function of , which are calculated numerically and by using the fitting , (b) . equation (a)



N



(18) Table IV shows the numeric values of the coefficients in (18) for the three ID-FDTD schemes. Fig. 4 shows the comparisons of the computed and the fitting values of and . In Fig. 5, the errors of dispersion are compared as a function of for the optimal and fitting values of and . Also, the sensitivity of the isotropy of the numerical dispersion is investigated to the number of the significant figure (digit) of the weighting factors. It can be observed that the maximum error is strongly dependent on the number of the significant figure. Since the overall error is very low, however, any “number of the significant figures” and fitting values can be practically applied. It can be numerically observed that the wideband property of the proposed ID-FDTD scheme is very similar to that of the 2D ID-FDTD scheme since the weighting factors are insensitive to the CPW as seen in the Fig. 4, and so the wideband property is mainly affected by the scaling factors that are the same as for the 2D case [9]. Therefore, the ID-FDTD scheme shows a narrow band performance, which is similar to other low-dispersion schemes [13].

1=p3

Fig. 5. The maximum errors of the normalized phase velocity of the ID-FDTD . The errors are scheme III are compared as a function of . is fixed as computed varying the significant figures of the optimal weighting factors.

NS

3D ID-FDTD is numerically analyzed. Since is less than unity, the following inequality can be obtained from (13): (19) Therefore, the maximum time step

can be estimated as

IV. STABILITY ANALYSIS AND COMPUTATIONAL COMPLEXITY Since the dispersion relation for the 3D case is more complicated than that of the 2D case [9], the stability condition for the

(20)

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TABLE IV THE COEFFICIENTS OF THE CURVE-FITTING FUNCTION FOR THE OPTIMAL VALUES OF  AND

Since the maximum time step is dependent on variables such and , the maximum value of as can be predicted by varying and . The intervals of and are fixed as , but can theoretically have any value belonging to . Here, is assumed as 3000. From the stability condition of the 2D ID-FDTD can be deduced scheme [9], a heuristic expression of . Here, is the maximum time as step for the Yee scheme. Fig. 6 shows the comparison of numerically estimated and as a function of . The two results are in excellent agreement. The maximum time step can be increased around 34.8% higher for ID-FDTD scheme III than for the Yee scheme. To compute the numerical complexity of the proposed ID-FDTD algoin (10) can be rithm, for example, the update equation of rewritten in a compact form as in (21) shown at the bottom of the page, where . The other field quantities can be written in the same fashion as (21). The computational complexities of the proposed ID-FDTD schemes are compared in Table V. Also, the computation times for one wavelength propagation of a sinusoidal signal are compared in Table V, which were computed using by the Yee and the three ID-FDTD schemes. Due to the increased time step, the total computation time is increased 2.38 times in spite of much increase of the arithmetic operations (3 and 9 times increases

Fig. 6. The maximum time steps are compared as a function of N . The time steps are estimated numerically and calculated by the heuristic equation, 1=(1  ).

0

0

of the number of multiplication and addition, respectively) for ID-FDTD scheme III. ID-FDTD schemes I and III have the same number of the numerical complexity, and the complexity of ID-FDTD scheme II is reduced. Hence, for a moderate size problem, the ID-FDTD scheme II can be the best ID-FDTD scheme, and for a very large problem, the ID-FDTD scheme III can provide the most accurate results.

(21)

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TABLE V COMPARISON OF THE NUMERICAL COMPLEXITY AND COMPUTATION TIME OF 3D ID-FDTD AND YEE FDTD SCHEMES FOR ONE PERIOD PROPAGATION OF A SINUSOIDAL SIGNAL. THE SIZE OF THE COMPUTATIONAL DOMAIN IS 501x 501y 501z WITH 1x = 1y = 1z = 1. N = 10 IS USED

2

2

It can be observed that ID-FDTD scheme III can produce more accurate results than the Yee scheme. The small discrepancy between the ID-FDTD results and the exact results is caused by the stair-case approximation of the sphere. Even using the large cell, however, the accuracy of the ID-FDTD scheme is sufficient. It can also be observed that ID-FDTD schemes I and II can provide almost identical accuracy to that of scheme III, due to the small size of the sphere. VI. CONCLUSION

Fig. 7. Comparisons of the co-pol. scattered field (E ) by a dielectric sphere computed by the Yee scheme, ID-FDTD scheme III, and Mie series. The radius of the sphere is 5.033  , and the observation line is 6.5  away from the center of the sphere in the forward direction. Here,  is the free-space wavelength. The dielectric constant is 6 j 0:1. (a) magnitude, (b) phase.

0

V. NUMERICAL EXAMPLE To verify the proposed ID-FDTD scheme, one scattering by a dielectric sphere is considered. The dielectric constant and the and , reradius of the sphere are given by is the wavelength in the free-space. For the spectively. Here, FDTD simulation, the cell-size and the Courant number are asand , respectively. The weighting factors sumed as in Table II and the scaling factors for lossy medium in [10] are used. For this problem, the exact eigen-series solution is known and compared [15]. The incident wave has an x-component, and propagates along the positive z-direction. The scattered fields , apart from the are computed along the observation line, center of the sphere in the forward direction, as seen in Fig. 7.

Three new 3D ID-FDTD schemes were proposed here based on the spatial ID-FD operator that is an extended version for the 2D case [9]. The same two scaling factors as for the 2D case are used to achieve isotropic dispersion and exact phase velocity/attenuation rate in a lossy medium [10]. Unlike for the 2D-FDTD scheme, however, two weighting factors are required to minimize the anisotropy dispersion. In result, seven different FDTD schemes can be formulated, including the Yee scheme, depending of the choice of the weighting factors. Among the seven versions of the proposed FDTD schemes, three schemes can show improved isotropy of the dispersion compared to that of the Yee scheme. Therefore, the properties of those three schemes were investigated in this paper. With a greater number of the non-zero weighting factors, the resulting dispersion is spatially more isotropic, but the numerical complexity also increases. For example, it is numerically observed that the computation time for ID-FDTD scheme III increases 2.36 times compared to that of the Yee scheme. However, the isotropy of scheme II and III are improved by more than and times compared to that of the Yee scheme, around respectively. Regardless of the higher numerical complexity of the scheme III, the scheme III can drastically improve the isotropy of the dispersion. Therefore, depending on the size of the problem, one of the three versions of the ID-FDTD scheme can be chosen. Based on the simulated computational time and the dispersion accuracy, it is recommended that ID-FDTD schemes II and III be used for moderate size problems, and for phase sensitive or very large problems, respectively. Additionally, the ID-FDTD schemes are also compared with known low-dispersion FDTD schemes such as Forgy’s and NS-FDTD schemes. The dispersion error of Forgy’s scheme is observed to be comparable to ID-FDTD scheme I and II, and the ID-FDTD scheme III to that of NS-FDTD scheme. The major difference between the proposed ID-FDTD and

KIM et al.: 3D ID-FDTD ALGORITHM: UPDATE EQUATION AND CHARACTERISTICS ANALYSIS

Forgy’s/NS-FDTD schemes are that the ID-FDTD schemes can satisfy the duality principle and so free from the long-term instability problem. Since two weighting factors should be determined based on one dispersion relation, the factors cannot be uniquely determined. Thus, two optimization techniques are used to calculate the weighting factors to minimize the spatial variation of the dispersion. It can be observed that the isotropy is not much affected by the optimization technique. For practical convenience, simple equations for the two optimal weighting factors were proposed that can produce comparable dispersion to that for the weighting factors computed by an optimization method. A heuristic equation of the maximum Courant number for the proposed FDTD scheme was deduced from the 2D case, and was numerically verified. The stability increases to around 1.20, 1.51, and 1.35 larger than that of the Yee scheme for ID-FDTD schemes I, II and III, respectively. To validate the proposed FDTD scheme, a dielectric sphere scattering problem was considered. In this simulation, near-field scattered fields were compared and were computed by the Yee scheme, ID-FDTD scheme III and the exact Mie series. The results of this example showed the superior accuracy of the proposed FDTD scheme to that of the Yee scheme. Also, the properties of the ID-FDTD scheme and other low-dispersion schemes are compared and the analogy and difference between the schemes are discussed. REFERENCES [1] K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, no. 3, pp. 302–307, 1966. [2] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Norwood, MA: Artech House, 2005. [3] T. Martin, “An improved near-to far-zone transformation for the finitedifference time-domain method,” IEEE Trans. Antennas Propag., vol. 46, no. 9, pp. 1263–1271, 1998. [4] J. Fang, “Time domain finite difference computation for Maxwell’s equations,” Ph.D. dissertation, Univ. California, Berkeley, CA, 1989. [5] E. Forgy and W. Chew, “A time-domain method with isotropic dispersion and increased stability on an overlapped lattice,” IEEE Trans. Antennas Propag., vol. 50, no. 7, pp. 983–996, 2002. [6] J. Cole, “A high accuracy FDTD algorithm to solve microwave propagation and scattering problems on a coarse grid,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 9, pt. 1, pp. 2053–2058, 1995. [7] J. Cole, “A high-accuracy realization of the Yee algorithm using nonstandard finite differences,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 6, pp. 991–996, 1997. [8] J. Cole, “High-accuracy Yee algorithm based on nonstandard finite differences: New developments and verifications,” IEEE Trans. Antennas Propag., vol. 50, no. 9, pp. 1185–1191, 2002. [9] I. Koh et al., “Novel explicit 2-D FDTD scheme with isotropic dispersion and enhanced stability,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pt. 2, pp. 3505–3510, 2006. [10] I. Koh, H. Kim, and J. Yook, “New scaling factors of 2-D isotropic-dispersion finite difference time domain (ID-FDTD) algorithm for lossy media,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 613–617, 2008.

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[11] W. Kim, I. Koh, and J. Yook, “3-D FDTD Scheme with isotropic dispersion for lossless media,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 2006, pp. 3765–3768. [12] K. Shlager et al., “Relative accuracy of several finite-difference timedomain methods in two and three dimensions,” IEEE Trans. Antennas Propag., vol. 41, no. 12, pp. 1732–1737, 1993. [13] K. Shlager, J. Schneider, and S. Lockheed-Martin, “Comparison of the dispersion properties of several low-dispersion finite-difference timedomain algorithms,” IEEE Trans. Antennas Propag., vol. 51, no. 3, pp. 642–653, 2003. [14] Z. Bi et al., “A new finite-difference time-domain algorithm for solving Maxwell’s equations,” IEEE Microw. Guided Wave Lett., vol. 1, no. 12, pp. 382–384, 1991. [15] N. Morita, N. Kumagai, and J. Mautz, Integral Equation Methods for Electromagnetics. Boston: Artech House, 1990.

Woo-Tae Kim was born in Seoul, Korea. He received the B.S. and M.S. degrees in electrical and electronics engineering from Yonsei University, Seoul, Korea, in 2001 and 2003, respectively, where he is currently working toward the Ph.D. degree. His research interests are in computational electromagnetics, computational bioeletromagnetics, and microwave circuits.

Il-Suek Koh (M’02) was born in Korea. He received the B.S. and M.S. degrees in electronics engineering from Yonsei University, Seoul, Korea, in 1992 and 1994, respectively, and the Ph.D. degree from the University of Michigan at Ann Arbor, in 2002. In 1994, he joined LG Electronics Ltd., Seoul, as a Research engineer. Currently, he is with Inha University, Incheon, Korea, as an Assistant Professor. His research interests include wireless communication channel modeling, and numerical and analytical techniques for electromagnetic field.

Jong-Gwan Yook (S’89–M’97) was born in Seoul, Korea. He received the B.S. and M.S. degrees in electronics engineering from Yonsei University, Seoul, Korea, in 1987 and 1989, respectively, and the Ph.D. degree from The University of Michigan, Ann Arbor, MI, in 1996. He is currently a Professor with the School of Electrical and Electronic Engineering, Yonsei University. His main research interests are in the areas of theoretical/numerical electromagnetic modeling and characterization of microwave/millimeter-wave circuits and components, design of radio frequency integrated circuits (RFIC) and monolithic microwave integrated-circuit (MMIC), analysis and optimization of highfrequency high-speed interconnects, including RF microelectromechanical systems (MEMS), based on frequency as well as time-domain full-wave methods, and development of numerical techniques. Recently, his research team is developing various biosensors, such as carbon-nano-tube RF biosensor for nanometer size antigen-antibody detection as well as remote wireless vital signal monitoring sensors.

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Aperture Antenna Modeling by a Finite Number of Elemental Dipoles From Spherical Field Measurements Mohammed Serhir, Jean-Michel Geffrin, Amélie Litman, and Philippe Besnier

Abstract—A method to determine a distribution of a finite number of elementary dipoles that reproduce the radiation behavior of the antenna under test (AUT) from truncated spherical field measurements is proposed. It is based on the substitution of the actual antenna by a finite number of equivalent infinitesimal dipoles (electric and magnetic), distributed over the antenna aperture. This equivalent set of elementary dipoles is optimized using the transmission coefficient involving the spherical wave expansion of the measured field and using an appropriate matching method. Once the current excitation of each dipole is known, the radiated field of the antenna at different distances can be rapidly determined. Moreover, using an iterative simplification procedure, the number of equivalent dipoles is reduced, which eases the implementation of the antenna equivalent model in any existing electromagnetic code. The feasibility, the reliability and the accuracy of the method are shown using experimental data issued from the measurement of an X-band horn antenna, in two different measurement setups. Index Terms—Antenna measurement, antenna modeling, elemental dipoles, spherical wave expansion.

I. INTRODUCTION OR most of the commercial antennas, the geometric parameters and the material characteristics may not be accessible or may even be unknown. Measurements, such as nearfield (NF) techniques, are therefore the only way to determine accurately the radiation characteristics of the antenna. In particular, the recent development of multi-probe near-field system contributes to reduce the measurements duration, enabling realtime 3D complex characterization [1]. The issue is to derive from these measurements the radiation pattern of the antenna in its real environment. When the geometrical parameters and the material properties of the antenna are well-known, the antenna modeling in its operating environment is still a challenge using conventional

F

Manuscript received March 17, 2009; revised September 23, 2009. Date of manuscript acceptance October 14, 2009; date of publication January 26, 2010; date of current version April 07, 2010. This work was supported by the Centre National de la Recherche Scientifique (CNRS). M. Serhir was with the Institut Fresnel Universités Aix-Marseille, Ecole Centrale Marseille, CNRS Institut Fresnel, Campus de Saint Jérôme, 13013 Marseille, France. He is now with the Laboratoire des Signaux et Systèmes, SUPELEC, Gif-sur-Yvette 91192, France (e-mail: [email protected]). J.-M. Geffrin and A. Litman are with the Institut Fresnel Universités AixMarseille, Ecole Centrale Marseille, CNRS Institut Fresnel, Campus de Saint Jérôme, 13013 Marseille, France. P. Besnier is with the IETR-INSA Institute of Electronics and Telecommunications of Rennes (I.E.T.R), I.N.S.A., Rennes 35043, France. Digital Object Identifier 10.1109/TAP.2010.2041157

full-wave techniques (finite elements method, finite difference time domain, or method of moments). This is due to the fact that the meshing criterion depends on the finest and the smallest element of the antenna. Consequently, the computation cost increases rapidly with increasing antenna size relative to wavelength. In this paper, we propose a hybrid three-step approach. Firstly, the radiation pattern of the antenna under test is measured in a completely controlled environment (anechoic chamber). Secondly, from these measurements, a simple equivalent model is achieved in terms of elementary equivalent sources that reproduce the antenna radiation pattern. Finally, this equivalent model is incorporated into a commercial electromagnetic (EM) algorithm to assess the behavior of the modelized antenna in different environments (operating conditions). In the literature, many authors have proposed different strategies where they determine currents/charges distributions that radiate an electromagnetic field matching the AUT radiation. In [2]–[6] an electric field integral equation method (EFIE) relating the measured electric field (tangential components) to equivalent electric/magnetic currents on the aperture of the antenna has been proposed. The EFIE is solved using the conjugate gradient method. This method has mainly been used to perform near-field to far-field transformation and for antenna diagnosis applications [7], [8]. In [9]–[11], the authors used the genetic algorithm (GA) technique to substitute the AUT by a set of infinitesimal dipole sources. These dipoles are distributed inside the volume enclosing the AUT. The infinitesimal dipoles have been selected due to their implementation simplicity in any EM code, and the GA demonstrates its ability to figure out the type (electric or magnetic), the position, the orientation and the excitation of each dipole. The limitation of this method appears when dealing with complex antennas (electrically large structures), which require a large number of equivalent dipoles, as well as convergence issues and prohibitive CPU computing time. In [12] the authors have developed a modeling technique, based on the equivalence principle. The AUT is substituted by a set of equivalent dipoles distributed over the minimum sphere circumscribing the antenna (closed surface). The excitations of the antenna equivalent dipoles are derived from the AUT transmission coefficients and the equivalent dipoles spatial distribution is well-adapted to build up a well-conditioned matrix system. Computations with EM simulation data illustrate the accuracy of the method. Nevertheless, these equivalent

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SERHIR et al.: APERTURE ANTENNA MODELING BY A FINITE NUMBER OF ELEMENTAL DIPOLES

sources distribution is not optimum for large radiating structures, particularly, for aperture antennas (horn antennas). The number of equivalent sources grows considerably when the sources are distributed over the antenna minimum sphere. Thus, the equivalent model looses its simplicity when implemented in an EM code. In the present paper, an improvement of the analysis of [12] is presented. The antenna equivalent current sources are placed over a planar surface with finite extent in front of the antenna aperture and only tangential dipoles are considered. For this purpose, spherical field measurement data is used to determine the spherical wave expansion (SWE) of the horn antenna radiated field. Then, taking into account the antenna geometrical a priori information, and by adopting a convenient equivalent sources spatial distribution over the planar surface, the SWE of the antenna radiated field is written in terms of tangential electric and magnetic dipoles placed over the antenna aperture. Hence, one can represent a large structure using a finite number of equivalent dipoles, wherein the excitations of the dipoles are correlated to the antenna transmission coefficients. The most significant aspect studied in the present paper is the reliability and the accuracy of the method when using experimental data issued from the measurement of a real antenna (horn antenna) in the far-field region. Firstly, using measurement data over a sphere and secondly using truncated measurement data. The viability of the proposed modeling method is determined by comparison with experimental measurement data, where we have to deal with the measurement errors, the position inaccuracies and the signal to noise ratio of the measurement system. Once the equivalent model is achieved, a simplification procedure is presented. This procedure aims to find a finite number of elementary dipoles whose radiation pattern fits accurately the actual AUT radiation pattern by eliminating the insignificant current sources and dipoles. Then, the EM field can be rapidly computed at any distances. In this paper a comparison between the SWE and the proposed modeling technique in terms of computation time is presented. To asses the overall procedure, the radiation pattern of an X-band horn antenna has been measured over a truncated sphere using a single probe spherical range measurement in the large anechoic chamber of the Institut Fresnel [13]. The antenna radiation pattern in that measurement setup is measured in the far field region. Once the equivalent model is defined, we compute the radiation pattern at a different distance. In particular, the estimated pattern is compared to another set of measurements performed this time with a single probe planar measurement setup, which corresponds to the new anechoic chamber of the Institut Fresnel. The paper is structured as follows: In Section II, following a review of the spherical wave expansion formulation, the basis of the proposed technique is detailed. Section III presents the modeling process applied to the X-band horn antenna radiating at the frequency of 12 GHz. In order to illustrate the robustness of the proposed modeling technique we use truncated experimental measurement data using a part of the measurement sphere to validate the efficiency of the method. The setups, as well as the measurement procedure, are described in Section III. Finally, some concluding remarks are outlined in Section IV. All

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Fig. 1. Original horn antenna (top) and the equivalent current sources distribu(bottom). tion over the equivalent surface S

theoretical quantities are expressed in the S.I. rationalized unit system with time dependence. II. CONSTRUCTION OF THE EQUIVALENT MODEL The goal of the proposed modeling technique is to find the of equivalent current excitation of a given number sources that reproduce the antenna radiation pattern. In particular for a horn antenna, these sources are placed over the planar (Fig. 1). We make surface representing the horn aperture use of the theoretical development, detailed in [12], regarding the new spatial distribution of the equivalent sources adopted in this work. Contrarily to [12], in this paper, we define a current source as the combination of four tangential co-localized short dipoles: 2 electric ones and 2 magnetic ones. The th current ), is associated to a coordinate source (for system , where the origin coincides with the position of this source in the AUT global coordinate system . The choice of this current source composition is justified by the fact that using two orthogonal electric (magnetic) dipoles, we make sure that whatever the dipole orientation over the (AUT) aperture is, the dipole can be represented by two orthogonal components. The aim of this work is to use the AUT transmission coefficients to determine the

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excitation of the equivalent model sources. These coefficients are determined from truncated measurement data in contrast to [12], where they were computed from a complete sphere of synthesized data (using a simulation software). For this reason, we recall the formulation of the spherical wave expansion (SWE) of the radiated field.

in the spherical coordinate system as

associated with

(2)

A. Spherical Wave Expansion In a source free region, outside the minimum sphere circumscribing the AUT, the SWE of the radiated field in the spherical coordinate system associated to the global coordinate , where the origin coincides system with the centre of the measurement sphere (Fig. 1), is expressed in terms of truncated series of spherical vector wave functions [14] as

In the global spherical coordinate system we have with

associated

(3) From (2) and (3) it follows that the field in the global spherical is given by coordinate system (4)

(1) and corresponds to the wave where the intrinsic admittance of the medium, number, are the spherical wave coefficients (transmission coefficients), and represent the power-normalized spherdepends ical vector wave functions. The truncation number on the antenna dimensions and the operating frequency [14]. , Using the compact index such as we simplify the SWE expression where . These coefficients are uniquely determined from the knowledge of the tangential components of either or on the measurement sphere (due to the uniqueness theorem). In [15], [16] truncated near-field measurement data has been shown to coefficients. Once the are be sufficient to calculate the determined, the field outside the measurement sphere is completely characterized by (1). Our aim is to determine an equivalent model of a given anusing experimental tenna from its transmission coefficients data. We take advantage of the compactness of the SWE expresfor electric sion for the field radiated by a short dipole ( or magnetic dipole) and we relate the field radiated by an anspherical vector wave functions to the tenna described by set of infinitesimal dipoles using translational and rotational addition theorems. B. Short Dipole Excitation In what follows, the superscripts “ ” and “ ” stand for the electric and magnetic dipoles, respectively. The subscripts “ ” and “ ” denote polarized dipoles, respectively. The subscript “ ” designates the th current source located at the position in . radiated by an infinitesimal The field tric dipole located at the origin of

-directed elecis expressed

Detailed formulations and expressions for and -dipoles are provided in [12]. Accordingly, the th current source is composed of 4 co-localized tangential dipoles, this will be characterized by 4 trans. These coefficients mission coefficients are proportional to the dipoles length and to the current exci. tations We associate to the th current source the row vector where

(5) are determined using The coefficients translations and rotations of the vector spherical wave functions related to each dipole from to [see Appendix in [12]]. C. Antenna Modeling Method In order to determine the excitation of each elemental dipole composing the equivalent model, we have to define the adequate matching method. This can only be done in the global coordinate system. This is why we have used translational and rotational addition theorems [14], [17], [18] to express the field radiated by the th current source located at in the antenna global coordinate system . current sources Then, the superposition of all the expressed in is identified with field . In the same way as [12] we have to solve the linear equations (6)

SERHIR et al.: APERTURE ANTENNA MODELING BY A FINITE NUMBER OF ELEMENTAL DIPOLES

where,

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Since

and . The superscript “ ” denotes the matrix transpose. The lsqr routine from MatLab (least square data fit) [19] has been used to solve the over-determined matrix equation . Once Equation (6) is solved, the equivalent model characterization is finalized and each current excitation is fully computed.

(9) and identifying (7) and (8), we obtain

D. Equivalent Model Simplification The equivalent model of the antenna can be furthermore simplified by reducing the number of infinitesimal dipoles. In [12] authors present an iterative process which aims at reducing the number of dipoles based on the evaluation of the dipoles’ radiation power. In this paper we have improved the elimination process of the dipoles having a non significant radiation power by eliminating as well the current sources having an insignificant radiation power. Let us first recall the elimination procedure of the dipoles having a too small radiation power. are fully deBy solving (6), the vector rows termined. Then, for each current source, the power contribution of each dipole in the total radiated power is evaluated. Let define the normalized row vector power contribution

where

Using different thresholds , the number of dipoles can be reduced. First, the number of current sources and their spatial position are chosen and a solution of (6) is built with the contribution of all the dipoles as depicted in Section II.C. By cal, culating the normalized row vectors are we identify the dipoles whose corresponding terms in . A new solution of (6) is then computed without less than these negligible dipoles. This procedure is pursued as long as no more negligible dipoles are detected. When the power con, the tribution of each dipole is greater than the threshold power contribution of each current source in the antenna radiation power is evaluated. The next paragraph describes the elimination of the current sources with negligible radiation power. The current sources have to reproduce the antenna radiation pattern. In particular, the superposition of the current sources radiated power has to be equal to the radiated power of the antenna . Let us define the radiated power by the antenna and the current sources, respectively by

(7)

(10) Equation (9) is justified by the fact that wherever an infinitesimal dipole is placed in the global coordinate system, its radifor each current ated power is invariable. Let now define sources such that

where

While supposing that the antenna radiated power is equidistributed over the equivalent current sources , we can , the contribution (weighting) of evaluate, by calculating the th current source in the radiation power. Afterward, we can are less isolate and eliminate current sources for which than the given threshold . Finally, a solution of (6) is recomputed without taking into account these insignificant current sources and dipoles. III. RESULTS A. Spherical Measurement Setup To illustrate the effectiveness of the proposed methodology, the modeling process is applied to a commercial X-band horn antenna, (Fig. 2) operating at the frequency of 12 GHz. The waveguide dimensions are 124 mm 23 mm and the aperture . The associated minimum size is 73.5 mm 50 mm sphere circumscribing the antenna has a radius of mm . This antenna is measured using the spherical field system in the anechoic chamber of the Centre Commun des Resources Micro-ondes of Marseille (CCRM) managed for this topic by Institut Fresnel researchers, which is presented in Fig. 3 [13]. The tangential complex components of the electric field of this antenna have been measured at the distance . To carry out the equivalent current sources model, the measurement data are collected over a spher. The field samical surface pling is of 3 both in and coordinates, with 56 120 measurement points. B. Full Spherical Measurement Data Set

(8)

The measurement data matrix is completed (with null field values) in order to fulfil the measurement over a sphere . Based on these data and using the

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Fig. 2. The AUT description: an X-band horn antenna with the exciting waveguide.

Fig. 3. Measurement setup using the single probe spherical near field range at CCRM (left). Description of the measured horn antenna (AUT) position in the measurement coordinate system (right).

Fig. 4. Magnitude of the AUT transmission coefficients calculated using the : m, for measurement data collected over the spherical surface R  and ' .

0   165

0   357

= 1 79

matrix method [15] we compute the antenna transmission coefficients. These are presented in Fig. 4. The SWE is truncated at , such that transmission coefficients are considered for the current configuration. The AUT equivalent model is composed of 10 10 cur. These sources are chosen to rent sources separated by of size be distributed uniformly over a planar domain 90 mm 90 mm positioned at the antenna aperture plane mm (Fig. 3). From a theoretical point of view, the aperture plane should be infinite. However, it is possible to truncate it as the radiated power of the antenna, and consequently, the equivalent dipoles, is concentrated in a finite region in front of the antenna aperture. Once the transmission coefficients of the antenna are known and the equivalent current sources positions defined, we solve

Fig. 5. Magnitude of the E-field components E and E in dB (V/m) at the measurement distance R : m. In (a) and (b) the measured field radiated by the actual horn antenna and in (c) and (d) the field radiated by the antenna equivalent model.

= 1 79

(6) and we compute the current excitation of each dipole. Thereafter, we rapidly calculate the E-field at different distances in straightforward manner using the equivalent dipoles radiation. The E-field radiated by the AUT equivalent model is compared to the actual antenna radiation pattern in the measurement in Fig. 5 at the measurearea ment distance. The measured field and the field obtained from the equivalent model are in a good agreement. In addition, the radiation pattern obtained from the equivalent model fits very well with the one resulted from the SWE method as presented in Fig. 6. Also, as shown in Fig. 5, the field radiated by the equivalent model tends to cancel the ripples observed in the measured field. These ripples are principally caused by measurement errors as for example: multiple reflections between the AUT and the measurement probe, the interaction between the antenna and its mounting system . C. Truncated Spherical Measurement Data Set In order to further verify the effectiveness of the proposed method, a comparative study is presented by determining the horn antenna equivalent model using a truncated measurement data set. For the comparison, the previously used measurement data (X-band horn antenna) have been truncated to the region of significant radiated field with angular steps of . This resulted in 51 51 field points. Then, the measurement data matrix is completed (with null field values) in order to fulfil the measurement over a complete sphere. Using these measurement data, the antenna transmission coefficients

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Fig. 8. Magnitude of the E-field components E and E in dB (V/m) at the : m. In (a) and (b) the measured field rameasurement distance R diated by the actual horn antenna and in (c) and (d) the field radiated by the antenna equivalent model determined using truncated measurements.

= 1 79

Fig. 6. Comparison of co-polarized and cross-polarized components of the : m. The comparison includes: fields at the measurement distance R the AUT measurement, the field calculated using the SWE and the field radiated by the equivalent model. (a) For  and (b) for ' .

= 1 79

= 90

= 180

Fig. 7. Magnitude of the AUT transmission coefficients calculated using the : m, for measurement data collected over the spherical surface R  and ' .

15   165

105   255

= 1 79

are recomputed. The SWE is truncated at and the resulting transmission coefficients are presented in Fig. 7. While considering the same equivalent current sources spatial distribution, we solve (6) and we determine the excitation of each equivalent dipole regarding the new calculated transmission coefficients (truncated measurement data). At the measure, in the area ment distance

, the comparison between the radiation patterns of the actual AUT and the equivalent model are presented in Fig. 8. As it can be seen, the radiation pattern of the equivalent model fits very well with the actual field. Moreover, the E-field of the equivalent model is in good agreement with the E-field resulted from the SWE as presented in Fig. 9. In Fig. 10 we present the equivalent dipoles distriresulting from different thresholds bution over % % % . As it can be seen the region where the magnitude of the equivalent dipoles is significant coincides with the horn aperture (dashed rectangle). The horn antenna being off-centred during the measurement, this can justify the fact that we have considered an equivalent aperture , whose dimensions are greater than the horn actual aperture. Nevertheless, the equivalent dipoles magnitude is consistent with the actual E-field in the aperture of the antenna, since the significant equivalent dipoles are concentrated in the aperture area (dashed rectangle). In addition, using different % % , the polarization of thresholds the equivalent dipoles (electric and magnetic) is improved. In other words, the actual antenna E-field polarization is not purely vertical. This corresponds to the equivalent model comprising all dipoles. However, eliminating non-significant dipoles tends to enhance the E-field polarization purity. Consequently, substituting the antenna by an equivalent model with %, and implementing this model in an EM code, we make sure that the polarization of the radiated field is a closer to a purely vertical polarization (the cross-polarization and . is then completely cancelled for

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Fig. 10. The magnitude of the equivalent dipoles distributed over the equivalent for the filtering thresholds P P %, 2% and 4% surface (left) magnetic dipoles. (right) Electric dipoles. The dashed rectangle represents the AUT actual aperture.

(S )

=

=0

In Table I, we show the composition of each simplified equivalent model corresponding to different thresholds, the initial number of equivalent current sources is reduced twice for % compared with the initial current sources distribution.

equivalent dipoles distribution tends to deteriorate the condition number of . Since the equivalent model is intended to be implemented in an EM code, its simplicity (number of dipoles) is very important. As a matter of fact, we have to compromise between the accuracy of the equivalent model and the conditioning of . The spacing criterion adopted for the example presented in the paper , is a good compromise. The results obtained here show that the antenna equivalent model . Further work is accurate for the criterion will be required to determine whether this criterion is applicable for various arrangements of dipoles and to broader range of antennas (different sizes and geometries).

D. Discussion: The Number of the Antenna Equivalent Sources

IV. VALIDATION

Fig. 9. Comparison of co-polarized and cross-polarized components of the : m. The comparison includes: fields at the measurement distance R the AUT measurement, the field calculated using the SWE and the field radiated by the equivalent model based on truncated spherical measurement. (a) For  and (b) for ' .

= 1 79

= 90

= 180

The number of the equivalent dipoles and their spatial distribution depend on the antenna aperture size and the antenna equivalent model targeted accuracy. In fact, the spacing between sources is directly correlated with the accuracy of the antenna equivalent model and has a direct effect on the condition number of the matrix in (6). We have tried out different spacing between equivalent sources and two general remarks arise. to distribute Using the Nyquist criterion the equivalent dipoles over the antenna aperture, the matrix in (6) is very well-conditioned. However, the antenna equivalent model, based on this criterion, is not accurate. Otherwise, , using a smaller spacing, for example the accuracy of the antenna equivalent model is improved but the number of equivalent dipoles increases. Furthermore, this

A. The Planar Measurement Setup and Experimental Validation Process The performances of the previously described simplification process are evaluated by comparing the simplified equivalent models E-fields with the actual radiation pattern over a line cm, cm cm, ) at a distance ( of 30 cm from the horn antenna aperture which is closer than the measurement made for the equivalent model determination (1.79 m). To achieve this measurement, we have exploited the new measurement setup of the Institut Fresnel which consists in a single probe planar measurement system in an anechoic environment (Fig. 11). The planar scanner has a total dual scan length of 2.1 m in X and Y direction placed in an anechoic environment. The same

SERHIR et al.: APERTURE ANTENNA MODELING BY A FINITE NUMBER OF ELEMENTAL DIPOLES

TABLE I EQUIVALENT MODELS AFTER THE SIMPLIFICATION PROCEDURE FOR P

Fig. 11. Measurement setup using the single probe planar near field range at Institut Fresnel.

horn antenna is used in the transmit mode. The probe, which is mounted on one of the two carriages of the linear scanner, consists of an open ended wave guide. The probe is connected to a vector network analyzer (VNA) and the magnitude and the phase of the antenna radiated field are measured. A computer monitors the probe displacement over the line in front of the antenna and acquires the data provided by the VNA. In Fig. 12 we present the Ez field radiated by different simpli% % % % fied AUT equivalent models compared to the measured Ez field and the field estimated using the SWE method over a line at a distance of 30 cm from the aperture. As it can be seen the agreement is excellent. In addition, we present in Fig. 12 the computation time needed to calculate Ez at the measurement distance (30 cm from the AUT aperture). We show that the proposed simplification procedure eliminates insignificant current sources and dipoles while preserving a good accuracy. Once the equivalent model is fully defined the E field is rapidly estimated at different distances. Generally, the computation time and the accuracy needed to derive the E field radiated by a simplified equivalent model depend on the chosen threshold. This flexibility allows one to make a compromise between the accuracy (computation time) and the complexity (number of considered equivalent current sources). B. The SWE Versus the AUT Equivalent Model The SWE is a very accurate way to describe the radiated field by means of truncated summation of spherical wave functions and the accuracy of the SWE cannot be questioned. In addition, a convenient choice of the truncation number can help to cancel the interaction between the antenna and the mounting system during the measurement, to filter out the rapid variation of the

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= 0; 1; 2; 3 AND 4 %

Fig. 12. The magnitude of the Ez field measured at the distance of 30 cm from the AUT aperture compared with the Ez fields calculated from the SWE and the simplified equivalent models.

electric field caused by the mounting system or by the interaction between the antenna under test and the probe. Moreover, the use of the SWE to assess the antenna equivalent model allows to reduce of the number of the linear equations to consider in (6). In fact, the horn antenna radiated field modelled in the paper is completely characterized by spherical wave coefficients. Consequently we have to solve a with 6048 equations. If the SWE were linear system not used, the number of equations would increase to 13440, the number of data points on the measurement sphere . Thus, rewriting the transmission coefficients in terms of equivalent dipoles using the proposed method, enhanced with the simplification procedure, can be a good way to express the whole information contained in the transmission coefficients with a limited number of dipole excitations. The rearrangement of the SWE into a multiple SWE expressed in different local coordinate systems (using a priori information), which is the principle of our method, is a time consuming procedure (translational and rotational addition theorems). Furthermore, if we want to achieve the accuracy of the SWE with a trunwe need to use a cation number large number of equivalent current sources (large number of unknowns and ill-posed problem). Consequently, the modeling procedure is time consuming for large structures. Once the AUT equivalent model is achieved, the computation of the radiation pattern at different distances from the AUT is faster than using the SWE. This is due to the fact that, the number of equivalent dipoles is smaller than the number of the AUT transmission coefficients. In addition, implementing the equivalent model in an EM code, one is able to calculate the E-field propagation properties

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in complex and inhomogeneous environments. As an example, one can study the antenna radiation pattern through an interface between the free space and a homogeneous media. V. CONCLUSION The presented method aims to substitute the measured AUT by a set of infinitesimal dipoles regularly distributed over the antenna aperture surface, simply from the knowledge of the measured field over a truncated spherical surface. Based on the orthogonal properties of the spherical vector wave functions, the presented matching method has been shown to be a convenient tool to define a set of elementary structures (with ) from the antenna transmission coefficients. The accuracy of the equivalent model depends on the current sources distribution over an equivalent surface of finite extent which makes the method flexible. This is one of the advantages of the proposed modeling procedure. We have shown from the presented example (X-band horn antenna) that from the collected tangential electric field, the equivalent principle can be applied to an equivalent planar surface of finite size. The equivalent model can reproduce rapidly the actual radiation pattern at different distances from the considered antenna. The overall procedure has been assessed by exploiting two different measurement systems, a spherical one and a planar one, both being in anechoic environments. ACKNOWLEDGMENT The authors would like to thank the reviewers for their careful review and helpful comments. REFERENCES [1] J. C. Bolomey, B. J. Cown, G. Fine, L. Jofre, M. Mostafavi, D. Picard, J. P. Estrada, P. G. Friederich, and F. L. Cain, “Rapid near-field antenna testing via arrays of modulated scattering probes,” IEEE Trans. Antennas Propag., vol. 36, pp. 804–814, Jun. 1988. [2] P. Petre and T. K. Sarkar, “Planar near-field to far-field transformation using an equivalent magnetic current approach,” IEEE Trans. Antennas Propag., vol. 40, pp. 1348–1356, Nov. 1992. [3] P. Petre and T. K. Sarkar, “Planar near-field to far field transformation using an array of dipole probes,” IEEE Trans. Antennas Propag., vol. 42, pp. 534–537, Apr. 1994. [4] T. K. Sarkar and A. Taaghol, “Near-field to near/far-field transformation for arbitrary near-field geometry utilizing an equivalent electric current and MoM,” IEEE Trans. Antennas Propag., vol. 47, pp. 566–573, Mar. 1999. [5] F. Las-Heras and T. K. Sarkar, “A direct optimization approach for source reconstruction and NF-FF transformation using amplitude-only data,” IEEE Trans. Antennas Propag., vol. 50, pp. 500–510, Apr. 2002. [6] F. Las-Heras, M. R. Pino, S. Loredo, Y. Alvarez, and T. K. Sarkar, “Evaluating near-field radiation patterns of commercial antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 2198–2207, Aug. 2006. [7] Y. Alvarez, F. Las-Heras, and M. R. Pino, “Reconstruction of equivalent currents distribution over arbitrary three-dimensional surfaces based on integral equation algorithms,” IEEE Trans. Antennas Propag., vol. 55, pp. 3460–3468, Dec. 2007. [8] C. Cappellin, A. Frandsen, and O. Breinbjerg, “On the comparison of the spherical wave expansion-to-plane wave expansion and the sources reconstruction method for antenna diagnostics,” Prog. Electromagn. Res., PIER 87, pp. 245–262, 2008. [9] J. R. Pérez and J. Basterrechea, “Antenna far-field pattern reconstruction using equivalent currents and genetic algorithms,” Microw. Opt. Technol. Lett., vol. 42, no. 1, pp. 21–25, July 2004. [10] T. S. Sijher and A. A. Kishk, “Antenna modeling by infinitesimal dipoles using genetic algorithms,” Progr. Electromagn. Res., PIER 52, pp. 225–254, 2005.

[11] S. M. Mikki and A. A. Kishk, “Theory and applications of infinitesimal dipole models for computational electromagnetics,” IEEE Trans. Antennas Propag., vol. 55, pp. 1325–1337, May 2007. [12] M. Serhir, P. Besnier, and M. Drissim, “An accurate equivalent behavioral model of antenna radiation using a mode-matching technique based on spherical near field measurements,” IEEE Trans. Antennas Propag., vol. 56, pp. 48–57, Jan. 2008. [13] C. Eyraud, J. Geffrin, P. Sabouroux, P. C. Chaumet, H. Tortel, H. Giovannini, and A. Litman, “Validation of a 3D bistatic microwave scattering measurement setup,” Radio Sci., vol. 43, pp. RS4018–, 2008. [14] J. E. Hansen, Spherical Near-Field Antenna Measurements. London, U.K.: Peregrinus, 1988. [15] M. Mekki-Kaidi, D. Lautru, F. Bancet, and V. F. Hanna, “A matrix inversion technique for the spherical modal decomposition field solution applied on the characterization of antenna in their environment,” Microw. Opt. Technol. Lett., vol. 41, no. 5, pp. 336–341, Jun. 2004. [16] P. Koivisto and J. C.-E. Sten, “On the influence of incomplete radiation pattern data on the accuracy of a spherical wave expansion,” Progr. Electromagn. Res., PIER 52, pp. 185–204, 2005. [17] A. R. Edmonds, Angular Momentum in Quantum Mechanics, 3rd ed. Princeton, NJ: Princeton Univ. Press, 1974. [18] J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres—part I: Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag., vol. AP-19, pp. 378–390, May 1971. [19] MatLab User’s Guide.

Mohammed Serhir was born in Casablanca, Morocco in 1981. He received the diplôme d’ingénieur degree from Ecole Mohammadia d’Ingénieurs (EMI), Rabat, Morocco, in 2003 and the Ph.D. degree in electronics from the National Institute of Applied Sciences at Rennes, INSA de Rennes, France, in 2007. His research interests include spherical wave expansion technique, spherical near-field antenna measurements in harmonic and time domains, and the development of numerical methods. After a C.N.R.S. Postdoctoral position at the Institut Fresnel, he is currently working at the Laboratoire des Signaux et Systèmes, SUPELEC, Gif-sur-Yvette, France, in a Postdoctoral position.

Jean-Michel Geffrin received the Ph.D. degree in physics from the University of Paris XI, France, in 1993. He worked for ten years as a Research Engineer in France, where he developed specific antennas and experimental setups for measuring targets radiation pattern. In 2002, he joined the Institut Fresnel, Université Aix-Marseille, Ecole Centrale Marseille, France, to reinforce the hyperfrequency experimentalist team and has contributed to the constitution of the second database of scattered fields proposed by the Institut Fresnel to the inverse problem community.

Amélie Litman was born in France in 1972. She received the Ph.D. degree in applied mathematics from the University of Paris XI, Paris, France, in 1997. During 1997–1998, she was with the Eindhoven University of Technology, Eindhoven, The Netherlands, working in a postdoctoral position. From 1998 to 2002, she was with Schlumberger, France, where she worked on the development of inversion algorithms for oil prospecting. In October 2002, she joined Institut Fresnel, UMR 6133 Centre National de la Recherche Scientifique-Universités Aix Marseille, Marseille, France, as an Assistant Professor. Her research interests include forward and inverse scattering techniques.

Philippe Besnier received the diplôme d’ingénieur degree from Ecole Universitaire d’Ingénieurs de Lille (EUDIL), Lille, France, in 1990 and the Ph.D. degree in electronics from the University of Lille, in 1993. He was with the Laboratory of Radio Propagation and Electronics, Centre National de la Recherche Scientifique (CNRS), University of Lille, as a Researcher from 1994 to 1997. Since 2002, he has been with the Institute of Electronics and Telecommunications of Rennes, Rennes, France, where he is currently a researcher at CNRS heading EMC-related activities such as EMC modeling, electromagnetic topology, reverberation chambers, and near-field probing.

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Invasive Weed Optimization and its Features in Electromagnetics Shaya Karimkashi, Student Member, IEEE, and Ahmed A. Kishk, Fellow, IEEE

Abstract—A new numerical stochastic optimization algorithm, inspired from colonizing weeds, is proposed for Electromagnetic applications. This algorithm, invasive weed optimization (IWO), is described and applied to different electromagnetic problems. The linear array antenna synthesis, the standard problem used by antenna engineers, is presented as an example for the application of the IWO. Compared to the PSO, The features of the IWO are shown. As another application, the design of aperiodic thinned array antennas by optimizing the number of elements and at the same time their positions is presented. By implementing this new scenario, thinned arrays with less number of elements and lower sidelobes, compared to the results achieved by genetic algorithm (GA) for the same aperture dimensions, are obtained. Finally, the IWO is applied to a U-slot patch antenna to have the desired dualband characteristics. Index Terms—Antenna arrays, aperiodic arrays, microstrip patch antenna, optimization method, thinned arrays.

optimizers like PSO, but also is capable of handling some new electromagnetic optimization problems. The main purpose of this paper is to introduce the desirable attributes and new features of the IWO for Electromagnetic problems. Of course, the efficiency of this optimization method compared to the other optimizers depends on the problem and choosing control parameters. Below, we first represent the proposed IWO algorithm and its desirable features. Then, by conducting several array antenna synthesis problems, including linear and thinned array antennas, the efficiency and specific features of this new algorithm are shown. Finally, the method is employed in designing a U-slot microstrip patch antenna fed by an L-probe to have the desired reflection coefficient for dual-band applications. II. IWO

I. INTRODUCTION A. The Inspiration Phenomenon LECTROMAGNETIC designing problems usually involve several parameters which are non-linearly related to the objective functions. In order to solve these problems efficiently, evolutionary optimization algorithms have been considered and successfully applied to electromagnetic problems. Among these optimizers, genetic algorithm (GA) [1] and particle swarm optimization (PSO) [2] have received considerable attentions by the electromagnetic community due to their efficiency and simplicity [3]–[6]. In addition, other optimization methods including Ant Colony Optimizer (ACO) [7] and simulated annealing (SA) [8] have shown high capability of searching for global minimum in electromagnetic optimization problems [9]–[13]. Here, a new optimization algorithm, invasive weed optimization (IWO) and mainly some of its new features are introduced by illustrating its applications to various electromagnetic problems. This numerical stochastic optimization algorithm, inspired from weed colonization, was first introduced by Mehrabian and Lucus in 2006 [14]. It is shown that this optimizer not only in certain instances outperforms other

E

Manuscript received April 20, 2009; revised July 27, 2009. Date of manuscript acceptance October 01, 2009; date of publication February 05, 2010; date of current version April 07, 2010. This work was supported in part by the National Science Foundation under Grant ECS-524293. The authors are with the Department of Electrical Engineering, University of Mississippi, University, MI 38677 USA (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2041163

The IWO, inspired from the phenomenon of colonization of invasive weeds in nature, is based on weed biology and ecology. It has been shown that capturing the properties of the invasive weeds, leads to a powerful optimization algorithm. The behavior of weed colonization in a cropping field can be explained as follows: Weeds invade a cropping system (field) by means of dispersal and occupy opportunity spaces between the crops. Each invading weed takes the unused resources in the field and grows to a flowering weed and produces new weeds, independently. The number of new weeds produced by each flowering weed depends on the fitness of that flowering weed in the colony. Those weeds that have better adoption to the environment and take more unused resources grow faster and produce more seeds. The new produced weeds are randomly spread over the field and grow to flowering weeds. This process continues till the maximum number of weeds is reached on the field due to the limited resources. Now, only those weeds with better fitness can survive and produce new weeds. This competitive contest between the weeds causes them to become well adapted and improved over the time. B. Algorithm Before considering the algorithm process, the new key terms used to describe this algorithm should be introduced. Table I shows some of these terms. Each individual or agent, a set containing a value of each optimization variable, is called a seed. Each seed grows to a flowering plant in the colony. The meaning of a plant is one individual or agent after evaluating its fitness.

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TABLE I SOME OF THE KEY TERMS USED IN THE IWO

Therefore, growing a seed to a plant corresponds to evaluating an agent’s fitness. To simulate the colonizing behavior of weeds the following steps, pictorially shown in Fig. 1, are considered. parameters (variables) that need to be 1. First of all, the optimized should be selected. Then, for each of these variables in the -dimensional solution space, a maximum and minimum value should be assigned (defining the solution space). 2. A finite number of seeds are being randomly dispread over the defined solution space. In other words, each seed takes a random position in the N-dimensional problem space. Each seed’s position is an initial solution, containing N values for the N variables, of the optimization problem (initialize a population). 3. Each initial seed grows to a flowering plant. That is, the fitness function, defined to represent the goodness of the solution, returns a fitness value for each seed. After assigning the fitness value to the corresponding seed, it is called a plant (Evaluate the fitness of each individual). 4. Before the flowering plants produce new seeds, they are ranked based on their assigned fitness values. Then, each flowering plant is allowed to produce seeds depending on its ranking in the colony. In other words, the number of seeds each plant produces depends on its fitness value or ranking and increases from the minimum possible seeds , to its maximum, . Those seeds that production, solve the problem better correspond to the plants which are more adapted to the colony and consequently produce more seeds. This step adds an important property to the algorithm by allowing all of the plants to participate in the reproduction contest (Rank the population and reproduce new seeds). 5. The produced seeds in this step are being dispread over the search space by normally distributed random numbers with mean equal to the location of the producing plants and varying standard deviations. The standard deviation (SD) at the present time step can be expressed by

(1) is the maximum number of iterations. and are defined initial and final standard deviations, respectively and is the nonlinear modulation index. Fig. 2 shows the standard deviation (SD) over the where

Fig. 1. Flow Chart showing the IWO algorithm.

Fig. 2. Standard deviation over the course of the run.

course of a run with 100 iterations and different modulation indexes. It can be seen that the SD is reduced from the initial SD to the final SD with different velocities. The algorithm starts with such a high initial SD that the optimizer can explore through the whole solution space. By increasing the number of iterations, SD value is decreased gradually to search around the local minima or maxima to find the global optimal solution (dispersion). 6. After that all seeds have found their positions over the search area, the new seeds grow to the flowering plants and then, they are ranked together with their parents. Plants with lower ranking in the colony are eliminated to reach . It is the maximum number of plants in the colony, obvious that the number of fitness evaluations, the population size, is more than the maximum number of plants in the colony (competitive exclusion). 7. Survived plants can produce new seeds based on their ranking in the colony. The process is repeated at step 3 till either the maximum number of iteration is reached or the fitness criterion is met (repeat). C. Selection of Control Parameter Values Among the parameters affect the convergence of the algorithm three parameters, the initial SD, , the final SD, , and the nonlinear modulation index, , should be tuned carefully in order to achieve the proper value of the SD in each iteration, according to (1). A high initial standard deviation should be chosen to allow the algorithm to explore the whole search area, aggressively. It seems that the IWO works well if the initial SD is set around a few percent (1 to 5 percent) of the

KARIMKASHI AND KISHK: INVASIVE WEED OPTIMIZATION AND ITS FEATURES IN ELECTROMAGNETICS

dynamic range of each variable. The final SD should be selected carefully to allow the optimizer to find the optimal solution as accurate as possible. A finer local optimum solution can be achieved by decreasing this parameter. However, it should be noticed that tuning the final SD much smaller than the precision criteria of the optimization variables, doesn’t improve the final error level and may deteriorate the convergence rate of the optimization. Therefore, the final SD in each dimension should be selected based on the precision effect of that variable on the objective function. It was shown that the value of nonlinear modulation index has a considerable effect on the performance of IWO [14]. It was suggested that the best choice for is 3. Besides (1), other functions to describe the standard deviation over the optimization process were considered. However, simis the best choice. ulation results showed that (1) with Maximum and minimum numbers of seeds are the two other important parameters needed to be selected. Based on different examples, it can be concluded that selecting the maximum number of seed between 3 and 5 leads to a good performance of the optimizer. Moreover, the minimum number of seeds is set to zero for all examples. The maximum number of plants is another parameter that should be chosen in the IWO. Parametric studies show that increasing this parameter not necessarily increases the performance of the algorithm. It was found that the best performance can be achieved for many problems when the maximum number of plant is set between 10 and 20. III. IWO FEATURES One important property of the IWO is that it allows all of the agents or plants to participate in the reproduction process. Fitter plants produce more seeds than less fit plants, which tends to improve the convergence of the algorithm. Furthermore, it is possible that some of the plants with the lower fitness carry more useful information compared to the fitter plants. This algorithm, the IWO, gives a chance to the less fit plants to reproduce and if the seeds produced by them have good finesses in the colony, they can survive. Another important feature of IWO is that weeds reproduce without mating. Each weed can produce new seeds, independently. This property adds a new attribute to the algorithm that each agent may have different number of variables during the optimization process. Thus, the number of variables can be chosen as one of the optimization parameters in this algorithm. Optimizing the number of variables gives such an interesting feature to the optimization that can handle some new electromagnetic design problems. The effectivity of this kind of optimization for designing aperiodic thinned array antennas is shown in Section V. Finally, comparing some aspects of the IWO with two common and standard optimizers, GA and PSO, can clarify some features of this new algorithm. Provided that the number of iterations and the population size are considered as common requirements for all evolutionary algorithms, the initial and final standard deviation, nonlinear modulation index, and maximum and minimum number of seeds are the parameter of the IWO need to be tuned. In the GA, crossover and mutation rates

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and in the PSO, inertial weight, , cognitive rate, , social , should be controlled rate, , and the maximum velocity, to achieve the desired convergence. It has been shown that the choice of boundary conditions and also the maximum velocity are critical in convergence of the PSO algorithm [15]–[17]. Moreover, in the case of GA, both crossover and mutation rates affects the convergence of the problem [18]. The effect of these tuning parameters on the GA and PSO convergences are difficult to perceive, but by tuning the critical parameters in the IWO, the initial and final SD, a high-level control in the convergence and accuracy of the algorithm is achieved [19], [20]. In addition, the IWO shows a high stability with different boundary conditions. IV. ARRAY ANTENNA DESIGN PROBLEMS In this section, both the IWO and PSO are applied to the problem of synthesizing the far-field radiation patterns of linear array antennas. The consideration focuses on the optimizing of array antennas to achieve the desired radiation patterns given by elements, user defined functions. For an array antenna with separated by a uniform distance , the normalized array factor is given by (2) Where are amplitude coefficients, is the angle from the is the maximum value of normal to the array axis, and the magnitude of the array factor. is assumed to be , where is the wavelength. Comparisons are made between the performances of the IWO and the PSO in achieving desired radiation patterns. In the case of the IWO, restricted and invisible boundary conditions are two possible choices. The restricted boundary condition relocates the particle on the boundary that the particle hits. However, the invisible boundary condition allows a particle to stay outside the solution space while the fitness evaluation of that particle is skipped and a bad fitness value is assigned to that errant particle. For the case of the PSO, the boundary conditions including invisible (IBC), reflective (RBC), absorbing (ABC), damping (DBC), invisible/reflecting and invisible/damping are tested [16]. In addition, the velocity-clipping technique, showing good performance in the PSO, for different values are implemented [9], [15]. The same number of population size and iterations are chosen for different algorithms. The population size is fixed to 40 for both algorithms. It should be noted that in the case of the IWO, the number of population is fixed by choosing the maximum number of plant population and also minimum and maximum number of seeds. In the coming examples, the maximum number of plants is fixed to 10 and the number of seeds increases linearly from 0 to 5. In the case of PSO, both the and the social rate are set to 2.0 and the cognitive rate inertial weight is varied linearly from 0.9 to 0.2 as suggested in [9], [17]. It should be also pointed out that various realizations of the same experiment produced results that are close to each other. All results reported are the average of 50 independent runs of the PSO or IWO algorithms and found to be sufficient.

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TABLE II IWO PARAMETER VALUES FOR THE LINEAR 40-ELEMENT ARRAY OPTIMIZATION

TABLE III COMPARISON OF AVERAGE NUMBER OF FITNESS EVALUATIONS REQUIRED PER SUCCESSFUL RUN IN THE PSO AND THE IWO ALGORITHMS FOR THE LINEAR 40-ELEMENT ARRAY OPTIMIZATION

Fig. 3. Amplitude-only synthesis for a linear 40-element array (a) radiation pattern obtained using IWO, (b) convergence curves for the IWO and PSO with different boundary conditions. The maximum velocity limit is changed for the PSO, (c) convergence curves for the IWO with restricted boundary condition and different values of initial and final standard deviations.

A. Optimizing Sidelobe Patterns In this section a linear 40-element array is considered to achieve the desired radiation pattern by optimizing the amplitude coefficients. The objective pattern is to obtain side lobe levels less than a tapered sidelobe mask that decreases linearly

from to . The beamwidth of the array pattern is 11 and the number of sampling points is 359. It should be pointed out that “Don’t exceed criterion” is utilized in the formulation of the objective function. That is, an error will be reported only if the obtained array factor exceeds the desired sidelobe levels. Fig. 3(a) shows the desired and obtained radiation patterns achieved by the IWO. Since the desired envelope is symmetric, we exploit the symmetry of the current distribution. Thus, the number of optimization parameters reduces to the half of the array elements. The parameters used for the IWO are summarized in Table II. The performance of the IWO compared to the PSO for different boundary conditions is shown in Fig. 3(b). It can be seen that the performance of the PSO is dramatically changed by choosing either different boundary conditions or changing the maximum velocity. Moreover, the algorithm is trapped in a local minimum when the absorbing boundary condition is used while IWO achieves better performance for both invisible and restricted boundary conditions. The PSO algorithm tested by some other boundary conditions [16], [17] doesn’t show any better performances. The average numbers of fitness evaluations required per successful run for both the IWO and PSO with different boundary conditions are shown in Table III. It can be seen that the IWO is faster than the PSO to achieve the same optimization goal for this problem. It should be mentioned that the 50 independent runs of the same experiment for each curve of the IWO algorithm are closer too each other compared to those for the PSO. These results are removed for brevity. The performance of the IWO for different standard deviations is shown in Fig. 3(c). Different initial and final standard deviations are tried for IWO with the restricted boundary condition to evaluate the performance of this algorithm. It can be observed that by changing these parameters, the performance of the algorithm is slightly changed. Thus, by applying different boundary conditions or different standard deviation parameters, the IWO shows more stability compared to the PSO.

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TABLE IV COMPARISON OF AVERAGE NUMBER OF FITNESS EVALUATIONS REQUIRED PER SUCCESSFUL RUN IN PSO AND IWO ALGORITHMS FOR THE LINEAR SHAPED BEAM SYNTHESIS OPTIMIZATION

sidelobes region. Therefore, the objective is to obtain sidelobes levels less than the mask in the sidelobes region and main beam equal to the mask in the main beam region. The beamwidth of the array pattern is 20 degrees and the number of sampling points is 719. The desired and obtained radiation patterns by using the IWO are shown in Fig. 4(a). The same optimization parameters, shown in Table II, are used for this synthesis problem, except the number of iterations which is set to 2000. The convergence curves for both the IWO with different boundary conditions and the PSO with reflective boundary condition (RBC) and different maximum velocities are shown in Fig. 4(b). It can be seen that the IWO convergence curves for both restricted and invisible boundary conditions converge to the same level. However, In the case of the PSO, by varying the maximum velocity limit, the performance of algorithm dramatically changes. Although in some cases the PSO is faster in convergence compared to the IWO, it traps in local minima. In addition, the 50 various realizations of the same experiment for each curve of the IWO algorithm are closer too each other compared to those for the PSO. Table IV shows the average numbers of fitness evaluations per successful run for both the IWO and PSO. The effect of varying the initial and final standard deviations on the convergence of the IWO with invisible boundary condition is shown in Fig. 4(c). It can be seen that by varying the initial SD, the convergence rate of the algorithm is improved and compete with the results obtained by PSO in the first number of iterations. However, neither the initial nor the final SD has any critical effect on the final error level. The IWO appears to be more stable since by applying different boundary conditions or different initial or final standard deviation values, the convergence speed or the level of the cost function doesn’t change too much. Therefore, the IWO doesn’t need much effort on tuning the parameters. Fig. 4. Amplitude and phase synthesis for a linear 50-element array (a) radiation pattern obtained using IWO, (b) convergence curves for the IWO with different boundary conditions and PSO with reflective boundary condition (RBC) and different maximum velocity limit, (c) convergence curves for the IWO with invisible boundary condition and different number of initial and final standard deviations.

B. Shaped Beam Synthesis The amplitude and phase optimization of a linear 50-element array antenna to achieve the desired radiation pattern is considered. Shaping the main beam requires minimizing the absolute difference between the desired and obtained radiation pattern. Meanwhile, the “Don’t Exceed” criterion is considered in the

V. THINNED ARRAY ANTENNA In this section, thinned planar array antennas are considered as the next optimization problem to show the effectivity and some special features of the IWO. By some modifications in the IWO, the number of elements and the position of those elements can be optimized which results in a new scenario for developing thinned arrays. By applying this scenario, planar thinned arrays with less number of elements and higher efficiencies are obtained. Thinned arrays, generally produced by removing certain elements from a fully populated half wavelength spaced array, are

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usually designed to generate low sidelobe levels. Different optimization algorithms including GA, PSO, Simulated Annealing and Ant Colony have been applied to remove the elements in such a way to have the lowest possible sidelobe levels [19]–[26]. Although, the thinned arrays obtained by using these algorithms produce low sidelobes levels, it has been shown that by considering the aperiodic arrangements, lower sidelobes levels can be achieved. This can be done by optimizing the inter-element spacing of periodic arrays or already thinned arrays to have lower sidelobes levels [20], [21], [27]–[30]. In this section, by some modifications in the IWO, the number of elements and at the same time their locations, the inter-element spacing, are optimized. It is shown that by using this algorithm, capable of optimizing such a problem, lower sidelobes levels with less number of elements can be achieved. Fewer elements for a given aperture mean reducing the cost and weight of the antenna system. It should be pointed out that the array is uniformly excited (all elements have identical current amplitude and phase). The advantage of uniform amplitude excitation is clear from the point of view of the feed network. A. Modified IWO As it was mentioned in Section II, in the IWO, each weed (agent) may have different number of variables during the optimization process. By taking this feature of the algorithm, different number of variables for each agent can be considered during the optimization. This modified IWO works similar to the routine explained in Section II, some modifications, however, should be made in the algorithm process to take the number of elements as an optimization parameter. In this modified version of the IWO, each agent, corresponding to an array antenna, has different number of elements. Thus, the fitness value of each agent is calculated based on the number of elements and the position of each element in that agent. Similar to the general IWO algorithm, each flowering plant produces new seeds based on its ranking in the colony. That is, the new arrays appear in the colony. However, the reproduction process is modified to have different number of elements for each produced array antenna. In the reproduction process, each element in the array is removed and then reproduces some new elements in that array. The number of new elements produced by each old element is defined to be a constant value in each iteration. Then, these new elements are being dispread over the aperture by normally distributed random numbers with mean equal to the location of the producing element and varying standard deviations. The standard deviation (SD) is defined similar to (1) where it starts from a large value, called initial SD, and by increasing the number of iterations; decreases gradually to a small value, called final SD. Without any limitations on the reproduced elements, the number of elements increases dramatically. Moreover, the distance between elements should be controlled not to have elements very close to each other. In order to overcome these problems, each new produced element is allowed to be located on the aperture if it is not closer than a predefined value (usually half wavelength or the size of the antenna element of the array) to any of the other elements already located on the aperture.

By choosing a relatively large value for the initial SD, the new elements are dispread over the aperture and the possibility of different number of elements over the aperture are tested. Then, by decreasing the SD to a small value, the position of each element on the aperture is optimized. Therefore, the number of elements and the location of each element are optimized. It should be noted that in this modified version of IWO the whole process explained in Section II is carried out. Meanwhile, the modified reproduction process is taken into account for each agent. B. Planar Thinned Array Examples As the first design problem of thinned arrays, a rectangular planar array with the aperture of is considered. The and objective is to minimize the maximum SLL in the planes. This problem is selected to compare the obtained result with the results in [19] and [28]. In [19] a 20 10 element planar array with a half a wavelength distance between uniformly spaced elements was thinned using GA by turning off some elements in that aperture. The optimal solution is a thinned array with 108 turned on elements on the rectangular aperture in [19, Fig. 7]. The optimized SLLs are equal to plane and in plane [19, Fig. 9]. The fitness value of the optimal solution, defined as the sum of max. The same problem is imum SLLs in both planes, is considered in [28] by optimizing the inter-element spacing between 108 elements of the obtained thinned array in [19], using a modified real GA to achieve lower SLLs. The optimal solution , and SLLs [28, Fig. 6] shows a lower fitness value, and in and , equal to respectively [28, Fig. 5]. In order to optimize this problem by using the IWO and based on the described method, the number of elements and their positions are optimized to obtain the lowest SLLs at the desired planes. The normalized array factor of a planar array with elements is given by

(3) where and are the locations of elements in and direction, respectively. This equation assumes that the array lies in the - plane. Since the desired pattern is symmetric about the -axis and -axis, a quarter of the aperture is considered to reduce the number of optimization parameters to the quarter of the array elements. The minimum distance between elements is assumed to be half wavelength. It should be noted that the amplitude coefficients, , are assumed to be 1. Set up of the IWO algorithm for solving this problem is summarized in Table V. The final SD is chosen to be a small value to optimize the location of each element with a high precision. An averaging of five runs is considered and found to be sufficient. The best thinned array obtained is presented in Table VI. Fig. 5(a) shows the radiation patterns of the obtained thinned and planes. The fitness value, the array in both and the sum of maximum SLLs in both planes, is

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TABLE V IWO PARAMETERS FOR THE THINNED ARRAY OPTIMIZATION PROBLEMS

TABLE VI THE COORDINATES OF THE ARRAY ELEMENTS IN WAVELENGTH: i

(x ; y )

TABLE VII THE COORDINATES OF THE ARRAY ELEMENTS IN WAVELENGTH: i

(x ; y )

=0

Fig. 5. Optimized thinned array antenna for reduction of SLL in the ' and ' planes, (a) radiation patterns in the ' and ' , (b) the array configuration of the optimized thinned array, (c) 3D radiation pattern.

= 90

=0

= 90

obtained SLLs are in plane and in plane. The array configuration of the thinned array for the upper right quarter of the aperture is depicted in Fig. 5(b) (compare with [28, Fig. 6] and [19, Fig. 7]). 18 elements (72 elements for the whole aperture) are the optimized number of elements to achieve the lowest SLLs. Comparing these results with those in [19] and [28] it can be concluded that by employing this

algorithm, much lower SLLs at both planes are achieved with more than 25% saving on the number of elements. Another optimization problem is the reduction of SLLs in all planes. It can be seen from Fig. 5(b) that most of the elements are around the and axes. Such an array configuration produces high SLLs at some other planes as shown in Fig. 5(c). In order to reduce the SLLs in all the planes, we decided to define , the fitness function as the maximum SLL in and planes to have more elements at the central part of the aperture. This fitness function helps to have less computation and avoid an expensive optimization process. The same optimization parameters shown in Table V are chosen for this problem. Table VII represents the obtained thinned array con, figuration. The array radiation pattern cuts in and for the best optimal solution are shown in Fig. 6(a). The array configuration, shown in Fig. 6(b), consists of 80 elements (for the whole aperture) distributed on the aperture. It is observed that more elements are located at the central part of the aperture as it was expected. Fig. 6(c) is the 3D radiation pattern of this array. Though low SLLs are obtained at the three cuts shown in Fig. 6(a), SLLs haven’t decreased effectively in the other planes. In order to have low SLLs in all planes, the fitness function is defined as the maximum SLLs in all the planes. The same optimization parameters are selected for this problem. The result of optimization is a thinned planar array with 92 elements (for the whole aperture) depicted in Table VIII. Fig. 7(a) shows the array configuration on a quarter of the aperture. The radiation pattern of this array is shown in Fig. 7(b) where the maximum . comparing this results to that of [28], where SLL is GA is used to minimize the SLLs for 100 elements sparse array [28, Figs. 7 and 8], one can see that the IWO results lower SLL in [28]) with less number of elements. ( VI. DUAL-BAND U-SLOT PATCH ANTENNA To demonstrate the applicability of the IWO in electromagnetics, the design of a U-slot patch antenna [31], [32] to have the desired dual-band characteristics is considered. This concept

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Fig. 7. Optimized thinned array antenna for reduction of SLL in all ' planes, (a) The array configuration of the optimized thinned array, (b) 3D radiation pattern. TABLE VIII THE COORDINATES OF THE ARRAY ELEMENTS IN WAVELENGTH: i

=0 = 45

Fig. 6. Optimized thinned array antenna for reduction of SLL in the ' , ' and ' planes, (a) radiation patterns in the ' ,' and ' , (b) the array configuration of the optimized thinned array, (c) 3D radiation pattern.

= 45 = 90

= 90

=0

was introduced in [33], [34] where a U-slot in the patch fed by an L-probe produces notches within the matching band. Fig. 8 shows the configuration of the antenna structure. The L-probe feeding technique is used to have a wideband patch antenna [35], [36] and then the U-slot is cut on the patch to introduce notches, resulting in dual-band operation. The length (L) and the and height of width (W) of the patch and also the position are predefined. Then, the optimization of seven L-probe , , , , and are required. other parameters, ,

(x ; y )

Since these parameters are not independent, their ranges should be chosen carefully. The purpose of the optimization is to achieve the desired reflection coefficient within the matching band, 2–4 GHz. The IWO is linked to a MoM program that simulates the antenna reflection coefficient at 20 points within the frequency range. The fitness function is defined as the summation of differences between the relative values of the desired and obtained reflection coefficient at all 20 frequencies. The objective is to have reflection coefficient at 2.4 GHz and 3.3 GHz frequencies and zero at the other frequencies. The reason for choosing as the desired reflection coefficient is that decreasing this value results in a very low value in one frequency but relatively high at the other one. The restricted boundary condition is applied to this optimization problem. The maximum number of plant population is selected to be 10 and the number of seeds is varied linearly from

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Fig. 8. The configuration of the U-slot patch antenna fed by an L-probe with top view (top) and side view (lower).

TABLE IX FINAL DIMENSIONS OF THE OBTAINED U-SLOT PATCH ANTENNA IN MM

Fig. 9. U-slot patch antenna fed by an L-probe (a) the obtained reflection coefficient, and (b) the convergence curve.

4 to zero. The nonlinear modulation index is set to 3 over 100 iterations. The optimized parameters of the U-slot patch antenna are shown in Table IX. After simulating the antenna with more number of frequencies within the frequency range, the reflection coefficient shown in Fig. 9(a) is achieved. Fig. 9(b) illustrates the convergence curve of IWO algorithm which is the normalized curve of the fitness function in dB. It can be seen that the desired reflection coefficient within the frequency range of the U-slot antenna is achieved. VII. CONCLUSION A numerical stochastic optimization algorithm based on the weed ecology was introduced for electromagnetic applications. The IWO algorithm is capturing the properties of the invasive weeds, which led to a powerful optimization algorithm. By applying the IWO to the array antenna synthesis problems, the performance of this algorithm was investigated. It was shown that in certain instances the IWO outperforms the PSO in the convergence rate as well as the final error level. Moreover, the performance of the IWO for different boundary conditions and tuning parameters was evaluated. From the simulation results, it was observed that this algorithm is very stable and efficient against different parameter values. The IWO was also utilized to design aperiodic planar thinned array antennas by optimizing the number of elements and at the same time their positions. It was shown that by using this technique, thinned arrays with less number of elements and lower sidelobes levels, compared to

the results already achieved from other methods were obtained. Also, the IWO was applied to the design of a U-slot patch antenna fed by an L-probe to have a dual band performance with the desired reflection coefficient. ACKNOWLEDGMENT The authors wish to thank the reviewer for his valuable suggestions. REFERENCES [1] J. H. Holland, “Genetic algorithm,” Scie. Amer., pp. 62–72, Jul. 1992. [2] J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” presented at the IEEE Conf. Neural Networks IV, Piscataway, NJ, 1995. [3] R. L. Haupt, “An introduction to genetic algorithms for electromagnetics,” IEEE Antennas Propag. Mag., vol. 37, pp. 7–15, Apr. 1995. [4] D. S. Weile and E. Michielssen, “Genetic algorithm optimization applied to electromagnetics: A review,” IEEE Trans. Antennas Propag., vol. 45, pp. 343–353, Mar. 1997. [5] D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 397–407, Mar. 2004. [6] J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans Antennas Propag., vol. 52, no. 2, pp. 397–407, Feb. 2004. [7] M. Dorigo, V. Maniezzo, and A. Colorni, “Ant system: Optimization by a colony of cooperating agents,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 26, no. 1, pp. 29–41, Feb. 1996. [8] R. H. J. M. Otten and L. P. P. P. van Ginnekent, The Annealing Algorithm. Boston/Dordrecht/London: Kluwer Academic, 1989. [9] S. Mikki and A. A. Kishk, “Quantum particle swarm optimization for electromagnetics,” IEEE Trans, Antennas Propag., vol. 54, pp. 2764–2775, Oct. 2006.

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[10] Y. L. Abdel-Majid and M. M. Dawoud, “Accurate null steering in linear arrays using tabu search,” in Proc. 10th international Conf. Antennas Propag., Apr. 1997, vol. 1, pp. 365–369. [11] W. G. Weng, F. Yang, and A. Elsherbeni, “Linear antenna array synthesis using Taguchi’s method: A novel optimization technique in electromagnetics,” IEEE Trans Antennas Propag., vol. 55, no. 3, pp. 723–730, Mar. 2007. [12] E. Rajo-lglesias and O. Quevedo-Teruel, “Linear array synthesis using an ant-colony-optimization-based algorithm,” IEEE Antennas Propag. Mag., vol. 49, no. , pp. 70–79, Apr. 2007. [13] C. M. Coleman, E. J. Rothwell, and J. E. Ross, “Investigation of simulated annealing, ant-colony optimization, and genetic algorithms for self-structuring antennas,” IEEE Trans Antennas Propag., vol. 52, no. 4, pp. 1007–1014, Apr. 2004. [14] A. R. Mehrabian and C. Lucas, “A novel numerical optimization algorithm inspired from weed colonization,” Ecol. Inform., vol. 1, no. 4, pp. 355–366, Dec. 2006. [15] S. Mikki and A. A. Kishk, “Improved particle swarm optimization technique using hard boundary conditions,” Microw. Opt. Tech. Lett., vol. 46, no. 5, pp. 422–426, Sep. 2005. [16] S. Xu and Y. Rahmat-Samii, “Boundary conditions in particle swarm optimization revisited,” IEEE Trans Antennas Propag., vol. 52, no. 3, pp. 760–765, Mar. 2007. [17] S. Mikki and A. A. Kishk, “Hybrid periodic boundary condition for particle swarm optimization,” IEEE Trans Antennas Propag., vol. 55, no. 11, pp. 3251–3256, Nov. 2007. [18] , Y. Ramat-Samii and E. Michielssen, Eds., Electromagnetic Optimization by Genetic Algorithms. New York: Wiley, 1999. [19] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 42, no. 7, Jul. 1994. [20] R. L. Haupt, “Optimized element spacing for low sidelobe concentric ring arrays,” IEEE Trans Antennas Propag., vol. 56, no. 1, pp. 266–268, Jan. 2008. [21] N. Jin and Y. Rahmat-Samii, “Advances in particle swarm optimization for antenna designs: Real-number, binary, single-objective and multiobjective implementations,” IEEE Trans Antennas Propag., vol. 55, no. 3, pp. 556–567, Mar. 2007. [22] K. V. Deligkaris, Z. D. Zaharis, D. G. Kampitaki, S. K. Goudos, I. T. Rekanos, and M. N. Spasos, “Thinned planar array design using boolean PSO with velocity mutation,” IEEE Trans Magn., vol. 45, no. 3, pp. 1490–1493, Mar. 2009. [23] C. A. Meijer, “Simulated annealing in the design of thinned arrays having low sidelobe levels,” in Proc. South African Symp. Communications and Signal Processing, 1998, pp. 361–366. [24] D. J. O’Neill, “Element placement in thinned arrays using genetic algorithms,” in Proc. Oceans Engineering for Today’s Technology and Tomorrows Preservation, 1994, vol. 2, pp. 301–306. [25] O. Quevedo-Teruel and E. Rajo-Iglesias, “Ant colony optimization in thinned array synthesis with minimum sidelobe level,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 349–352, 2006. [26] S. Mosca and M. Ciattaglia, “Ant colony optimization to design thinned arrays,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 2006, vol. 1, pp. 4675–4678. [27] M. G. Bray, D. H. Werner, D. W. Boeringer, and D. W. Machuga, “Optimization of thinned aperiodic phase array using genetic algorithm to reduce grating lobes during scanning,” IEEE Trans Antennas Propag., vol. 50, no. 12, pp. 1732–1742, Dec. 2002. [28] K. Cheng, X. Yun, and C. Han, “Synthesis of sparse planar array using modified real genetic algorithm,” IEEE Trans. Antennas Propag., vol. 55, no. 4, Apr. 2007. [29] T. G. Spence and D. H. Werner, “Thinning of aperiodic antenna arrays for low side-lobe levels and broadband operation using genetic algorithms,” in Proc. IEEE Antennas and propagation Society Int. Symp., 2006, pp. 2059–2062. [30] M. M. Khodier and C. G. Christodoulou, “Linear array geometry synthesis with minimum sidelobe level and null control using particle swarm optimization,” IEEE Trans. Antennas Propag., vol. 53, no. 8, Apr. 2005. [31] T. Huynh and K. F. Lee, “Single-layer single-patch wideband microstrip antenna,” Electron. Lett., vol. 31, no. 16, pp. 1310–1312, 1995. [32] K. F. Tong, K. M. Luk, K. F. Lee, and R. Q. Lee, “A broadband U-slot rectangular patch antenna on a microwave substrate,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 954–960, 2000. [33] K. F. Lee, S. L. Yang, and A. A. Kishk, “Dual- and multiband U-slot patch antennas,” IEEE Trans. Antennas Wireless Propag. Lett., vol. 7, pp. 645–647, 2008.

[34] K. F. Lee, S. L. Yang, and A. A. Kishk, “The versatile U-slot patch antenna,” in Proc. 3rd Eur. Conf. on Antennas and Propagation, Mar. 23–27, 2009, pp. 3312–3314. [35] C. L. Mak, K. M. Luk, K. F. Lee, and Y. L. Chow, “Experimental study of a microstrip patch antenna with an L-shaped probe,” IEEE Trans. Antennas Propag., vol. 48, no. 5, pp. 777–783, May 2000. [36] Y. X. Guo, C. L. Mak, K. M. Luk, and K. F. Lee, “Analysis and design of L-probe proximity fed-patch antennas,” IEEE Trans. Antennas Propag., vol. 49, pp. 145–149, Feb. 2001.

Shaya Karimkashi was born in Tehran, Iran. in 1980. He received the B.S. degree in electrical engineering from K. N. Toosi University of technology, Tehran, and the M.S. degree in electrical engineering from University of Tehran, in 2003 and 2006, respectively. Currently he is working toward Ph.D. degree at the University of Mississippi, University. His research interests include array antennas, focused antennas, reflector antennas, optimization methods in EM and microwave measurement techniques.

Ahmed A. Kishk received the B.S. degree in electronic and communication engineering from Cairo University, Cairo, Egypt, in 1977, and the B.S. degree in applied mathematics from Ain-Shams University, Cairo, Egypt, in 1980, and the M.Eng and Ph.D. degrees from the University of Manitoba, Winnipeg, Canada, in 1983 and 1986, respectively. From 1977 to 1981, he was a Research Assistant and an Instructor at the Faculty of Engineering, Cairo University. From 1981 to 1985, he was a Research Assistant in the Department of Electrical Engineering, University of Manitoba, where, rom December 1985 to August 1986, he was a Research Associate Fellow. In 1986, he joined the Department of Electrical Engineering, University of Mississippi, as an Assistant Professor. He was on sabbatical leave at Chalmers University of Technology, Sweden during the 1994–1995 academic years. He is now a Professor at the University of Mississippi (since 1995). His research interest includes the areas of design of millimeter frequency antennas, feeds for parabolic reflectors, dielectric resonator antennas, microstrip antennas, EBG, artificial magnetic conductors, soft and hard surfaces, phased array antennas, and computer aided design for antennas. He has published over 200 refereed journal articles and 27 book chapters. He is a coauthor of the book Microwave Horns and Feeds (London, U.K., 1994; New York: 1994) and a coauthor of chapter 2 in Handbook of Microstrip Antennas (London, U.K., 1989). Dr. Kishk received the 1995 and 2006 outstanding paper awards for papers published in the Applied Computational Electromagnetic Society Journal. He received the 1997 Outstanding Engineering Educator Award from Memphis section of the IEEE. He received the Outstanding Engineering Faculty Member of the Year in 1998 and 2009, Faculty Research Award for outstanding performance in research on 2001 and 2005. He received the Award of Distinguished Technical Communication for his paper published in the IEEE Antennas and Propagation Magazine, 2001. He also received The Valued Contribution Award for outstanding Invited Presentation, “EM Modeling of Surfaces with STOP or GO Characteristics—Artificial Magnetic Conductors and Soft and Hard Surfaces” from the Applied Computational Electromagnetic Society. He received the Microwave Theory and Techniques Society Microwave Prize 2004. He is a Fellow member of IEEE since 1998 (Antennas and Propagation Society and Microwave Theory and Techniques), a member of Sigma Xi society, a member of the U.S. National Committee of International Union of Radio Science (URSI) Commission B, a member of the Applied Computational Electromagnetics Society, a Fellow member of the Electromagnetic Academy, and a member of Phi Kappa Phi Society. He was an Associate Editor of the IEEE Antennas and Propagation Magazine from 1990 to 1993 and is now an Editor. He was a Co-editor of the Special Issue on Advances in the Application of the Method of Moments to Electromagnetic Scattering Problems in the ACES Journal, was an Editor the during 1997, and was Editor-in-Chief from 1998 to 2001. He was the Chair of Physics and Engineering Division of the Mississippi Academy of Science (2001–2002). He was a Guest Editor of the Special Issue on Artificial Magnetic Conductors, Soft/Hard Surfaces, and Other Complex Surfaces of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, January 2005.

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Radio-Wave Propagation Into Large Building Structures—Part 1: CW Signal Attenuation and Variability William F. Young, Member, IEEE, Christopher L. Holloway, Fellow, IEEE, Galen Koepke, Member, IEEE, Dennis Camell, Member, IEEE, Yann Becquet, and Kate A. Remley, Senior Member, IEEE

Abstract—We report on our investigation into radio communications problems faced by emergency responders in disaster situations. A fundamental challenge to communications into and out of large buildings is the strong attenuation of radio signals caused by losses and scattering in the building materials and structure. Another challenge is the large signal variability that occurs throughout these large structures. We designed experiments in various large building structures in an effort to quantify continuous wave (CW) radio-signal attenuation and variability throughout twelve large structures. We carried radio frequency transmitters throughout these structures and placed receiving systems outside the structures. The transmitters were tuned to frequencies near public safety, cell phone bands, as well as ISM and wireless LAN bands. This report summarizes the experiments, performed in twelve large building structures. We describe the experiments, detail the measurement system, show primary results of the data we collected, and discuss some of the interesting propagation effects we observed. Index Terms—Emergency responder communications, large building radio frequency propagation, radio-frequency propagation measurements.

I. INTRODUCTION

W

HEN emergency responders enter large structures (e.g., apartment and office buildings, sports stadiums, stores, malls, hotels, convention centers, warehouses) communication to individuals on the outside is often impaired. Cell phone and mobile-radio signal strength is reduced due to attenuation caused by propagation through the building materials and scattering by the structural components [1]–[8]. Also, the large amount of signal variability throughout the structures causes degradation in communication systems. Here, we report on a National Institute of Standards and Technology (NIST) project to investigate the communications

Manuscript received December 18, 2008; revised August 10, 2009; accepted September 25, 2009. Date of publication February 05, 2010; date of current version April 07, 2010. This work was supported in part by the U.S. Department of Justice, Community-Oriented Police Services through the NIST Public-Safety Communications Research Laboratory. W. F. Young is with Sandia National Laboratories, Albuquerque, NM 87185 USA (e-mail: [email protected]). C. L. Holloway, G. Koepke, D. Camell, Y. Becquet, and K. A. Remley are with the National Institute of Standards and Technology, Boulder, CO 803053328 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.2010.2041142

problems faced by emergency responders (firefighters, police and medical personnel) in disaster situations involving large building structures. The goal is to create a large body of statistical data in the open literature for improved communication system development and design. For example, these results are useful to technology advancement initiatives focused on improving public safety communications such as [9]–[11]. As part of our effort, we are investigating the propagation and coupling of radio waves into large building structures. This paper is the first part of a two-part series covering radio frequency propagation into large structures. Part 2 summarizes results from [12] where we investigate time domain and modulated signal propagation behavior into some of these same structures. The complete results for this set of measurements are found in [13]. The experiments reported here were performed in twelve different large building structures and are essentially measurements of the reduction in radio signal strength caused by penetration into the structures. These structures include four office buildings, two apartment buildings, one hotel, one grocery store, one shopping mall, one convention center, one sports stadium, and one oil refinery. In order to study the radio characteristics of these structures at the various frequencies of interest to emergency responders, we chose frequencies near public-safety and cell phone bands, as well as ISM and wireless LAN bands (approximately 50 MHz, 150 MHz, 225 MHz, 450 MHz, 900 MHz, 1.8 GHz, 2.4 GHz, 4.9 GHz). The experiments performed here are referred to as “radio mappings.” This involved carrying transmitters (or radios) tuned to various frequencies throughout the twelve structures while recording the received signal at sites located outside the building. The reference level for these data was a direct, unobstructed line-of-sight signal-strength measurement with both transmitters and receiver external to the different structures. The purpose of the radio-mapping measurements was to investigate how the signals at the different frequencies couple into the structures, and to determine the field strength variability throughout the structures. Several previous papers, e.g., [14]–[17], have provided results on radio mapping in buildings. Others such as [18]–[27] provide results on building penetration for a few of the frequency bands discussed here. Reference [28] outlines a measurement campaign at 2.4 GHz and 5.2 GHz carried out in support of a overall system design effort, while [26] and [27] focus on measurements for public safety. With respect to the

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TABLE I PUBLIC SAFETY COMMUNITY FREQUENCIES

TABLE II CELLULAR PHONE FREQUENCIES

TABLE III FREQUENCY BANDS USED IN THE EXPERIMENTS

breadth of our measurement campaign, [29] provides a review of similar radio propagation measurements up to 1990 within our bands of interest, and also proposes a building classification scheme. Our effort is different is several ways: (1) a wider range and number of frequencies are covered per building, (2) multiple large building structures are investigated, and (3) the walk-through path and receive site locations are selected to emulate an emergency response scenario. Hence, this work builds upon previous work designed to accurately characterize wireless RF propagation in some key environments by adding much needed data to building categories 5 and 6 in [29], and to the general body of radio propagation measurements into and within buildings. This paper is organized as follows. Section II provides more detail on the frequencies, the transmitters, and the automated measurement system used in the radio-mapping measurements. Section III describes the different structures and details our experimental procedures for each structure. Section IV presents representative data from the complete set of results in [13]. Finally, Section V summarizes the results of these experiments and discuss some of the interesting propagation effects observed. II. EXPERIMENTAL PARAMETERS AND EQUIPMENT This section covers the frequency bands, transmitter descriptions, and the receiver configurations used in the experiments. A. Frequency Bands An overview of the frequencies used by the public-safety community nationwide (federal, state, and local) is given in Table I, which shows a broad range of frequencies ranging from 30 MHz to 4.9 GHz. The modulation scheme has historically been analog FM, but this is slowly changing to digital as Project 25 radios come online [9]. The modulation bandwidth in the VHF and UHF bands has been 25 kHz, but due to the need for additional communications channels in an already crowded spectrum, most new bandwidth allocations are 12.5 kHz. The older bandwidth allocations will gradually be required to move to narrow bandwidths to increase the user density even further. The crowded spectrum and limited bandwidth are also pushing the move to higher frequency bands in order to support new data-intensive technologies. The cellular phone bands are summarized in Table II. As shown in Table I, frequencies currently used by public safety and other emergency responders and cellular telephones

are typically below 2 GHz. New frequency allocations and systems including higher frequencies (e.g., around 4.9 GHz) will become increasingly important in the future. We chose eight frequency bands below 5 GHz, from about 50 MHz to 4.9 GHz. These include four VHF bands typically used for analog FM voice, one band used for multiple technologies (analog FM voice, digital trunked FM, and cellular telephone), one band near the digital cellular telephone band, and wireless LAN bands. In designing an experiment to investigate the propagation characteristics into large buildings at these different frequency bands, we chose frequencies very close, but not identical, to the above bands. If frequencies were chosen in the public safety or commercial land-mobile bands, interference to the public safety and cellular systems could possibly occur. Conversely, these existing systems could interfere with our experimental setup. In addition, obtaining frequency authorizations in these bands for our experiments would have been problematic due to the intense crowding of the spectrum. To circumvent these issues, we were able to receive temporary authorization to use frequencies in the U.S. government frequency bands adjacent to these public safety bands. Table III lists the frequency bands that were used in the experiments. The lower four bands correspond to the frequencies used by the public-safety community; 902 MHz can be associated with several services including both public safety and cellular phones; 1830 MHz is near the digital cellular phone; 2450 MHz covers a wireless LAN band; 4900 MHz represents a new public safety band. The exact frequencies varied, depending on which city the experiments were performed in, to avoid RF interference. B. Transmitters The design requirements for the transmitters used in the experiments discussed here were that they should (1) transmit at the frequencies listed in the tables above, (2) operate continuously for several hours, and (3) be portable. For the four lower frequency bands (the VHF/UHF publicsafety bands), off-the-shelf amateur radios were modified. The

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Fig. 1. 49 to 448 MHz transmitter shown in (a),(b), and (c); 902 to 4900 MHz transmitter shown in (d). Fig. 2. Receiver equipment and setup. The initial nine experiments used antennas in (a), while the last three experiments used the antennas in (b).

modifications included (a) reprogramming the frequency synthesizer to permit transmitting at government frequencies, (b) disabling the transmitter time-out mode in order to allow for continuous transmission, and (c) connecting a large external battery pack. The modified radios and battery packs were placed in durable orange plastic cases for mobility, and the antennas were mounted on the outside of the cases. Fig. 1(a) and (b) show the final arrangement of modified transmitters used for the lower four frequency bands and Fig. 1(c) shows a closed case. Commercial transmitters already in plastic protective cases, depicted in Fig. 1(d), were available for the higher frequency bands (900 MHz, 1800 MHz, 2.4 GHz, and 4.9 GHz). The antennas had omnidirectional patterns and gains of approximately 0 dBi for the frequency bands to 1.8 GHz, and 3–5 dBi for the 2.4 and 4.9 GHz bands. C. Receiving Antenna and Measurement System The experiments have taken place over multiple years from 2003 to present, and in conjunction with other experiments [5]–[8], [12]. This created some differences in the data collection process over time. The first four experiments collected data for all frequencies during a single walk-through, while the remaining experiments consisted of a separate walk-through for each frequency. Radio mapping of multiple frequencies on a single walk-through led to approximately 7 s between samples at each frequency. The initial single frequency per walk-through experiments enabled a data collection rate of less than 1.5 s between samples, and further refinements reduced the time between samples to approximately 0.2 s in more recent experiments. The main components and connections of the receiving system are sketched in Fig. 2. There were two main configuration variations. For the first series of experiments, we assembled four antennas on a 4-meter mast, as illustrated in Fig. 2(a). The radio-frequency output from each antenna was fed through a

4:1 broadband power combiner. This arrangement gave us a single input to the portable spectrum analyzers, which could then scan over all the frequencies of interest without switching antennas. The four antennas were chosen to be optimal (or at least practical) for each of the frequency bands we were measuring. The selected antennas were an end-fed vertical omnidirectional antenna for 50 MHz with a gain of 4.6 dBi, a log-periodic-dipole-array (LPDA) used for the 160 MHz, 225 MHz, and 450 MHz bands with a gain of 6.7–7.1 dBi and a beamwidth of 60 degrees, and Yagi-Uda arrays for 900 MHz and 1830 MHz with gains of 15 and 13 dBi and beamwidths of 35 degrees, respectively. This assembly could then be mounted on a fixed tripod at one of the receive sites, or it could be inserted into a modified garden cart for portable measurements (see Fig. 2(a)). In addition, the receiving sites contained a spectrum analyzer, computer, associated cabling (see Fig. 2(d)), and in some cases, a global positioning system (GPS) receiver (see Fig. 2(c)). The second series of experiments used the omnidirectional discone antennas in Fig. 2(b), with the same receiver configuration as in Fig. 2(d) and antenna heights of 2 to 3 meters. The discone antennas had a gain of 2 dBi and a beamwidth of approximately 45 degrees. Horn antennas in Fig. 2(b) were used for the 2.4 GHz and 4.9 GHz frequency bands in later experiments. The gain was 10 dBi and a beamwidth of approximately 45 degrees at the frequencies of interest. As shown in Fig. 2, the measurement system consisted of a portable spectrum analyzer controlled by a graphical programming language. The software ran parallel processes of collecting, processing, and saving the data for post-collection processing. The data were continuously read from the spectrum analyzer and stored in data buffers. These buffers were read and processed for each signal and displayed for operator viewing.

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TABLE IV LIST OF STRUCTURES IN THE EXPERIMENTS

The processed data were then stored in additional buffers to be re-sorted and saved to a file on disk. The sampling rate of the complete measurement process was the major factor in how much spatial resolution we had during radio-mapping experiments and the time resolution for recording the signals. (We also had some flexibility in our walking speed.) Over the course of this multiple year effort, we used three different models of spectrum analyzers, with different sampling rate capabilties. As mentioned above, the slow sampling rate for the first four experiments primarily arose from collecting data for all frequencies during a single walk-through. For the remaining eight experiments, only a single frequency was collected during each walk-through, resulting in an immediate increase in sampling rate by a factor of approximately eight. Further software and parameter optimization allowed the latter experiment sampling rates of between 0.2 to 0.3 s.

Fig. 3. Pictures of the Philadelphia sports stadium.

III. BUILDING DESCRIPTIONS Table IV lists the large structures tested, which include four office buildings, two apartment buildings, one hotel, one grocery store, one shopping mall, one convention center, one sports stadium, and one oil-refinery. This section briefly describes three of the twelve different structures, while descriptions and pictures of all the structures are given in [13]. A. Philadelphia Sports Stadium The nearly circular stadium was constructed of reinforced concrete, steel, and standard interior finish materials. The stadium was scheduled for demolition via implosion later that week. Fig. 3 shows the stadium with some of the implosion preparations and partial demolition of different sections. As depicted in these figures, the stadium had multiple levels with large open areas. The exterior perimeter of the stadium was approximately 805 m (1/2 mile). Since this structure was also scheduled for implosion, significant demolition was already completed when we arrived; all plumbing fixtures, most glass windows and doors, and other contents had been removed.

Fig. 4. Plan view of the Philadelphia sports stadium (Veteran’s Stadium).

Material had been judiciously removed from certain structural parts of the lower levels including stairwells and elevator shafts to facilitate a proper collapse during the implosion. Fig. 4 shows a plan of the structure and approximate locations of the receive site. Measurements performed during and after the implosion are reported in [7].

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Fig. 5. Pictures of the Washington DC Convention Center. The two lower pictures show two of the three receive sites. Fig. 7. Pictures of the “interior” of the Commerce City oil refinery. The picture in the bottom right-hand corner was taken from the top of a tower.

tion center. During radio mapping experiments the transmitters were carried throughout the convention center at various levels. Measurements were performed with the receiving antennas polarized in the vertical direction (with respect to the ground). The transmitter labels (TX) on Fig. 6 locate the static placement for the implosion experiment discussed in [8]. C. Commerce City, Colorado Oil Refinery

Fig. 6. Topological view of the Washington DC Convention Center. Note that receive site 3 is located at the southeast corner.

B. Washington DC Convention Center This massive two-level structure was constructed of reinforced concrete, steel, and standard interior finish materials. Fig. 5 shows the exterior of the convention center and two of the three receive sites. The convention center had two large levels with three levels of offices. This structure was scheduled for demolition, and significant demolition was already completed when we arrived; all plumbing fixtures, most glass windows and doors, and other contents had been removed. Three fixed receiving sites, depicted in Fig. 6 were located on the perimeter of the convention center. Receiving site RX 1 was placed approximately 23 m (75 ft) from the northwest perimeter of the convention center. Receiving site RX 2 was placed approximately 23 m (75 ft) from the northeast perimeter of the convention center. Receiving site RX 3 was placed approximately 15 m (50 ft) from the southeast corner of the conven-

The refinery (Fig. 7), was basically an outdoor facility with several intricate piping systems. Measurements were performed during daytime hours and, as a result, people were moving throughout the refinery. During the measurement, the transmitters were carried throughout the dense piping systems and driven around the large metal storage tanks. The results here are only for the walk through the dense piping; [13] includes the results for the path through the storage tank area. Fig. 8 depicts the path of the walk-through and the two receive sites. For the radio mapping experiments, two fixed receiving sites (as described above) were assembled on the south side and north side of the refinery complex (see Fig. 8), approximately 100 m and 30 m from the piping structures. The arrows in the figure represent the actual paths that were walked. Measurements were performed with the receiving antennas polarized in the vertical direction. As the received signals were recorded, the location of the transmitters in the refinery complex was also recorded. IV. SUMMARY OF EXPERIMENTAL RESULTS A. Data Presentation and Analysis In the complete report [13] the measured data are presented from several perspectives. Plots of the radio-mapping data, normalized to a reference point, are provided for the investigated

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Fig. 9. Plots of the received signal level for the Philadelphia Stadium radiomapping.

Fig. 8. Path of walk through the Commerce City oil refinery. The transmitters were carried on a round trip path up a tower from the base at position 8.

TABLE V 2.4 GHz FREQUENCY BAND STATISTICS

Fig. 10. Plots of the received signal level at receive site 3 during the DC Convention Center radio-mapping.

The mean and standard deviation are found from

frequencies, and the radio-mapping data statistics are also included. [13] contains a series of tables that summarizes the processed data by frequency bands and buildings. This allows easier comparisons between results from different building or structure types. Table entries include a building or structure identifier, the , used actual frequency tested, the specific reference value to normalize that particular data set, and the median, mean , and standard deviation , for the normalized data. For example, Table V replicates the 2.4 GHz table in [13]. The spectrum analyzers and laptops collect the raw received , but the subsequent processing is performed on the power normalized received power . Only the samples collected during the actual walk-throughs were used, i.e., the samples collected while in the process of covering the prescribed path. The normalized power is calculated as (1)

(2) and

(3) respectively. Figs. 9 – 11 show radio-mapping data for the three buildings discussed above. The labels on the vertical dashed lines indicate the approximate physical location within the structure for the sample point. For the case of the stadium, (Fig. 9), the data were collected on a single walk-through for all six of the measured frequencies, and hence the vertical location markers line up almost exactly. Since the convention center and the oil refinery used a separate walk-through for each frequency, the locations

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Fig. 11. Plots of the received signal level at the North receive site for the Commerce City oil refinery radio-mapping.

Fig. 12. Histogram and statistics for the Philadelphia stadium. 3 dB bin widths are used for histogram calculation. Note that the data were normalized during the initial collection process; hence all reference values are 0 dBm.

are not always precisely lined up. Thus, for some of the locations a box or region is outlined to include the corresponding sample point across all frequencies. For example, in Fig. 11 the “top of tower” box includes the sample point corresponding to the top of the tower for all eight frequencies. Figs. 12 – 14 are histograms for the walk-through data shown in Figs. 9 – 11. These histograms may be viewed as approximating the probability density function (PDF) associated with the data normalized to the reference value (ref.). Also included on the plots are the corresponding mean , and standard deviation . B. Discussion of Aggregate Results It is impractical to present all the experimental results here, due to the vast amount of data collected from multiple receivers at the twelve different sites. However, the next three graphs

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Fig. 13. Histogram and statistics for the DC Convention Center. 1 dB bin widths are used for histogram calculation.

Fig. 14. Histogram and statistics for the oil refinery. 1 dB bin widths are used for histogram calculation.

are helpful in examining some general trends in the complete set of collected data. (We excluded the first receiver site for the NIST laboratory in Boulder due to an excessive amount of data collected at the noise floor caused by limitations of the receive equipment.) Fig. 15 plots statistics calculated on the standard deviation across the results from all twelve buildings on a per-frequency basis. In other words, the standard deviation is calculated for a data set generated at a particular receiver site for a specific building at a single frequency, e.g., DC Convention Center at 162 MHz. The results in Fig. 15 are taken from the aggregate of all the experiments. The mean and median of are calculated by including all the values at a particular frequency, e.g., 49 MHz. The maximum and minimum refer to a single radio-mapping, e.g., 49 MHz at the DC Convention Center receive site 3. These standard deviation results show that the median and mean are between 11 and 14 dB across all frequencies. Thus, the average variability is fairly constant across the

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Fig. 15. Statistics of the standard deviation calculated per frequency across all twelve structures.

Fig. 16. Mean values of scenarios listed in Table VI.

TABLE VI SELECTED TEST SCENARIOS

Fig. 17. Difference between mean values and reference level for scenarios listed in Table VI.

frequency bands. Note that only three structures were tested at 2.445 and 4.9 GHz. For a coarse insight into the attenuation behavior, we examine mean values for eleven of the test scenarios listed in Table VI. (A limited number of frequencies were covered for the Phoenix office building, and thus no scenario is taken from that experiment set.) Each scenario uses the receive site at the structure location that typically provided the greatest range of received signal levels so as to reduce the impact of the test equipment dynamic range on the statistics. Note that the combination of the reference value and the dynamic limitations of the test equipment can skew the statistics. [22] uses median values instead of mean values to reduce the impact of equipment limitations. Fig. 16 shows the mean values of the various frequencies for eleven scenarios listed in Table VI. The receiver noise floor provides an approximate value for the usable dynamic range. To obtain a rough idea of how close the mean signal is to the noise floor, we compute the difference between the mean signal level and the noise floor. Results for the eleven scenarios are shown in Fig. 17. In general, a greater difference implies less impact on the statistical results due to measured values at or below the

noise floor. For example, the results for scenario six indicate mean signal levels very near the receiver noise floor, as indicated by a difference of less than 10 dB for all the frequencies. In contrast, scenario ten indicates at least a 23 dB separation between the mean signal level and the noise floor for all frequencies. Fig. 18 provides the median values for the eleven scenarios listed in Table VI, which are quite similar to the results in Fig. 16. C. Discussion on General Trends The discussion here is based both on the data presented in this paper, as well as additional data presented in [13]. Any conclusion based on material unique to either publication is duly noted. First, examination of the standard deviation results in Fig. 15 yields some interesting insights. In particular, we see that the average standard deviation for the six lower frequency bands ranges from 11.8 to 13.4 dB . Only the three latter experiments collected data at 2.445 GHz and 4.9 GHz, but those results also lie in the same range at 13.5 dB and 12.95 dB, respectively. Across all eight frequency bands, the range of maximum

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Fig. 18. Median values of scenarios listed in Table VI.

varies from 16.8 to 22.3 dB, with 22.3 dB occurring at 900 MHz. The minimum values range from 3.9 to 9.6 dB, with 3.9 dB occurring at 1.8 GHz. Second, in almost all cases, the median value for the power received is 2 to 3 dB less than the average power received. The trend is clearly evident in the median and mean values listed in Figs. 12 – 14, as well as additional similar plots in [13]. This suggests that the median value may be a better measure for representing the general behavior of the building, or at least a more conservative performance measure. Use of the median is also suggested in [22]. Third, the histograms (which approximate PDFs) and the empirical cumulative density functions (see Figs. 12 – 14 and [13]) do not appear to follow any typical density function. To some extent, each individual test case will generate a unique density. However, the data do not seem to suggest an obvious average or best approximation density that could be used to collectively represent the buildings. One of the likely contributing factors is that the data collected do not have the free-space path loss due to distance removed from the data. Rician and Rayleigh distributions are used in the context of data with the path loss impact already removed from the data. However, for an emergency responder in a large building, the impact due to the size of the structure can be as important as the losses due to local scattering and multipath. Another possible contributor to the variability in distributions is that the receive sites and antenna are located as representative of an emergency response situation, i.e., close to the structures with an antenna less than 5 m above the ground. This is significantly different than the often used cell-tower measurement configuration.

V. CONCLUSION Differences in building types make it difficult to provide very strong general conclusions based on the collected data. However, we can observe some general trends in the data. The average standard deviation was between approximately 12 to 13.5

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dB across the tested frequencies, which is indicative of a consistent variability in the received signal level. However, the differences in the histograms do not indicate an obvious density function for designing communication systems for such environments. Since the experiments were intended as representative of actual emergency responder behavior, (e.g., a firefighter walking through the stairwells and hallways), the radio mapping results clearly point out the challenges in designing a communication system that is well-suited to a wide range of large structures. This paper is based on the fourth in a series of reports detailing experiments performed by NIST in order to better understand the emergency responder’s radio propagation environment. The first three reports covered implosion experiments performed in three different building structures. In addition, ongoing NIST experiments are simulating and testing the performance of ad-hoc networks in various building structures. The ultimate goal of this paper and supporting reports is to provide system designers and wireless communication engineers with greater insight into this challenging RF propagation environment.

REFERENCES [1] “Statement of Requirements: Background on public safety wireless communications,” The SAFECOM Program, Dept. Homeland Security, 2004, vol. 1. [2] M. Worrell and A. MacFarlane, Phoenix Fire Dept. Final Report,” Phoenix Fire Department Radio System Safety Project, 2004 [Online]. Available: http://www.ci.phoenix.az.us/FIRE/radioreport.pdf [3] “9/11 Commission Report,” National Commission on Terrorist Attacks Upon the United States, 2004. [4] “Final report for September 11, 2001 New York World Trade Center terrorist attack,” Wireless Emergency Response Team (WERT), 2001. [5] C. L. Holloway, G. Koepke, D. Camell, K. A. Remley, D. F. Williams, S. Schima, S. Canales, and D. T. Tamura, “Propagation and detection of radio signals before, during and after the implosion of a thirteen story apartment building,” Boulder, CO, 2005, NIST Tech. Note 1540. [6] C. L. Holloway, G. Koepke, D. Camell, K. A. Remley, and D. F. Williams, “Radio propagation measurements during a building collapse: applications for first responders,” in Proc. Int. Symp. Advanced Radio Tech., Boulder, CO, Mar. 2005, pp. 61–63. [7] C. L. Holloway, G. Koepke, D. Camell, K. A. Remley, D. F. Williams, S. Schima, and D. T. Tamura, “Propagation and detection of radio signals before, during and after the implosion of a large sports stadium (Veterans’ Stadium in Philadelphia),” Boulder, CO, 2005, NIST Tech. Note 1541. [8] C. L. Holloway, G. Koepke, D. Camell, K. A. Remley, D. F. Williams, S. Schima, M. McKinely, and R. T. Johnk, “Propagation and detection of radio signals before, during and after the implosion of a large convention center,” Boulder, CO, 2006, NIST Tech. Note 1542. [9] “APCO Project 25 Standards for Public Safety Digital Radio,” APCO International, Aug. 1995 [Online]. Available: http://www.apcointl.org/ frequency/project25/information.html [10] P. Whitehead, “The other communications revolution [TETRA standard],” IEE Rev., vol. 42, no. 4, pp. 167–170, Jul. 1996. [11] C. Edwards, “Wireless – Building on tetra,” Eng. Technol., vol. 1, no. 2, pp. 32–36, May 2006. [12] K. A. Remley, G. Koepke, C. L. Holloway, C. Grosvenor, D. Camell, J. Ladbury, R. T. Johnk, D. Novotny, W. F. Young, G. Hough, M. D. McKinley, Y. Becquet, and J. Korsnes, “Measurements to support modulated-signal radio transmissions for the public-safety sector,” Boulder, CO, 2008, NIST Tech. Note 1546. [13] C. L. Holloway, W. Young, G. Koepke, D. Camell, Y. Becquet, and K. A. Remley, “Attenuation of radio wave signals coupled into twelve large building structures,” Boulder, CO, 2008, NIST Tech. Note 1545.

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[14] D. M. J. Devasirvatham, C. Banerjee, R. R. Murray, and D. A. Rappaport, “Four-frequency radiowave propagation measurements of the indoor environment in a large metropolitan commercial building,” in Proc. GLOBECOM, Phoenix, AZ, Dec. 2–5, 1991, pp. 1282–1286. [15] K. M. Ju, C. C. Chiang, H. S. Liaw, and S. L. Her, “Radio propagation in office building at 1.8 GHz,” in Proc. 7th IEEE Int. Symp. on Personal, Indoor and Mobile Radio Communications, Taipei, Oct. 15–18, 1996, pp. 766–770. [16] T. N. Rubinstein, “Clutter losses and environmental noise characteristics associated with various LULC categories,” IEEE Trans. Broadcasting, vol. 44, no. 3, pp. 286–293, Sep. 1998. [17] J. H. Tarng and D. W. Perng, “Modeling and measurement of UHF radio propagating through floors in a multifloored building,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 144, no. 5, pp. 359–363, Oct. 1997. [18] L. P. Rice, “Radio transmission into buildings at 35 and 150 mc,” Bell Syst. Tech. J., pp. 197–210, Jan. 1959. [19] E. Walker, “Penetration of radio signals into building in the cellular radio environment,” Bell Syst. Tech. J., vol. 62, no. 9, Nov. 1983. [20] W. J. Tanis and G. J. Pilato, “Building penetration characteristics of 880 MHz and 1922 MHz radio waves,” in Proc. 43th IEEE Veh. Technol. Conf., Secaucus, NJ, May 18–20, 1993, pp. 206–209. [21] L. H. Loew, Y. Lo, M. G. Lafin, and E. E. Pol, “Building penetration measurements from low-height base stations at 912, 1920, and 5990 MHz,” National Telecommunications and Information Administration, 1995, NTIA Rep. 95-325. [22] A. Davidson and C. Hill, “Measurement of building penetration into medium buildings at 900 and 1500 MHz,” IEEE Trans. Veh. Technol., vol. 46, pp. 161–168, Feb. 1997. [23] A. F. De Toledo, A. M. D. Turkmani, and J. D. Parsons, “Estimating coverage of radio transmissions into and within buildings at 900, 1800, and 2300 MHz,” IEEE Personal Commun., vol. 5, no. 2, pp. 40–47, Apr. 1998. [24] E. F. T. Martijn and M. H. A. J. Herben, “Characterization of radio wave propagation into buildings at 1800 MHz,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 122–125, 2003. [25] A. Chandra, A. Kumar, and P. Chandra, “Propagation of 2000 MHz radio signal into a multistoryed building through outdoor-indoor interface,” in Proc. IEEE 14th Personal, Indoor and Mobile Radio Communications PIMRC 2003, Sep. 2003, vol. 3, pp. 2983–2987. [26] R. J. C. Bultitude, Y. L. C. de Jong, J. A. Pugh, S. Salous, and K. Khokhar, “Measurement and modelling of emergency vehicle-to-indoor 4.9 GHz radio channels and prediction of IEEE 802.16 performance for public safety applications,” in Proc. IEEE 65th Vehicular Technology Conf. VTC2007-Spring, Apr. 2007, pp. 397–401. [27] M. Karam, W. Turney, K. Baum, P. Satori, L. Malek, and I. OuldDellahy, “Outdoor-indoor propagation measurements and link performance in the VHF/UHF bands,” in Proc. IEEE 68th Vehicular Technology Conf. VTC 2008-Fall, Sep. 2008, pp. 1–5. [28] S. R. Saunders, K. Kelly, S. M. R. Jones, M. Dell’Anna, and T. J. Harrold, “The indoor-outdoor radio environment,” Electron. Commun. Eng. J., vol. 12, no. 6, pp. 249–261, Dec. 2000. [29] D. Molkdar, “Review on radio propagation into and within buildings,” Inst. Elect. Eng. Proc.-H, vol. 38, no. 1, pp. 61–73, Feb. 1991. William F. Young (M’06) received the B.S. degree in electronic engineering technology from Central Washington University, Ellensburg, in 1992, the M.S. degree in electrical engineering from Washington State University at Pullman, in 1998, and the Ph.D. degree from the University of Colorado at Boulder, in 2006. Since 1998, he has worked for Sandia National Laboratories in Albuquerque, NM, where he is currently a Principal Member of the Technical Staff. His work at Sandia includes the analysis and design of cyber security mechanisms for both wired and wireless communication systems used in the National Infrastructure and the Department of Defense. He has also been a Guest Researcher at the National Institute of Standards and Technology in Boulder, CO, from 2003 to 2009, and is working on improving wireless communication systems for emergency responders. His current research interests are in electromagnetic propagation for wireless systems, and the impacts of the wireless channel on overall communication network behavior.

Christopher L. Holloway (S’86–M’92–SM’04– F’10) was born in Chattanooga, TN, on March 26, 1962. He received the B.S. degree from the University of Tennessee at Chattanooga in 1986, and the M.S. and Ph.D. degrees from the University of Colorado at Boulder, in 1988 and 1992, respectively, both in electrical engineering. During 1992, he was a Research Scientist with Electro Magnetic Applications, Inc., Lakewood, CO. His responsibilities included theoretical analysis and finite-difference time-domain modeling of various electromagnetic problems. From fall 1992 to 1994, he was with the National Center for Atmospheric Research (NCAR), Boulder. While at NCAR his duties included wave propagation modeling, signal processing studies, and radar systems design. From 1994 to 2000, he was with the Institute for Telecommunication Sciences (ITS), U.S. Department of Commerce in Boulder, where he was involved in wave propagation studies. Since 2000, he has been with the National Institute of Standards and Technology (NIST), Boulder, CO, where he works on electromagnetic theory. He is also on the Graduate Faculty at the University of Colorado at Boulder. Dr. Holloway was awarded the 2008 IEEE EMC Society Richard R. Stoddart Award, the 2006 Department of Commerce Bronze Medal for his work on radio wave propagation, the 1999 Department of Commerce Silver Medal for his work in electromagnetic theory, and the 1998 Department of Commerce Bronze Medal for his work on printed circuit boards. His research interests include electromagnetic field theory, wave propagation, guided wave structures, remote sensing, numerical methods, and EMC/EMI issues. He is currently serving as Co-Chair for Commission A of the International Union of Radio Science and is an Associate Editor for the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. He was the Chairman for the Technical Committee on Computational Electromagnetics (TC-9) of the IEEE Electromagnetic Compatibility Society from 2000–2005, served as an IEEE Distinguished lecturer for the EMC Society from 2004–2006, and is currently serving as Co-Chair for the Technical Committee on Nano-Technology and Advanced Materials (TC-11) of the IEEE EMC Society.

Galen Koepke (M’94) received the B.S.E.E. degree from the University of Nebraska, Lincoln, in 1973 and the M.S.E.E. degree from the University of Colorado at Boulder, in 1981. He is an NARTE Certified EMC Engineer. He has contributed, over the years, to a wide range of electromagnetic issues. These include measurements and research looking at emissions, immunity, electromagnetic shielding, probe development, antenna and probe calibrations, and generating standard electric and magnetic fields. Much of this work has focused on TEM cell, anechoic chamber, open-area-test-site (OATS), and reverberation chamber measurement techniques along with a portion devoted to instrumentation software and probe development. He now serves as Project Leader for the Field Parameters and EMC Applications program in the Radio-Frequency Fields Group. The goals of this program are to develop standards and measurement techniques for radiated electromagnetic fields and to apply statistical techniques to complex electromagnetic environments and measurement situations. A cornerstone of this program has been National Institute of Standards and Technology (NIST), work in complex cavities such as the reverberation chamber, aircraft compartments, etc.

Dennis Camell (M’XX) received the B.S. and M.E. degrees in electrical engineering from the University of Colorado, Boulder, in 1982 and 1994, respectively. From 1982 to 1984, he worked for the Instrumentation Directorate, White Sands Missile Range, NM. Since 1984, he has worked on probe calibrations and EMI/EMC measurements with the Electromagnetics Division, National Institute of Standards and Technology (NIST), Boulder, CO. His current interests are measurement analysis (including uncertainties) in various environments, such as OATS and anechoic chamber, and development of time domain techniques for use in EMC measurements and EMC standards. He is involved with several EMC working standards committees and is chair of ANSI ASC C63 SC1.

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Yann Becquet, photograph and biography not available at the time of publication.

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Kate A. Remley (S’92–M’99–SM’06) was born in Ann Arbor, MI. She received the Ph.D. degree in electrical and computer engineering from Oregon State University, Corvallis, in 1999. From 1983 to 1992, she was a Broadcast Engineer in Eugene, OR, serving as Chief Engineer of an AM/FM broadcast station from 1989–1991. In 1999, she joined the Electromagnetics Division, National Institute of Standards and Technology (NIST), Boulder, CO, as an Electronics Engineer. Her research activities include metrology for wireless systems, characterizing the link between nonlinear circuits and system performance, and developing methods for improved radio communications for the public-safety community. Dr. Remley was the recipient of the Department of Commerce Bronze and Silver Medals and an ARFTG Best Paper Award. She is currently the Editor-inChief of IEEE Microwave Magazine and Chair of the MTT-11 Technical Committee on Microwave Measurements.

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Radio-Wave Propagation Into Large Building Structures—Part 2: Characterization of Multipath Kate A. Remley, Senior Member, IEEE, Galen Koepke, Member, IEEE, Christopher L. Holloway, Fellow, IEEE, Chriss A. Grosvenor, Member, IEEE, Dennis Camell, Member, IEEE, John Ladbury, Member, IEEE, Robert T. Johnk, Member, IEEE, and William F. Young, Member, IEEE

Abstract—We report on measurements that characterize multipath conditions that affect broadband wireless communications in building penetration scenarios. Measurements carried out in various large structures quantify both radio-signal attenuation and distortion (multipath) in the radio propagation channel. Our study includes measurements of the complex, wideband channel transfer function and bandpass measurements of a 20 MHz-wide, digitally modulated signal. From these, we derive the more compact metrics of time delay spread, total received power and error vector magnitude that summarize channel characteristics with a single number. We describe the experimental set-up and the measurement results for data collected in representative structures. Finally, we discuss how the combination of propagation metrics may be used to classify different propagation channel types appropriate for public-safety applications. Index Terms—Attenuation, broadband radio communications, building penetration, digital modulation, emergency responders, error vector magnitude, excess path loss, received power, time-delay spread, vector network analyzer, vector signal analyzer, wireless signals, wireless system measurements, wireless telecommunications.

I. INTRODUCTION

T

O aid in the development of standards that support reliable wireless communications for emergency responders such as those discussed in [1], the National Institute of Standards and Technology (NIST) has embarked on a project to acquire data on radio-wave propagation in key emergency-responder and public-safety environments. In past work [2]–[4], measurements were made in buildings scheduled for implosion to simulate collapsed-building environments. The focus of current work [5], [6] is to study the penetration of radio waves into

Manuscript received December 18, 2008; revised August 10, 2009; accepted September 25, 2009. Date of publication January 22, 2010; date of current version April 07, 2010. This work was supported in part by the U.S. Department of Justice, Community-Oriented Police Services through the NIST Public-Safety Communications Research Laboratory. K. A. Remley, G. Koepke, C. L. Holloway, C. Grosvenor, D. Camell, J. Ladbury, and W. F. Young are with the National Institute of Standards and Technology, Boulder, CO 80305 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). R. T. Johnk was with the National Institute of Standards and Technology, Boulder, CO 80305 USA. He is now with the Institute for Telecommunication Sciences, Boulder, CO 80305 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.2010.2041143

large buildings where difficult radio reception is often encountered because of signal attenuation and variability. Our measurement set-up is intended to simulate a response scenario, where an incident command vehicle is located near a structure and a mobile unit is deployed inside. In contrast to the many existing studies on cell- or trunked-radio systems, in our case we focus on ground-based point-to-point radio communications. Our goal is to provide a large body of measurement data, acquired in key responder environments, to the open literature (see [7] for a complete list) for improved communication system development and design and to aid in technically sound standards development. A second goal is to share our methodology so that responder organizations and others may carry out these characterizations as desired. A third goal of this program is to provide measurement data that will be useful for verification of network simulations of emergency responder radio links. Such simulations are being developed by NIST, among others [8]. Much work has been published describing measurement characterization of multipath in the radio-propagation environment. Various figures of merit are often used to describe multipath effects, including excess path loss, frequency selectivity, time delay spread, bit error rate (BER) and its variants, and/or error vector magnitude (EVM). Most of these publications (for example, [9]–[13] and references cited therein) describe measurements intended to simulate communications via cellular telephone or other wireless systems that rely on a fixed base station whose antenna is positioned high above the ground and a mobile user located at ground level. Only a few publications describe measurements that simulate point-to-point radio-communication scenarios, such as those utilized in many emergency-responder scenarios. Examples include sections of [10], [14] and references cited therein, and [15]. Part 1 of this work [16] summarizes the statistics of received-signal-strength measurements under single-frequency excitation in twelve large public building structures, based on NIST Technical Note 1545 [5]. In that work, a spectrum analyzer was used to measure the relative received signal strength at a fixed receiver location as a portable transmitter was carried throughout the buildings. The mean and standard deviation of the measurements in the structures were calculated. Frequencies included bands near licensed public-safety bands and cell phone bands including 49 MHz, 160 MHz, 225 MHz, 450 MHz, 900 MHz, 1.8 GHz, and a limited set of data at 2.4 GHz and 4.95 GHz. In this paper, we focus on additional parameters relevant to successful transmission of modulated signals including wideband channel frequency response, excess path loss, time delay spread, channel power and error vector magnitude. We focus on

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carrier frequencies over 1 GHz, focusing on frequency bands around 2.4 GHz and 4.95 GHz to investigate the differences in transmission between existing wireless systems in the unlicensed 2.4 GHz industrial/scientific/medical (ISM) frequency band (which is sometimes used by public-safety organizations) and systems proposed for use in the licensed public-safety frequency band covering 4.94 GHz to 4.99 GHz [17], [18]. In NIST Technical Note 1546 [6], we conducted tests in four representative environments that are notoriously difficult in terms of radio reception for emergency responders. These are a multi-story apartment building, an oil refinery, a long corridor in an office building typical of many commercial facilities, and a subterranean tunnel. Here we provide representative results of the characterization of multipath in the propagation channel and discuss key findings from [6]. We extend the work of [6] to demonstrate how compact “summary” metrics that quantify the channel characteristics with a single number, may be used to classify propagation channels for responder applications. Summary metrics are often easier and more inexpensive to acquire than are extensive frequency-domain measurements. By classifying propagation channel types, responders may better know how to deploy wireless systems, and standards development organizations may develop better performance metrics and verification tests for wireless technology. These data may also be used to support general building or scenario classifications such as those presented in [9], [10]. In Section II, we discuss the instrumentation, calibrations, and post-processing methods we used to collect the various data. In Section III, we show representative measured data collected at two of the four large structures reported in [5], [6]: an 11-story apartment building and an oil refinery. We summarize the propagation effects that we observed and draw some conclusions on the characteristics of the propagation channels seen in the structures. Occasionally product names are specified solely for completeness of description, but such identification constitutes no endorsement by the National Institute of Standards and Technology. Other products may work as well or better. II. MEASUREMENTS FOR MODULATED-SIGNAL CHANNEL CHARACTERIZATION A. Overview In the study of [6], we conducted three types of measurements: single-frequency received-power using a communications receiver; frequency response data over a very broad frequency band at fixed points in the propagation environment using a vector network analyzer (VNA); and modulated-signal measurements using a vector signal analyzer (VSA). The single-frequency data are similar to those reported in [5] and are not discussed here. From the VNA data, we determined the wideband frequency response of the propagation channel and, from this, the excess path loss (the loss that exceeds that measured in a free-space environment [19], [20]). After transforming these data to the time domain, we calculated the root-mean-square (RMS) time-delay spread. The RMS delay spread is a figure of merit that quantifies the time it takes for reflections in a received signal to die out. Using the VSA, we also collected modulated-signal measurements associated

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Fig. 1. Wideband measurement system based on a vector network analyzer. Frequency-domain measurements, synchronized by the optical fiber link, are transformed to the time domain in post-processing. This enables determination of excess path loss, time-delay spread, and other figures of merit important in characterizing broadband modulated-signal transmissions.

with broadband digitally modulated signals at 2.4 GHz and 4.95 GHz. From these, we found the error vector magnitude (EVM) in each environment. EVM is a figure of merit that describes the level of distortion in received, demodulated symbols of a digitally modulated signal. In the following sections, we describe these measurements. B. Wideband Vector Network Analyzer Measurements 1) Measurement Set-Up: We measured the wideband frequency response and time-delay characteristics of the propagation channel using a measurement system based on a vector network analyzer, shown in Fig. 1. This instrument collects data over a very wide frequency range, from 25 MHz to 18 GHz for the system we used. This system, also described in [19], lets us measure the complex transfer function of the channel, including frequency-selective characteristics. By taking the inverse Fourier transform of the measured transfer function, the power delay profile and RMS delay spread of the channel are found in post processing. The VNA acts as both transmitter and receiver in this system. The transmitting section of the VNA steps through the frequencies a single frequency at a time. The signal is amplified and fed to a transmitting antenna, as shown in Fig. 1. The received signal is returned to the VNA via a fiber-optic cable. Transmitting the received signal along the fiber optic cable back to the VNA eliminates the loss and phase changes that would be associated with RF coaxial cables between the receive antenna and the transmit antenna, allowing characterization of the complex radio channel. One advantage of this system is that it provides a high dynamic range relative to true time-domain-based measurement instruments. In Fig. 1, the system is configured for a line-of-sight reference measurement. In practice, the transmitting and receiving antennas may be separated by significant distances, although they must remain tethered together by the fiber-optic link. While directional horn antennas are shown in Fig. 1, omnidirectional antennas were also used in our measurements. Omnidirectional antennas are most often used in public-safety applications. However, high-frequency measurements often benefit from the use of directional antennas to maximize gain in a specific direction.

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Emergency response agencies may use directional antennas in these situations as well. We made measurements in two frequency bands: a low band that ranged from 100 MHz to 1.2 GHz, and a high band, that ranged from 1 GHz to 18 GHz. We used omnidirectional transmit and receive antennas for the lower frequency band and, for the measurements reported on here, a dual-ridge-guide (DRG) directional transmit antenna and omnidirectional receive antenna for the higher frequency band (1 GHz to 18 GHz). The beamwidth of the omnidirectional antennas is approximately 40 to 50 in the vertical direction. The beamwidth of the DRG at the low end at varies with frequency, from almost at 18 GHz, necessitating reorientation of about 1 GHz to the antenna to track the receiver’s position in certain structures such as the apartment building. It is well known that in propagation environments where the transmitting antenna is located in the midst of highly reflecting objects, the use of a directional horn antenna can reduce the apparent multipath by eliminating reflections from behind. Our studies in an automobile manufacturing plant [21] demonstrated this. For the measurements presented here, strong reflectors (the building structures) were located in front of the transmitting antenna with very few scatterers behind. Thus, we anticipate the use of directional antennas introduced only small deviations to the measured multipath effects. To make a measurement, the vector network analyzer is first calibrated by use of standard techniques where known impedance standards are measured. The calibration enables us to correct for the response of the fiber-optic system, amplifiers, and any other passive elements and electronics used in the measurement. We also high-pass filter our measurements in post processing to suppress the large, low-frequency oscillation that occurs in the optical fiber link. For the measurements reported here, the VNA-based measurement system was set up with the following parameters: the initial output power was set between 15 dBm to 13 dBm. The gain of the amplifier and the optical link and the system losses resulted in a received power level no more than 0 dBm. An intermediate-frequency (IF) averaging bandwidth of around 1 kHz was used to average the received signal. We typically recorded 6401 points per frequency band and chose the number of bands recorded in each measurement to avoid aliasing of the signal. We report here on our high-band measurements ranging from 1 GHz to 18 GHz, which were taken by measuring 48003 points for a total of three bands. Low-frequency measurements from 100 MHz to 1.2 GHz are reported in [6]. The dwell time per point and the frequency spacing was approximately 25 was approximately 379 kHz. 2) Wideband Frequency Response and Path Loss: Our wide, band measurements provide a channel transfer function typically is derived from the measured transmiswhere sion parameter . To find the frequency-dependent path loss between the transmit and receive antennas, we first com, where is a free-space reference pute from the transmit antenna. The made at a known distance use of a ratio to find the path loss enables us to calibrate out the antenna response of the system. We correct the measurements for the free-space path loss between the transmit

Fig. 2. Reference measurement at three meters for a dual-ridge guide horn antenna, transformed to the time domain. The waveform shows the antenna response, the ground-bounce response and the spurious environmental effects.

antenna and the reference location by dividing by . To find the excess path loss, we additionally reduce the total path loss by the expected free-space path loss over the overall separation distance between the transmit and receive antennas. by . To do this, we divide the measurement of Equivalently, we can multiply by . The distance may be measured or estimated from maps, depending on the environment. As stated earlier, this provides the loss in excess of that which would be measured at the same distance in free space. We note that communication engineers typically think of excess path loss as a single frequency or narrowband measurement. However, the VNA measurements provide a much richer data set because they include both magnitude and phase information over a broad frequency range. The reference measurement is made at a specified distance and may be acquired either during field tests or from a laboratory measurement. In the field, the measurement includes environmental effects, and we use time-domain gating to minimize reflections on the free-space reference. If we are not able to gate out the reflections satisfactorily, the reference measurement is made in a laboratory facility such as an anechoic chamber or an open-area test site. In this case, we use the same antennas and measurement system as were used in the field. For the measurements shown below, we used a two-meter reference made in the field. We chose this distance to balance the need to be in the antenna far field of our lowest frequency of interest (1 GHz for the results reported here), while keeping the reference measurement as free from environmental reflections as possible. As an example, Fig. 2 shows the time-domain response for a reference measurement using a pair of dual-ridged-guide antennas separated by 3 m. In Fig. 2, the reference measurement, transformed to the time domain, is shown with all environmental effects. The reference measurement is gated (windowed) from 20 ns to 32 ns to isolate the antenna response, which was determined previously in a separate measurement. The frequencydomain responses for the reference measurement would show

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C. Modulated-Signal Measurements Along with the single-frequency measurements described in [5], [6] and the wideband VNA measurements described above, a third set of tests involved measuring the waveforms of digitally modulated signals. From these, we plotted the spectra of the signals, and calculated the received power and the error vector magnitude associated with a given propagation channel. EVM gives an indication of the distortion introduced into a digitally modulated signal as it passes through a propagation channel. Mathematically, this can be expressed as

(2)

Fig. 3. Power-delay profile for a building propagation measurement. Important parameters for a measured signal are the peak level, the maximum dynamic range, the mean delay, and the RMS delay spread.

a noisier trace when environmental effects are included, compared with a smoother trace for isolated antennas. The gated response is what we would see if the antenna were measured in a free-space environment, free from environmental reflections. In the following, we present graphs of the inverse excess path loss in decibels. Graphs of path loss would have positive ordinates and increase with distance. Thus, we refer to the curves in our graphs as “penetration.” This terminology is commonly used in studies of electromagnetic shielding. 3) RMS Delay Spread: The time-domain representation of the signal was calculated from the excess path loss data in post processing. From this we found the RMS delay spread. RMS delay spread is calculated as the square root of the second central moment of the power-delay profile of a measured signal [22]–[24]. Fig. 3 shows the power-delay profile for a representative building propagation measurement. The peak level usually occurs when the signal first arrives at the receiving antenna, although in high multipath environments we sometimes see the signal build up over time to a peak value and then fall off. A common rule of thumb is to calculate the RMS delay spread from signals at least 10 dB above the noise floor of the measurement [24]. For the measurements described in the following sections, we defined the minimum dynamic range to be approximately 40 dB below the peak value, although this value was reduced for lower signal levels. For the illustrative measurement shown in Fig. 3, we extended the window down to 70 dB below the peak value. Whether we use a 40 dB or a 70 dB threshold, the RMS delay spread does not change appreciably due to the almost constant slope of the power decay curve. can be defined as The RMS delay spread

(1) In (1), is defined as the average value of the power-delay profile in the defined dynamic range window, and is the variance of the power-delay profile within this window.

is the normalized symbol in a stream of where is the ideal normalized constellameasured symbols, symbol, and is the number of unique tion point for the symbols in the constellation. The fractional form of EVM given in (2) is often represented as a percentage. The algorithm used to find EVM was built into the receiver. The modulated signal used as excitation was based on orthogonal frequency-division multiplexing (OFDM) 802.11a/g, as specified by the IEEE 802.11a-1999 standard [25]–[27]. OFDM is used in wireless local-area networks (WLANs) and in the public-safety band at 4.95 GHz. In the latter, OFDM signals may be transmitted in a 10 MHz wide channel using the 802.11j standard, instead of the 20 MHz wide channel utilized in 802.11a. Our demodulator was able to measure only the 802.11a standard. As a consequence, this is what we report on in the following sections. The OFDM multiplexing scheme was developed to provide immunity to interference. Data are encoded onto 52 narrowband, frequency-multiplexed subcarriers. For strong received signals, data are transmitted up to a maximum of 54 megabits per second (Mbps). As the received signal strength decreases or the level of multipath increases, the data rate decreases to compensate for the decrease in signal-to-noise ratio. In the tests reported here, we force the signal generator to transmit either a slow-data-rate quadrature-phase-shift-keyed (QPSK) or a highdata-rate 64-quadrature-amplitude-modulated (64QAM) signal. This lets us assess the impact of multipath for a given channel when different modulation schemes are used. For the measurements described in the following sections, a vector signal generator was used to create the digitally modulated signals. The signal generator was mounted on a rolling cart and moved through the various propagation environments. We used omnidirectional antennas where possible to mimic those used by emergency responders. In some environments reported here, we were limited by the dynamic range of the receiver and so we used directional, dual-ridge-guide horn antennas. We used a vector signal analyzer to acquire the signals. The VSA maintains the phase of the measured frequency components relative to one another and enables measurements of complex distortion in digitally modulated signals, including EVM. The VSA used had a 36 MHz measurement bandwidth. No correction for system effects such as antenna gain, system electronics, or system impedance mismatch was conducted. Thus,

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these measurements are not absolute, but are relative to the measurement configuration we used. The VSA was adjusted for minimum received-signal distortion before each set of measurements. This entailed performing an internal calibration followed by a range adjustment under line-of-sight conditions. III. MEASUREMENT RESULTS AND IDENTIFICATION OF CHANNEL CLASSIFICATIONS A. Overview As stated in the introduction, in [6] we collected measurements in four large public structures, two of which are reported on here. By observing key features from graphs of the measurements, we are able to classify distinct propagation effects depending on the distance and type of structural obstruction between the transmitter and receiver. The propagation effects illustrated here are specifically relevant to point-to-point communications used by most emergency response organizations. The primary features we observed to make our classifications include (i) the excess path loss, (ii) the amount and structure of frequency-selective distortion over the modulation bandwidth (in this case, the spectrum covering 2.4 GHz and 4.95 GHz 10 MHz), (iii) the RMS delay spread, (iv) the received channel power, and (v) the error vector magnitude. We illustrate how the summary metrics (iii)–(v) can be used to gain a similar insight into the channel as the more complete frequency-domain measurements (i) and (ii). We illustrate these concepts using measured results from an apartment building and an oil refinery. The latter was discussed in detail in [5] with respect to the use of single-frequency statistics to classify channels. The oil refinery was also discussed in [28] and [29] as part of overviews on the difficulties faced by public-safety practitioners and wireless sensor networks, respectively, in high-multipath environments. Here, we discuss how a combination of the metrics above can help to identify classes of propagation environments that may be important to the emergency response community. Again, the reader is referred to [6] for a more complete discussion.

Fig. 4. The 11-story, concrete, steel, and brick apartment building where the NIST measurements were made.

B. Apartment Building

Fig. 5. Layout of a typical floor of the 11-story apartment building. Circles show the test positions where measurements were made. The receiver site shown is approximately 60 m east of the building.

We carried out propagation measurements at an 11-story apartment building located in Boulder, Colorado in October 2007. The building, shown in Fig. 4, is constructed of reinforced concrete, steel, and brick. It contains standard interior finish materials. The building was fully furnished and occupied during the experiments. Measurements were performed during daytime hours, so people were moving throughout the building during the experiments. The layout of each floor of the apartment building was T-shaped, with two elevators near the junction of the T. This is illustrated in Fig. 5. The hallway along top of the T was approximately 20 m long and the body of the T was approximately 50 meters long. Our receiver site was located approximately 60 m from the building in a parking lot, also shown in Fig. 5. Both VNA and VSA measurements were made every 5 meters, on Floors 2 and 7, as illustrated in Fig. 5. This apartment building was chosen because it has several features in common with the building described in the Apartment Fire Scenario of the SAFECOM Statement of Require-

ments [1], including concrete construction, stairwells at the ends of the hallways, apartments off a main corridor with outsidefacing windows, and the need for single- or two-wall radio-wave penetration. The Apartment Fire Scenario of [1] deals with a fire response on the second floor of such an apartment building. The apartment building propagation environment consisted entirely of non-line-of-sight (NLOS) propagation paths because all of the received signals penetrated through the outside walls of the structure and at least one interior wall. The penetration measurement examples in Fig. 6 show a predominantly monotonic roll-off with frequency. These graphs show excess path loss from 1 GHz to 18 GHz made at test position 2 (Fig. 6(a)), where the only obstructions between transmitter and receiver are the building walls and windows, and test position 5 (Fig. 6(b)), where a metallic elevator obstructs the signal path. The building penetration decreases with frequency indicating a channel that includes attenuation due to signal penetration

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Fig. 7. Measured spectra across a 30 MHz bandwidth in the apartment building. (a) Test point 2, Floor 2; (b) test point 5, Floor 2. Top graphs for each location represent measurements for a carrier frequency of 2.4 GHz and bottom graphs for a carrier frequency of 4.95 GHz. Note that transmissions on other channels within the 2.4 GHz band are visible on the graphs.

Fig. 6. Measured building penetration 1/(excess path loss), in dB, on Floor 2 (top, dark curves) and Floor 7 (middle, lighter curves) of the apartment building for frequencies from 1 GHz to 18 GHz at (a) position 2 and (b) position 5. The lower curves represent the noise floor of the VNA-based measurement system. Channel parameters such as loss and delay spread are calculated only where the signal is significantly in excess of the noise floor. A 200 MHz moving average filter has been applied to the data.

through some type of building material. Building materials typically exhibit a monotonic increase in attenuation with frequency, changing on the order of decibels over several decades of frequency. The fact that the penetration curves (where the free-space path loss has been removed) in Fig. 6(a) measured on Floors 2 and 7 practically overlay indicates that the attenuation due to building penetration was similar on both floors. The difference in penetration between Floors 2 and 7 when the receiver is located behind the elevator indicates that more complicated, angle-dependent multipath scattering is involved in this propagation path. Fig. 6 shows peaks and nulls that vary rapidly with frequency. These indicate multipath in the environment in addition to the building penetration effects. Because all of the received signal components arrive on non-line-of-sight paths, the phase relationship between them is random, indicating a Rayleigh distribution for the received signal. Note that the peaks and nulls

that have structure (at higher frequencies) occur when the signal level approaches noise floor of the measurement system and do not give us information on the propagation channel. We neglect data whose level is not significantly in excess of the noise floor in our calculations of excess path loss and RMS delay spread. Given that this is a multipath channel, it is useful to determine whether the multipath distortion is narrowband—where all frequency components in the modulation bandwidth are similarly affected by reflections, resulting in “flat” fading—or wideband—where peaks and nulls occur within the modulation bandwidth of the excitation, resulting in “frequency-selective” fading. Typically, wideband distortion is harder to overcome for radio receivers because different signal components are affected by the channel differently. Note that the definition of wideband distortion for a particular environment will be dependent on the excitation signal and the receiver used. The VSA measurements exemplified in Figs. 7(a) and (b) indicate whether wideband distortion is present in the frequency band of interest. Here, we see the spectrum of the signal plotted versus frequency for the OFDM digitally modulated signal (dark curve) and for a multisine signal designed to simulate the RF statistics of the digitally modulated signal (lighter, dashed curve). The multisine is easier to generate and characterize using standard RF measurement instrumentation [30].

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Fig. 8. Received power averaged across a 30 MHz bandwidth at 13 positions on two different floors of the 11-story apartment building for frequencies from 1 GHz to 18 GHz. Measurements were not recorded at all locations, as shown by the missing data points.

The graphs in Fig. 7 show the received spectrum at test positions 2 and 5 on Floor 2. In each figure, the measurements in the top graph had a center frequency of 2.4 GHz and the bottom graph had a center frequency of 4.95 GHz. Note the additional signals received in the 2.4 GHz frequency band, both above and below the OFDM signal. These are from other 2.4 GHz wireless devices operating nearby. Emergency response organizations using unlicensed frequency bands may be susceptible to interference from other sources [17]. The VSA measurements show that the channel is generally frequency flat when simple building penetration through windows and concrete walls occurs; for example, at test position 2. However, frequency-selective distortion can be seen when the transmitter is behind the elevators or stairwell doors; for example, at test position 5. A simple classification scheme for the propagation channels encountered in this type of building structure may be derived from the two frequency-domain measurements illustrated in Figs. 6 and 7. For the majority of the locations we tested, the channel could be classified as “simple building penetration,” consisting of flat fading combined with decade-scale, frequency- and elevation-dependent path loss. The frequency and elevation dependence are indicated by the reduced overall signal level for the 4.95 GHz measurements, and for higher floors, respectively. In the location behind the elevator, we may classify the channel as “building penetration combined with frequency-selective fading.” The latter, illustrated by the spectral variations shown in Fig. 7(b), is not the dominant channel classification in this type of building. We next consider three summary metrics that can be used to derive the same classifications for the propagation channels as were derived from the frequency-domain representations above. Fig. 8 shows the received power of the OFDM signals averaged across a 30 MHz bandwidth. These graphs illustrate the frequency- and elevation-dependent path loss. When compared to the 2.4 GHz signal on Floor 2, the received power for the 4.95 GHz carrier frequency was on the order of 15 dB to 20 dB lower, and the received power for the 2.4 GHz carrier on Floor 7 was around 10 dB lower. To classify the fading characteristics, we consider the RMS delay spread. Fig. 9 shows the RMS delay spread, computed for

Fig. 9. RMS delay spread (ns) at 13 positions on two different floors of the 11-story apartment building made using a directional transmit antenna for frequencies between 1 GHz to 18 GHz.

Fig. 10. EVM on Floor 2 of the apartment building for a QPSK-modulated OFDM signal for carrier frequencies of 2.4 GHz (circles) and 4.95 GHz (triangles). The thick black line represents an EVM of 15%. Measurements on Floor 7 were weak enough that it was difficult to determine the EVM.

the entire high-frequency band, at all test positions on Floors 2 and 7. We see an increase in RMS delay spread when the receiver is blocked by the metallic elevator shaft, as expected. We see only a moderate increase in the RMS delay spread between the different floors, which is consistent with our classification of the majority of test positions as simple building penetration (attenuation as opposed to multipath). The error vector magnitude plots shown in Fig. 10 illustrate a significantly lower EVM for the 2.4 GHz signal than for the 4.95 GHz signal. We see an increase in EVM at 2.4 GHz when the transmitter is shadowed by the elevator. Received signals whose EVM is high will have difficulty maintaining a link. An EVM of 15% is marked by the thick black line on the graph to denote channels that may be unusable. Because the dynamic range of the VSA is not as high that of the VNA-based measurement system, we were unable to obtain meaningful EVM measurements on Floor 7. However, we expect that the channel characteristics on Floor 7 are similar to those on Floor 2, based on the RMS delay spread measurements shown in Fig. 9. The received power measurements again show a frequency and elevation dependence. RMS delay spread and EVM indicate that the multipath environment introduces both narrowband- and wideband-distortion. Whether we use the frequency measurements directly or a combination of summary metrics,

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Fig. 12. Layout of the test positions for the excess path loss and modulatedsignal measurements in the oil refinery complex. The test positions were located under dense overhead piping and metallic structures, in most cases several stories high.

Fig. 11. Modulated-signal measurements made at an oil refinery: (a) transmitting unit consisting of a vector signal generator and omnidirectional antenna on a mobile cart inside dense piping in the facility. Photograph (b) shows the omnidirectional receiving antenna, indicated by an arrow, located on top of the ITS van.

analysis tells us we are dealing with a combination of simple building penetration in a multipath environment. This combination is representative of many similar structures where signals do not penetrate too deeply within a building. C. Oil Refinery A second example of the use of multiple metrics for propagation channel classification is illustrated by measurements conducted at a large oil refinery in Commerce City, Colorado in March 2007. As with the measurements at the apartment building, we chose this facility to simulate a response scenario (the Chemical Plant Explosion) in the SAFECOM Statement of Requirements [1]. The refinery is an outdoor facility covering many hectares in area, with intricate piping systems. We carried out tests primarily in an area of dense piping that forms a tunnel-like structure, shown in Fig. 11(a). We studied the propagation from a location outside the piping tunnel, shown in Fig. 11(b), to within the tunnel. Even though the site was outdoors, the dense piping was a significant barrier to radio communications and the propagation channel may be thought of as involving structure penetration. Measurements were made at locations in the oil refinery indicated in Fig. 12 for both VNA and VSA measurements. One antenna (transmit antenna for the VNA measurements, receive

Fig. 13. Penetration 1/(excess path loss), in dB, measured at two locations in the oil refinery covering frequencies from 1 GHz to 18 GHz at (a) test position 6 (LOS) and (b) test position 8 (NLOS).

antenna for the VSA measurements) was located outside the piping complex on top of a mobile test van owned by the Institute for Telecommunication Sciences (ITS), a sister Department of Commerce organization at the Boulder Labs Site. This

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Fig. 15. Total received power averaged across a 30 MHz bandwidth for OFDM signals measured in the oil refinery for test positions 1 to 12.

Fig. 14. Examples of the signal spectra derived from the VSA measurements of the modulated signal at (a) test position 6 and (b) test position 8 in the oil refinery. The top graphs represent measurements made at a carrier frequency of 2.4 GHz and the bottom graphs are for a carrier frequency of 4.95 GHz. These examples show 64QAM modulated signals.

antenna was vertically polarized. The photo in Fig. 11(b) shows the omnidirectional antenna mounted on the mast. As may be expected, the many metallic surfaces in this environment introduced significant multipath into the signals we measured. Fig. 13 shows representative measurements of the penetration for frequencies from 1 GHz to 18 GHz. Between test positions 2 and 6, the overhead piping created a tunnel-like environment where the multipath has significant structure. We see distinct frequency resonances at these positions, even though the receiving antenna had a line-of-sight condition with the transmitting antenna. Once the receiving antenna turned the corner, for test positions 7 and higher, no line-of-sight condition existed. The received signal took on a rapidly varying, Rayleigh, appearance. We again consider the spectrum of the OFDM signal to assess the frequency-selective distortion in this environment. Representative VSA measurements are shown in Fig. 14. Our data showed only moderate frequency-selective distortion at test positions 2 to 6 (test position 6 is shown in Fig. 14(a)). This indicates that the extremely deep, frequency-resonant nulls in the penetration measurements of Fig. 13 are spaced widely enough that the modulated signal experiences multipath distortion that is flat across the bandwidth. Once the transmitter turns the corner, however, the signal experiences additional

Fig. 16. RMS delay spread in the oil refinery at test positions 1 to 12. The curve with circles represents frequencies from 25 MHz to 1.2 GHz and the curve with inverted triangles represents frequencies from 1 GHz to 18 GHz.

multipath that does introduce frequency-selective distortion, as shown in Fig. 14(b). Based on the frequency-domain measurements above, the propagation channel in the oil refinery can be classified as “low attenuation, frequency flat” for the LOS conditions and “high attenuation, frequency-selective” for the NLOS conditions. We next study the use of the summary metrics total received power, RMS delay spread, and EVM to assess the same channel. Fig. 15 shows the received power averaged across the modulation bandwidth of the OFDM signals. The signals with a carrier frequency of 4.95 GHz were approximately 5 dB to 10 dB lower than those with a carrier frequency of 2.4 GHz. When the receiver turned the corner into a NLOS condition, the received power levels dropped by around 15 dB for the 2.4 GHz signal and around 20 dB for the 4.95 GHz signal. Results were similar for both QPSK and 64QAM modulation formats, because the output power was the same for both transmissions. A graph of the RMS delay spread in Fig. 16 shows short-duration values in the line of sight condition, and longer values after the receiver has turned the corner. The increase in delay spread is more significant for the low frequency band (100 MHz to 1.4 GHz) than for the high frequency band (1 GHz to 18 GHz). The EVM graphs in Fig. 17 show that once the transmitter turns the corner, the EVM increases significantly. Before the turn, the EVM remains below 10% for both modulation formats.

REMLEY et al.: RADIO-WAVE PROPAGATION INTO LARGE BUILDING STRUCTURES—PART 2: CHARACTERIZATION

Fig. 17. EVM at the oil refinery for QPSK- and 64QAM-modulated OFDM signals for carrier frequencies of 2.4 GHz (circles and squares) and 4.95 GHz (triangles and inverted triangles.)

Even though significant multipath exists, this channel is probably usable. Thus, using the frequency-domain metrics or the summary metrics, the oil refinery environment can be described as a channel that is low attenuation and flat fading under line-of-sight conditions, and high-attenuation with frequency-selective fading when in non-line-of-sight conditions. Additionally (not shown here), at the lower frequencies we saw attenuation due to lossy waveguide effects, where frequencies below the “cut-off” frequency of the tunnel were significantly attenuated [6], [31]. IV. CONCLUSION We studied radio-wave propagation in representative environments of interest to the public-safety community. We used measurement configurations that replicate response scenarios, including point-to-point communications at human height, and receive sites located where response vehicles may be placed. Our study focused on measurements of quantities of interest in design, testing, and standards development for broadband, modulated-signal transmissions. These measurements included wideband penetration 1/(excess path-loss), in dB, measurements at specific locations in each structure and measurements of digitally modulated signals under the OFDM protocol at carrier frequencies of 2.4 GHz and 4.95 GHz. From the wideband path-loss measurements, we calculated channel power and RMS delay spread. From the modulated-signal measurements, we calculated error vector magnitude. Here we reported on the results for an 11-story apartment building and an oil refinery. These tests show that a number of factors are needed to reliably define the propagation environment. Use of a combination of summary metrics such as received signal strength, RMS delay spread, and error vector magnitude can provide a fairly complete picture of the propagation environment, necessary for reliable deployment of radio systems, especially those employing wideband digital modulation. Because these summary measurements may be less expensive and complex to perform, use of summary metrics can save time and money. In buildings where only one or two walls separate the transmit and receive antennas, such as the apartment building

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shown here, response organizations can expect to encounter simple building penetration with narrowband and, occasionally, wideband fading. These effects are expected to be common to environments that superficially appear quite different from each other. For certain propagation mechanisms such as simple building penetration, single-frequency signal-strength or channel power measurements may be sufficient to characterize the environment. In cases where multipath conditions exist, measurements that cover multiple frequencies and metrics such as RMS delay spread or EVM can offer additional insight. It is clear that one critical factor for successful radio communications is knowledge of both the basic environmental characteristics (highly reflective, as in the industrial facility, or lossy building materials, such as windows and brick in the apartment building). A second factor is an understanding of the transmitted signal itself. If the signal is broadband, it is important to understand the sensitivity of the modulation format to flatand frequency-selective multipath and attenuation. Finally, it is necessary to select the appropriate measurement metrics to be able to assess the environment for a given application. These procedures, as we have tried to illustrate here, are not difficult, but they do take some fore knowledge of the communication scenario. Consideration of these concepts will help to enable reliable communications for the public-safety community. Finally, we direct the reader to [5], [6] for data on other large structures. REFERENCES [1] “SAFECOM Statement of Requirements,” vol. 1, ver. 1.2 [Online]. Available:http://www.safecomprogram.gov/NR/rdonlyres/8930E37CC672-48BA-8C1B-83784D855C1E/0/SoR1_v12_10182006.pdf, Section 3.3 and Section 3.5 [2] C. L. Holloway, G. Koepke, D. Camell, K. A. Remley, D. F. Williams, S. A. Schima, S. Canales, and D. T. Tamura, “Propagation and detection of radio signals before, during, and after the implosion of a 13-story apartment building,” Nat. Inst. Stand. Tech. Note 1540, May 2005. [3] C. L. Holloway, G. Koepke, D. Camell, K. A. Remley, D. F. Williams, S. A. Schima, S. Canales, and D. T. Tamura, “Propagation and detection of radio signals before, during, and after the implosion of a large sports stadium (Veterans’ Stadium in Philadelphia),” Nat. Inst. Stand. Tech. Note 1541, Oct. 2005. [4] C. L. Holloway, G. Koepke, D. Camell, K. A. Remley, S. A. Schima, M. McKinley, and R. T. Johnk, “Propagation and detection of radio signals before, during, and after the implosion of a large convention center,” Nat. Inst. Stand. Tech. Note 1542, Jun. 2006. [5] C. L. Holloway, W. F. Young, G. Koepke, K. A. Remley, D. Camell, and Y. Becquet, “Attenuation of radio wave signals coupled into twelve large building structures,” Nat. Inst. Stand. Tech. Note 1545, Apr. 2008. [6] K. A. Remley, G. Koepke, C. L. Holloway, C. Grosvenor, D. Camell, J. Ladbury, R. T. Johnk, D. Novotny, W. F. Young, G. Hough, M. D. McKinley, Y. Becquet, and J. Korsnes, “Measurements to support modulated-signal radio transmissions for the public-safety sector,” Nat. Inst. Stand. Tech. Note 1546, Aug. 2008. [7] [Online]. Available: http://www.nist.gov/eeel/electromagnetics/rf_ fields/wireless.cfm [8] [Online]. Available: http://www.antd.nist.gov/seamlessandsecure. shtml [9] D. Molkdar, “Review on radio propagation into and within buildings,” Inst. Elect. Eng. Proc.-H, vol. 38, no. 1, pp. 61–73, Feb. 1991. [10] S. R. Sounders, K. Kelly, S. M. R. Jones, M. Dell-Anna, and T. J. Harrold, “The indoor-outdoor radio environment,” Inst. Elect. Eng. Elect. Comm. Eng. J., pp. 249–261, Dec. 2000. [11] P. Papazian, “Basic transmission loss and delay spread measurements for frequencies between 430 and 5750 MHz,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 694–701, Feb. 2005.

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[12] E. S. Sousa, V. M. Jovanovic, and C. Daigneault, “Delay spread measurements for the digital cellular channel in Toronto,” IEEE Trans. Veh. Tech., vol. 43, no. 4, pp. 837–847, Nov. 1994. [13] G. Calcev et al., “A wideband spatial channel model for system-wide simulations,” IEEE Trans. Veh. Tech., vol. 56, no. 2, pp. 389–403, Mar. 2007. [14] J. R. Hampton, N. M. Merheb, W. L. Lain, D. E. Paunil, R. M. Shurfor, and W. T. Kasch, “Urban propagation measurements for ground based communication in the military UHF band,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 644–654, Feb. 2006. [15] R. J. C. Bultitude, Y. L. C. de Jong, J. A. Pugh, S. Salous, and K. Khokhar, “Measurement and modeling of emergency vehicle-toindoor radio channels and prediction of IEEE 802.16 performance for public safety applications,” IET Comm., vol. 2, no. 7, pp. 878–885, Aug. 2008. [16] W. F. Young, C. L. Holloway, G. Koepke, D. Camell, Y. Becquet, and K. A. Remley, “Radio wave signal propagation into large building structures—Part 1: CW signal attenuation and variability,” IEEE Trans. Antennas Propag., vol. 58, no. 4, p. , Apr. 2010. [17] T. L. Duomi, “Spectrum considerations for public safety in the United States,” IEEE Comm. Mag., vol. 44, no. 1, pp. 30–37, Jan. 2006. [18] K. Balachandran, K. C. Budka, T. P. Chu, T. L. Duomi, and J. H. Kang, “Mobile responder communication networks for public safety,” IEEE Comm. Mag., vol. 44, no. 1, pp. 56–64, Jan. 2006. [19] B. Davis, C. Grosvenor, R. T. Johnk, D. Novotny, J. Baker-Jarvis, and M. Janezic, “Complex permittivity of planar building materials measured with an ultra-wideband free-field antenna measurement system,” Nat. Inst. Stand. Technol. J. Res., vol. 112, no. 1, pp. 67–73, Jan.–Feb. 2007. [20] M. Riback, J. Medbo, J. Berg, F. Harryson, and H. Asplund, “Carrier frequency effects on path loss,” in Proc. 63rd IEEE Vehic. Technol. Conf., 2006, vol. 6, pp. 2717–2721. [21] K. A. Remley, G. Koepke, C. Grosvenor, R. T. Johnk, J. Ladbury, D. Camell, and J. Coder, NIST Tests of the wireless environment in automobile manufacturing facilities Nat. Inst. Stand. Tech. Note 1550, Oct. 2008. [22] J. C.-I. Chuang, “The effects of time delay spread on portable radio communications channels with digital modulation,” IEEE J. Sel. Areas Comm., vol. SAC-5, no. 5, pp. 879–889, Jun. 1987. [23] Y. Oda, R. Tsuchihashi, K. Tsuenekawa, and M. Hata, “Measured path loss and multipath propagation characteristics in UHF and microwave frequency bands for urban mobile communications,” in Proc. 53rd IEEE Vehic. Technol. Conf., May 2001, vol. 1, pp. 337–341. [24] J. A. Wepman, J. R. Hoffman, and L. H. Loew, Impulse response measurements in the 1850–1990 MHz band in large outdoor cells NTIA Rep. 94-309, Jun. 1994. [25] IEEE Standard for Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications: High-Speed Physical Layer in the 5 GHz Band, IEEE Standard 802.11a-1999. [26] IEEE Standard for Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications: Higher-Speed Physical Layer Extension in the 2.4 GHz Band, IEEE Standard 802.11b-1999. [27] M. D. McKinley, K. A. Remley, M. Myslinski, J. S. Kenney, D. Schreurs, and B. Nauwelaers, “EVM calculation for broadband modulated signals,” in 64th ARFTG Conf. Dig., Orlando, FL, Dec. 2004, pp. 45–52. [28] K. A. Remley, G. Koepke, C. L. Holloway, C. Grosvenor, D. G. Camell, and R. T. Johnk, “Radio communications for emergency responders in high-multipath outdoor environments,” in Proc. Int. Symp. Advanced Radio Tech., Boulder, CO, Jun. 2008, pp. 106–111. [29] K. A. Remley, G. Koepke, C. L. Holloway, D. Camell, and C. Grosvenor, “Measurements in harsh RF propagation environments to support performance evaluation of wireless sensor networks,” Sensor Rev., vol. 29, no. 3, 2009. [30] N. B. Carvalho, K. A. Remley, D. Schreurs, and K. G. Gard, “Multisine signals for wireless system test and design,” IEEE Microw. Mag., pp. 122–138, Jun. 2008. [31] K. A. Remley, G. Hough, G. Koepke, D. Camell, C. Grosvenor, and R. T. Johnk, “Wireless communications in tunnels for urban search and rescue robots,” in Proc. Performance Metrics for Intelligent Systems Workshop (PerMIS), Aug. 2008, pp. 236–243, NIST Special Publication 1090.

Kate A. Remley (S’92–M’99–SM’06) was born in Ann Arbor, MI. She received the Ph.D. degree in electrical and computer engineering from Oregon State University, Corvallis, in 1999. From 1983 to 1992, she was a Broadcast Engineer in Eugene, OR, serving as Chief Engineer of an AM/FM broadcast station from 1989–1991. In 1999, she joined the Electromagnetics Division, National Institute of Standards and Technology (NIST), Boulder, CO, as an Electronics Engineer. Her research activities include metrology for wireless systems, characterizing the link between nonlinear circuits and system performance, and developing methods for improved radio communications for the public-safety community. Dr. Remley was the recipient of the Department of Commerce Bronze and Silver Medals and an ARFTG Best Paper Award. She is currently the Editor-inChief of IEEE Microwave Magazine and Chair of the MTT-11 Technical Committee on Microwave Measurements.

Galen Koepke (M’94) received the B.S.E.E. degree from the University of Nebraska, Lincoln, in 1973 and the M.S.E.E. degree from the University of Colorado at Boulder, in 1981. He is an NARTE Certified EMC Engineer. He has contributed, over the years, to a wide range of electromagnetic issues. These include measurements and research looking at emissions, immunity, electromagnetic shielding, probe development, antenna and probe calibrations, and generating standard electric and magnetic fields. Much of this work has focused on TEM cell, anechoic chamber, open-area-test-site (OATS), and reverberation chamber measurement techniques along with a portion devoted to instrumentation software and probe development. He now serves as Project Leader for the Field Parameters and EMC Applications program in the Radio-Frequency Fields Group. The goals of this program are to develop standards and measurement techniques for radiated electromagnetic fields and to apply statistical techniques to complex electromagnetic environments and measurement situations. A cornerstone of this program has been National Institute of Standards and Technology (NIST), work in complex cavities such as the reverberation chamber, aircraft compartments, etc.

Christopher L. Holloway (S’86–M’92–SM’04– F’10) was born in Chattanooga, TN, on March 26, 1962. He received the B.S. degree from the University of Tennessee at Chattanooga in 1986, and the M.S. and Ph.D. degrees from the University of Colorado at Boulder, in 1988 and 1992, respectively, both in electrical engineering. During 1992, he was a Research Scientist with Electro Magnetic Applications, Inc., Lakewood, CO. His responsibilities included theoretical analysis and finite-difference time-domain modeling of various electromagnetic problems. From fall 1992 to 1994, he was with the National Center for Atmospheric Research (NCAR), Boulder. While at NCAR his duties included wave propagation modeling, signal processing studies, and radar systems design. From 1994 to 2000, he was with the Institute for Telecommunication Sciences (ITS), U.S. Department of Commerce in Boulder, where he was involved in wave propagation studies. Since 2000, he has been with the National Institute of Standards and Technology (NIST), Boulder, CO, where he works on electromagnetic theory. He is also on the Graduate Faculty at the University of Colorado at Boulder. Dr. Holloway was awarded the 2008 IEEE EMC Society Richard R. Stoddart Award, the 2006 Department of Commerce Bronze Medal for his work on radio wave propagation, the 1999 Department of Commerce Silver Medal for his work in electromagnetic theory, and the 1998 Department of Commerce Bronze Medal for his work on printed circuit boards. His research interests include electromagnetic field theory, wave propagation, guided wave structures, remote sensing, numerical methods, and EMC/EMI issues. He is currently serving as Co-Chair for Commission A of the International Union of Radio Science and is an Associate Editor for the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. He was the Chairman for the Technical Committee on Computational Electromagnetics (TC-9) of the IEEE Electromagnetic Compatibility

REMLEY et al.: RADIO-WAVE PROPAGATION INTO LARGE BUILDING STRUCTURES—PART 2: CHARACTERIZATION

Society from 2000–2005, served as an IEEE Distinguished lecturer for the EMC Society from 2004–2006, and is currently serving as Co-Chair for the Technical Committee on Nano-Technology and Advanced Materials (TC-11) of the IEEE EMC Society.

Chriss A. Grosvenor (M’91) was born in Denver, CO. She received the B.A. degree in physics and M.S. degree in electrical engineering from the University of Colorado, Boulder, in 1989 and 1991, respectively. In 1990, she joined the Electronics and Electrical Engineering Laboratory, National Institute of Standards and Technology (NIST), Boulder, CO. Her work at NIST includes design and analysis of large stripline cavities for materials measurements as well as a large 77 mm diameter coaxial system and a 60 GHz Fabry-Perot resonator. She has worked in the noise temperature project and assembled the 1 to 12 GHz automated systems and repackaged the 30 and 60 MHz noise temperature measurement systems. She joined the time-domain project in 2002 and has worked on measurements of shielding effectiveness of aircraft including the orbiter Endeavour. She has authored papers in all of these technical areas. Ms. Grosvenor was awarded the Bronze Medal for her work with the orbiter Endeavour.

Dennis Camell (M’94) received the B.S. and M.E. degrees in electrical engineering from the University of Colorado, Boulder, in 1982 and 1994, respectively. From 1982 to 1984, he worked for the Instrumentation Directorate, White Sands Missile Range, NM. Since 1984, he has worked on probe calibrations and EMI/EMC measurements with the Electromagnetics Division, National Institute of Standards and Technology (NIST), Boulder, CO. His current interests are measurement analysis (including uncertainties) in various environments, such as OATS and anechoic chamber, and development of time domain techniques for use in EMC measurements and EMC standards. He is involved with several EMC working standards committees and is chair of ANSI ASC C63 SC1.

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John Ladbury (M’92) was born Denver, CO, 1965. He received the B.S.E.E. and M.S.E.E. degrees in signal processing from the University of Colorado, Boulder, in 1987 and 1992, respectively. Since 1987 he has worked on EMC metrology and facilities with the Radio Frequency Technology Division, National Institute of Standards and Technology (NIST), Boulder, CO. His principal focus has been on reverberation chambers, with some investigations into other EMC-related topics such as time-domain measurements and probe calibrations. He was involved with the revision of RTCA DO160D and is a member of the IEC joint task force on reverberation chambers. Mr. Ladbury has received three “best paper” awards at IEEE International EMC symposia over the last six years.

Robert T. Johnk, photograph and biography not available at the time of publication.

William F. Young (M’06) received the B.S. degree in electronic engineering technology from Central Washington University, Ellensburg, in 1992, the M.S. degree in electrical engineering from Washington State University at Pullman, in 1998, and the Ph.D. degree from the University of Colorado at Boulder, in 2006. Since 1998, he has worked for Sandia National Laboratories in Albuquerque, NM, where he is currently a Principal Member of the Technical Staff. His work at Sandia includes the analysis and design of cyber security mechanisms for both wired and wireless communication systems used in the National Infrastructure and the Department of Defense. He has also been a Guest Researcher at the National Institute of Standards and Technology in Boulder, CO, from 2003 to 2009, and is working on improving wireless communication systems for emergency responders. His current research interests are in electromagnetic propagation for wireless systems, and the impacts of the wireless channel on overall communication network behavior.

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Numerical Investigations of and Path Loss Predictions for Surface Wave Propagation Over Sea Paths Including Hilly Island Transitions Gökhan Apaydin, Member, IEEE, and Levent Sevgi, Fellow, IEEE

Abstract—Surface wave propagation along multi-mixed-paths with irregular terrain over spherical Earth in two-dimension (2D) is discussed. For the first time in the literature, sea-land-sea (island) transition problem including non-flat (hilly) islands is investigated systematically. A finite element method based multi-mixed path surface wave virtual propagation predictor tool FEMIX is developed for this purpose. FEMIX, tested and calibrated against analytical ray-mode reference models accommodated with the Millington curve fitting approaches, is shown to be capable of modeling propagation along sea surface having multiple hilly islands over MF (300 kHz – 3 MHz) and HF (3 – 30 MHz) bands. Index Terms—Cauchy-type boundary condition, finite difference method, finite element method (FEM), hilly island, impedance boundary condition, island transition, Millington effect, mixed-path propagation, path loss, ray-mode methods, surface waves.

I. INTRODUCTION

S

URFACE wave propagation has long been one of the important options for medium and long range communication at MF (0.3–3 MHz) and HF (3–30 MHz) bands. The traditional MF/HF broadcast and communication systems, HF and VHF radars, as well as novel digital systems such as intelligent transportation systems, Digital Radio Mondiale (DRM), etc., necessitate understanding, analytical and/or numerical modeling of, and developing reliable simulators for, the propagation characteristics over lossy Earth’s surface along realistic propagation paths including irregular terrain profiles through inhomogeneous atmosphere. Early analytical formulations were mostly based on ray-mode representations in two-dimension (2D), over smooth spherical Earth through homogeneous (standard) atmosphere [1]–[17]. Numerical approaches of the last couple of decades have extended their application area to model the effects of vertical and/or horizontal refractivity variations and irregular terrain profiles onto the surface wave propagation. An important surface wave propagation phenomena still needs to be further discussed is the mixed path propagation Manuscript received June 08, 2009; revised September 08, 2009; accepted September 15, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. G. Apaydin is with the Electrical and Electronics Engineering Department, Zirve University, Gaziantep 27260, Turkey (e-mail: [email protected]. tr). L. Sevgi is with Electronics and Communications Engineering Department, Dogus University, Istanbul, Turkey (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2041169

problem. Surface wave path loss prediction over realistic propagation paths having land-sea, sea-land, and sea-land-sea (island) transitions including land irregularities (hilly islands) is still a challenging problem. Available analytical models, recommended by the ITU (International Telecommunication Union), use the surface wave representation of Norton [2] (which is good within the line-of-sight (LOS)) or the surface guided mode model of Wait [8] (which is better beyond the LOS, in the shadow region), or their hybrid forms (see, for example, [18]–[20] for the historical reviews and interesting hybridization approaches). The Millington curve fitting approach, endorsed in the ITU recommendations [1], accounts for multi-mixed but smooth path propagation scenarios [9], [10] therefore it cannot take terrain irregularities into account. In his 1998 (last) paper [16], J. R. Wait discussed the ancient and modern history of groundwave propagation modeling. Starting from Nicholas Tesla who stated that waves were bounded in some fashion by the Earth-air boundary, Wait discussed early analytical contributions of Zenneck and Sommerfeld and mathematical derivations of the surface waves. He then summarized Norton and Millington contributions and mixed-path propagation effects. Finally, he ended up with outlining some experimental studies of Millington and King confirming all analytical approximate models. Unfortunately, these are all valid for the smooth spherical Earth and cannot take terrain irregularities (e.g., island heights) into account. There are only a few attempts in modeling mixed-path transitions with terrain irregularities. Furutsu developed a mathematical model based on the Green’s function representation [5], (but island heights are non-physical) Monteath [11] used electromagnetic (EM) compensation theorem, and Ott [12] formulated his model via the Volterra integral equation technique. To the best of authors’ knowledge there are only a few data showing the path losses versus range along mixed-paths including hilly islands (see, for example, [13] or [17, p. 40, Fig. 3.17]). Their 2002 paper [21] dedicated to J. R. Wait, L. Sevgi, F. Akleman, and L. B. Felsen reviewed groundwave propagation modeling strategies from early problem-matched analytical formulations to direct frequency- and time-domain numerical techniques. L. Sevgi also summarized strategies and challenges in groundwave propagation modeling in his paper [22] in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION Special Issue dedicated to L. B. Felsen. A few, highly effective virtual propagation prediction tools have also been presented together with these studies [23]–[26]. These virtual tools were shown to handle multi-mixed path propagation scenarios but

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APAYDIN AND SEVGI: NUMERICAL INVESTIGATIONS OF AND PATH LOSS PREDICTIONS

the investigation of signal attenuation due to the hilly islands along sea/ocean paths was missing. L. Sevgi and F. Akleman also introduced finite-difference time-domain (FDTD)- and transmission line matrix (TLM)-based groundwave propagators based on the sliding window approach [23], [24] which are capable of handling irregular terrain and atmospheric refractivity effects as well as impedance (Cauchy)-type boundary condition (BC). They are applicable for short ranges and require intensive signal processing techniques in order to extend the propagation range. Yet, surface wave propagation over multi-mixed irregular paths has not been implemented with these methods. T. Ishihara and his group recently conducted several measurements about surface wave mixed-path propagation effects in the MF and lower HF bands in the Kanto area including the city of Tokyo and the Tokyo bay [27], [28]. They showed and confirmed that the surface wave mixed-path theory agrees very well with the experimental results. Nevertheless, height irregularities were not taken into account in those studies, either. Among the others, the parabolic equation (PE) technique [29]–[34] was the most attractive groundwave propagator due to its robustness, low memory requirements and fast implementation. Leontovich and Fock [6] are pioneers who described the use of PE for EM wave propagation in a vertically inhomogeneous medium. However, this approach has become famous after the introduction of the Fourier split step (SSPE) technique by Tappert [30] who implemented acoustic wave propagation in the ocean. The SSPE algorithm was then applied to the scalar PE associated with EM propagation through Earth – troposphere waveguides. Since then, the PE technique has been improved, accommodated with many auxiliary tools and applied to variety of complex propagation problems. Although its multi-mixed path capability has been shown in a few PE-based publications (see, for example, [33] for the discrete mixed Fourier transform – DMFT), this method has been widely used in VHF and above (basically at microwaves) especially to investigate wave attenuation, ducting, anti-ducting conditions due to daily, monthly, as well as yearly atmospheric variations. PE-based wave attenuation (systematic) modeling due to hilly island transitions along multi-mixed sea/ocean paths at lower frequencies is still missing. In his 2007 paper, P. D. Holm presented [35] a wide-angle parabolic wave equation solution, which is based on the shift-map and finite-difference techniques. He showed that finite-difference based PE solution (allowing irregular terrain profiles), where the standard PE is modified into the so-called Claerbout equation, take propagation angles up to 40 –45 from the paraxial direction into account effectively. Both firstand second-order solutions with respect to the terrain slope effect can be modeled in this approach. Unfortunately, Holm’s study includes only wedge-type homogeneous terrain irregularity applications and comparisons against geometrical theory of diffraction (GTD) solutions. The finite element method (FEM) [36] has also been used in developing PE based numerical propagators. Initially, FEM based PE models were applied to underwater acoustic propagation prediction problems (e.g., see [37]). A few FEM based PE models have appeared in EM propagation modeling for the last

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decade [38]–[40], but a systematic investigation with characteristic applications and reliable comparisons against SSPE was given in [41]. There, FEM-based 2D groundwave propagator LINPE was developed and tested/calibrated against analytical exact (reference) data as well as the SSPE propagator. FEMbased surface wave propagation package SURFPE was also introduced [42], which accounts for the signal attenuation along multi-mixed path propagation paths through homogeneous atmosphere. The SURFPE package was tested against ITU curves as well as the Millington curve fitting method. All those tests showed that FEM-based propagator is highly capable of predicting groundwave attenuation caused by atmospheric refractivity changes and irregular, lossy terrain along multi-mixed propagation paths. This paper discusses surface wave attenuation in the MF/HF bands due to successive isolated islands aligned along sea/ocean propagation paths. This is the first time in the literature where systematic investigation of hilly island effects is illustrated, its physical phenomenon is described through 3D visualizations, data is presented for possible comparisons. A novel FEM-based numerical surface wave propagation prediction simulator FEMIX is developed for this purpose. FEMIX, coded in Matlab can handle surface wave propagation over spherical lossy Earth’s surface with multi-mixed paths including irregular terrain profiles (i.e., island heights). Numerical investigations of and path loss predictions for the surface wave propagation over sea paths including hilly island transitions necessitate a model that accounts for irregular terrain effects, impedance(Cauchy-) type BCs along multi-mixed path propagation segments and atmospheric variations which are all incorporated in the FEMIX predictor. It should be noted that Helmholtz equation in 2D with appropriate boundary conditions models a two-way, full-wave propagation problem which takes into account both forward and backward propagated waves. It can numerically be discretized and solved either of the finite difference, finite element or method of moments approaches which yields a closed form matrix system with a number of unknowns equal to the number of equations. Unfortunately, their application is limited to low frequencies and very short ranges because of huge matrix operations. The PE, on the other hand, models a one-way, forward propagation problem and can numerically be discretized and solved via step by step marching iterative representations from source to receiver either of the finite difference, finite element or method of moments approaches as well as discrete Fourier transformation algorithms. This is why PE algorithms are widely used in propagation prediction problems. The PE method has also been used to represent two-way propagation problems [43]–[47]; all of which are based on the Fourier split step algorithms. To the best of authors’ knowledge, there are no FEM based two-way PE propagation models. The FEM modeling of the two-way propagation problems is beyond the scope of this manuscript. II. THE 2D PARABOLIC WAVE EQUATION The standard parabolic wave equation may also be obtained from 2D Helmholtz equation by separating the rapidly varying phase term to obtain an amplitude factor, which varies slowly in range when the direction of propagation is predominantly along

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Fig. 1. The 2D surface wave propagation scenario: Source is injected as a . Homogeneous atmosphere surface coupled mode. Maximum height is X with the inclusion of Earth’s curvature (i.e., +117 M Unit=km) is assumed ]. Artificial loss is introduced between [X -2X ]. A between [0-X window function is also used to eliminate artificial reflections from the upper boundary. Paraxial boundary and maximum terrain slope are also shown. The ] in height and [0-Z ] in range. propagation region covers a 2D area [0-2X

the -axis (i.e., paraxial direction), and under dependence, is given as

time

(1) where and corresponds to electric and magnetic field components for horizontal and vertical polarizations, respectively, depending on the type of the problem, and and stand for transverse (i.e., height above ground) and longitudinal (i.e., the direction of propagation) is the refractive coordinates, respectively (see Fig. 1); index, and is the free space wavenumber. Note that, vertically polarized surface waves propagate above the Earth’s surface without significant attenuation and establish the link between the transmitter and receiver at LF/MF/HF bands at long ranges beyond the LOS well into the diffraction (shadow) region. On the other hand, horizontally polarized surface waves attenuate very rapidly as the range increases, therefore only vertically polarized waves are used for the surface wave propaor ). gation in these frequencies (i.e., The 2D propagation scenario is completed by choosing appropriate transverse and longitudinal BCs. With the flat-Earth assumption, the transverse BC of ground (along ) is (2) Here , become constants for perfectly electrical conducting (PEC) surface and ( , 2) results in Dirichlet (horizontal polarization) and Neumann (vertical polarization) BCs, respectively. The Cauchy type BC is intro, and , duced with for horizontal and vertical polarization, in terms of the conductivity respectively, ( ) and relative permittivity ( ) of the ground at range . The radiation BC along is (3)

Although PE in 2D describes one-way propagation and cannot take backscattered waves into account, this is not a serious restriction for propagation engineers who investigate waves emanating from a transmit antenna and reaching a receiver. It should be noted that, the solution of (1) in 2D yields waves that are attenuated by the square root of distance as they propagate away from the transmitter (i.e., cylindrical wave spread for an infinite-length line source which allows the reduction of the 3D wave equation into 2D). Therefore, the user should divide the field strength results by the square root of distance in order to obtain path loss versus range variations in a realistic (3D) environment [16]. There is no natural boundary upward in height therefore waves propagating upwards are either go to infinity or bent down because of refractivity variations. In either case the vertical numerical computation space must be abruptly terminated at a certain height. This termination introduces artificial reflections therefore proper treatment is required above a height of interest and these artificial downward reflections must be eliminated or attenuated for a level which is much less than the lowest signal levels. Irregular terrain modeling can be implemented in the PE via several different mathematical approaches and it is possible for the user to choose the appropriate one for the problem. The staircase approximation of the range-dependent terrain profile is the best and easiest in order to handle the DBC since neither analytical terrain function nor slope values are required; only the terrain height at each range step is needed. When terrain height changes corner diffraction is ignored and the field is simply set to zero on vertical nodes falling on and inside the terrain. The NBC necessitates use of the coordinate transformation [33]. The CBC over irregular terrain profiles may be handled via the coordinate transformation plus adding surface parameters. Horizontal and/or vertical refractivity variations (i.e., ) which cause surface and/or elevated duct formations may be implemented in the PE model. Moreover, the Earth’s curvature can be included by replacing with where is the effective Earth’s radius ( ). It is customary to use refrac( ) or modified refractivity tivity ( ) with the height given in kilometers. is dimensionless, but is measured in “N units” for convenience [6]. For the standard atmosphere decreases by about 40 N increases by about 117 M unit/km (although unit/km while standard atmosphere defines exponentially decreasing refractive index, this could be approximated as being linear for low altitudes). Based on the assumptions and approximations made there, the validity range and accuracy limits of the PE-based propagators should be well-understood. First of all, backscattered fields cannot be taken into account in the PE model (second derivative with respect to range is eliminated). This means, interaction of forward scattered waves with their backscattered contributions caused by the discontinuities along the propagation path (e.g., in front of hills and along valleys) cannot be observed in the PE propagators. Secondly, the results of the PE propagators are not accurate in short regions at high altitudes since these

APAYDIN AND SEVGI: NUMERICAL INVESTIGATIONS OF AND PATH LOSS PREDICTIONS

regions violate the requirement of being inside the paraxial region. Roughly speaking, the range of the observer should be at least five to ten times greater than the heights of the transmitter and the observer. For example, for a transmitter located 500 m above the ground the results of the PE propagators can be accurate at ranges beyond 2.5 km – 5 km, which satisfy a vertical propagation angle of less than 5 to 10 (vertical coverage may be extended up to 30 –40 degrees by using wide-angle PE models). III. A NOVEL FEM-BASED MULTI-MIXED PATH SURFACE WAVE PROPAGATOR The idea of FEM-based formulation of the parabolic equation is to divide the transverse domain between ground and selected maximum height into sub domains (called elements), use approximated field values at the selected discrete nodes in the vertical domain, and propagate longitudinally by the application of Crank-Nicholson technique, based on improved Euler method [41]. Multiplying (1) by a smooth test function while considering BC given in (2) and integrating from to we have (4) shown at the bottom of the page, and then using integrating by parts rule yields (5), shown at the bottom of the page. The last two terms of (5) should be taken into consideration according to the BCs. The open boundary upward in height can be modeled by using an artificial lossy layer with the help of Hanning window [33] in order to eliminate rewith the apflection coming from upward. Replacing proximated solution

(6)

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is the number of elements, indicates the coefwhere ficients of unknown function with the help of linear piecewise Lagrange polynomials as (7a) (7b) where stands for the elements between nodes and and with for , 2; (5) replacing test function becomes (8), shown at the bottom of the page. with Kronecker for , for ); then we delta ( have (9) at the bottom of the page. The matrix form of (9) is (10) or for

,

, 2, and

, 2 with (11) (12) (13)

is to be kept in mind with boundary conditions as for each step in , so on. The matrix provides to determine the ground properties such as land or sea for the FEM and the matrix is determined once the refractivity index is given. For example in terms of and

(4)

(5)

(8)

(9)

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the slope of modified refractivity ( ) in M unit/km, the square of modified refractivity index can be approximated as (14) and these elemental matrices between nodes tained as

and

are ob-

(15) (16) (17) shows the grid width. Using where Crank-Nicholson approximation based on improved Euler where is the number of nodes in method for the horizontal domain, the new coefficients for the next step are (18) with the given land/sea properties for each step and multiplying and eliminate the derivative terms by using differential by (10), one obtains

(19) which yields an unconditionally stable system and accurate . The coefficients method with the discretization error at are generated from Gaussian of the initial field antenna pattern specified by its height, beamwidth, and tilt. Although Crank-Nicholson gives fast solution, it should be noted that this method has disadvantages since oscillation . occurs for large The implementation of different BCs can be explained as follows. Assume there are 5 nodes vertically (i.e., fields are speciand the matrices fied at 5 discrete heights). Then become

(20)

) in order to satisfy The first coefficient must be zero ( the DBC on Earth’s surface. Therefore, the first node and first rows/columns of the matrices are not required. The required coare then computed using (20) which efficients . The initial coefficients are not known in the results in . NBC but the implementation of this BC also yields is replaced In the numerical implementation of the CBC, (i.e., the last term in (9)). with IV. FEMIX AND NUMERICAL IMPLEMENTATION The FEM-based surface wave propagator uses the initial field , generated from a Gaussian antenna pattern specified at by its height, vertical beamwidth, and the tilt angle. Note that, Crank-Nicholson based longitudinal marching uses the NBC at in (19). each range and the CBC is satisfied with the use of Although not necessary, the antenna is located on the surface to better couple surface waves in a short range. Crank-Nicholson and approach is fast but, both height and range step sizes, , respectively, should be chosen as small as necessary to overcome numerical oscillation problems (FEM-based discrete PE equations can be found in [41]). The procedure and steps in applying the FEMIX propagator are as follows (see Fig. 1). ] is injected into the • An initial height profile for [0FEMIX algorithm. A 1D complex array is used to build , where ( ) is the the initial profile sampled height coordinate. This array contains amplitude and phase of the fields at each height. • The initial profile is replaced with the FE basis functions with the help of linear piecewise Lagrange polynomials ] (an [36] and the vertical region is extended to [0artificial lossy layer is inserted between [ ]). • This profile is propagated longitudinally from to using Crank-Nicholson approach. Using the new height profile as the initial profile for the next step and applying the same procedure for the second time yields the height . The procedure profile at the second range step is applied repeatedly and vertical field profiles are stored at each range step until the propagator reaches the desired . range • At each range step the BC on the surface is satisfied manually. The DBC and NBC are satisfied without using the . In addition, the first column and row of mamatrix trices should be eliminated for the DBC since the initial node is always 0. On the other hand, the effect of boundary for the condition is included while using the matrix CBC. • The problem at hand has a vertically semi-open propagation region therefore an abrupt truncation is required at certain height, which means strong artificial reflections will occur if not taken care of [33]. These non-physical reflections can be removed by using a windowing function, perfectly matched layer (PML) termination, or locating an artificial absorbing layer above the height of interest. Artificial layer plus windowing is applied in FEMIX. • Waves detached from the surface are absorbed in the artificial lossy layer as they propagate. In order to increase absorption and eliminate artificially reflections a window function is also applied to the vertical field profile at

APAYDIN AND SEVGI: NUMERICAL INVESTIGATIONS OF AND PATH LOSS PREDICTIONS

each range step. Up to 40 dB suppression of these undesired, non-physical reflections from the top layer has been achieved for the parameters used in the examples presented below. • Coordinate transformation is used to handle irregular terrain effects [32]–[35]. Irregular terrain can be introduced or a terrain file either via a terrain height function which contains range height data pairs. The terrain effect is then included by the modification of the refractivity and BCs which includes first and second derivative of the terrain function (analytically or numerically). Note that, only forward scattering effects are taken into account in the PE with this implementation. V. MULTI-MIXED PATH SURFACE WAVE PROPAGATION AND AVAILABLE ANALYTICAL MODELS Available analytic models for the surface wave propagation, recommended by the ITU, are the ray-optical plus surface wave model of Norton [2] and the surface guided mode model of Wait [8]. The Millington curve fitting approach, endorsed and used in the ITU recommendations [1]), is used to account for the multi-mixed-path propagation effects [20]. The surface wave path for a given transmitter receiver separation, in terms of transmit and receive powers ( and ), is defined as [3]

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(25) Here, is the complementary error function. The funcintroduces an attenuation which depends on the range, tion frequency, and electrical parameters of the ground. At ranges up has a value to 10–100 wavelengths from the transmitter, of nearly unity therefore (23) is called non-attenuated surface wave, causes an exponential decay at medium ranges, and varies inversely with the square of the distance. The effects of the antenna heights can be taken into account as extra losses via height gain functions [20]. B. Wait’s Surface Wave Representation The Wait formulation restructures the spectral integral as a series of normal modes propagating along the Earth’s surface. in terms of height Wait expressed the attenuation function gain functions for the vertical component of electric field [8] under flat-Earth assumption as (26) with the same

given in (23), and the attenuation function

(27)

(21) , is the field strength for the where transmitter received power at an arc distance . For a (i.e., for a short electric dipole with a dipole moment of Am), the path loss is calculated from

and are the source and observation heights, , . Here, is the effective Earth’s radius and is the refractive index at Earth’s surface. The suris given as face impedance where

(22)

(28)

The path loss prediction is then reduced to calculating/predicting field strength at a given distance.

For the standard atmosphere with the inclusion of the Earth’s curvature, the transverse mode functions are solutions of the Airy equation [25]

A. Norton Surface Wave Contribution The Norton formulation [2] extracts a ray-optical asymptotic approximation from a wavenumber spectral integral representation under the standard atmosphere assumption. Since the space wave cancels out at long ranges (and/or transmitter/receiver on or close to the ground) it is sufficient to use Norton’s vertical field component only under flat-Earth assumption [20], [22] (23) is the reference where , field over the PEC flat-Earth at the distance , is the dipole moment, , is relative complex dielectric constant of the ground, and . The surface wave attenuation function is defined as (24)

(29) which satisfy the CBC on the surface (30) and the radiation condition at

.

C. The Millington Curve Fitting Method For the multi-mixed path propagation prediction the Millington curve-fitting method endorsed by the ITU and described in [16] is used. The recursive equations of the Millington Method [3] are (31)

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(32) (33)

(34) and are the fields along direct (source-to-rewhere ceiver) and reverse (receiver-to-source) paths and , respecrepresents field strength at a range segment tively. Here, over the homogeneous medium. The total field along the multi-mixed propagation path at the receiver is then obtained via the interpolation of the direct and reverse electric fields as . VI. CHARACTERISTIC SCENARIOS AND CANONICAL TESTS FEMIX is a surface wave path loss prediction package that can be used for variety of complex propagation scenarios. This section is devoted to these scenarios. The aim is multifold; (i) to show the efficiency and accuracy of the FEMIX package, (ii) to visualize the surface wave propagation along multi-mixed paths including irregular lossy terrain effects, (iii) for the first time in the literature, to systematically investigate the effects of hilly islands, and (iv) to present curves which may serve comparison data for the other propagation modelers and/or measurers. The first example is given in Fig. 2. Here, a 40 km homo, geneous (sea) path with electrical parameters: is taken into account. A surface wave is directly coupled onto the surface at the initial range and is propagated longitudinally via the FEMIX propagator. The initial surface wave is a half-Gaussian beam with its maximum on the surface. The top figure shows the signal versus range/height variations in 3D for the visualization of the physical phenomena occurring there. At the bottom, path loss versus range at 10 MHz is shown. The flow of the wave energy is shown by thick curved arrows. Observe that most of the energy detaches from Earth’s surface as the wave propagates longitudinally (caused because of the Earth’s curvature) and only a small portion follows the surface as the surface wave. Excellent agreement between Norton-Wait hybrid method and FEMIX-predicted data is clearly observed. The discrepancy at short ranges is because of the artificial location of the surface wave, but the main reason is the introduction of the Earth’s curvature in the FEMIX from the initial range. In reality, Earth’s curvature may be negligible at short ranges (up to a few kilometers at MF/HF) and the flat-Earth assumption the attenuation function is valid. This is also observed in given in (24). This is tested in the FEMIX. Standard atmosphere for the first few kilometers, and 117 is taken as M unit/km thereafter, and the discrepancy at short ranges is observed to remove. Second example in Fig. 3 belongs to a 40 km homogeneous , having path with electrical parameters: a 10 km long, 250 m high Gauss-shaped hill. Signal versus range/height 3D plot produced with FEMIX is given on the top figure. Path loss versus range at 10 MHz is given at the bottom.

= 5S m

= 80

= , : (Top) 3D Fig. 2. A 40 km homogeneous path with  Signal versus range/height produced with FEMIX, (bottom) path loss versus range at 10 MHz. Solid: Millington; dashed: FEMIX. Surface wave is directly coupled onto the sea surface. Basic wave energy flow is shown by thick curved arrows.

= 5S m

= 80

Fig. 3. A 40 km homogeneous path with  having = , a 10 km long, 250 m high Gauss-shaped hill: (top) 3D Signal versus range/height produced with FEMIX, (bottom) path loss versus range at 10 MHz. Solid: Millington; dashed: FEMIX.

Note that the Millington method does not take irregular terrain effects into account (because Norton-Wait analytical formulations are valid only for the smooth spherical Earth), so the difference between the two curves shows the effects of the hill. Surface wave is directly coupled onto the sea surface. Observe that the FEMIX accounts for the attenuation of the surface wave along irregular paths. The surface wave energy accumulates in the front slope of the hill and signal strength increases. A sharp decrease at the back slope of the hill and a slight recovery effect beyond is also observed. Propagation along the same 40 km homogeneous path but with a different shaped (a half-sinusoidal) hill is given in Fig. 4 , , hill (Electrical parameters: , hill ). On top, signal versus range/ height 3D variations produced with the FEMIX is shown. At the bottom, path loss versus range at 10 MHz is plotted. Again, surface wave is directly coupled onto the sea surface. Observe that

APAYDIN AND SEVGI: NUMERICAL INVESTIGATIONS OF AND PATH LOSS PREDICTIONS

= 5S m

= 80

Fig. 4. A 40 km homogeneous path with  = , having a 10 km long, 250 m high half-sinusoidal hill: (Top) 3D signal versus range/height produced with FEMIX, (bottom) path loss versus range at 10 MHz. Solid: Millington; dashed: FEMIX.

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Fig. 6. (Top) 3D Signal versus range/height produced with FEMIX at 10 MHz, (bottom) path loss versus range over a 3-segment 40 km mixed path (a 10 km long island is 15 km away from the transmitter) at 5 MHz, 10 MHz, and 15 MHz : = , ; sea:  = , ). Solid: FEMIX; (island:  dashed: FDMIX.

= 0 002 S m

= 10

= 5S m

= 80

Fig. 7. The 3D Signal versus range/height produced with FEMIX at 10 MHz over a 3-segment 40 km mixed path (a 10 km long, 250 m high Gauss-shaped ; island is 15 km away from the transmitter, island:  : = , sea:  ). = ,

= 5S m

= 5S m

= 80

Fig. 5. A 40 km homogeneous path with  having = , a 10 km long, 250 m high full-sinusoidal hill: (top) 3D signal versus range/height produced with FEMIX, (bottom) path loss versus range at 10 MHz. Solid: Millington; dashed: FEMIX.

half-sinusoidal hill is fully convex up in the front and down at the back so surface wave energy almost fully detaches (without accumulating in the front slope of the hill), and no signal recovery occurs behind the hill. Fig. 5 belongs to the same 40 km homogeneous path propagation but over a full-sinusoidal hill (Electrical parameters: , , hill , hill ). This scenario may represent a tsunami wave on the sea/ocean with these electrical parameters, or a mountain on a propagation path over land if electrical parameters are changed to land values. On top, 3D signal versus range/height plot is shown. Path loss versus range at 10 MHz is presented at the bottom. Observe that full-sinusoidal hill is partially concave/convex at both sides so surface wave energy accumulates in the front slope of the hill and signal strength increases; a sharp decrease at the back slope of the hill and a slight recovery effect beyond is also observed. The visuals of Figs. 3–5 show how energy accumulates in the front slope of the hill. It requires concave/convex irregularity in general to accumulate energy.

= 80

= 0 002 S m

= 10

The first path loss predictions along multi-mixed paths are given in Fig. 6. In this scenario, there is a 10 km-long island at a distance of 15 km from the transmitter. Electrical parameters of , , and , the sea and land are , respectively. On top, signal versus range/height 3D plot produced with the FEMIX at 10 MHz is given. At the bottom, path loss versus range at different frequencies is plotted and compared with the finite difference multi-mixed (FDMIX) path technique of [35]. As observed in these figures, the surface wave detaches from the surface as it propagates because of the Earth’s curvature. A small portion follows the surface as the surface wave. When the surface wave reaches the sealand discontinuity an extra sharp detachment (energy tilt up) occurs. This tilt up explains sharp attenuation first mentioned by Millington. Signal recovery also occurs at the land-sea discontinuity. Very good agreement between the FDMIX and FEMIX result is clearly observed. The discrepancy at short ranges is because of the artificial location of the surface wave. Further investigations are essential before making speculations on which result is more accurate. Fig. 7 belongs to the same 40 km 3-section path with a Gaussian shaped hilly island. The 3D Signal versus range/height plot produced with the FEMIX at 10 MHz is given here. Wave detachment over the first sea path, energy tilt-up at the sea-land discontinuity, and the tilt up when the wave hits the front slope

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Fig. 8. Path loss versus range, produced with FEMIX, over a 3-segment 30 km mixed path (a 10 km long, 250 m high Gauss-shaped island is 15 km away from the transmitter) at 5 MHz, 10 MHz, 15 MHz, and 20 MHz (island:  : = , ; sea:  = , ).

0 002 S m

= 10

=5S m

= 80

=

Fig. 9. Path loss versus range over a 3-segment 44 km mixed path having a 7 km long, Gauss-shaped island at a distance of 28 km away from the transmitter at 10 MHz. This example is given for a comparison against data available in the literature. Solid: Millington; dashed: FEMIX, dots: data read from [17, p. 40, ; : = , Fig. 3.17] produced via Ott’s method (island:  sea:  ). = ,

= 5S m

of the island are clearly observed in the 3D plot. Observe that the FEMIX accounts for the attenuation of the surface wave along the first sea path, the sharp increase in the attenuation at the sea-land discontinuity, signal recovery in the front slope of the island, the additional signal attenuation at the back slope of the island, and finally the signal recovery at the land-sea discontinuity. Path losses versus range over the scenario given in Fig. 7 are given in Fig. 8 for different frequencies. As expected, path loss increases when frequency increases. Observe the sharp attenuation at the sea-land discontinuity and signal recovery at land-sea discontinuity; also the signal accumulation in the front slope of the island, and additional signal attenuation at the back slope of the island. Curves presented in Fig. 8 need verification. A comparison with the data available in the literature is given in Fig. 9 for this purpose. Here, path loss versus range over a 3-segment 44 km mixed path having a 7 km long, 250 m high Gauss-shaped island at a distance of 28 km away from the transmitter at 10 MHz is , ; shown (Electrical parameters: Island: , ). Dots show the data read from [17, p. Sea: 40, Fig. 3.17] produced via the Ott’s method. Very good agreement is shown in the figure. The clarification of whether Ott’s method over estimates or the FEMIX under estimates signal recovery at the front slope of the island needs further investigations and comparisons (even measurements). Another comparison is given in Fig. 10. Here, the FEMIX package is compared against the finite-difference parabolic equation package FDMIX developed using equations given in [35]. The example belongs to path loss versus range variations over a 3-segment 40 km mixed path (a 10 km long, 250 m high Gauss-shaped island is 15 km away from the transmitter) at three different frequencies. Very good agreement between the FEMIX and FDMIX predictions is clearly observed. Note

= 0 002 S m

= 80

= 10

Fig. 10. Path loss versus range over a 3-segment 30 km mixed path having a 10 km long, 250 m high Gauss-shaped island at a distance of 15 km away from the transmitter at 5 MHz, 10 MHz, and 15 MHz. Solid: FEMIX; dashed: FDMIX (island:  ; sea:  = , ). : = ,

= 0 002 S m

= 10

=5S m

= 80

that, these plots are obtained using the same discretization parameters. The discrepancy at long distances behind the island can be removed if parameters are optimized for each package. Path losses versus range over the same scenario given in Fig. 7 for different island heights are given in Fig. 11. As observed, energy accumulation in the front slope and deep signal loss at the back slope of the island hill increase when the hill height increases.

APAYDIN AND SEVGI: NUMERICAL INVESTIGATIONS OF AND PATH LOSS PREDICTIONS

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Fig. 13. Path loss versus range over a 3-segment 40 km mixed path (5 km, 10 km, 15 km, and 20 km long, 250 m high Gauss-shaped islands are 15 km ; away from the transmitter) at 10 MHz (island:  : = , ). Dots: all sea; others: FEMIX. sea:  = ,

=5S m

Fig. 11. Path loss versus range over a 3-segment 30 km mixed path (Gaussshaped islands with different heights are 15 km away from the transmitter) at : = , ; sea:  = , ). 10 MHz (island:  Dots: Millington; others: FEMIX.

= 0 002 S m

= 10

= 5S m

= 80

Fig. 12. 3D Signal versus range/height produced with FEMIX for the surface wave propagation along a 3-segment 40 km mixed path (a 10 km long, different height Gauss-shaped island is 15 km away from the transmitter) at 10 MHz (Island:  ; sea:  = , ). Note that : = , vertical scales are different.

= 0 002 S m

= 10

= 5S m

= 80

The 3D plots for these scenarios are given in Fig. 12 which assists to explain visually the physical phenomenon occurring there. Note that vertical scales are different. Observe surface wave propagation, energy detachment because of the Earth’s curvature, energy tilt-up at the sea-land discontinuity, and energy accumulation in the front slope of the island. The effects of the island length onto the surface wave path loss variations are given in Fig. 13. Here, path loss versus range over a 3-segment 40 km mixed path with a Gauss-shaped hilly islands

= 80

= 0 002 S m

= 10

at 10 MHz is shown (island lengths are 5 km, 10 km, 15 km, , . and 20 km, island Observe that path loss increases when the length of the island increases, as expected. Figs. 14 and 15 belong to the effects of hill-shapes onto the multi-mixed path losses. In Fig. 14, a full-sinusoidal-island is used. In Fig. 15, a half-sinusoidal-island is used. In both figures, on top, 3D Signal versus range/height produced with the FEMIX are given. At the bottom, path losses versus range at 10 MHz are shown. Observe wave detachment over the first sea path, energy tilt-up at the sea-land discontinuity and the tilt up wave hitting the hill of the island. Observe that, full-sinusoidal hilly island causes attenuation with a sharp increase at the sea-land discontinuity, signal recovery in the front slope of the island, the additional signal attenuation at the back slope of the island, and finally the signal recovery at the land-sea discontinuity. But, there is no energy accumulation in the front slope and signal recovery at the back slope for the half-sinusoidal hilly island. Also, path loss over the sea beyond the island is much more for this type of islands. Finally, Figs. 16–18 belong to various multi-island scenarios. Here, results for a 5-segment 50 km long mixed path with various islands are shown. Note that FEM-based PE modeling requires optimization of the discretization parameters for different frequencies. First of all the selection of the initial antenna pattern is important to excite/couple the surface wave. It directly affects the maximum height and the number of height nodes used in PE computations. The better the surface wave coupling the longer the propagation range. This directly affects numerical computations because of the artificial reflections from the top boundary. Secondly, the range step should be optimized. Large steps cause phase errors and introduce non-physical oscillations; small steps increase the computation burden. Optimum parameters used in the presented , at 5 and 10 MHz; figures are: , at 15 and 20 MHz.

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Fig. 14. (Top) 3D signal versus range/height produced with FEMIX, (bottom) path loss versus range over a 3-segment 40 km mixed path (a 10 km long, 250 m high full-sinusoidal island is 15 km away from the transmitter) at 10 MHz (is: = , ; sea:  ). Solid: Millington; = , land:  dashed: FEMIX.

Fig. 17. (Top) 3D signal versus range/height produced with FEMIX, (bottom) path loss versus range over a 5-segment 50 km mixed path (10 km long, 250 m high and 15 km long, 500 m high two islands, respectively, 10 km and 25 km ; away from the transmitter) at 10 MHz (island:  : = , ). Solid: Millington; dashed: FEMIX. Sea:  = ,

Fig. 15. (Top) 3D signal versus range/height produced with FEMIX, (bottom) path loss versus range over a 3-segment 40 km mixed path (a 10 km long, 250 m high half-sinusoidal island is 15 km away from the transmitter) at 10 MHz (Is; sea:  ). Solid: Millington; = , land:  : = , dashed: FEMIX.

Fig. 18. (Top) 3D Signal versus range/height produced with FEMIX, (bottom) path loss versus range over a 3-segment 50 km mixed path (25 km long island, 15 km away from the transmitter with 250 m high and 500 m high Gauss-shaped ; sea:  = , islands) at 10 MHz (island:  : = ,  ). A Gaussian shaped antenna pattern with 5 vertical beamwidth is located 500 m above the sea surface. The antenna is tilted 1 downwards. Solid: Millington; dashed: FEMIX.

= 0 002S m

= 0 002S m

= 10

= 10

= 5S m

= 5S m

= 80

= 80

=5S m

= 80

= 0 002 S m

= 80

= 0 002 S m

= 10

= 10

= 5S m

VII. CONCLUSION

Fig. 16. (Top) 3D signal versus range/height produced with FEMIX, (bottom) path loss versus range over a 5-segment 50 km mixed path (10 km long, 500 m high Gauss-shaped and 5 km long smooth islands, respectively, 15 km and 35 km away from the transmitter) at 10 MHz (island:  : = ,  ; Sea:  = , ). Solid: Millington; dashed: FEMIX.

= 10

=5S m

= 80

= 0 002 S m

Surface wave path loss prediction over spherical Earth’s surface along multi-mixed paths including irregular terrain/island profiles is discussed and, for the first time in the literature, a systematic investigation of this highly old, challenging problem is given. A novel FEM-based surface wave multi-mixed path propagator is developed and used for the prediction of surface wave attenuation along multi-mixed paths including irregular islands profiles. The effects of hilly island transitions are shown through variety of propagation scenarios over a wide frequency range (i.e., MF and HF bands). Tests against ITU curves as well as the Millington package introduced before [16] are also given. Sharp attenuation before and strong recovery after the island in the transmitted signal, known as the Millington effect, is numerically illustrated. Also, the increase (decrease) in signal strength in front

APAYDIN AND SEVGI: NUMERICAL INVESTIGATIONS OF AND PATH LOSS PREDICTIONS

(back) slope of the hilly island is illustrated through characteristic examples. REFERENCES [1] “Groundwave Propagation Curves for Frequencies Between 10 kHz and 30 MHz,” International Telecommunications Union, 1992, ITU-R, Recommendations P-368-7. [2] K. A. Norton, “The propagation of radio waves over the surface of the earth and in the upper atmosphere-Part I,” Proc. IRE, vol. 24, no. 10, pp. 1367–1387, Oct. 1936. [3] G. Millington, “Ground-wave propagation over an inhomogeneous smooth earth,” Proc. IEE (London), vol. 96, no. 39, pp. 53–64, Mar. 1949. [4] H. Bremmer, “The extension of Sommerfeld’s formula for the propagation of radio waves over a flat earth, to different conductivities of the soil,” Physica, vol. 20, no. 1-6, pp. 441–460, 1954. [5] K. Furutsu, “On the excitation of the waves of proper solutions,” IRE Trans. Antennas Propag., vol. 7, no. 5, pp. 209–218, Dec. 1959. [6] V. A. Fock, Electromagnetic Diffraction and Propagation Problems. New York: Pergamon Press, 1965. [7] K. Furutsu, “A systematic theory of wave propagation over irregular terrain,” Radio Sci., vol. 17, no. 5, pp. 1037–1050, 1982. [8] J. R. Wait, Electromagnetic Waves in Stratified Media. New York: Pergamon Press, 1962. [9] J. R. Wait and L. C. Walters, “Curves for ground wave propagation over mixed land and sea paths,” IEEE Trans. Antennas Propag., vol. 11, no. 1, pp. 38–45, Jan. 1963. [10] D. A. Hill and J. R. Wait, “Ground wave propagation over a mixed path with an elevation change,” IEEE Trans. Antennas Propag., vol. 30, no. 1, pp. 139–141, Jan. 1982. [11] G. D. Monteath, Applications of the Electromagnetic Reciprocity Principle. Tarrytown, New York: Pergamon Press, 1973. [12] R. H. Ott, “A New Method for Predicting HF Ground Wave Attenuation Over Inhomogeneous, Irregular Terrain,” Office of Telecommunications/Institute for Telecommunication Sciences, Boulder, CO, 1971, OT/ITS Research report 7. [13] R. H. Ott, “An alternative integral equation for propagation over irregular terrain, II,” Radio Sci., vol. 6, no. 4, pp. 429–435, Apr. 1971. [14] R. H. Ott, L. E. Vogler, and G. A. Hufford, “Ground-wave propagation over irregular inhomogeneous terrain: Comparison of calculations and measurements,” IEEE Trans. Antennas Propag., vol. 27, no. 2, pp. 284–286, Mar. 1979. [15] S. W. Marcus, “A hybrid (finite difference surface Green’s function) method for computing transmission losses in an inhomogeneous atmosphere over irregular terrain,” IEEE Trans. Antennas Propag., vol. 40, no. 12, pp. 1451–1458, Dec. 1992. [16] J. R. Wait, “The ancient and modern history of EM ground-wave propagation,” IEEE Antennas Propag. Mag., vol. 40, no. 5, pp. 7–24, Oct. 1998. [17] N. M. Maslin, HF Communications: A Systems Approach. London: Pitmann Publishing, Taylor & Francis E-Library, 2005. [18] L. Sevgi and L. B. Felsen, “A new algorithm for ground wave propagation based on a hybrid ray-mode approach,” Int. J. Numer. Modeling, vol. 11, no. 2, pp. 87–103, Mar. 1998. [19] L. Sevgi, “A mixed-path groundwave field-strength prediction virtual tool for digital radio broadcast systems in medium and short wave bands,” IEEE Antennas Propag. Mag., vol. 48, no. 4, pp. 19–27, Aug. 2006. [20] L. Sevgi, “A numerical Millington propagation package for medium and short wave DRM systems field strength predictions,” IEEE Broadcast Technol. Society. News., vol. 14, no. 3, pp. 9–11, 2006. [21] L. Sevgi, F. Akleman, and L. B. Felsen, “Groundwave propagation modeling: problem-matched analytical formulations and direct numerical techniques,” IEEE Antennas Propag. Mag., vol. 44, no. 1, pp. 55–75, Feb. 2002. [22] L. Sevgi, “Groundwave modeling and simulation strategies and path loss prediction virtual tools,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1591–1598, Jun. 2007. [23] F. Akleman and L. Sevgi, “A novel finite-difference time-domain wave propagator,” IEEE Trans. Antennas Propag., vol. 48, no. 5, pp. 839–841, May 2000.

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[24] M. O. Ozyalcin, F. Akleman, and L. Sevgi, “A novel TLM-based timedomain wave propagator,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1680–1682, Jul. 2003. [25] L. Sevgi, Complex Electromagnetic Problems and Numerical Simulation Approaches. Piscataway, NJ: IEEE Press/Wiley, 2003. [26] L. Sevgi, C. Uluisik, and F. Akleman, “A matlab-based twodimensional parabolic equation radiowave propagation package,” IEEE Antennas Propag. Mag., vol. 47, no. 4, pp. 164–175, Aug. 2005. [27] T. Kawano, K. Goto, and T. Ishihara, “Analysis of ground wave propagation over land-to-sea mixed-path by using equivalent current source on aperture plane,” IEICE Trans. Electron., vol. E92-C, no. 1, pp. 46–54, 2009. [28] T. Kawano, K. Goto, and T. Ishihara, “Ground wave propagation in an inhomogeneous atmosphere over mixed-paths,” IEICE Trans. Electron., vol. E90-C, no. 2, pp. 288–294, Feb. 2007. [29] F. Akleman and L. Sevgi, “A novel MoM- and SSPE-based groundwave-propagation field-strength prediction simulator,” IEEE Antennas Propag. Mag., vol. 49, no. 5, pp. 69–82, Oct. 2007. [30] F. D. Tappert, “The parabolic approximation method,” in Wave Propagation and Underwater Acoustics, J. B. Keller and J. S. Papadakis, Eds. Berlin and New York: Springer-Verlag, 1977, pp. 224–287. [31] A. E. Barrios, “Parabolic equation modeling in horizontally inhomogeneous environments,” IEEE Trans. Antennas Propag., vol. 40, no. 7, pp. 791–797, Jul. 1992. [32] A. E. Barrios, “A terrain parabolic equation model for propagation in the troposphere,” IEEE Trans. Antennas Propag., vol. 42, no. 1, pp. 90–98, Jan. 1994. [33] M. F. Levy, Parabolic Equation Methods for Electromagnetic Wave Propagation. London, U.K.: The Institution of Electrical Engineers, 2000. [34] D. J. Donohue and J. R. Kuttler, “Propagation modeling over terrain using the parabolic wave equation,” IEEE Trans. Antennas Propag., vol. 48, no. 2, pp. 260–277, Feb. 2000. [35] P. D. Holm, “Wide-angle shift-map PE for a piecewise linear terrain-a finite-difference approach,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2773–2789, Oct. 2007. [36] J.-M. Jin, The Finite Element Method in Electromagnetics. New York: Wiley, 2002. [37] D. Huang, “Finite element solution to the parabolic wave equation,” J. Acoust. Soc. Am., vol. 84, no. 4, pp. 1405–1413, Oct. 1988. [38] V. M. Deshpande and M. D. Deshpande, “Study of electromagnetic wave propagation through dielectric slab doped randomly with thin metallic wires using finite element method,” IEEE Microwave Wireless Comp. Lett., vol. 15, no. 5, pp. 306–308, May 2005. [39] K. Arshad, F. A. Katsriku, and A. Lasebae, “An investigation of tropospheric radio wave propagation using finite elements,” WSEAS Trans. Commun., vol. 4, no. 11, pp. 1186–1192, Nov. 2005. [40] K. Arshad, F. A. Katsriku, and A. Lasebae, “An investigation of tropospheric wave propagation over irregular terrain and urban streets using finite elements,” in Proc. 6th WSEAS Conf. on Telecommunications and Informatics, Dallas, TX, Mar. 2007, pp. 105–110. [41] G. Apaydin and L. Sevgi, “The split step Fourier and finite element based parabolic equation propagation prediction tools: canonical tests, systematic comparisons, and calibration,” IEEE Antennas Propag. Mag., to be published. [42] G. Apaydin and L. Sevgi, “FEM-based surface wave multi-mixed-path propagator and path loss predictions,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1010–1013, Aug. 2009. [43] M. D. Collins and R. B. Evans, “A two-way parabolic equation for acoustic backscattering in the ocean,” J. Acoust. Soc. Amer., vol. 91, no. 3, pp. 1357–1368, Mar. 1992. [44] M. D. Collins, “A two-way parabolic equation for elastic media,” J. Acoust. Soc. Amer., vol. 93, no. 4, pp. 1815–1825, Apr. 1993. [45] J. F. Lingevitch, M. D. Collins, M. J. Mills, and R. B. Evans, “A two-way parabolic equation that accounts for multiple scattering,” J. Acoust. Soc. Am., vol. 112, no. 2, pp. 476–480, Aug. 2002. [46] M. J. Mills, M. D. Collins, and J. F. Lingevitch, “Two-way parabolic equation techniques for diffraction and scattering problems,” Wave Motion, vol. 31, no. 2, pp. 173–180, Feb. 2000. [47] O. Ozgun, “Recursive two-way parabolic equation approach for modeling terrain effects in tropospheric propagation,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2706–2714, Sep. 2009.

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Gökhan Apaydin (M’08) received the B.S., M.S., and Ph.D. degrees in electrical and electronic engineering from Bogazici University, Istanbul, Turkey, in 2001, 2003, and 2007, respectively. He was employed as a Teaching and Research Assistant by Bogazici University from 2001 to 2005 and a project and Research Engineer by Applied Research and Development, University of Technology, Zurich, Switzerland, from 2005 and 2010. Since 2010, he has been with Zirve University, Turkey. He has been working several research projects on electromagnetic scattering, the development of finite element method for electromagnetic computation, propagation, wireless communications, positioning, radio frequency identification applications, digital signal processing, filter design, wavelets and related areas. He has (co)authored ten journal and 15 conference papers, and many technical reports at the University of Applied Science Zurich.

Levent Sevgi (M’99–SM’02–F’09) received the Ph.D. degree from the Polytechnic Institute of New York, in 1990. Prof. Leo Felsen was his advisor. He was with Istanbul Technical University (1991–1998), TUBITAK-MRC, Information Technologies Research Institute (1999–2000), Weber Research Institute/Polytechnic University in New York (1988–1990), Scientific Research Group of Raytheon Systems, Canada (1998 – 1999), Center for Defense Studies, ITUV-SAM (1993 –1998, 2000–2002). Since 2001, he has been with Dogus University. He has been involved with complex electromagnetic problems and systems for more than 20 years. His research study has focused on propagation in complex environments, analytical and numerical methods in electromagnetic, EMC/EMI modeling and measurement, radar and integrated surveillance systems, surface wave HF radars, FDTD, TLM, FEM, SSPE, and MoM techniques and their applications, RCS modeling, bio-electromagnetics. He is also interested in novel approaches in engineering education, teaching electromagnetics via virtual tools. He also teaches popular science lectures such as science, technology and society. Prof. Sevgi is the writer/editor of the “Testing ourselves” Column in the IEEE Antennas and Propagation Magazine, a member of the IEEE Antennas and Propagation Society Education Committee, and the “Scientific Literacy” column writer of the IEEE Region 8 Newsletter.

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On the Effective Low-Grazing Reflection Coefficient of Random Terrain Roughness for Modeling Near-Earth Radiowave Propagation DaHan Liao, Member, IEEE, and Kamal Sarabandi, Fellow, IEEE

Abstract—An investigation on the effects of terrain roughness on near-ground radiowave propagation is featured. In spite of the fact that a variety of analytical and numerical routines have been proposed by many workers for the general treatment of the scattering properties of rough surfaces, much disagreement remains in the solution of the problem for near-grazing scenarios. In striving to analytically describe the near-grazing propagation of signals from 2D and 3D radiators, an existing volumetric polarization current-based perturbation approach is exploited in this work to formulate closed-form expressions for the coherent rough surface reflection coefficients. Although the perturbation approach was originally intended for analyzing the scattering coefficients of a ground with scale of roughness much smaller than the wavelength, it is shown through Monte Carlo simulations that the effective reflection coefficients reported herein are applicable for near-ground height) path-loss prediction even when the surface variation ( is on the order of a wavelength or more. Index Terms—Monte Carlo simulations, near-earth radiowave propagation, rough surfaces, volumetric perturbation method.

complete replacement of the rough surface with a smooth surface at the original surface’s physical mean height, for now the height and effective height is a function of both surface correlation length. Although numerical models such as those prescribed in [1], [2] have proven to be efficient simulators in dealing with the near-ground channel, it is also convenient to quantitatively capture the aforementioned observations—which have not been sufficiently addressed and explained in existing literature—in analytical formulations. Owing to the random multiple-scattering processes inherent in an undulating terrain environment, a radio signal has both coherent and incoherent components, each contributing to the total channel transfer characteristics. In considering the effects of random roughness, an equivalent coherent reflection coefficient can be produced—according to physical optics, for source and observation points located sufficiently away from the surface—by supplementing the Fresnel reflection coefficient for a flat surface, , with a correction factor

I. INTRODUCTION

(1)

OR propagation over a rough terrain, the physical statistical properties of the surface (height profile probability density function, surface autocorrelation function or roughness spectrum) have a direct impact on the statistics of the propagating signal. When the transmitter and receiver are close to each other, the line-of-sight (LOS) space wave (when it is unobstructed) from the transmitting antenna provides the primary contribution to the total received signal, as the coherent reflection from the underlying rough surface is reduced by the random scattering effects. However, over a long distance, as the propagation path approaches the grazing condition, in accordance with the Rayleigh criterion, the surface appears electrically smooth again and coherent cancellation between the direct and ground scattered signal is re-established. These qualitative observations are consistent with numerical simulation results presented in previous works [1], [2]; specifically, as it has been shown in [1], height the far field propagation loss increases with surface as expected but also shows considerable dependence on the surface correlation length. Furthermore, at grazing propagation, it is no longer proper to calculate coherent signal statistics by a

F

Manuscript received January 12, 2009; revised August 08, 2009; accepted September 25, 2009. Date of publication January 26, 2010; date of current version April 08, 2010. The authors are with the Radiation Laboratory, The University of Michigan, Ann Arbor, MI 48109-2122 USA (e-mail: [email protected]; saraband@eecs. umich.edu). Digital Object Identifier 10.1109/TAP.2010.2041310

The ensemble average can be calculated based on the probability distribution function (PDF) of the surface profile ; for instance, assuming a normally distributed surface profile, the correction factor reduces to the Ament approximation, (with denoting the grazing angle of propagation with respect to the normal and the surface height), which has been incorporated into ray-tracing routines by other workers for predicting reflection loss due to surface roughness. Note that Ament derived his result from an integral equation approach formulated for a PEC surface on which the value of the induced surface current is estimated to be independent of surcan also be reached face elevation [3]; the same result for using the Kirchhoff, or tangent-plane, approximation [4]. Another commonly used form of the correction factor has been derived by Miller and Brown, who originally considered the reflection effects of an ocean surface modeled as a collection of sinusoidal waves with Gaussian distribution in amplitude and uniform distribution in phase [5], [6]; although the Miller-Brown (with as approximation, the modified Bessel function of the first kind of order zero), has been shown to be in better agreement with experimental results [7] (as compared to the Ament approximation), the theoretical validity of applying such a PDF specific to ocean surfaces for terrain propagation problems has yet to be investigated. and as given While being simple to implement, both above are not valid for grazing angle propagation as (1) (and

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the Kirchhoff approximation itself) does not include terrain selfshadowing and Norton wave effects. The fraction of the surface that is illuminated by the incident rays can be estimated and the surface PDF can be modified to include a shadowing factor before insertion into (1) [8]; the subsequent expression for the effective reflection coefficient, however, maybe in a complicated integral form that is dependent upon the choice of an “optimal” shadowing PDF [9]. Alternatively, instead of defining the effective reflection coefficient directly, an equivalent impedance for the rough surface can be calculated based on the surface statistical properties [10]. The shadowing PDF and the equivalent impedance approaches have been shown to provide improvement over the traditional Ament and Miller-Brown formulations; however, these approaches still do not consider the implications of surface wave effects and have not been tested for near-ground propagation scenarios. Extension of the classic Sommerfeld propagation problem to the 2D finitely-conducting, rough surface has been made in [11], but the technique folheight lowed therein limits the region of solution validity to . In this study, in order to arrive at an analytical representation for the effects of terrain roughness on the near-ground channel, a new closed-form expression for the effective reflection coefficient is presented. The basis of the derivation is founded on the perturbation approach applied to a volumetric integral equation as introduced by Sarabandi and Chiu [12]–[14] for remote sensing applications involving modeling the general scattering coefficients of rough surfaces with inhomogeneous dielectric profiles. Here, expressions for the coherent effective reflection coefficient are derived for the 2D problem; the accuracy of these expressions is validated using the numerical simulator outlined in [2] for 2D excitation sources of vertical and horizontal polarizations. Extensions to—and simulation results for—3D sources are also provided. II. THEORETICAL BACKGROUND The perturbation approach set forth in [12] is briefly reviewed here; for a detailed discussion of the derivation of this method, the reader is referred to the original work. In formulating the response of the dielectric surface, a volumetric integral equation is first formed, relating the scattered field to the fictitious polarization current induced in the rough layer region ; after taking the Fourier transform of the expansion of the integral equation about the surface elevation , with the assumption that the polarization current can be written as a perturbation series in terms of the perturbation parameter , an iterative set of relations is generated from which the current can be found to arbitrary order. Essentially, the zeroth-order current gives the induced current of the flat , and higher-order terms represent interface geometry the corrections necessary to account for the presence of surface . Though the validity of the fluctuations about the mean at , it original formulations in [12] has been confirmed for is shown here that, in the context of near-grazing propagation, the reflection coefficient derived herein is even applicable for height) on the order of or more; the surface variations ( main reason for such an extended region of validity has its roots

in the fact that only the normal component of the wave vector ) “sees” the surface roughness. Also, it should (i.e., be noted that in contrast to other existing perturbation methods (e.g., small perturbation method—SPM [15]), the perturbation series of interest here is expanded in terms of the volumetric current instead of the surface current or tangential fields; therefore, this method offers the advantage of being readily modifiable for the analysis of a surface with an inhomogeneous vertical dielectric profile. A. Horizontal Polarization—2D With an excitation of the form , following the procedure delineated in [12], the Fourier transform of the polarization current within the rough layer, in the 2D case, can be shown to reduce to the following forms: (2) where each order of the current is given by (3) (4) and (5)

(6) , as defined in [12], is taken as the convolution of the and function with itself times. The field response can then be calculated from the polarization current using the Fourier transforms of the flat-interface, , and that of the half-space dyadic Green’s function, surface profile,

(7) in which is the horizontal polarization reflection coefficient . Taking the ensemble for a flat interface located at

LIAO AND SARABANDI: ON THE EFFECTIVE LOW-GRAZING REFLECTION COEFFICIENT OF RANDOM TERRAIN ROUGHNESS

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average of (7), after much manipulation and making use of the identities (8) (9) where is the power spectral density of the surface the total coherent scattered field to the second order reduces to

,

(10) with

Fig. 1. Propagation geometry; the rough terrain is characterized by relative dielectric constant  and normalized variation profile f (x), which has zeromean and stationary statistics.

not include the branch point near , and then another ), is expressible in terms changing of variable of parabolic cylinder functions. Therefore, the lower order terms are in the series expansion for

(11) (12) (13) The function

(16)

is given by (14)

where (17)

since it can Note that (11) is essentially accurate up to be shown that the coherent averages of the odd-order terms in is zero-mean Gaussian. (7) is zero—assuming In general, functions of the form (14) can be evaluated numerically in this study. Assuming a Gaussian correlation func, where tion for the surface, i.e., is the two-point correlation length of the surface, the integral in (14) is fast convergent, especially for large correlation lengths, as most of the contribution to the integral comes from the path . When is not close to the branch point , an near analytical approximation to (14) can be found simply by first replacing the first term in the integrand with its Taylor series expansion, and then integrating the resulting expression exactly using the identity

(18) and are the zeros of , ; and denotes the parabolic cylinder function. In deriving (16), a second order Taylor series expansion is used for the integral component and for the aforementioned term not containing containing , the following simplification the branch point . As can be used: (19) Fig. 2 shows the validity of expression (16). B. Vertical Polarization—2D

(15) . is For near-grazing propagation, evaluation of (14) at needed; in deriving a closed-form formulation for the integral, a more complete procedure [16], [17] must be taken owing to the presence of the branch point at . (For realistic ground conditions, here it is assumed that the branch point does not lie near the real axis.) After multiplying the numerator and denominator , the integral is separated into two of the integrand by —which is regular over the encomponents: one containing —which, tire path of integration; and the other containing after some manipulations (namely, after the changing of vari, expanding the part of the integrand that does able

Following a procedure similar to that given in the last section, but now with excitation as , the polarization current simplifies to (20)

(21)

(22)

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Fig. 2. Comparison between (14) and (16): (a) k c  i; f .

= 2+

= 300 MHz

= 2 ; (b) k

c

= 40 ;

1

Fig. 3. Convergence as functions of for perturbation solution applied to flat interface geometry: (a) percent error in field amplitude and (b) error in phase as compared to exact Sommerfeld solution; d : ; z : ; z : ; the source is a TM line source; distance between source and .

=21

100

=35 =25 observation point =

where

(23) (24)

is the same as (6). From (7)–(9), the effective and can reflection coefficient defined by be shown to be

(25) is the Fresnel reflection coefficient for vertical poin which larization for a flat surface located at and

(26)

(27) Approximate analytical forms can also be found for the functions (26) and (27), as it has been done for (14). Note that in

LIAO AND SARABANDI: ON THE EFFECTIVE LOW-GRAZING REFLECTION COEFFICIENT OF RANDOM TERRAIN ROUGHNESS

Fig. 4. Total field (E ) of TM line source located above a dielectric random surface with k 1 =  and k c = 10 ; d = 2; x = 200; z = 4:9; z = 4:3;  = 2 + i; f = 300 MHz.

contrast to the 3D case shown below, for non-oblique (to -axis) incidence, no cross-polarization component for the polarization current is generated; thus, no depolarization effects are observed for the 2D problem. Unlike the results from geometrical optics approximations (the Ament and Miller-Brown formulations), (11) and (25) show that the amount of correction to the reflection coefficient is dependent on the polarization. Also, the correction factor is a complex number as opposed to a purely real number as specified by geometrical optics; the complete statistics of the rough surface are taken into account through the parameter and the func. tion III. SIMULATION RESULTS In order to validate the accuracy and convergence of the perturbation-derived effective reflection coefficients as given in the previous section, the perturbation method is applied to the constant perturbation function , resulting in a flat interface problem with a well-defined exact solution; thus, in this case, the deterministic coherent field in (7) can be found to arbitrary order . Fig. 3 shows the error in the computed scattered

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Fig. 5. Total field of TE line source located above a dielectric random surface with k 1 = 2 and k c = 40 ; d = 4; x = 200; z = 8:4; z = 8;  = 2 + i; f = 300 MHz.

field as functions of perturbation order for a TM line radiator located above a half-space with a varying interface as specified by the parameter . Since the wave vectors responsible for the interaction at grazing angle are almost entirely parallel to the horizontal plane of the surface (i.e., ), the variation of the polarization current in is small; therefore, even in the presence of significant roughness (large ), the first few orders of the current are adequate in capturing the scattering characteristics of the surface. It should also be pointed out that although the method originally derives the perturbation series for the volumetric polarization current in terms of , the series could as well, where be re-written as an expansion in is the angle of incidence. (For example, this can be seen by realizing that the second order terms in and can .) Consequently, at grazing angle , be written in the series quickly converges even for large . Note that for near-earth propagation, since the direct wave and the zeroth order reflection term ( term) almost cancel, the second term) is very important for properly generorder term ( ating the Norton surface wave component despite the fact that its magnitude by itself is small compared to that of the zeroth order reflection term.

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Fig. 6. Total field of TE line source located above a dielectric random surface  and k c ; d ; x ; z ; z ; with k  i; f .

Fig. 7. Total field of TM line source above a rough surface as functions of surface correlation length and distance; ; d ; x ; z i; f . z ; 

A. Validation With Monte Carlo Simulations

and denotes the location of the source; the scattered field due to the ground appears as

1=2 = 40 = 8 + 6 = 300 MHz

=4

= 200

=6

=6

In this section, the effective reflection coefficients derived in the previous section are employed for characterizing the near-grazing radiation properties of 2D sources located above a rough surface. The radiators of interest here include the electric and magnetic line sources, free space fields of which are given by

=8

= 2+

= 300 MHz

1=1

=4

= 200

=

(31) For observation points in the far field, the second order asymptotic method as introduced in [18] can be applied for the evaluation of (31). After the transformation , saddle point integration leads to

(28)

with (29) (30)

(32)

LIAO AND SARABANDI: ON THE EFFECTIVE LOW-GRAZING REFLECTION COEFFICIENT OF RANDOM TERRAIN ROUGHNESS

Fig. 8. Variation of (k

) as functions of correlation length and w

where ; the expression in (11) and (25) without the the saddle point is

(a) 

is term; and

(33)

Figs. 4–6 show the comparison of the total fields computed from the perturbation solution and from Monte Carlo simulations; very good agreement is seen in both the field amplitude and phase. The variation of the signal intensity as function of surface correlation length is shown in Fig. 7; for constant , the signal intensity decreases with , consistent with the discussions in [1]; this result can be attributed to the shadowing effect, the dependence upon of which is more apparent at grazing angles; this is seen in Fig. 8, which plots as function of the saddle point angle; similar curves and in the vertical polarization case can for be generated. From the numerical results shown, it can be surmised that the presented method does not suffer the shortcoming of classical SPM, which infamously breaks down at grazing angle for vertical polarization in propagation and scattering problems

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= 2 + i; (b)  = 8 + 6i.

[19]. It is expected that the coherent reflection coefficient derived from SPM would not lead to meaningful results as relevant for the problem of interest in the present context for vertical polarization. For horizontal polarization, on the other hand, SPM may work for near-ground propagation, but the authors are not aware that validations of such a claim exist; this could be a subject for a future study. B. Extension to 3D Having confirmed the accuracy of the perturbation-derived, rough-surface effective reflection coefficients in the 2D case, their complementary 3D forms are presented in this section. Since the analytical procedure is analogous to that of the 2D case, only the direct results are shown below. For a horizontally-polarized incident wave —where , , the effective reflection coefficient is

(34)

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where

In the 3D case, cross-polarization current and field components are also generated; for horizontally-polarized incidence, the coefficient of the vertically-polarized field is

(41) (35)

with

and, for a surface with Gaussian height distribution and correction function, (36) For

(42)

a vertically-polarized incident wave —where , the coefficient becomes

(43)

(44) (37) with

Similarly, for vertically-polarized incidence, the coefficient of the horizontally-polarized field is

(45) (38)

with

(46)

(39)

(40)

(47)

LIAO AND SARABANDI: ON THE EFFECTIVE LOW-GRAZING REFLECTION COEFFICIENT OF RANDOM TERRAIN ROUGHNESS

Fig. 9. Excess path-loss of (a) vertical dipole and (b) horizontal dipole (x ^-directed) located above a dielectric random surface with k 1 = 2 ; d = 4; z = 8:4; z = 8;  = 2 + i; f = 300 MHz; observation point is above the y -axis.

The radiation pattern of a dipole source can be computed by inserting the effective reflection coefficients listed above into the half-space asymptotic formulations (9)–(17) from [18]; in order to justify this procedure, it is important to mention that all the forms of are functions of only; this can , be seen with the change of variable , , , and subsequently realizing that the equivalent forms for are expressible in the azimuthal parameters as . Figs. 9 and 10 show the propagation loss for the dipole source (positioned on the -axis) as functions of distance and surface statistics. IV. CONCLUSION In view of the limited practicality of existing closed-form expressions for analyzing the scattering characteristics of random rough surfaces as relevant to the near-earth radio channel, a new set of expressions for the coherent effective reflection coefficients is presented in this work. The basis of the derivation is founded upon the volumetric current-based perturbation

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Fig. 10. Excess path-loss of vertical dipole located above a dielectric random surface with d = 4; z = 8:4; z = 8;  = 2 + i; f = 300 MHz: (a) constant correlation length at k c = 10; (b) constant rms slope.

approach originally prescribed in [12]. In applying the results of this study to compute the radiation of 2D and 3D sources, it is noted that even in the presence of significant roughness, a coherent cancellation still occurs between the LOS signal and the ground scattered signal in the far field, establishing a ground wave important in long-distance transmission. The signal strength also has a strong dependence on surface correlation statistics; however, for most realistic terrain surfaces of interest, it is seen that the roughness may only weaken the ground wave by up to 5–6 dB as compared to the smooth interface case. As the initial step in the perturbation approach assumes that the plane wave providing the excitation is impinging upon the rough surface from free space, the formulations introduced for the effective reflection coefficients are valid only for transmitter . Nevlocations located above the mean surface height ertheless, because of the generality of the outlined perturbation approach, it is conjectured here that a similar procedure can be applied in deriving the polarization current and the scattered signal for NLOS (non-line-of-sight) propagation paths and even for when the radio terminals are embedded within the ground.

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In addition, few modifications are needed in adapting the formulations to solve the problem in which a ground with an inhomogeneous or stratified dielectric profile is situated below the rough layer region; the essential adjustment is done by replacing the flat-interface reflection coefficients within the expressions for the polarization currents with their equivalent counterparts for the inhomogeneous profile. Undertakings in the areas as described above will be conducted as part of our future studies. REFERENCES [1] D. H. Liao and K. Sarabandi, “An approximate numerical model for simulation of long-distance near-ground radiowave propagation over random terrain profiles,” presented at the Military Communications Conf. (MILCOM 2007), 2007. [2] D. H. Liao and K. Sarabandi, “Simulation of near-ground long-distance radiowave propagation over terrain using Nyström method with phase extraction technique and FMM-acceleration,” IEEE Trans. Antennas Propag., vol. 57, pp. 3882–3890, Dec. 2009. [3] W. S. Ament, “Toward a theory of reflection by a rough surface,” Proc. IRE, pp. 142–146, 1953. [4] A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering. Englewood Cliffs, N.J.: Prentice Hall, 1991. [5] R. M. Brown and A. R. Miller, “Geometric-optics theory for coherent scattering of microwaves from the ocean surface,” NRL Rep. 7705, 1974. [6] A. R. Miller, R. M. Brown, and E. Vegh, “New derivation for the roughsurface reflection coefficient for the distribution of sea wave elevations,” Inst. Elect. Eng. Proc.-H, vol. 131, pp. 114–116, Apr. 1984. [7] L. Boithias, Radio Wave Propagation. New York: McGraw-Hill, 1987. [8] D. M. Milder, “Surface shadowing at small grazing angle,” Waves in Random Media, vol. 13, 2003. [9] C. Bourlier, N. Pinel, and V. Fabbro, “Illuminated height PDF of a random rough surface and its impact on the forward propagation above oceans at grazing angles,” Eur. Space Agency, Tech. Rep. ESA SP626 SP, 2006. [10] F. G. Bass and I. M. Fuks, Wave Scattering From Statistically Rough Surfaces. New York: Pergamon, 1979. [11] A. Ishimaru, J. D. Rockway, Y. Kuga, and S.-W. Lee, “Sommerfeld and Zenneck wave propagation for a finitely conducting one-dimensional rough surface,” IEEE Trans. Antennas Propag., vol. 48, pp. 1475–1484, Sep. 2000. [12] K. Sarabandi and T. Chiu, “Electromagnetic scattering from slightly rough surfaces with inhomogeneous dielectric profiles,” IEEE Trans. Antennas Propag., vol. 45, pp. 1419–1430, Sep. 1997. [13] K. Sarabandi and T. Chiu, “Electromagnetic scattering from slightly rough surfaces with inhomogeneous dielectric profile,” in Proc. Int. Geoscience and Remote Sensing Symp., 1996, vol. 4, pp. 2122–2124. [14] T. Chiu and K. Sarabandi, “Propagation of electromagnetic waves near the surface of a half-space dielectric with rough interface,” in Proc. IEEE Antennas Propag. Society Int. Symp., 1999, vol. 2, pp. 1402–1405. [15] S. O. Rice, “Reflection of electromagnetic wave by slightly rough surfaces,” Commun. Pure Appl. Math., vol. 4, pp. 351–378, 1951. [16] N. Bleistein, “Uniform asymptotic expansion of integrals with stationary point near algebraic singularity,” Commun. Pure Appl. Math., vol. 19, pp. 353–370, 1966. [17] N. Bleistein and R. A. Handelsman, Asymptotic Expansion of Integrals. New York: Holt, Rinehart, and Winston, 1975. [18] D. H. Liao and K. Sarabandi, “Near-earth wave propagation characteristics of electric dipole in presence of vegetation or snow layer,” IEEE Trans. Antennas Propag., vol. 53, pp. 3747–3756, Nov. 2005. [19] D. E. Barrick, “Grazing behavior of scatter and propagation above any rough surface,” IEEE Trans. Antennas Propag., vol. 46, pp. 73–83, Jan. 1998.

DaHan Liao (S’05–M’09) was born in Canton, China. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from The University of Michigan, Ann Arbor, in 2003, 2005, and 2008, respectively. From 2003 to 2008, he was a Graduate Student Research Assistant at the Radiation Laboratory, The University of Michigan, Ann Arbor. Currently, he is carrying out his research at the U.S. Army Research Laboratory, Adelphi, MD. His research interests include physics-based near-earth wave propagation modeling and simulation, electromagnetic scattering, and computational electromagnetics. Dr. Liao was a recipient of the MIT Lincoln Laboratory Fellowship in 2006. He was a finalist of the student paper competition at the USNC/URSI National Radio Science Meeting in 2005 and 2006 and at the IEEE IGARSS Conference in 2009.

Kamal Sarabandi (S’87–M’90–SM’92–F’00) received the B.S. degree in electrical engineering from 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, Ann Arbor, in 1989. He is the Rufus S. Teesdale Professor of Engineering and Director of the Radiation Laboratory in the Department of Electrical Engineering and Computer Science at the University of Michigan. His research areas of interest include microwave and millimeter-wave radar remote sensing, metamaterials, electromagnetic wave propagation, and antenna miniaturization. He has 22 years of experience with wave propagation in random media, communication channel modeling, microwave sensors, and radar systems and is leading a large research group including two research scientists, 12 Ph.D. and 2 M.S. students. He has graduated 31 Ph.D. and supervised numerous postdoctoral students. He has served as the Principal Investigator on many projects sponsored by NASA, JPL, ARO, ONR, ARL, NSF, DARPA and a larger 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 170 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 420 papers and invited presentations in many national and international conferences and symposia on similar subjects. Dr. Sarabandi is a member of NASA Advisory Council appointed by the NASA Administrator. 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. He also served as a Vice President of the IEEE Geoscience and Remote Sensing Society (GRSS) and a member of the IEEE Technical Activities Board Awards Committee. He is serving on the Editorial Board of the IEEE PROCEEDINGS, and served as Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and the IEEE Sensors Journal. He is a member of Commissions F and D of URSI and is listed in American Men & Women of Science Who’s Who in America and Who’s Who in Science and Engineering. 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, AMTA’06, URSI GA 2008) have received student paper awards.

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Truncated Gamma Drop Size Distribution Models for Rain Attenuation in Singapore Lakshmi Sutha Kumar, Yee Hui Lee, Member, IEEE, and Jin Teong Ong, Member, IEEE

Abstract—A model that is less sensitive to errors in the extreme small and large drop diameters, the gamma model with central moments (3, 4 and 6), is proposed to model the rain drop size distribution of Singapore. This is because, the rain rate estimated using measured drop size distribution shows that the contributions of lower drop diameters are small as compared to the central drop diameters. This is expected since the sensitivity of the Joss distrometer degrades for small drop diameters. The lower drop diameters are therefore removed from the drop size data and the gamma model is redesigned for its moments. The effects of the removal of a particular rain drop size diameter on the specific rain attenuation (in dB) and the slant-path rain attenuation calculations with forward scattering coefficients for vertical polarization are analyzed at Ku-band, Ka-band and Q-band frequencies. It is concluded that the sensitivity of the Joss distrometer although affects the rain rate estimation at low rain rates, does not affect the slant path rain attenuation on microwave links. Therefore, the small drop diameters can be ignored completely for slant path rain attenuation calculations in the tropical region of Singapore. Index Terms—dead-time problem, drop diameters, gamma distributions, rain, rain attenuation, rain drop size distribution.

I. INTRODUCTION

R

AIN drop size distribution model is required for the evaluation of microwave propagation attenuation due to rain. The prediction of rain attenuation is very important to communication engineers since the microwave attenuation caused by rain limits the performance of the microwave link. Rain attenuation is much more severe in the tropical and equatorial regions due to higher precipitation rates. In the design of both terrestrial and earth-satellite communication links, detailed knowledge of the drop size distribution (DSD) becomes imperative for the calculation of the rain-induced attenuation. In 1943, Laws and Parsons examined the relationship between raindrop size and rain intensity [1]. For a long time, the exponential DSD has been the most widely used analytical parameterization for the raindrop size distribution: (1)

where is the number of drops per unit volume per unit interval of drop diameter and the parameters and can be determined experimentally. Marshall and Palmer [2] suggested Manuscript received January 23, 2009; revised September 11, 2009; accepted October 18, 2009. Date of publication February 02, 2010; date of current version April 07, 2010. L. S. Kumar and Y. H. Lee are with Nanyang Technological University, Singapore 639798, Singapore (e-mail: [email protected]; [email protected]). J. T. Ong is with C2N Pte. Ltd, Singapore 199098, Singapore (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2042027

m mm and mm where that is the rainfall rate in mm/hr. However, subsequent DSD measurements have shown that the exponential distribution does not capture rain DSDs and a more general function is necessary. Although some authors [3] have considered using a lognormal distribution, Ulbrich [4] and Willis [5] have suggested that the DSD is best modeled by a gamma distribution (which has the exponential distribution as a special case) though Smith [6] recently has argued that the exponential distribution is preferred because of its simplicity. In this paper, the use of the gamma distribution for modeling the DSD in the tropical and equatorial regions will be examined. Before considering how best to estimate the parameters of a gamma DSD, we need to understand the limitations of the Joss distrometer (JWD), which is the instrument used to measure DSD in this study. JWD tends to underestimate the number of small drops during a heavy rain event because of ringing of the Styrofoam cone when it is hit by the rain drops. This is known as the distrometers’ dead time. To correct for it, the correction matrix, supplied by the manufacturer is used [7]. In the presence of numerous large rain drops during intense tropical rain events of mm/hr, drop sizes smaller than 1.0 mm are underrepresented [8]. This problem is due to an automatic threshold circuitry that monitors the ambient noise level to reject spurious pulses. However, under intense rain, the high noise level of the drops themselves is interpreted as ambient noise and small-drop signals are rejected. The larger drops produce longer dead times and therefore, requires greater correction. However, if there are no drops in a given bin, the correction matrix does not add any drops to the bin. Rather, it modifies the DSD and increases the high moments of the drop size such as rain rate significantly. This is a problem of the correction matrix, and thus, many users choose not to implement it [8], [9]. At the largest drop size end, drops larger than 5.0–5.5 mm in diameter cannot be resolved at their true size; rather, they are assigned to the largest size bin. The sensitivity degradation of the JWD has been discussed in the comparative studies carried out between the JWD and other drop size measurement instruments in [7] and [10]. In the present study, the dead-time correction has been applied using the software provided by Distromet, Inc. The correction is intended to correct up to 10% of the accuracy. The distrometer was installed on the rooftop of a 50 m high building where the environmental noise is minimal. The DSD over Singapore was studied previously by Li et al. [11] and Ong and Shan [12], [13]. Li et al. [11], Ong and Shan [12] have proposed a modified gamma model for DSD. In [13], they modeled the rain drop size distributions by the lognormal model. In their paper, Singapore lognormal and gamma models

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are compared with the results from different regions in [13]. In both the papers [12], [13], the 0 , 1 and 2 nd moments and the 0 , 2 and 3 moments are used to represent the modified gamma model. However, for the distrometer data, the use of the 0 moment is impossible and the use of the 1st moment is not advisable, since the number of drops with diameters less than is not known [14]. Many authors [6], [8], [9], [15], [16] prefer to work with central moments, since JWD has degraded sensitivity at small drop diameters. Kozu and Nakamura [15], and Tokay and Short [8] used the 3 , 4 and 6 moments (MM346). Smith [6], [16] suggested 2 , 3 and 4 moments to model gamma DSD (MM234). Ulbrich and Atlas [17] took , and into account the maximum value for drop diameters, used the 2 , 4 and 6 moments (MM246). Their method allows for truncation of the DSD at the large diameter end of the spectrum due in part to instrumental effects. Timothy [18] used 3 , 4 and 6 moments to represent Singapore’s DSD using lognormal model. He observed a significant decrease in drop density in small drop bins from visual inspection of Singapore’s data (year 1997–1998 data). In recent years, it has become common for DSD to be represented by a normalized DSD [19], [20]. The normalization allows the shapes of different DSDs from different rain regimes to be compared relative to the total liquid water content and the mean drop size. The concept of normalized gamma distribution was first introduced by Willis [5] and revisited by Illingworth and Blackman [19] in 2002. Normalized gamma DSD has been used by Testud et al. [20]. Generally normalization is used to remove the dependence of and in the gamma DSD, allowing comparison of the shape of the distributions at different rainfall rates. Caracciolo [9] used the higher order moments 4 , 5 and 6 to form a gamma model (MM456) which is less sensitive to small drop diameters. She selected these moments because the higher moments are less dependent on small drops which are underestimated by Joss distrometer due to dead time problem. This model is especially effective at higher rain rates, where the dead time effect is severe. Smith [16] pointed out that the bias is stronger when higher order moments are used. Brawn [14] compared the gamma models using different combinations of moments (MM346, MM234, MM246 and MM456) along with his new procedure. He found that MM456 model deviates more from the measured data compared to the gamma models using lower moment combinations, MM234, MM246 and MM346. This conclusion is similar to Smith’s [16]. But a model which is less sensitive to small drop diameters is required as stated by Caracciolo [9]. Therefore, Brawn [14] suggested that an interesting alternative to develop a less sensitive model is by completely ignoring the counts in the lower order bins. This research work started with the modeling of rain drop size distribution in the tropical region using different models, such as lognormal, gamma and exponential models. The three mentioned models describe the larger drop diameters well. However, the exponential model tends to overestimate the smaller drop diameters whereas the gamma and the lognormal models follow the distrometers’ measured DSD at the smaller drop diameters. Since the measured DSD experiences the dead time effect at the smaller drop diameters, the accurate modeling of the small drop

diameters needs to be carefully examined. Therefore, by taking Brawn’s recommendation to ignore the counts in the lower order bins, the gamma model can be used as a universal model including the tropical region of Singapore. In this paper an investigation is performed to study the validity of ignoring the counts in the lower order bin as proposed by Brawn for rain attenuation calculations. This is done by removing the small drop size bins consecutively starting from the first bin and redesigning the gamma models using the remaining bins for each bin removal. The deviations in the redesigned gamma models as compared to the actual gamma model are studied. This will aid in the understanding of the importance of small rain drop sizes to the rain attenuation of the terrestrial and earth-satellite communication links. Based on these redesigned models, the importance of small rain drop sizes on communication links at different frequencies is examined. II. METHODOLOGY The measurement of rain drop size distribution has been conducted since August 1994 at the Nanyang Technological University (1 21 N, 103 41 E), Singapore (NTU). The rain data is collected from August 1994 to September 1995, excluding June and July 1995 using a “Joss-type” Distrometer RD-69. Rain rates are calculated from measured DSD for seven rain rates, 1.96 mm/hr, 4.20 mm/hr, 10.45 mm/hr, 22.80 mm/hr, 66.54 mm/hr, 120.30 mm/hr and 141.27 mm/hr. The Joss distrometer has an integration time of one minute, therefore, rain rate calculated from the distrometers’ measured data is directly used. The contribution of individual bins is found in order to check the effect of the dead time problem at different rain rates. In our previous work [21], gamma model is used to find the contribution of individual bins and it is reported that the error in rain rate calculations from the removal of drop diameters below 0.77 mm is small as compared to the removal of the larger drop diameters. In this paper, the DSD is described by a three-parameter gamma distribution. The 3 , 4 and 6 moments are proposed to model gamma DSD (MM346) in Singapore as suggested by Kozu and Nakamura [15]. In order to form a gamma DSD model which is less sensitive to small drop diameters, consecutive bins are removed, meaning; initially, only bin 1 is removed, and then bins 1 and 2 are removed, followed by the removal of bins 1 to 3 and bins 1 to 4 starting from the smallest drop size diameter. Moments are calculated from the remaining bins in each case. In this way, four different truncated gamma models are designed. Mean square errors are calculated to compare the modeled DSDs with the measured drop size distribution. The study continues by examining the contribution of particular rain drops on the specific rain attenuation of microwave signals using forward scattering coefficients for a vertically polarized wave at frequencies 11 GHz, 28 GHz, 38 GHz and 48 GHz. T-Matrix code is used to calculate the forward scattering coefficients which are more accurate over various drop shapes and radio frequencies. The Ku-band frequency of 11 GHz [18], representative of the INTELSAT 602 satellite and the Ka-band frequency of 28 GHz [22], representative of the IPSTAR satellite is used in this study. A number of sources [23] have identified 38 GHz

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TABLE I THRESHOLDS OF DROP SIZE BINS AND MEASURED RAIN DROPS FROM JWD AT SEVEN RAIN RATES

as the largest growth area in the supply of fixed radio-relay systems as lower frequency bands become congested. The 48 GHz [24] band has been allocated world-wide for fixed service with High Altitude Platform Stations (HAPS). The main drawback in HAPS is the troposphere effects on the propagation, particularly rain attenuation that may limit the link availability. Therefore, in the Q-band, 38 GHz and 48 GHz are selected for this study. exceeding 1% of the time is Slant-path rain attenuation calculated using ITU-R P.618-9 recommendations [25] for all the truncated gamma models at 11 GHz, 28 GHz, 38 GHz and 48 GHz. The first step in applying this method is to obtain the rainfall intensity for the area of interest, integrated over one minute, which is exceeded for 0.01% of the time. of 122 mm/hr is used in [26] for the NTU The averaged is 126.65 mm/hr for the site from year 1990 to 1996. year 2000, 105.58 mm/hr for 2005 to 2006 and 111.18 mm/hr for 2007 to 2008 also at NTU site. Therefore, 120.30 mm/hr is in this paper since it is one of the rain rates taken as the exceeding 1% of the considered. Slant-path rain attenuation time is calculated for all the four frequencies at this rain rate. Changes in slant-path rain attenuations are calculated by comparing bins removed MM346 gamma models with the actual gamma model. This paper presents the contribution of drop size on specific rain attenuation for different frequencies. The changes in gamma models due to the truncation of lower bins are analyzed along with slant-path rain attenuation calculations. The results from this analysis are useful for the

prediction of rain attenuation at Ku-band, Ka-band and Q-band communication links in the tropical region. III. DATA ANALYSIS AND MEASUREMENT A. Measured DSD The Distrometer is capable of measuring the drop diameters mm with an accuracy of %. It ranging from 0.3 mm to distinguishes between drops with time interval of about 1 ms. The total number of drops with diameters ranging from 0.3 mm mm is divided into 20 different bins for 1 minute integrato tion time [27]. It is possible to calculate the rain rate from the measured number of drops for a minute. Seven different minutes are selected from the rain events that occurred on 26th February 1995. The measured drop counts at different bins from the JWD for those seven minutes are listed along with the thresholds for the drop size bins in Table I. As can be seen in Table I, as the rain rate increases, the dead time problem can be observed from the lack of drops in the lower bins (labeled as “X”). This will be analyzed in detail in is a representative diameter the results section. The diameter for the th bin. Generally, the mean value of range is calculated in (2). (2) where and are the lower and upper diameter value of that th bin respectively. The representative diameter is used

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for computing DSD, rain rate and specific rain attenuation. The number of raindrops, , in the th bin with diameters, , in the are collected over a sample area of range of mm with an integration time of sec determined by the JWD. The rain rate (in mm/hr) can be calculated from the measured data by

size, operating frequency and wave polarization. The scattering calculations were performed using the T-Matrix method [29] which enables the computation of the complex forward scattering coefficients, considering spherical drops at 20 C with the shape according to the Pruppacher-Pitter model [30] for and polarizations. The specific rain attenuation for and polarizations and in dB/km are given by

(3) The measured rain drop size distribution [12] can be expressed by

(m

mm

(12)

)

(4) where is the terminal velocity of rain drop in m/s from Gunn and Kinzer [28].

and are complex forward scattering cowhere efficients (in units of m) for horizontal and vertical polarization is the number of drops per unit volume per respectively, is the drop size interval unit drop diameter in m mm in mm and is the wavelength. D. The ITU-R Rain Attenuation Model

B. Gamma Modeled DSD Gamma model is usually expressed in the form (5) is the number of rain drops per cubic meter per where is the rain drop diameter millimeter diameter (m mm ), and are parameters to be determined through the (mm), measured DSD. The estimates from the method of moments are obtained by equating a sufficient number of measured moments to the corresponding theoretical moments. The th experimental moment is expressed by

(13)

(8)

The elevation angle, , is considered from 10 to 90 in steps of 10 . The trend of slant path rain attenuation is not similar at different frequencies as the elevation angle changes. High Altitude Platform Stations [32] use the full range of elevation angles for their different coverage zones (urban, suburban and rural area coverage). Therefore, different elevation angles are considered in order to study the variations in slant-path rain attenuation at the four frequencies. is then calculated as exThe effective slant path length plained in ITU-R P.618-9 (refer to [25] for details) using the (km), elevation earth station height above mean sea level angle , latitude and the rain height . Using the effective slant path length, and the calculated in (12), the attenuis calculated as ation exceeding 0.01% of the year

, the gamma DSD parame-

(14)

(6) where is the number of samples and is the particle number is obtained through the concentration and is equal to experimental data. The th theoretical moments can be written as (7) By equating the theoretical and experimental moments, it is derived that

and Using ters are obtained [15] as follows:

The ITU-R P.618-9 [25] gives a step by step procedure to estimate the long-term statistics of the slant-path rain attenuation at a given location for frequencies up to 55 GHz. The earth station is assumed to be at the rooftop of a building in NTU, Singapore at location (1 21 N, 103 41 E) and the earth station height is measured using a Symmetricom XL-GPS receiver. The rain is calculated in (13) using the 0 isotherm height data height , given by ITU-R P.839-3 [31]

(9) (10) (11)

The estimated attenuation to be exceeded for other percentages of an average year, in the range 0.001% to 5%, is determined from the attenuation to be exceeded for 0.01% for an average year using (15)

C. Specific Rain Attenuation Specific attenuation due to a rain path depends on the rain rate, shape of the rain drops, distribution of the rain drops

(15) where

when

% in this study.

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Fig. 2. Normalized deviation (%) for each individual bin removal. Fig. 1. Drop size distribution for seven rain rates.

IV. RESULTS A. Rain Rate and DSD 1) Analysis of Measured DSD: Fig. 1 illustrates the DSD obtained from the measured data using (4) for seven one minute rain rates 1.96 mm/hr, 4.20 mm/hr, 10.45 mm/hr, 22.80 mm/hr, 66.54 mm/hr, 120.30 mm/hr and 141.27 mm/hr. Logarithmic in the vertical axis. The DSD scale is used to represent increases initially with the diameter of the rain drop and then decreases for all the rain rates. The sensitivity of the distrometer at different bins can be checked by the number of rain drops in the measured data at different rain rates. The lower bins which are marked as “X” in Table I have no rain drops in the measured data. It is clear that the number of lower bins which are affected by dead time problem increase with rain rate. There are zero rain drops in the first bin at 22.80 mm/hr and 66.54 mm/hr and in the first two bins at 120.30 mm/hr. It is also observed that the first three bins shows low number of counts compared to the higher bins at all the rain rates even after the dead time correction is applied. At 141.27 mm/hr, there are no rain drops in the first 3 bins and only five rain drops at the 4th bin. However, common amongst all rain rates is that, the fifth bin always has a reasonable number of rain drop counts. As stated in the introduction, the numbers of rain drops in the lower 4 bins are severely affected by the dead time problem at high rain rates. In order to study the significance of lower bins which are more erroneous due to the dead time problem, the rain rate estimation is taken. The contribution of individual bins in rain rate estimation is found using measured data in the following section. 2) Contribution of Individual Drop Diameters: The rain rates are calculated using (3) from the measured DSD. Then the rain rate contribution of individual bin is removed one by one, starting from the first bin to the last bin (containing rain drops) for the measured DSD. The rain rate with individual bins removed is calculated for each case. The difference between the measured rain rate and the rain rate with the th bin

removed is then calculated in order to study the significance of individual bins with its corresponding range of rain drop size diameters. This will provide information on the importance and contribution of each drop diameter range to the overall rain rate. The normalized deviation (%) is calculated using the true rain rate and the rain rate with the th bin removed using the following equation

%

(16)

where is found from (3) and ( th bin removed) is th term removed from the also found from (3) but with summation. Fig. 2 shows a normalized deviation (%) using (16) for each bin removal at the seven rain rates considered. As seen from Fig. 2, the contribution of bins in rain rate measurement increases gradually with the removal of bin 1 to the middle bins and then decreases for all rain rates. As the rain rate increases, the bin that has the major contribution increases. Therefore, it is clear that at higher rain rates, the contribution of larger drops is more significant in rain rate estimation as is expected. At the lower rain rate of 4.20 mm/hr, from Table I, the highest mm), number of rain drop count appears at bin 7 ( however, the most significant bin in rain rate estimation is bin mm). Even though the highest number of rain 8( drop count appears at bin 7, the most significant bin in rain rate estimation is bin 11 at 10.45 mm/hr. This shows that the contribution of rain drops to the overall rain fall rate is not only dependent on the number of drop counts and drop diameter but also the distribution. At the low rain rate of 1.96 mm/hr, most drop counts are in bin 6, with the number of drop counts distributed between bins 4 to 8 (Table I). The most significant bin at this rain rate is bin 7 (Fig. 2). At the high rain rate of 141.27 mm/hr, the drops are quite equally distributed over bins 6 to 18. This results in a number of bins (above bin 10) showing significant contribution to the overall rain rate. This confirms that the contribution to the overall rain rate is dependent on both the drop diameter and drop count distribution.

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Fig. 3. Normalized deviation (%) for consecutive bins removal.

The (%) is very small at rain rates from 1.96 mm/hr to 10.45 mm/hr for the first 2 bins. The contributions of 3 and 4 bins are more significant at these rain rates. The total (%) produced by the first 4 bins is 8.45, 3.75 and 1.71 at 1.96 mm/hr, 4.20 mm/hr and 10.45 mm/hr respectively. After(%) increases with the removal of the succeswards, the sive bins. The total normalized deviation produced by the first 4 bins is less than 1% at the higher rain rates from 22.80 mm/hr to 141.27 mm/hr. In Fig. 3, the percentage of normalized deviation is calculated, but this time, instead of removing one bin at a time, consecutive bins are removed starting from the smallest drop size diameter. This enables us to study the contribution of the range of the rain drop size diameters to the overall rain rate. As expected from Fig. 3, when the number of bins removed increases, the deviation increases correspondingly. The point of this study is to find the bin after which there is a sharp increase in deviation. When the first 1 to 2 bins are removed, the deviation is minimal for all the considered rain rates. However, when bins 1 to 3 are removed, there is a sudden increase in the deviation for rain rates 1.96 mm/hr and 4.20 mm/hr. After that, with the consecutive removal of bins 1 to 4, 1 to 5 and so on, the change in deviation is significant for all the rain rates considered. This gives a clear indication that the first 2 bins are affected severely by the distrometers’ dead time problem at rain rates from 1.96 mm/hr to 10.45 mm/hr. For higher rain rates of 22.80 mm/hr and above, the significant increase in deviation starts from the consecutive removal of bins 1 to 5. Therefore, (%) calculations that the first two it is concluded from mm/hr) bins can be neglected for lower rain rates ( and the first 4 bins can be neglected for higher rain rates mm/hr). ( From the above analysis, it can be assumed that the dead time problem is severe at the lower four bins. Therefore, taking into account of both the lower and higher rain rates, the first four bins are removed consecutively from bin 1 to bin 4 to redesign the gamma models in the following section. These truncated gamma models will be used in the specific rain attenuation and slant-path rain attenuation calculation in the following sections.

Fig. 4. Truncated gamma models with actual gamma model and measured DSD. (a) 4.20 mm/hr, (b) 66.54 mm/hr.

B. Truncated Gamma Models In order to obtain the gamma model less sensitive to lower bins; the first bin is removed and the moments are calculated from the remaining bins using (7). Then, the gamma model is redesigned using (9) to (11); next, the first two bins are removed and the moments are calculated using the remaining bins and using these moments the gamma model is redesigned; similarly, gamma models are redesigned for the removal of the first 3 bins and the first 4 bins. Fig. 4 shows the redesigned gamma models at 4.20 mm/hr and 66.54 mm/hr. From Fig. 4, it can be seen that, in general, all the gamma models fit well with the measured data at both 4.20 mm/hr and 66.54 mm/hr with the exception of one redesigned gamma model. This is where 4 bins are removed before the gamma model is redesigned at the rain rate of 4.20 mm/hr as shown in Fig. 4(a). There is slight deviation at the lower and higher drop diameters. The redesigned gamma models show the same trend as 4.20 mm/hr at the rain rates

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TABLE II MEAN SQUARE ERROR (%) FOR TRUNCATED GAMMA MODELS

1.96 mm/hr, 10.45 mm/hr and 22.80 mm/hr, where the removal of the first 4 bins shows a slight deviation at the lower and higher drop diameters. The deviation at the large diameter end is minimal at 66.54 mm/hr for the 4 bins removed model, however this model also deviates with the measured data at the lower diameters as shown in Fig. 4(b). All the gamma models fit well with the measured data at 120.30 mm/hr and 141.27 mm/hr at all drop diameters. The accuracy of the gamma models can be evaluated by calculating mean square error as a percentage using the formula %

(17) where is the number of bins in the measured DSD with data. MSE (%) is calculated using (17) for all the rain rates. Table II shows the MSE (%) for all the rain rates with different number of bins removed. The second column shows the MSE (%) of the actual MM346 model. The cells which are marked as “X” in the Table II have zero rain drops in the measured data. Therefore, the removal of that bins does not change the MSE (%). From Table II, the removal of bins from 1 to 4 will not make much difference for all the higher rain rates of 66.54 mm/hr and above. At 22.80 mm/hr, from Table II, the MSE (%) of the MM346 gamma model itself is high at 24.33%. This high MSE (%) is because the measured DSD at 22.80 mm/hr deviates more from the gamma model as compared to the DSD at other rain rates, especially at the lower drop size bins. The removal of bins from 1 to 3 increases the MSE from 24.33% for the actual gamma model to 32.11% for the redesigned gamma model, a difference of about 8%. Although the absolute MSE is large, the increase in percentage due to the removal of the bins is similar to that at other rain rates. For the lower rain rates of 1.96 mm/hr, 4.20 mm/hr and 10.45 mm/hr, the removal of the first 3 and 4 bins introduce higher deviations in MSE (%). It can be concluded that the removal of the first 4 bins with mean drop diameters of less than 0.77 mm mm/hr) whereas can be done for the higher rain rates (

Fig. 5. Specific rain attenuation at different frequencies using gamma model.

only lower 2 bins with mean drop diameters less than 0.55 mm mm/hr) and yet the can be removed at lower rain rates ( accuracy of the redesigned model is not affected. However, for terrestrial and earth-satellite communication links, the contributions of different rain drop diameters at different frequencies are important. The importance of small drop diameters increases at lower rain rates, it is necessary to check their contribution to the rain attenuation calculations especially at high frequencies. This is because, as frequency increases, the drop diameters relative to the wavelength becomes comparable, therefore, the attenuation caused by these small rain drops becomes significant. In the following section, the rain attenuation contribution of the rain drop diameters is studied. C. Specific Rain Attenuation The specific rain attenuation (in dB/km) is calculated using (12) at different rain rates for the frequencies 11 GHz, 28 GHz, 38 GHz and 48 GHz. Fig. 5 shows the specific attenuation in dB/km at different frequencies for both horizontal and vertical polarization using the MM346 gamma model. The specific rain attenuation increases with both the rain rate and the frequency

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TABLE III SPECIFIC RAIN ATTENUATION (DB/KM) USING FORWARD VERTICAL SCATTERING COEFFICIENTS FOR TRUNCATED GAMMA MODELS

increase. As seen in Fig. 5, the specific rain attenuation for vertical polarization is smaller than horizontal polarization at all frequencies. This is because, as the size of the rain drops increase, their shape tends to change from spherical to oblate spheroids. Furthermore [33)], rain drops may also be inclined (canted) to the horizontal because of vertical wind gradients. Thus, the depolarization due to rain can significantly depends on canting and tilt angle with drop vibration effects. The rain induced depolarization has been studied previously [34] by calculating the differential attenuation and the differential phase. The specific differential attenuation [29] is the difference between the specific rain attenuation for horizontal and vertical polarization and the differential propagation phase is defined as the difference of the imaginary part of the propagation constants of horizontally and vertically polarized waves. The cross-polar discrimination [33] (signal moved to orthogonal polarization/ signal in original polarization) will be different for horizontal and vertical polarization and in absolute value (linear units) will be higher for horizontal polarization (because horizontal polarization will be more attenuated). At low frequencies [34], as expected from the Rayleigh theory, the attenuation and the phase shift is greater in the horizontal polarization compared to the vertical polarization. Therefore, forward scattering coefficients for vertical polarization are used to calculate the specific rain attenuation for the redesigned gamma models. The specific rain attenuation contributions by individual drop diameters are calculated using vertical forward scattering coefficients at all the rain rates for all the frequencies. Fig. 6 shows the specific rain attenuation contribution of each bin for the four frequencies at 120.30 mm/hr. It is clear from Fig. 6 that the highest specific rain attenuation contribution moves to lower drop diameters as the frequency increases. Similarly, the highest attenuation contribution of all the other rain rates also moves to lower drop diameters as frequency increases. The smaller wavelengths of higher frequencies are comparable with the rain drop diameters, therefore, results in more attenuation at smaller drop diameters as frequency increases. Small drops tend to be Rayleigh scatterers. Larger drops due to small number and reduced increase of scattering

Fig. 6. Specific rain attenuation contributions at different frequencies using gamma model at 120.30 mm/hr.

coefficients become less significant [33]–[35]. Therefore, contribution is so a question of number of drops and wavelength. Most striking effects of bigger drops at high frequency are the reduction of differential phase that can be observed in depolarization measurements [34], [35]. The reversal in sign of the differential phase at high frequencies is a purely resonance phenomenon in which large drops produce negative differential phase outweighing the positive contribution from smaller ones. Major drops contribute mostly for differential attenuation. As a whole, less deformed smaller drops make a greater relative contribution to the total specific rain attenuation at high frequencies. Table III shows the calculated specific rain attenuation (dB/km) for the redesigned gamma models with 1, 2, 3 and 4 bins removed using (12) at the Q-band frequencies, 38 GHz and 48 GHz, at 0 elevation angle. Only specific rain attenuation for the Q-band is shown since they have the maximum change in attenuation. The cells which are marked as “X” in Table III have no rain drops in the measured DSD. The specific rain attenuation increases with both the rain rate and the frequency increase for all the truncated gamma models.

KUMAR et al.: TRUNCATED GAMMA DSD MODELS FOR RAIN ATTENUATION IN SINGAPORE

Fig. 7. Slant-path rain attenuation at the four frequencies using gamma model (120.30 mm/hr). at R

The point to note from Table III is that there is not much deviation in attenuation values if any of the lower 4 bins are removed for the redesigned gamma models at any rain rate. Although this change in specific rain attenuation is small, it is worthwhile to evaluate the slant-path rain attenuation values using (15), which are considered next. For slant-path rain attenuation calculations, the specific rain attenuation at different elevation angles are calculated at the four frequencies using forward scattering coefficients for vertically polarized waves for the redesigned gamma models. It is important to note that the specific rain attenuation increases with the increase of elevation angle for vertically polarized waves. D. Slant-Path Rain Attenuation The slant-path rain attenuation (in dB) exceeding 1% of the time is calculated using (13) to (15), with mm/hr, for all the redesigned gamma models at the four frequencies. This is for practical application purposes. From [36], the exceeding calculated slant-path rain attenuation (in dB) exceeding 0.1% of the time for a 0.01% of the time and coastal region like Calabar in Nigeria, where the highest avmm/hr, the erage annual accumulation results in a and are as high as 37.8 dB and 17 dB for 19.45 GHz (Ka-band), and 19.6 dB and 8.4 dB for 12.675 GHz (Ku-band) respectively. These rain attenuation values may exceed the fade margins of practical systems. Similarly, the calculated slant-path rain attenuation values for Singapore at these percentages of time are well above the feasible fade margins especially at high (in frequencies. Therefore, the slant-path rain attenuation dB) exceeding 1% of the time is selected for the study of the truncated gamma models. Fig. 7 shows the calculated slant-path rain attenuation using the gamma model at the four frequencies. As frequency increases, slant-path rain attenuation also increases. The increase in elevation angle decreases the slant-path rain attenuation since the slant-path length decreases. This is clear from Fig. 7 that attenuation decreases with the decrease in path length (increase in elevation angle) from the elevation angle 10 to 50 at the three

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higher frequencies. Above 60 , the slant-path rain attenuation increases again. The increase in slant-path rain attenuation at higher elevation angles may be due to the presence of convective rain cells [37]. These relatively small cross-sectional areas of intense rain extend from, or often above, the accepted freezing level to the ground. Therefore, the slant-path with the highest elevation will have its entire path virtually through the column of very heavy rain. In contrast, the slant-path with the lowest elevation due to its long path length may pass through a number of rain cells. However, because the horizontal extent of a cell is relatively small, the total rainfall in the path may be less than those for a higher elevation angle, resulting in less signal attenuation [37]. The calculated slant-path rain attenuation values show that there are insignificant changes for the redesigned gamma models with bins removed from actual gamma model at the four frequencies. The absolute difference between the actual gamma modeled and the redesigned gamma modeled slant-path rain attenuation is calculated by (18)

(18) is the slant-path rain attenuation exwhere ceeding 1% of time calculated from actual gamma model and (other ga) is the slant-path rain attenuation exceeding 1% of time calculated from redesigned gamma models with bins removed. There are no rain drops in the first two bins at 120 mm/hr; therefore, their slant-path attenuation changes are not plotted. Fig. 8 shows the slant-path rain attenuation changes for the redesigned gamma models with bins 3 and 4 removed as compared to the actual gamma model at 38 GHz and 48 GHz. As seen in Fig. 8, the changes in slant-path rain attenuation increase at both the frequencies especially at the lower and higher elevation angles for the redesigned gamma models with 4 bins removed from actual gamma model. From Fig. 8(a), it can also be seen that the removal of the first 4 bins results in less than 0.03 dB change in slant-path rain attenuation at 38 GHz. However as shown in Fig. 8(b), at 48 GHz, the removal of the first 4 bins results in a high change in slant-path rain attenuation of around 0.07 dB at the elevation angle of 90 . This shows that, as frequency increases, the relative contribution from the smaller drop size increases. At 120.30 mm/hr, there are few counts in bin 4; however, as the wavelength of the higher frequency is comparable with the drop diameter, their contribution to the slant-path rain attenuation increases. At the drop diameter 0.66 mm, the drop sizes are one twelfth of and one tenth of at 38 GHz and 48 GHz respectively. Since the change in slant-path rain attenuation are below 0.07 dB at all the frequencies for the redesigned gamma model with 4 bins removed, it can be concluded that, since the dynamic range of a satellite system is generally above 1 dB, the removal of the first 4 bins will not affect the satellite communication system for all frequencies within the Ku-band, Ka-band and Q-band. This validates Brawn’s recommendations of ignoring the counts in erroneous bins and the truncated gamma models can be used for

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dation to ignore the small drop diameters due to the dead time problem is valid for the calculation of slant-path rain attenuation of microwave links in Singapore. Therefore, the truncated gamma models using 3 , 4 and 6 moments, with the first 4 bins removed can be used for DSD modeling and rain attenuation calculations at Singapore. ACKNOWLEDGMENT The authors would like to thank Prof. M. Thurai (Colorado State University, USA) for providing the T-Marix code used in this paper. REFERENCES

Fig. 8. Slant-path rain attenuation changes for truncated gamma models compared to the actual gamma model. (a) 38 GHz, (b) 48 GHz.

DSD modeling and rain attenuation calculations. The analysis of DSD’s will hold for rain attenuation calculations, however, other uncertainties such as DSD variations along the path, temperature, bright band are also important factors to be considered. V. CONCLUSION This paper finds the contribution of drop size diameters from the calculation of rain rate using measured data for seven one minute rain rates. Gamma model with 3 , 4 and 6 moments is used to model DSD of Singapore in order to minimize the dead time problem. Truncated gamma models (redesigned gamma model) are designed with removal of lower bins. Specific rain attenuation (dB/km) and slant-path rain attenuation (in dB) exceeded for 1% of time are calculated at 11 GHz, 28 GHz, 38 GHz and 48 GHz using forward scattering coefficients for vertical and horizontal polarization using the gamma model. Slant-path rain attenuation changes show that the redesigned gamma models with the first 4 bins removed can be used at Ku-band, Ka-band and Q-band frequencies in Singapore mm/hr). This shows that Brawn’s recommen(

[1] J. O. Laws and D. A. Parsons, “The relation of raindrop-size to intensity,” Trans. Amer. Geophys. Union, vol. 24, pp. 452–460, 1943. [2] J. S. Marshall and W. M. Palmer, “The distribution of raindrops with size,” J. Meteor., vol. 5, pp. 165–166, 1948. [3] G. Feingold and Z. Levin, “The lognormal fit to raindrop spectra from frontal convective clouds in Israel,” J. Clim. Appl. Meteor., vol. 25, pp. 1346–1363, 1986. [4] C. W. Ulbrich, “Natural variation in the analytical form of the raindrop size distribution,” J. Climate Appl. Meteor., vol. 22, pp. 1764–1775, 1983. [5] P. T. Wills, “Functional fits to some observed drop size distributions and parameterization of rain,” J. Atmos. Sci., vol. 41, pp. 1648–1661, 1984. [6] P. L. Smith, “Raindrop size distributions: Exponential or gamma—Does the difference matter?,” J. Appl. Meteor., vol. 42, pp. 1031–1034, 2003. [7] B. E. Sheppard and P. I. Joe, “Comparison of raindrop size distribution measurements by a Joss-Waldvogel disdrometer, a PMS 2DG spectrometer, and a POSS Doppler radar,” J. Atmos. Ocean. Technol., vol. 11, pp. 874–887, 1994. [8] A. Tokay and D. A. Short, “Evidence from tropical raindrop spectra of the origin of rain from stratiform versus convective clouds,” J. Appl. Meteor., vol. 35, pp. 355–371, 1996. [9] C. Caracciolo, F. Prodi, A. Battaglia, and F. Porcu, “Analysis of the moments and parameters of a gamma DSD to inferprecipitation properties: A convective stratiform discrimination algorithm,” Atmos. Res., vol. 80, pp. 165–186, 2008. [10] C. Caracciolo, F. Prodi, and R. Uijlenhoet, “Comparison between Pludix and impact/optical disdrometers during rainfall measurement campaigns,” Atmos. Res., vol. 82, pp. 137–163, 2006. [11] L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “A gamma distribution of raindrop sizes and its application to Singapore’s tropical environment,” Microw. Opt. Tech. Lett., vol. 7, pp. 253–257, 1994. [12] J. T. Ong and Y. Y. Shan, “Modified gamma model for Singapore raindrop size distribution,” in Proc. Int. Geoscience and Remote Sensing Symp. IGARSS’97, Singapore, Aug. 3–8, 1997, vol. 4, pp. 1757–1759. [13] J. T. Ong and Y. Y. Shan, “Rain drop size distribution models for Singapore—Comparison with results from different regions,” in Proc. 10th Int. Conf. on Antennas and Propagation, Apr. 14–17, 1997, pp. 2.281–2.285, paper no. 436. [14] D. Brawn and G. Upton, “Estimation of an atmospheric gamma drop size distribution using disdrometer data,” Atmos. Res., vol. 87, pp. 66–79, 2008. [15] T. Kozu and K. Nakamura, “Rainfall parameter estimation from dualradar measurements combining reflectivity profile and path-integrated attenuation,” J. Atmos. Ocean. Technol., vol. 8, pp. 259–271, 1991. [16] P. L. Smith, D. V. Kliche, and R. W. Johnson, “The bias in moment estimators for parameters of drop size distribution functions: Sampling from gamma distributions,” presented at the AMS 32nd Conf. on Radar Meteorology, 15R.5, Albuquerque, NM, Oct. 24–29, 2005. [17] C. W. Ulbrich and D. Atlas, “Rain microphysics and radar properties: Analysis methods for drop size spectra,” J. Appl. Meteor., vol. 37, pp. 912–923, 1998. [18] K. I. Timothy, J. T. Ong, and E. B. L. Choo, “Raindrop size distribution using method of moments for terrestrial and satellite communication,” IEEE Trans. Antennas Propag., vol. 50, no. 10, pp. 1420–1424, 2002. [19] A. J. Illingworth and T. M. Blackman, “The need to represent raindrop size spectra as normalized gamma distributions for the interpretation of polarization radar observations,” J. Appl. Meteor., vol. 41, pp. 286–297, 2002.

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[20] J. Testud, S. Oury, R. Black, P. Amayenc, and X. Dou, “The concept of “normalized” distribution to describe raindrop spectra: A tool for cloud physics and cloud remote sensing,” J. Appl. Meteorol., vol. 40, no. 6, pp. 1118–1140, 2000. [21] Y. H. Lee, S. Lakshmi, and J. T. Ong, “Rain drop size distribution modelling in Singapore—Critical diameters,” presented at the 2nd Eur. Conf. on Antenna and Propagation (EuCAP 2007), Edinburgh, U.K., Nov. 2007, Conf. paper no. 0678. [22] J. X. Yeo, Y. H. Lee, and J. T. Ong, “Ka-band satellite beacon attenuation and rain rate measurements in Singapore—Comparison with ITU-R models,” presented at the IEEE AP-S Int. Symp. on Antennas and Propagation, Jun. 2009. [23] S. Dynes and T. Gordon, “38 GHz fixed links in telecommunications networks,” in Proc. Inst. Elect. Eng. Colloq. on Exploiting the Millimetric Wavebands, London, Jan. 7, 1994, pp. 5/1–5/4. [24] S. Zvanovec, P. Piksa, M. Mazanek, and P. Pechac, “A study of gas and rain propagation effects at 48 GHz for HAP scenarios,” EURASIP J. Wireless Commun. Networking, vol. 2008, Article ID 734216, 7 pages. [25] Propagation Data and Prediction Methods Required for the Design of Earth-Space Telecommunication Systems 2007, Recommendation ITU-R P.618-9. [26] J. T. Ong and C. N. Zhu, “Rain rate measurements by a rain gauge network in Singapore,” Electron. Lett., vol. 33, no. 3, pp. 240–242, 1997. [27] Distrometer RD-69 Instruction Manual 1993, Distromet Ltd. [28] R. Gunn and G. D. Kinzer, “The terminal velocity of fall for water droplets in stagnant air,” J. Atmos. Sci., vol. 6, no. 4, pp. 243–248, 1949. [29] M. Thurai, V. N. Bringi, and A. Rocha, “Specific attenuation and depolarization in rain from 2-dimensional video distrometer data,” IET Microw. Antennas Propag., vol. 1, no. 2, pp. 373–380, 2007. [30] H. R. Pruppacher and R. L. Pitter, “A semi empirical determination of the shape of cloud and raindrops,” J. Atmos. Sci., vol. 28, pp. 86–94, 1971. [31] “Rain Height Model for Prediction Methods,” 2001, Recommendation ITU-R P.839-3. [32] “Preferred Characteristics of Systems in the FS Using High Altitude Platforms Operating in the Bands 47.2–47.5 GHz and 47.9–48.2 GHz,” 2000, Recommendation ITU-R F.1500. [33] L. J. Ippolito, “Radio propagation for space communications systems,” Proc. IEEE, vol. 69, no. 6, pp. 697–727, Jun. 1981. [34] G. O. Ajayi, I. E. Owolabi, and A. Adimula, “Rain induced depolarization from 1 GHz to 300 GHz in a tropical environment,” Int. J. Infrared Millimeter Waves, vol. 8, no. 2, pp. 177–197, 1987. [35] K. Aydin and S. Daisley, “Rainfall rate relationships with propagation parameters (attenuation and phase) at centimeter and millimeter wavelengths,” in Proc. Int. Geosc. Remote Sensing Symp. (IGARSS 2000), 2000, vol. 1, pp. 177–179. [36] J. S. Ojo, M. O. Ajewole, and S. K. Sarkar, “Rain rate and rain attenuation prediction for satellite communication in Ku and Ka bands over Nigeria,” Progr. Electromagn. Res. B, vol. 5, pp. 207–223, 2008. [37] B. J. Bowthorpe, F. B. Andrews, C. J. Kikkert, and P. L. Arlett, “Elevation angle dependence in tropical region,” Int. J. Satellite Commun., vol. 8, pp. 211–221, 1990.

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Lakshmi Sutha Kumar received the B.Eng. degree from Bharadhidasan University, Thirichirappalli, India, in 1994, and the M.Tech. degree from Vellore Institute of Technology, Vellore, India, in 2005. She is currently working toward the Ph.D. degree at Nanyang Technological University, Singapore. From 1995 to 1998, she worked as a Lecturer at Bharadhidasan University and, from 1998 to 2002, at Pondicherry University, India. Her research interest includes microwave and millimeter-wave propagation and the study of the effects of rain on performance of microwave terrestrial and satellite communications.

Yee Hui Lee (S’96–M’02) received the B.Eng. (Hons) and M.Eng. degrees in electrical and electronics engineering from the Nanyang Technological University, Singapore, in 1996 and 1998, respectively, and the Ph.D. degree from the University of York, York, U.K., in 2002. Since July 2002, she has been an Assistant Professor at the School of Electrical and Electronic Engineering, Nanyang Technological University. Her interest is in channel characterization, rain propagation, antenna design, electromagnetic band gap structures, and evolutionary techniques.

Jin Teong Ong received the B.Sc. (eng.) degree from London University, London, U.K., the M.Sc. degree from University College, London, U.K., and the Ph.D. degree from Imperial College, London. He was with Cable and Wireless Plc, from 1971 to 1984. He was an Associate Professor at the School of EEE, Nanyang Technological (now Nanyang Technological University), Singapore, from 1984 to 2005, an Adjunct Associate Professor from 2005 to 2008, and Head of the Division of Electronic Engineeringm from 1985 to 1991. He is presently the Director of research and technology of C2N Pte. Ltd., a company set-up to provide consulting services in Wireless and broadcasting systems. His research and consulting interests are in antenna and propagation—in systems aspects of satellite, terrestrial and free space optical systems including the effects of rain and atmosphere; planning of broadcast services; intelligent transportation system; EMC/I and frequency spectrum management. Prof. Ong is a member of the Institution of Engineering and Technology.

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Comparison of TE and TM Inversions in the Framework of the Gauss-Newton Method Puyan Mojabi, Student Member, IEEE, and Joe LoVetri, Senior Member, IEEE Abstract—The Gauss-Newton inversion method in conjunction with a regularized formulation of the inverse scattering problem is used to invert transverse electric (TE) and transverse magnetic (TM) data. The utilized data sets consist of experimental data provided by the Institut Fresnel as well as synthetic data. The TE inversion outperformed the TM inversion when utilizing near-field scattering data collected using only a few transmitters and receivers. However, very little difference was found between TE and TM inversions when using far-field scattering data. It is conjectured that the reason for the better performance of the near-field TE result is that the near-field TE data contains more information than the near-field TM data at each receiver point. In all cases considered herein, the TE inversion required equal or fewer iterations than the TM inversion. The per-iteration computational complexity of both TE and TM inversions is discussed in the framework of the Gauss-Newton inversion method. Actual costs are consistent with the computational complexity analysis that is given. Index Terms—Gauss-Newton method, inverse problems, microwave imaging, remote sensing.

I. INTRODUCTION

T

HE electromagnetic inverse scattering problem considered herein consists of determining the electric constitutive parameters, i.e., permittivity and conductivity, of an unknown object inside a bounded imaging domain located in a known background medium. The inversion is obtained from measured field data exterior to the imaging domain when it is irradiated by a number of known incident fields. It is well-known that the inverse scattering problem is ill-posed: the solution to the mathematical problem is not guaranteed to be unique for most measurement configurations and does not depend continuously on the measured data [1]. This ill-posedness is usually treated by employing different regularization techniques. The other difficulty in solving the inverse scattering problem is that it is nonlinear with respect to the unknown contrast. The nonlinearity of the problem has led to the development of various iterative techniques during the past two decades. These iterative techniques attempt to minimize an appropriately constructed cost-functional. Two approaches based on the formulation of the problem using two different cost-functionals have been successfully used to solve the inverse scattering problem. Manuscript received March 14, 2009; revised August 24, 2009. Date of manuscript acceptance October 18, 2009; date of publication January 26, 2010; date of current version April 07, 2010. This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, MITACS and CancerCare Manitoba. The authors are with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB R3T5V6, Canada (e-mail: pmojabi@ee. umanitoba.ca). 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.2010.2041156

The first approach, which includes the Gauss-Newton inversion (GNI) method, uses the conventional cost-functional which is based on the difference between the measured and predicted scattered data for a particular choice of the material parameters; see for example [2]–[13]. The conventional cost-functional is usually augmented by various regularization techniques. The second approach, which includes the modified gradient method (MGM) [14] and the Contrast Source Inversion (CSI) method [15], uses the same conventional cost-functional, formulated in terms of contrast sources, in the case of CSI, added to an error functional involving the domain integral equation which relates the fields inside the imaging domain to the constitutive parameters of the unknown object. This latter functional is formulated in terms of the contrast and contrast sources. Although researchers have developed full-vectorial 3D inversion algorithms [5], [16], [17] (also see the papers in Inverse Problems special section [18]), the 2D algorithms considered herein are also very important because of their use in existing experimental systems. For example, in the microwave biomedical imaging systems developed at Dartmouth College for breast cancer imaging, the data is collected in seven different planes and a 2D Transverse magnetic (TM) GNI algorithm is used to invert the data [7]. The usefulness of this 2D assumption for biomedical imaging has been verified in [19]. Various 2D TM inversion algorithms have been tested with experimental data whereas only a few 2D Transverse Electric (TE) inversion methods have been investigated against experimental data. The 2D TM problem can be formulated as a scalar problem for a single electric field component. This is not the case for 2D TE problems where both electric field components in the transverse plane need to be taken into account in the formulation which results in a more complex (i.e., vectorial) formulation compared to the TM case. It should be noted that TE problems can also be formulated as scalar problems for a single magnetic field component. However, for the TE inversion, it has been shown in [20] that inverting the integral equation of the two electric field components is more stable and has better performance than inverting the integral equation of the single magnetic field component. From a physical perspective, the TE-polarized case includes polarization charges at dielectric discontinuities, which are difficult to model numerically [21]. On the other hand, TE-polarized data may contain more useful information about the object of interest as it is based on two different components of the electric field as opposed to one in the TM-polarized case. Note that these two polarizations are physically uncoupled: they provide independent information about the object being imaged. This fact can be used to improve the reconstruction in tomographic configurations by either simultaneously inverting TE and TM data [22] or using a cascaded TE-TM algorithm [23], [24].

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MOJABI AND LOVETRI: COMPARISON OF TE AND TM INVERSIONS IN THE FRAMEWORK OF THE GAUSS-NEWTON METHOD

There are only a few reports on the inversion of TE experimental data (using any method). In the special edition of Inverse Problems dedicated to inversions of the first Fresnel data set [25], only two papers dealt with the single TE case data that was provided: the first one [26] was concerned with determining the shape of the conducting u-shaped scatterer and the second one [27] used the multiplicative regularized contrast source inversion (MR-CSI) method to reconstruct the dielectric contrast of this scatterer. In the second special edition from Inverse Problems dedicated to the second Fresnel data set [28], [29], which includes TE and TM data for four targets, only two contributions addressed the TE-polarized data: the first one [30] applied the MR-CSI method to reconstruct the constitutive parameters of all the unknown objects in the data set and in the second contribution [31], a TM inversion algorithm based on the Diagonal Tensor Approximation and the Contrast Source Inversion method (DTA-CSI) was applied to invert the TE-polarized data. This last contribution uses a calibration of the TE data in a way that, according to the authors, allows the use of the scalar TM inversion algorithm. In addition, a 2D TE bi-conjugate gradient inversion method is used in [24] to reconstruct buried objects from experimental TE scattering data. In [32] an iterative multi-scaling approach was applied to the single u-shaped metal target case from the first Fresnel data set, in both TE and TM illuminations. Most recently, a TE stochastic inversion algorithm which utilizes a priori information about the object of interest has been used to reconstruct the second Fresnel data set [33]. In this paper, the GNI method is applied to a regularized formulation of the inverse scattering problem for inverting the complete second TE Fresnel data set which are combinations of lossless dielectric and metallic cylinders. As the Fresnel data contains only far-field scattering data, we also show the performance of the TE inversion against near-field synthetic scattering data. These TE inversions are compared with the TM inversions of the same targets. The motivation for moving to the near-field is that it is postulated that near-field TE data may contain more information than near-field TM data. This does not hold in the , where far-field, because in the far-field assuming denotes the electric field and is shown in Fig. 1, is a good approximation for the TE case and is easily recoverable using measurements. In the near-field such an approximation is not valid and therefore two orthogonal field components need to be measured independently. This is difficult in practice and is one reason why 2D TE near-field microwave tomography systems have not been constructed in the past. It should be noted that the two orthogonal electric field components of TE near-field configurations can be extracted by measuring the single magnetic field component and then taking the derivative thereof. To compute an accurate derivative, magnetic field measurements must be performed in close proximities, which can cause difficulties in microwave tomography systems with co-resident antenna arrays (e.g., coupling between the co-resident antennas [34]). However, in TE far-field configurations, one can measure the single magnetic field component and then use a plane-wave approximation in order to extract the electric field from the magnetic field. The result of the present investigation may be useful for justifying the added cost of developing such systems. The main

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Fig. 1. Geometrical model of the inverse scattering problem (z^ is the unit vector pointing outside the x-y plane and ' ^ is the unit vector in the azimuthal direction).

contribution of this paper is to provide a quantitative comparison of TE and TM inversions of synthetic and experimental data sets for various cases including near-field and far-field imaging. This includes a comparison of computational complexity, image quality and convergence rate. The paper is organized as follows. The formulation of the mathematical problem is given in Section II. Brief descriptions of the utilized Gauss-Newton inversion method and the forward solver used are given in Sections III and IV, respectively. A discussion of the computational complexity of the utilized algorithm for the TE and TM inversions is presented in Section V. Sections VI and VII provides the reconstruction results for all targets in the data sets. Finally, the results will be summarized in Section VIII.

II. PROBLEM STATEMENT Consider a bounded imaging domain containing the object of interest and a measurement domain outside of (see Fig. 1). Let and denote position vectors in the plane and define the complex electric contrast which is to be determined as

(1) where is the background permittivity and is the permittivity inside the imaging domain at the point . In general, these permittivities are complex to allow the modeling of lossy material. The total electric field is represented by two rectangular , in the TE case, and only one components, , in the TM case. The scattered electric component, field is then defined as where denotes the incident field. Throughout the analysis, all material properties are taken to be non-magnetic: the permeability is taken as that of free-space, . The wavenumber of the background medium is denoted by . A time factor of is . The symbols implicitly assumed in this paper where and represent the radial frequency and time respectively.

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The so-called data equation in terms of the unknown contrast can be written as,

. The discretized inthe measured scattered field, i.e., verse scattering problem is formulated as the minimization of the least-squares data misfit

(2) (6) and is the dyadic Green’s function for the where background medium. Assuming the TE case, may be written as [35],

(3) is the 2D scalar Here is the 2D identity dyad and Green’s function for the homogeneous background. The symbol represents the gradient operator which is taken with respect . to the subscript variable. For the TM case, Note that the data equation is nonlinear with respect to the unknown contrast as the electric field inside the imaging domain is a function of . That is, the electric field inside the imaging domain is given via the domain equation,

where represents the simulated scattered field vector at the measurement points due to the predicted contrast and denotes the -norm on . III. GAUSS-NEWTON INVERSION METHOD The Gauss-Newton Inversion (GNI) method is based on Gauss-Newton optimization [36] where the nonlinear cost-functional is approximated with a quadratic form at the current iteration that ignores the second-order derivatives. The stationary point of the quadratic model is then chosen as the next iterate. Herein, the cost-functional to be minimized is chosen to be an additive-multiplicatively regularized form of , (6), to overcome the ill-posedness of the the data misfit inverse scattering problem. That is, we apply the GNI method to the following cost-functional [5], [37], [38], (7)

(4) . The inverse scattering problem may then be forwhere mulated as the minimization over of the least-squares data misfit cost-functional,

is the regularizer and is a small positive paramwhere eter which is determined by the user in an ad hoc way. The regularizer is chosen to be the -norm total variation of the contrast vector over the imaging domain. That is, is the discrete form of the cost-functional

(8) (5) where denotes the measured scattered field on and denotes the -norm on . It should be mentioned that for the TE inversion, we can also use the magnetic field integral equation which can be derived by taking the curl of both sides of (2) with respect to . However, we do not use the magnetic field formulation for the inversion as it has been shown in [20] that the TE inversion using the electric field formulation is more stable and has better performance than that using the magnetic field formulation. Herein, we consider a discrete nonlinear inverse scattering problem where the number of measured data is limited, say , is discretized into cells using and the imaging domain 2D pulse functions. Therefore, the measured scattered data on the discrete measurement domain is denoted by the complex and the contrast function is represented vector . We further assume that in by the complex vector is ordered in such a way that the TE case the vector it contains the -component of the measured scattered field, , at all observation points followed by the corre. However, in the TM sponding -component, consists only of the -component of case, the vector

where is the area of the imaging domain. We then minimize (7) using the Gauss-Newton method and the contrast vector at iteration is updated as (9) Here the step-length is a real positive number which is determined using a line search algorithm described below. The is found as in [5] by solving Gauss-Newton correction (10) is the simulated scattered field vector at the obserwhere denotes the vation points due to the contrast . The matrix Jacobian matrix containing the derivative of with respect to and evaluated at . The superscript stands for the complex conjugate transpose. The matrix represents the discrete form of where represents the Laplacian operator. Assuming that the contrast function is zero on the boundary of , the matrix is a negative definite matrix [38]. Therefore, the represents a positive definite matrix; thus, matrix

MOJABI AND LOVETRI: COMPARISON OF TE AND TM INVERSIONS IN THE FRAMEWORK OF THE GAUSS-NEWTON METHOD

ensuring a descent direction for the Gauss-Newton correction is calculated as [5], [36]. The positive parameter

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(4) by the contrast and formulating the domain equation in terms of the contrast sources as (13)

(11) where in (10) Note that the regularization weight is controlled by decreases throughout the iterations, because and that as is minimized, the regularization is lessened; thus, providing an adaptive regularization [38], [39] for the inversion algorithm. The details of the algorithm are not presented here but are available in [5]. is determined using a line search algoThe step-length rithm and is based on that described in [8] and [10]. In this line , and search algorithm, we start with the full step, i.e., check whether it satisfies,

(12) where is a small positive number (set to be ) and is the decrease rate of at in the direction of . If satisfies (12), we choose it as an appropriate step-length; otherwise we reduce the step-size along until we find a which satisfies (12). In this procedure, the function is approximated by a quadratic expression in terms of and a new candidate for the step-length is then found by minimizing this quadratic form. As in [8], the minimum possible value for is set to 0.1. If the step-length becomes less and terminate the line search algothan 0.1, we choose rithm. The inversion algorithm terminates if one of the following is less than three conditions is satisfied: (i) the data misfit ), (ii) the difference between a prescribed error (set to be two successive data misfits becomes less than a prescribed value ), or (iii) the total number of iterations exceeds (set to be a prescribed maximum (set to be 50 for the single-frequency inversion).

denotes the identity operator and

is defined as

(14) After finding the contrast sources from the discrete form of (13) can be using the CG-FFT algorithm, the total field inside simply calculated from (15) To further accelerate the forward solver, we have used the marching-on-in-source-position technique [5], [40] where an appropriate initial guess for the CG-FFT algorithm with respect to a specific transmitter position is obtained via an extrapolation of the fields corresponding to some previous transmitter positions. In this paper, the initial guess for the th transmitter to be used in the CG-FFT algorithm applied to the discrete form of (13) is written as

(16) where to efficients norm [5],

is the converged solution of (13) with respect transmitter. A closed-form expression for the cois available such that they minimize the following

(17) denotes the -norm on . For the first three transwhere mitters, we have used a zero initial guess.

IV. THE FORWARD SOLVER

V. THEORETICAL COMPUTATIONAL COMPLEXITY ANALYSIS

In each iteration of the GNI algorithm, the forward solver is called several times to compute the simulated scattered field at the observation points and its derivative with respect to the current estimate of the contrast. In addition, the line search algorithm requires calling the forward solver to evaluate (12). Therefore, having a fast forward solver is essential for this inversion algorithm. The forward solver is concerned with solving a linear well-posed system of equations which is solved using the Conjugate Gradient (CG) technique and accelerated using the Fast Fourier Transform (FFT). This is possible because of the convolutionary form of the domain operator when the imaging domain is discretized uniformly using pulse basis functions in the and directions. To accomplish a CG-FFT forward solver, a procedure similar to [16] is adopted where the domain equation, (4), is formulated in terms of the so-called contrast sources, . This can be done by multiplying defined as

A description of the per-iteration computational complexity of the utilized TE and TM GNI algorithms is now given. The following conventions are used: the number of transmitters is , denoted by , the total number of receiver positions by and the number of receiver positions per transmitter by . The number of CG iterations required for the TE and TM forward and , respectively. The number solvers are denoted by of CG iterations to find the Gauss-Newton correction in the TE and , respectively. The and TM cases are denoted by contrast function is discretized on a uniform grid using 2D and denoted by the contrast pulse functions , vector whose th component is represented by . A. Jacobian Matrix Each row of the Jacobian matrix will correspond to a combination of the scattered field at a receiver located at, say, and

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polarized along some direction, say, and due to one transmitter, say, the th transmitter. That is, one row for each individual datum of the collected data. The ordering of the rows will obviously depend on the ordering of this data, but the th element in such a row will correspond to the derivative of this . This element may be found scattered field with respect to using an adjoint formulation [41] as [5], [8]

(18) In the TE case, is the dyadic Green’s function for the inhomogeneous background, which is the predicted scatterer at the current GNI iteration, evaluated at the point due to the source at the point (also called the distorted dyadic Green’s function). In the TM case, where is the 2D scalar Green’s function for the inhomogeneous background. is the total field inside the imaging domain due to Also, transmitter and corresponding to the predicted contrast. the For our cases, the polarization direction is considered to be either or in the TE case and in the TM case. difFinding the distorted dyadic Green’s function for the ferent receiver positions requires calling the forward solver times in the TE case and times in the TM case. This is due to the fact that two different polarizations should be considered in the TE illumination while only one polarization is needed for the for TM illumination. The computational cost for finding calls of the forward solver different transmitter locations is for both TE and TM cases as the TE-polarized data is calibrated (or synthetically created) using an infinite magnetic line source directed in direction. In our implementation, the elements of the matrix , as given in (18), are not found explicitly as we only need to do the right , see (10). Therematrix-vector multiplication using and fore, the integration and the dot-product , as required ) operates on a vector of the in (18), is computed when (or proper size and will be considered in the computational complexity of finding the Gauss-Newton correction.

conforming vectors. Using (19) and (20), it can be concluded requires approximately that the matrix-vector multiplication operations in the TE case and operations in the TM case. The same conclusion can be drawn for multiplying the by an arbitrary vector of the correct size. Therefore, matrix , as required the computational cost of calculating in the TE case and in the TM in (10), is about case. The matrix for a rectangular imaging domain is a symmetric block Toeplitz matrix with Toeplitz blocks [42, p. 100], so its multiplication with a vector can be accelerated using is neglected the FFT; thus, the computational cost of . Therefore, the computacompared to that of tional cost for finding the Gauss-Newton correction is about for the TE case and for the TM case. Note that each iteration of the CG algorithm requires , the two matrix-vector multiplications. Assuming computational complexity of finding the correction in the TE case is almost four times more than that in the TM case. C. The Forward Solver The CG-FFT forward solver, applied to the discrete form of operator and its (13), requires the definition of the adjoint. In the TE case, this operator may be defined as [16]

(21) and in the TM case, as (22) where as

is the component of

. Also,

and

are defined

(23) (24)

B. The Gauss-Newton Correction in (10) using CG requires multiplying by a Solving vector and this requires approximately multiplications multiplications in the TM case. This in the TE case and can be explained as follows: in the TE case, the multiplication can be written of the Jacobian matrix with a vector as,

(19) and in the TM case as (20) where represents the matrix form of the -component of the distorted dyadic Green’s function. The operation denotes the elementwise product (Hadamard product) of two

As pointed out in [16], the discrete forms of both and can be computed by FFT routines. The discrete form of may be computed by multiplying a symmetric block Toeplitz matrix . However, the discrete form of with Toeplitz blocks with can be computed by multiplying a symmetric Toeplitz maand . Due to the fact that the matrix-vector multrix with tiplication by a symmetric block Toeplitz matrix with Toeplitz blocks is more expensive than that by a symmetric Toeplitz matrix [43], we ignore the computational complexity of finding compared to that of finding . Using this approximation, the computational cost of multiplying the discrete form of by an arbitrary vector of the correct size in the TE case is roughly two times of that in the TM case as the TE forward and whereas the TM solver requires the calculation of forward solver only requires the calculation of . Using the above approximation, it can also be shown that the computational complexity of multiplying the discrete form of with an arbitrary vector of the adjoint operator of

MOJABI AND LOVETRI: COMPARISON OF TE AND TM INVERSIONS IN THE FRAMEWORK OF THE GAUSS-NEWTON METHOD

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the correct size in the TE case is also two times of that in the TM case. Therefore, the per-iteration computational complexity of the TE CG-FFT algorithm, utilized in the forward solver, is roughly twice that of the TM case. D. Line Search The computational cost of the utilized line search algorithm is approximately equal to that of evaluating for the known background Green’s function and this is equal to times for both TE and TM cases. calling the forward solver As mentioned earlier, if the full step satisfies the condition (12), we choose it as an appropriate step-length. From our experience with the regularized cost functional (7), the full step mostly satisfies the condition (12); therefore, very few calls to this line search algorithm are made in the cases that we have run. This can be explained as follows. In the Gauss-Newton optimization, may lead to an increase in the cost-functhe correction , see (10), is not positive-definite, or tional if (i) (ii) the quadratic model of the nonlinear regularized cost-funcat is not a good approximation to [4]. As tional is positive pointed out in Section III, the matrix definite. Moreover, due to the use of adaptive regularization, the is maximum at early GNI iterations regularization weight where the predicted contrast can be very far from the true solution. Thus, at early GNI iterations, the quadratic model of is dominated by that of the regularizer. Noting that the regularizer is an -norm, the quadratic model of the regularized cost-functional has a good chance to be a good approximation at early GNI iterations. As the algorithm gets closer to of is lessened. Thus, the true solution, the regularization weight the quadratic model of the regularized cost-functional is dominated by that of the data misfit functional. Due to the fact that the predicted contrast is close to the true solution, the quadratic model of the regularized cost-functional has a good chance to . Therefore, the use of adapbe a good approximation of tive regularization will usually make the quadratic model of the . regularized cost-functional be a good approximation to VI. INVERSION RESULTS The inversion results from both synthetic and experimental data are now shown. To be able to compare the TE inversion with the TM inversion, we introduce an image error cost-functional defined as

(25) where is the final reconstruction, is the true contrast and denotes the -norm on . For the experimental data, is created according to the geometrical configurations and the average permittivity of the object being imaged. For the synthetic data, as the data is generated on a different grid than the one used in the GNI algorithm (to avoid inverse crime), the image error cost-functional (25) is calculated by interpolating onto a finer and finer mesh until the norm converged. For the synthetic data sets, all parameters of the forward solver are kept the same for TE and TM polarizations. We have also added 3%

Fig. 2. The exact contrast of the scatterer for the synthetic test case. (a) Real(). (b) Imag().

RMS additive white noise to the synthetic data set using the formula given in [44]. A. Synthetic Data: Concentric Squares We consider a similar test case which has been used in [15], [20], [45]. The scatterer consists of two concentric squares with ( is the wavean inner square having dimension of . length in the background medium) with a contrast of The inner square is surrounded by an exterior square having and contrast . The exact consides of trast profile is shown in Fig. 2. The frequency of operation is chosen to be 1 GHz and free space is assumed for the backconsists of a square ground medium. The imaging domain . We consider three different scenarios for having sides of collecting the data. In the first scenario, we choose 10 transon the measurement mitters and 10 receivers circle and in the second scenario, we choose 30 transmitters on . Therefore, the length and 30 receivers in the second scenario is 9 times that of of the vector in the first scenario. In these two scenarios, the transmitters and receivers are placed evenly on the measurement circle of radius . In the third scenario, we choose evenly placed 10 transmitters and 10 receivers . The on the measurement circle of radius forward data is then generated on a grid of 30 30 for both TE and TM polarizations. The transmitters for the TE and TM cases are the magnetic line source and electric line source reand components are colspectively. For the TE case, lected at the receiver positions whereas in the TM case, the component is collected. We will note that the synthetically collected data in the first and second scenarios may be considered as the near-field data whereas the collected data in the third scenario is at far-field. For the first scenario, the TE and TM inversions are shown in Fig. 3. As can be seen, both TE and TM inversions provide good reconstructions for the real part of the contrast profile. However, the TM inversion is not successful in reconstructing the imaginary part of the contrast: the inner square is unresolved in the imaginary part of the TM inversion. It should be noted that when the number of transmitters/receivers decreased to 8, the TE inversion also failed (not shown here) in reconstructing this target. The TE and TM inversions for the second scenario are shown in Fig. 4. In this case, both TE and TM inversions are successful in reconstructing real and imaginary parts of the contrast. For the third scenario which utilizes the same number of transmitters

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Fig. 4. Inversion of the synthetic data set (the second scenario: T = 30 and = 10 = R = 30) (a)–(b) TE case (c)–(d) TM case (e)–(f) cross-section at x = = 0R. (a) Real(). (b) Imag (). (c) Real(). (d) Imag (). (e) Real(). (f) ( ) Imag ().

Fig. 3. Inversion of the synthetic data set (the first scenario: T and R R ) (a)–(b) TE case (c)–(d) TM case (e)–(f) cross-section at x . (a) Real  . (b) Imag  . (c) Real  . (d) Imag  . (e) Real  . (f) Imag  .

0

=

= 10 () ( )

( )

( )

( )

and receivers as in the first scenario but located in far-field, the TE and TM inversions are shown in Fig. 5. In this case, the TE and TM inversions are very similar. The number of GNI iterations utilized to reconstruct this target and the value of in these three different scenarios are given in Tables I and II. That the TE inversion outperforms the TM inversion in the first scenario is probably due to the fact that the TE near-field data contains more information than the TM near-field data (the in the TE case is twice that in the TM length of the vector case). Noting that the measurement circle is in the near-field for this test case, it is expected that and provide non-redundant information. However, when the number of transmitters and receivers increases in the second scenario, the TM scattering data provides sufficient information to reconstruct the object with a reasonable accuracy while the TE inversion also provides a good reconstruction in this case. Comparing the inversion results for the first and third scenarios, we speculate that the TE far-field data does not provide extra information compared to the TM far-field data. B. The Second Fresnel Experimental Data Set For the second Fresnel data set [28], the transmitting and receiving antennas are both wide-band ridged horn antennas and are located on a circle with radius 1.67 m. The targets, see Fig. 6,

are all long circular cylinders and have no variations in the longitudinal direction. Both TE and TM polarizations are measured for each target where the background medium is free space. In the TM illumination, the -component of the total and incident electric fields are collected for different transmitter positions and frequencies. In the TE illumination, the -component of total and incident electric fields are measured (the direction is depicted in Fig. 1). The scattered field is obtained by subtracting the measured incident field from the measured total field. The scattered field is then calibrated by approximating the horn transmitting and receiving antennas by line transmitters and receivers (electric line source for TM illumination and magnetic line source for TE illumination). The calibration procedure adopted is that explained in [27] where a single calibration factor per transmitter is used: for each transmitter the calibration factor used is the ratio of the simulated incident field to the measured incident field for the receiver point opposite to the transmitter (this factor is used for all receiver points). Note that in the TE-polarized data provided by the Fresnel group, only , has been measured. one component of the electric field, i.e., and This component is then calibrated and converted to to be used by the inversion algorithm. The FoamDielInt and FoamDielExt targets are illuminated by 8 transmitters and the measured data is collected at 9 different frequencies from 2 GHz to 10 GHz with a step of 1 GHz at

MOJABI AND LOVETRI: COMPARISON OF TE AND TM INVERSIONS IN THE FRAMEWORK OF THE GAUSS-NEWTON METHOD

TABLE II IMAGE ERROR COST—FUNCTIONAL

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M()

= 10

Fig. 5. Inversion of the synthetic data set (the third scenario: T and R R and the transmitters/receivers are located in far-field) (a)–(b) TE case (c)–(d) TM case (e)–(f) cross-section at x . (a) Real  . (b) Imag  . (c) Real  . (d) Imag  . (e) Real  . (f) Imag  .

=

= 10 ( )

( )

( )

=0

( )

()

( )

TABLE I NUMBER OF GNI ITERATIONS REQUIRED FOR THE CONVERGENCE (MULTIPLE-FREQUENCY INVERSION)

241 points per transmitter. The FoamTwinDiel target is irradiated by 18 transmitters and the number of receivers and frequencies stays the same as two previous cases. For the FoamMetExt target, the numbers of transmitters and receivers are the same as those for FoamTwinDiel target but the object is illuminated at 17 different frequencies in the range of 2 GHz to 18 GHz with 1 GHz step. For all these targets, the imaging domain, , is a 15 cm 15 cm square region which is discretized into a uniform grid. For the single-frequency inversion, as the initial guess to the GNI method and for the we use multiple-frequency inversions, the frequency-hopping approach [46] is used where the reconstructed image from low-frequency

Fig. 6. The targets of the second Fresnel data set. (a) FoamDielInt. (b) FoamDielExt. (c) FoamTwinDiel. (d) FoamMetExt.

data is used as an initial guess for the inversion of high-frequency data. For the FoamMetExt target, we have limited the as maximum value of the imaginary part to be 4 at otherwise the imaginary part of the metal cylinder will become too high (on the order of 200), making the convergence of the forward solver difficult. Therefore, if the imaginary part of the contrast of this target becomes more than four, it is set to four. 1) Multiple-Frequency Inversion: The multiple-frequency inversion results for the Fresnel targets are shown in Figs. 7–10. For all these four targets, the TE and TM inversions have been successful in reconstructing the targets with a reasonable accuracy. For the FoamDielInt, FoamDielExt and FoamTwinDiel, the reconstructed imaginary parts of both TE and TM inversions are small which indicates that these three targets are lossless. For the FoamMetExt target, it can be seen that the shape of the dielectric cylinder is reconstructed well in the TE case whereas its shape in the proximity of the metallic cylinder is not reconstructed in the TM case. Also, for both polarizations the reconstructed real part of the metallic cylinder is close to zero whereas the imaginary part is indicated to be an object of high loss. The number of GNI iterations required to reconstruct these objects is listed in Table I which shows a faster convergence for the TE inversion of the Fresnel targets. The value of the image

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Fig. 7. FoamDielInt reconstruction (a)–(b) TE case (c)–(d) TM case. (a) Real(). (b) Imag(). (c) Real(). (d) Imag().

Fig. 8. FoamDielExt reconstruction (a)–(b) TE case (c)–(d) TM case. (a) Real(). (b) Imag(). (c) Real(). (d) Imag().

error cost-functional for TE and TM inversions of these targets is given in Table II which shows a relatively similar reconstructions for the TE and TM inversions. 2) Single-Frequency Inversion: To investigate the single-frequency inversion of the experimental Fresnel data for the TE case, we show the reconstruction results of FoamTwinDiel target at and compare it to the TM inversion at the same frequency. In both TE and TM inversions, we start the inveras the initial guess. The algorithm sion algorithm with converged after 7 iterations for the TE case and 25 iterations for the first iteration was for the TM case. The data misfit 0.3803 for the TE case and 0.3809 for the TM case. However, in the final reconstruction, the data misfit reduced to 0.0285 for

Fig. 9. FoamTwinDiel reconstruction (a)–(b) TE case (c)–(d) TM case. (a) Real(). (b) Imag(). (c) Real(). (d) Imag().

Fig. 10. FoamMetExt reconstruction (a)–(b) TE case (c)–(d) TM case. (a) Real(). (b) Imag(). (c) Real(). (d) Imag().

the TE case and 0.0266 for the TM case. The data misfit for different iterations of the inversion algorithm for both TE and TM inversions is shown in Fig. 11. The TE inversion, Fig. 12(a)–(b), overshoots the real part of the contrast for the external cylinder while the TM inversion, Fig. 12(c)–(d), is very close to the true contrast. The value of the image error cost-functional for the single-frequency TE and TM inversions of the FoamTwinDiel is given in Table II. As far as the computational complexity of the TE and TM inversions is concerned, the inversion codes have been written in object-oriented Matlab and all the computations are performed on a computer with a quad-core 2.66 GHz Intel processor and 2 GB of RAM. As an example, we consider the FoamDielInt target , , and . In the first where

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case to find the inhomogeneous Green’s function without using the marching-on-in-source-position technique. However, the update procedure took just 295 sec for the TE case and 53 sec for the TM case when this technique was used. It is important to note that for experimental tomographic systems where the receiver positions are the same as transmitter positions, which is the case for most practical microwave imaging systems currently in existence, computational savings can be made in updating the Green’s function of the inhomogeneous background using the already updated total field corresponding to each transmitter.

Fig. 11. The data misfit FoamTwinDiel target at f

F

for the single-frequency inversion of the .

= 6 GHz

= 6 GHz ( )

Fig. 12. Single-frequency reconstruction of FoamTwinDiel at f (a)–(b) TE case (c)–(d) TM case. (a) Real  . (b) Imag  . (c) Real  . (d) Imag  .

( )

()

( )

GNI iteration at we have and for the TE case whereas in the TM case, and . Finding the Gauss-Newton correction took about 320 sec for the TE case and 79 sec for the TM case. That is, finding the correction in the TE case is about 4 times more expensive than that in the TM case which matches the expected theoretical ratio. Also, for each transmitter, the forward solver took about 0.99 sec in the TE case and 0.31 sec in the TM case showing that the per-iteration computational complexity of the TE forward solver is about 2.4 times more than that of the TM case which is very close to the approximate theoretical ratio. Also, in the inversion of the FoamDielInt target, the line search algorithm was called once for each frequency in both polarizations. The computational cost can be significantly alleviated by using the marching-on-in-source-position technique [5], [40] and . For example, in the which essentially reduces first GNI iteration for the FoamDielInt target at , it took about 691 sec for the TE case and 114 sec for the TM

VII. THE GAUSS-NEWTON INVERSION WITH A KRYLOV SUBSPACE REGULARIZATION METHOD To verify that the above results are not due to the specific use of the additive-multiplicative regularization, we also use a Krylov subspace regularization method. Specifically, we use the conjugate gradient least squares (CGLS) regularization technique [7], [38], [47], as a Krylov regularization method, in conjunction with the GNI method. We will refer to this inversion algorithm as GNI-CGLS in this paper. We have three main reasons to use this specific regularization method. First, the CGLS regularization method provides a basic and simple regularization method. It can be shown that this regularization provides similar results to truncated singular value decomposition (TSVD) and standard-form Tikhonov regularization [42, pg. 50], [48, p. 146], [38], [49]; mainly due to the similarity between the Krylov subspace basis and the SVD basis. Second, the CGLS regularization provides computationally more efficient regularization compared to the TSVD and Tikhonov regularization. Third, an adaptive regularization parameter choice method for the CGLS regularization technique has been presented in [7] and successfully used for the inversion of experimental biomedical data such as the ones collected from human breast and forearm [7], [47]. The GNI-CGLS method was applied to the the first scenario of the synthetic data set and the inversion result is shown in Fig. 13. Similar to the inversion result using the GNI method with the additive-multiplicative regularizer shown in Fig. 3, the TE GNI-CGLS inversion outperforms the TM GNI-CGLS inversion. The TE GNI-CGLS inversion converged after 7 iterations whereas the TM GNI-CGLS inversion converged after 12 is 0.08 for the TE iterations. The image cost-functional GNI-CGLS inversion and 0.11 for the TM GNI-CGLS inversion. For the second and third scenarios of the synthetic data set as well as the Fresnel experimental data set, the inversion results from the GNI-CGLS method (not shown here) were very similar to those from the GNI method with the additive-multiplicative regularizer. Similar to the GNI method with the additive-multiplicative regularizer, the TE GNI-CGLS method requires equal or less iterations than the TM GNI-CGLS method. For example, , the for the single-frequency FoamTwinDiel case TE GNI-CGLS method converged after 15 iterations whereas the TM GNI-CGLS method converged after 27 iterations. The data misfit for different iterations of the GNI-CGLS method for both TE and TM polarizations is shown in Fig. 14. Similar to the GNI with the additive-multiplicative regularizer shown in

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Fig. 14. The data misfit FoamTwinDiel target at f

F

for the single-frequency inversion of the

= 6GHz using the GNI-CGLS method.

Fig. 15. The data misfit F for the single-frequency inversion of the FoamTwinDiel target at f using the GNI-CGLS method with the derivative-free line search algorithm.

= 6 GHz

where the TE and TM inversions converged in 14 and 28 iterations respectively. It can easily be seen that this convergence is very similar to the convergence of the GNI-CGLS method using the derivative-based line search algorithm shown in Fig. 14.

= 10

Fig. 13. Inversion of the synthetic data set (the first scenario: T and R R ) using the GNI-CGLS method (a)–(b) TE case (c)–(d) TM case (g)–(h) cross-section at y . (a) Real  . (e)–(f) cross-section at x (b) Imag  . (c) Real  . (d) Imag  . (e) Real  . (f) Imag  . (g) Real  . (h) Imag  .

=

( )

= 10 ( )

( )

=0 ( )

( )

( )

=0

( )

()

Fig. 11, the TE inversion converged faster than the TM inversion. To check the sensitivity of the convergence rate to the line search algorithm described in Section III, we have also used another line search technique. This line search algorithm uses the Matlab function fminsearch which is based on the simplex method [50]. As opposed to the line search algorithm presented in Section III, this method does not require the derivative of the cost-functional. Applying this line search algorithm at each iteration of the GNI method, the TE inversion required equal or less iterations than the TM inversion to converge. For example, the convergence of the GNI-CGLS method applied to the single-frefor both TE and TM quency FoamTwinDiel case polarizations using this line search algorithm is shown in Fig. 15

VIII. DISCUSSION AND CONCLUSION We have shown how the recently developed regularized cost functional [5] can be optimized using the Gauss-Newton method in conjunction with a CG-FFT forward solver accelerated by a marching-on-in-source-position technique and applied to the experimental and synthetic data sets in both TE and TM polarizations. For testing this approach, the experimental Fresnel data set was used to provide far-field scattering data and a synthetic data set was used to provide near-field and far-field scattering data. The TE inversions in all cases were compared with the TM inversions in terms of the reconstruction accuracy, convergence rate and theoretical computational complexity. For all Fresnel targets, the TE and TM inversions are very similar. This is probably due to the fact that the measured data is collected in the far-field where only one scalar field component in the TE is required to represent the electric field vector: case and in the TM case. Thus, in the far-field, splitting into and does not provide more information than the TM case. In the first scenario of the synthetic test and and the collected data case, where

MOJABI AND LOVETRI: COMPARISON OF TE AND TM INVERSIONS IN THE FRAMEWORK OF THE GAUSS-NEWTON METHOD

is in the near-field, the TE inversion provides more accurate reconstruction compared to the TM inversion. This is likely due to and provide non-redundant informathe fact that tion for the TE inversion whereas the TM inversion only utilizes field. However, when the number of transmitters and the receivers increases to 30 for the same test case, the TE and TM inversions provide similar results which verifies the fact that the TM inversion lacked enough information compared to the TE . Keeping the number of transcase when mitters and receivers as in the first scenario but placing them in the far-field (the third scenario), the TE and TM inversions result in a similar reconstruction. This is consistent with the similar performance of TE and TM inversions of Fresnel data set. In all cases considered in this paper, the TE inversion requires the same or fewer number of iterations than the TM inversion to converge (of course, for the same convergence criteria listed in Section III). The same observation has been reported in [32] where the TE Iterative Multi-Scaling Approach (IMSA) converged faster than the TM IMSA when the signal to noise ratio of the collected data was low. Also, in [51], it has been theoretically speculated that the TE inversion has a lower degree of nonlinearity compared to the TM case which may result in a faster convergence in the TE case. In addition, the actual computational cost of the TE and TM inversions were very close to the approximate theoretical ones presented in Section V. To verify these results using another regularization technique, we have also inverted these data sets using the CGLS regularization scheme. The conclusion from inversion results obtained from the GNI-CGLS method is consistent with that obtained from the GNI method with the additive-multiplicative regularizer. We have also used another line search algorithm which is a derivative-free method which resulted in a similar convergence compared to the derivative-based line search method. Considering all this numerical data, we speculate that the ultimate performance and convergence of the GNI algorithm applied to these data sets are highly dependent on the information content of the field, irrespective of the regularization and line search strategies. Thus, the TE inversion, which utilizes both rectangular components of the electric vector at each receiver position, may result in more accurate reconstruction than the TM inversion when utilizing near-field scattering data collected using only a few transmitters and receivers. This paper serves as a preliminary study to compare the performance of the scalar and vectorial inversions and may lead to a theoretical comparison between the performance of these two inversions. ACKNOWLEDGMENT The authors would like to thank Institut Fresnel, France, for providing the experimental data set. REFERENCES [1] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems. New York: Springer-Verlag, 1996. [2] W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted born iterative method,” IEEE Trans. Med. Imaging, vol. 9, no. 2, pp. 218–225, 1990. [3] N. Joachimowicz, C. Pichot, and J. P. Hugonin, “Inverse scattering: An iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag., vol. 39, no. 12, pp. 1742–1752, Dec. 1991.

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[27] R. F. Bloemenkamp, A. Abubakar, and P. M. van den Berg, “Inversion of experimental multi-frequency data using the contrast source inversion method,” Inverse Probl., vol. 17, pp. 1611–1622, 2001. [28] J.-M. Geffrin, P. Sabouroux, and C. Eyraud, “Free space experimental scattering database continuation: experimental set-up and measurement precision,” Inverse Probl., vol. 21, pp. S117–S130, 2005. [29] K. Belkebir and M. Saillard, “Testing inversion algorithms against experimental data: inhomogeneous targets,” Inverse Probl., vol. 21, pp. S1–S3, 2005. [30] A. Abubakar, P. M. van den Berg, and T. M. Habashy, “Application of the multiplicative regularized contrast source inversion method on TM- and TE-polarized experimental fresnel data,” Inverse Probl., vol. 21, pp. S5–S13, 2005. [31] C. Yu, L. P. Song, and Q. H. Liu, “Inversion of multi-frequency experimental data for imaging complex objects by a DTA-CSI method,” Inverse Probl., vol. 21, pp. S167–S178, 2005. [32] D. Franceschini, M. Donell, G. Franceschini, and A. Massa, “Iterative image reconstruction of two-dimensional scatterers illuminated by TE waves,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1484–1494, Apr. 2006. [33] O. Féron, B. Duchêne, and A. Mohammad-Djafari, “Microwave imaging of piecewise constant objects in a 2D-TE configuration,” Int. J. Appl. Electromagn. Mechan., vol. 26, pp. 167–174, 2007. [34] K. Paulsen and P. Meaney, “Nonactive antenna compensation for fixedarray microwave imaging—I. model development,” IEEE Trans. Med. Imag., vol. 18, pp. 496–507, Jun. 1999. [35] J. Ma, W. C. Chew, C. C. Lu, and J. Song, “Image reconstruction from TE scattering data using equation of strong permittivity fluctuation,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 860–867, Jun. 2000. [36] E. Chong and S. Zak, An Introduction to Optimization. New York: Wiley Interscience, 2001. [37] J. D. Zaeytijd and A. Franchois, “Three-dimensional quantitative microwave imaging from measured data with multiplicative smoothing and value picking regularization,” Inverse Probl., vol. 25, 2009. [38] P. Mojabi and J. LoVetri, “Overview and classification of some regularization techniques for the Gauss-Newton inversion method applied to inverse scattering problems,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2658–2665, Sep. 2009. [39] M. S. Zhdanov, Geophysical Inverse Theory and Regularization. Amsterdam: Elsevier, 2002. [40] A. G. Tijhuis, K. Belkebir, and A. C. S. Litman, “Theoretical and computational aspects of 2-D inverse profiling,” IEEE Trans. Geosci. Remote Sensing, vol. 39, pp. 1316–1330, 2001. [41] P. R. McGillivray and D. W. Oldenburg, “Methods for calculating Fréchet derivatives and sensitivities for the non-linear inverse problem,” Geophys. Prospect., vol. 38, pp. 499–524, 1990. [42] T. K. Jensen, “Stabilization Algorithms for Large-Scale Problems,” Ph.D. dissertation, Technical Univ. Denmark, Kongens Lyngby, Denmark, 2006. [43] P. C. Hansen, “Deconvolution and regularization with Toeplitz matrices,” Numer. Algorithms, vol. 29, pp. 323–378, 2002. [44] A. Abubakar, P. M. van den Berg, and S. Y. Semenov, “A robust iterative method for born inversion,” IEEE Trans. Geosci. Remote Sensing, vol. 42, pp. 342–354, Feb. 2004. [45] P. M. van den Berg, A. L. van Broekhoven, and A. Abubakar, “Extended contrast source inversion,” Inverse Probl., vol. 15, pp. 1325–1344, 1999.

[46] W. C. Chew and J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microw. Guided Wave Lett., vol. 5, pp. 439–441, Dec. 1995. [47] P. Mojabi and J. LoVetri, “Enhancement of the Krylov subspace regularization for microwave biomedical imaging,” IEEE Trans. Med. Imag., vol. 28, no. 12, pp. 2015–2019, Dec 2009. [48] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. Philadelphia, PA: SIAM Review, 1998. [49] P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Review, vol. 34, no. 4, pp. 561–580, Dec. 1992. [50] J. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optimization, vol. 9, no. 1, pp. 112–147, 1998. [51] O. M. Bucci, N. Cardace, L. Crocco, and T. Isernia, “2D inverse scattering: degree of nonlinearity, solution strategies and polarization effects,” Proc. SPIE, vol. 4123, pp. 185–193, 2000.

Puyan Mojabi (S’09) received the B.Sc. degree in electrical and computer engineering from the University of Tehran, Tehran, Iran, in 2002 and the M.Sc. degree in electrical engineering from Iran University of Science and Technology, Tehran, in 2004. Currently, he is working toward the Ph.D. degree at the University of Manitoba, Winnipeg, MB, Canada. His current research interests are computational electromagnetics and inverse problems.

Joe LoVetri (SM’09) was born in Enna, Italy, in 1963. He received the B.Sc. (with distinction) and M.Sc. degrees, both in electrical engineering, from the University of Manitoba, Winnipeg, MB, Canada, in 1984 and 1987, respectively, and the Ph.D. degree in electrical engineering from the University of Ottawa, Ottawa, ON, Canada, in 1991. From 1984 to 1986, he was an EMI/EMC Engineer at Sperry Defence Division, Winnipeg, Manitoba. From 1986 to 1988, he held the position of TEMPEST Engineer at the Communications Security Establishment, Ottawa. From 1988 to 1991, he was a Research Officer at the Institute for Information Technology, National Research Council of Canada. From 1991 to 1999, he was an Associate Professor in the Department of Electrical and Computer Engineering, the University of Western Ontario. In 1997/98, he spent a sabbatical year at the TNO Physics and Electronics Laboratory, The Netherlands. Since 1999, he has been a Professor in the Department of Electrical and Computer Engineering, University of Manitoba, and was Associate Dean, Research, from 2004 to 2009. His main interests lie in time-domain computational electromagnetics, modeling of electromagnetic compatibility problems, microwave tomography and inverse problems.

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Experimental Study of the Invariants of the Time-Reversal Operator for a Dielectric Cylinder Using Separate Transmit and Receive Arrays Matthieu Davy, Jean-Gabriel Minonzio, Julien de Rosny, Claire Prada, and Mathias Fink

Abstract—The decomposition of the time reversal operator (DORT method) is applied to electromagnetic waves in order to characterize a dielectric cylinder. It consists in determining the Time Reversal Invariants of the Time Reversal Operator. Here, this matrix is built from the inter-element responses between distinct transmit and receive arrays. In this paper experimental results obtained between 2 and 4 GHz are compared to a theoretical model, developed in an other paper (Minonzio et al., “Theory of the time-reversal operator for a dielectric cylinder using separate transmit and receive arrays,” IEEE Trans. Antennas Propag., vol. 57, pp. 2331–2340, 2009). The DORT method is then applied to an inverse problem to determine the diameter and the permittivity of the cylinder. It is shown experimentally that different experimental parameters can be estimated from the singular values of the time reversal operator. Index Terms—Antenna array processing, Electromagnetic inverse problem, electromagnetic scattering.

I. INTRODUCTION

P

RADA and Fink introduced the DORT method (French acronym for “Décomposition de l’Opérateur de Retournement Temporel”) in 1994. [1]. They proposed a method based on the analysis of the time reversal invariants (TRI) of the time-reversal operator (TRO) in order to detect and identify targets with active antenna arrays. This method has been first developed for ultrasonic applications with piezo-electric transducers arrays. It has shown its efficiency for several applications. In non-destructive evaluation, it leads to detection of flaws in cluttered solids, cracks in hollow cylinder, or defects on thin plates [2]–[4]. Several studies have also been used for detection of targets in shallow water [5]. More recently the DORT method has been applied in medical imaging [6], [7]. In those applications, the number of time reversal invariants is equal to the number of the small-hard targets (flaw, cracks, micro-calcifications, etc.) as the scattering can be considered isotropic. Chambers and Gautesen showed that with a small elastic scatterer, due to the anisotropy of the scatterer, the number of TRI can be up to four and weakly depends on Manuscript received March 12, 2009; revised September 14, 2009. Date of manuscript acceptance September 23, 2009; date of publication January 26, 2010; date of current version April 07, 2010. The authors are with the Institut Langevin, Laboratoire Ondes et Acoustique, Université Denis Diderot Paris 7, UMRS CNRS 7587, ESPCI, 75231 Paris cedex 05, 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.2010.2041154

) [8]. They the scatter characteristics (geometry, materials, provided the expression of the time reversal invariants in terms of monopolar and dipolar normal modes of vibrations. Later, Chambers derived the expression of TRI for small electromagnetic spheres [9] and ellipsoids [10]. For an extended scatterer, several invariants actually exist depending on both geometry and physical parameters. Komilikis et al. observed experimentally this phenomenon for a steel hollow cylinder [11]. Prada et al. applied the DORT method to an air-filled cylindrical shell embedded in water showing that each Lamb mode (elastic waves in plane) is associated with two invariants of the TRO [12]. Still in acoustics, Minonzio et al. studied the invariants for a large elastic cylinder or sphere when the antenna array has a limited aperture [13]. It was shown that the geometry of the array has an important impact on the TRI. A good agreement between theoretical and experimental results has been reported for a copper cylinder and steel sphere. The studies about the application of the DORT method to microwaves began almost ten years ago. Tortel et al. considered the role of different polarizations for dielectric scatterers [14]. They performed experiments with dielectric spheres in an anechoic chamber with a very large aperture complete array. They showed that the DORT method leads to a highly precise localization of the sphere in conjunction with a low sensitivity to noise. Later, Micolau and Saillard performed numerical experiments to detect and localize buried objects through TRI [15]. They showed that Time Reversal methods are more efficient than classic methods for the detection of buried objects because they take advantage of the scattering within the medium. Following a theoretical development in an accompanying article [16], this experimental study reports TRI for various dielectric cylinders using a configuration where the transmit and receive arrays are distinct. After recalling the DORT method principle in Section II and introducing the experimental setup in Section III, the experimental TRI are compared in Section IV to the theoretical ones, obtained in the companion paper. Then, in Section V, the non coherent and coherent back-propagations of the TRI are performed. They lead to the cylinder localization. In the last section, the DORT method is applied to an inverse problem and the different parameters of the cylinder like permittivity, diameter or location are determined from the TRI. II. TIME REVERSAL INVARIANTS AND SINGULAR VALUE DECOMPOSITION Here we briefly recall the DORT method. We consider two and antennas respectively. Let be the arrays of

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Fig. 1. Experiment configuration (a) and experimental setup (b).

response of the propagation at the frequency between anof the Rx array. tenna # of the Tx array and antenna # All those responses are used to build the inter-element-response matrix, denoted . As explained in the accompanying theoretical paper, the TRI are given by the singular value decomposition (SVD) of

(1) , corresponding to the Rx array, and , correwhere sponding to the Tx array, are the two singular vectors (SVe) . The direct link associated to the singular value (SVa) between the singular vectors and the Time Reversal Invariants is shown in the companion paper. III. EXPERIMENT CONFIGURATION Our experiments consist of using distinct transmit and receive arrays. This kind of configuration was first studied in acoustics [4]. Moura and Jin also performed electromagnetic TR experiments in such a configuration in order to detect a target in a highly cluttered environment with electromagnetic waves [17], [18]. Two horn antennas are mounted on a rail. Each horn slides across the rail axis ( -axis) (Fig. 1). The horns are set on a rotation stage in order to take aim towards the target at each position. The antennas are plugged to a N5230A PNA-L network analyser parameters is performed be(Agilent). The measurement of tween 2 and 4 GHz, which corresponds to the working bandmatrix is acquired by moving each width of the horns. The antenna ten times to build two virtual arrays of ten elements. The span of each array reaches 1.3 m and the distance between them is equal to 0.4 m. This spacing between each antenna poat a frequency of 3.75 GHz, where is sition is equal to

Fig. 2. Normalized experimental singular values for PVC cylinders versus ka for 3 diameters (2.5 cm, 3 cm and 4 cm). The three first theoretical singular values are plotted in bold, dashed and dashed-dot lines.

the wavelength. From the Nyquist-Shannon sampling theorem, it assures the optimal sampling of the field along the array. The target consists of a dielectric cylinder set up in front of an anechoic wall. The electric polarisation of the field generated by the two horns is parallel to the cylinder axis. The distance between the rail and the cylinder is measured by and dx represents the off axis distance between the cylinder and the centre of the two arrays. Experiments are performed with cylinders of cm, cm and three different bulk materials: PVC ( cm), Ertaflon ( cm) and Plexiglas ( cm). The cylinder lengths are rather larger than the beam elevation length. Therefore, the cylinders are considered as “infinite”. The horn antennas behave like waveguides. By measuring the response between two horns antennas facing each other, we have found that the equivalent phase velocity inside the horn antenna

DAVY et al.: EXPERIMENTAL STUDY OF THE INVARIANTS OF THE TRO FOR A DIELECTRIC CYLINDER

Fig. 3. Normalized experimental singular values for an Ertaflon cylinder of diameter 3 cm (a) and a Plexiglas cylinder of diameter 10 cm (b) versus ka.

equals m.s . This has been taken into consideration for the back-propagation of the TRI. matrix is acquired, the singular value decomOnce the position shown in (1) is performed at each angular frequency . In the following, for conciseness, the frequency dependence of the quantities is omitted. IV. EXPERIMENTAL TIME REVERSAL INVARIANTS A. Singular Values In the first experiment, three PVC cylinders of different diameters are studied: 2.5 cm, 3 cm and 4 cm. Their relative permittivity has been roughly estimated to 3 at 3 GHz. For each cylinder, the SVa are divided by their radius and plotted with respect to . Here stands for the wavenumber of the surrounding medium: . According to the companion paper, the singular values can be expressed as linear combination of the defined by weights of the projected harmonics

with

and

.

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Fig. 4. Modulus (a) and phase (b) of the first and second eigenvectors versus the antenna number at frequency 3.9 GHz, for the Tx array, in the case of PVC cylinder of diameter 4 cm.

The scattering coefficient is function of ka, while the responses are function of . Hence, introducing can be written as . It ima function plies that the singular values normalized by are functions of ka [16]. On Fig. 2, the experimental data are compared to the theoretical predictions. A very good agreement appears for the two , the second highest Sva,. However, at low frequencies SVa reaches the noise level. The third theoretical SVa is far below the noise level, making it undetectable in our experiment. In the case of Ertaflon [Fig. 3(a)], the third experimental SVa is stronger than the theoretical one. Nevertheless it is now frequency dependent. This effect can be interpreted as following: when the theoretical SVa is close to the noise level, the coupling with noise raises the experimental SVa. This is particularly . striking near We observe in Fig. 3(b) that the frequency dependence of the SVa for the Plexiglas cylinder is much more complex. As the cylinder diameter is larger than the ones considered before, the scattering shows strong resonances. It leads to crossings between singular values. Similar behavior has been observed for

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Fig. 5. Back-propagation of the SVe measured with the Plexiglas cylinder at 3.1 GHz using (3): first SVe (a), second SVe (b), third SVe (c) and fourth SVe (d).

electromagnetic scattering in [13]. This effect is confirmed with the DORT imaging technique in Section V. B. Singular Vectors In Fig. 4, the diameter of the cylinder is smaller than the resolution width. Indeed the size of focal spot is where is the distance of the cylinder from the array and the total span of cm, the arrays. Here, with m and m, the focal spot width equals 6.5 cm which is in agreement with results plotted in Fig. 6. In this regime, is almost constant in modulus and its we have shown that where is the distance between the phase is given by th antenna and the cylinder centre [19]. Due to the orthogoand , the second Sve vanishes with a nality between phase shift, between the 7th and the 8th antenna [Fig. 4(b)]. As the phase of SVe depends on the cylinder location, we shall see in the next section that a numerical back-propagation provides a precise location of the cylinder. V. DORT IMAGING As often done in former studies, it is possible to “back-propagate” numerically the singular vectors. Reminding that is to the Tx array, the simplest associated to the Rx array and

way to form an image with the singular vectors is to add incoherently their back-propagation on each point of space. The image created by the back propagation of the two th eigenvectors is then computed from

(2) where and correspond to the Green functions between the position and the Tx and Rx antennas respectively. is the Hermitian scalar product, i.e., . The notation We have shown in the theoretical paper that and are associated to monopolar focusing. The superposition of these two monopolar spots generates a single spot [Fig. 5(a)] as in and , each of [13]. As for the second SVe represented by them generates a dipolar focal spot. Their superposition gives rise to four main lobes [Fig. 5(b)]. Whereas the third SVa is clearly above the noise level, the interpretation of the third SVe pattern is less obvious. As for the fourth SVe, it is clearly dominated by noise. Nevertheless, due to the orthogonality between singular vectors, we observe that the intensity level of their corresponding images is almost null on the cylinder position.

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Fig. 6. Coherent back-propagation of the first (a) and second SVe (b) at 3.1 GHz. In figures (c) and (d), the coherent (in dashed-dot red lines) and incoherent (in bold blue lines) back-propagation of the focal spot along the x-axis are compared for the first SVe (c) and the second SVe (d).

When the back-propagation contributions are added incoherently, the focal spot width is given by the aperture of each Tx and Rx array, i.e., 13 cm. A way to increase the resolution consists of adding coherently the back-propagation contributions as shown in (4). However the sum have to be done with care because the singular vectors are determined with an arbitrary and are two singular vectors verifying phase. Indeed, . By multiplying each side by it becomes . Thus and are also singular values of . As the image must be physically computed from the back-propagation of * and , we introduce a phase to correct this uncertainty, such as the two contributions at the is worked focus interfere constructively. This relative phase out from the two numerical back-propagations of the transmit and receive singular vectors. It is equal to the phase difference of the two backpropagate fields at the focus. The image given by the coherent back-propagation becomes

(3) Now the focal spot width is given by the total aperture of the two arrays. Consequently, in our case, the focal spot width is divided

by 2 compared to the incoherent method and becomes about 6.5 cm. The experimental results are in good agreement with this approach (Fig. 6). Furthermore, the symmetric and antisymmetric nature of the two first Sve is recovered [Fig. 6(a) and (b)]. The first SVe is generally associated with a monopolar focus and the second SVe with a dipolar focus. However it is not always the case. In particular, when the target is resonating, crossings between Sva appear [Fig. 3(b)] leading to a switch between singular vectors. In Fig. 7, the normalized back-propagations of the first SVe (a) and the second SVe (b) are plotted in regard to the plots of the singular values with respect to . We observe that the first SVe is monopolar or bipolar depending on the SVa crossings. For example, in Fig. 7(a), the first SVa is monopolar below 2.8 and then becomes bipolar before reverting to for . monopolar for This effect is more complex concerning the back-propagation of the second SVe. Indeed, we have to take into account the crossings of the first, second and third SVa. VI. INVERSE PROBLEM As shown in the companion theoretical paper [16], the SVa depends on the characteristics of the cylinder. In this part, we

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Fig. 7. (a) Plot of the two first SVa and incoherent back-propagation of the first Sve U and V . The SVe are propagated at y = F (focal distance), with respect to x. (b) plot of the three first SVa and back-propagation of the second SVe. Vertical lines represent the crossings between the first and the second SVa (dash) and between the second and the third SVa (dash-dot).

TABLE I COMPARISON BETWEEN USING THE FIRST OR THE TWO FIRST SINGULAR VALUES FOR PVC CYLINDER TO DETERMINE CYLINDER CHARACTERISTICS. IN PARENTHESIS IS WRITTEN THE WIDTH OF THE ESTIMATED PARAMETER AT 90% OF THE MAXIMUM

Fig. 8. Grayscale images of M (X) with respect to the permittivity " and the = 1 (a) and diameter d for a PVC cylinder. Images are obtained for N N = 2 (b).

issue consists in finding the parameter vector minimizes the cost function

which

(4)

show that the SVa can be used to estimate the permittivity, the radius and the location of the cylinder. As the relationships between those different parameters are non-linear, they cannot be directly worked out from the plot of the singular values. We use a least mean square method to solve this inverse problem. The

is the number of SVa taken into account, i.e., 1 where represents the normalized mismatch or 2. The estimator between the theoretical and experimental SVa, integrated over the frequency range. The accuracy of the results increases with the bandwidth. In (4), we only consider the first and the second SVa because the third one is perturbed by noise. The difference between the theoretical and the experimental SVa are divided by the experimental SVa to give more weight to the second SVa. We use a simplex algorithm to minimize the cost function. Table I shows that our method provides accurate results.

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Taking into account the second SVa reduces significantly the mismatch between real and measured parameters. Analysis of function confirms this result. Two deviation paramethe ters for the 4 cm diameter PVC cylinder are plotted with respect to the permittivity and the diameter in Fig. 8. In Fig. 8(a), is computed with the first SVa only . In Fig. 8(b), the first two SVa are taken into account . We observe that the minimum of , i.e., the param. Indeed, eter estimation, is much sharper when the second Sva is more frequency dependant than the first one [Fig. 3(a)]. Those fluctuations that depend on the parameters improve the accuracy of the estimation of the parameters, as shown in Table I. cm by using the first For example, we have found cm by using the both, cm being SVa only, and the theoretical value. The relative error has thus been reduced to 2.25%, instead of 16% initially. VII. CONCLUSION Following a theoretical companion article, this study reports experimental measurements of the Time Reversal Invariants corresponding to dielectric cylinders observed by two distinct Tx and Rx arrays. A very good agreement has been observed between the experimental results and theoretical predictions. In order to localize the cylinder, we have back-propagated the singular vectors and compared incoherent and coherent sum methods. The DORT method has then been used to find out the radius and the permittivity through the inverse problem. For this purpose, we observe that the use of several Time Reversal Invariants greatly improves the accuracy of the estimation. This experimental method could possibly lead to a new approach in the characterization of extended objects. REFERENCES [1] C. Prada and M. Fink, “Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media,” Wave Motion, vol. 20, pp. 151–163, 1994. [2] E. Kerbrat, D. Clorennec, C. Prada, D. Royer, D. Cassereau, and M. Fink, “Detection of cracks in a thin air-filled hollow cylinder by application of the DORT method to elastic components of the echo,” Ultrasonics, vol. 40, pp. 715–720, 2002. [3] E. Kerbrat, R. K. Ing, C. Prada, D. Cassereau, and M. Fink, “The D.O.R.T. method applied to detection and imaging in plates using Lamb waves,” Review of Progress in Quantitative Nondestructive Evaluation. Ames, Iowa, 2001, vol. 20, pp. 934–940. [4] C. Prada, M. Tanter, and M. Fink, “Flaw detection in solid with the D.O.R.T. method,” in Proc. Ultrasonics Symp., 1997, vol. 1, pp. 679–683. [5] N. Mordant, C. Prada, and M. Fink, “Highly resolved detection and selective focusing in a waveguide using the D.O.R.T. method,” J. Acoust. Society Amer., vol. 105, pp. 2634–2642, 1999. [6] M. R. Burcher, A. T. Fernandez, and C. Cohen-Bacrie, “A novel phase aberration measurement technique derived from the DORT method: Comparison with correlation-based method on simulated and in-vivo data,” in Proc. IEEE Ultrasonics Symp., 2004, vol. 2, pp. 860–865. [7] J.-L. Robert, M. Burcher, C. Cohen-Bacrie, and M. Fink, “Time reversal operator decomposition with focused transmission and robustness to speckle noise: Application to microcalcification detection,” TJ. Acoust. Society Amer., vol. 119, pp. 3848–3859, 2006. [8] D. Chambers and A. K. Gautesen, “Time reversal for a single spherical scatterer,” J. Acoust. Society Amer., vol. 109, pp. 2616–2624, 2001. [9] D. H. Chambers and J. G. Berryman, “Analysis of the time-reversal operator for a small spherical scatterer in an electromagnetic field,” IEEE Trans. Antennas Propag., vol. 52, pp. 1729–1738, 2004.

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[10] D. H. Chambers and J. G. Berryman, “Target characterization using decomposition of the time-reversal operator: Electromagnetic scattering from small ellipsoids,” Inverse Probl., vol. 22, pp. 2145–2163, 2006. [11] S. Komilikis, C. Prada, and M. Fink, “Characterization of extended objects with the D.O.R.T. method,” in Proc. IEEE Ultrasonics Symp., 1996, vol. 2, pp. 1401–1404. [12] C. Prada and M. Fink, “Separation of interfering acoustic scattered signals using the invariants of the time-reversal operator. Application to Lamb waves characterization,” J. Acoust. Society Amer., vol. 104, pp. 801–807, 1998. [13] J.-G. Minonzio, F. D. Philippe, C. Prada, and M. Fink, Inverse Problem, vol. 24, p. 025014, 2008. [14] H. Tortel, G. Micolau, and M. Saillard, “Decomposition of the time reversal operator for electromagnetic scattering,” J. Electromagn. Waves Applicat., vol. 13, pp. 687–719, 1999. [15] G. Micolau and M. Saillard, “D.O.R.T. method as applied to electromagnetic subsurface sensing,” Radio Sci., vol. 38, 2003. [16] J. G. Minonzio, M. Davy, J. De Rosny, C. Prada, and M. Fink, “Theory of the time-reversal operator for a dielectric cylinder using separate transmit and receive arrays,” IEEE Trans. Antennas Propag., vol. 57, pp. 2331–2340, 2009. [17] J. M. F. Moura and J. Yuanwei, “Detection by time reversal: Single antenna,” IEEE Trans. Signal Processing, vol. 55, pp. 187–201, 2007. [18] J. M. F. Moura and J. Yuanwei, “Time reversal imaging by adaptive interference canceling,” IEEE Trans. Signal Processing, vol. 56, pp. 233–247, 2008. [19] J.-G. Minonzio, C. Prada, D. Chambers, D. Clorennec, and M. Fink, “Characterization of subwavelength elastic cylinders with the decomposition of the time-reversal operator: Theory and experiment,” J. Acoust. Society Amer., vol. 117, pp. 789–798, 2005.

Matthieu Davy was born in Paris, France, in 1983. He graduated from Ecole Centrale de Lille in 200 and received the M.S. degree in physical acoustics from the University of Valenciennes, in 2007. He is currently working toward the Ph.D. degree at the Institut Langevin, Laboratoire Ondes et Acoustique. His research include time-reversal of electromagnetic waves and array signal processing.

Jean-Gabriel Minonzio was born in Dijon, France, in 1978. He received the B.S. degree in engineering physics from Ecole Supérieure de Physique et de Chimie Industrielles de la ville de Paris (ESPCI), Paris, France, in 2003 and the M.S. and Ph.D. degrees in physical acoustics from University Paris 7, Denis Diderot Paris, in 2003 and 2006, respectively. At the Laboratoire Ondes et Acoustique, Paris, France, his research interests include time-reversal and scattering of acoustical and electromagnetic waves, array signal processing, underwater acoustics and target characterization. He currently works on elastic guided waves in bones at the Laboratoire d’Imagerie Paramétrique, Université Pierre et Marie Curie, Paris, France.

Julien de Rosny was born in 1972 in Conflans Sainte Honorine, France. He graduated from the Université Pierre et Marie Curie (University of Paris 6) with a B.S. degree in physics and received the M.S. and Ph.D. degrees in physics from University of Paris 6, in 1995 and 2000, respectively. In 2001, he worked as a Postdoctoral Researcher in the Marine Physical Laboratory at SCRIPPS, University of California, San Diego. He is currently a Researcher of CNRS (National Center for Scientific Research). He works at Laboratoire Ondes et Acoustique, Paris, France. His current research interests include signal processing, underwater acoustics, ultrasonics, wave propagation in complex media and timereversal of waves.

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Claire Prada was born in Paris in 1962. She graduated from Ecole Normale Superieure in 1987 and received the Ph.D. degree in physical acoustics in 1991, with a dissertation on acoustic time reversal mirrors She is currently working at Laboratoire Ondes et Acoustique in Paris as a Research Scientist employed by the Centre National de la Recherche Scientifique. Her research interest is in signal analysis techniques for arrays of transit receivers and application to nondestructive evaluation, medical imaging, or underwater acoustics.

Mathias Fink received the M.S. degree in mathematics in 1967, the Ph.D. degree in solid state physics in 1970, and the Doctorat es-Sciences degree in 1978 all from Paris University, France. His Doctorat es-Sciences research was in the area of ultrasonic focusing with transducer arrays for real-time medical imaging. He is a Professor of physics at the Ecole Superieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI), Paris, France, and at Paris 7 University (Denis Diderot), France. In 1990, he founded the

Laboratory Ondes et Acoustique at ESPCI. In 2002, he was elected at the French Academy of Engineering, in 2003 at the French Academy of Science and in 2008 at the College de France on the Chair of Technological Innovation. His current research interests include medical ultrasonic imaging, ultrasonic therapy, nondestructive testing, underwater acoustics, telecommunications, seismology, active control of sound and vibration, analogies between optics, quantum mechanics and acoustics, wave coherence in multiply scattering media, and time-reversal in physics. He has developed different techniques in acoustic imaging (transient elastography, supersonic shear imaging), wave focusing in inhomogeneous media (time-reversal mirrors), speckle reduction, and in ultrasonic laser generation. He holds more than 40 patents, and he has published more than 300 articles. Four start-up companies have been created from his research (Echosens, Sensitive Object, Supersonic Imagine and Time Reversal Communications)

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Wireless Communication for Firefighters Using Dual-Polarized Textile Antennas Integrated in Their Garment Luigi Vallozzi, Patrick Van Torre, Carla Hertleer, Hendrik Rogier, Senior Member, IEEE, Marc Moeneclaey, Fellow, IEEE, and Jo Verhaevert

Abstract—A compact wearable antenna system, completely made out of textile materials for integration into protective garments, is proposed. The system implements combined pattern and polarization diversity to improve the quality of the communication link. The performance of the on-body antenna system, integrated into a firefighter jacket worn by a test person, was investigated in an indoor measurement campaign. Several receiver diversity schemes and different combining techniques were evaluated in terms of bit error rate, signal-to-noise ratio and signal correlations. By comparing them to theoretical results, we demonstrate the reliability of the proposed system and the advantage of using diversity. Index Terms—Body-centric communications, firefighter, pattern-polarization diversity, textile antenna, wearable antenna system.

I. INTRODUCTION

W

EARABLE textile systems for use in body-centric communication create an interface between the wearer and the external world, by continuously monitoring the vital and activity functions of the body (i.e., temperature, heart rate, pressure), as well as the surrounding environment (i.e., humidity, temperature, etc.). On the one hand, once the data of the sensors integrated into the system are acquired, they need to be transmitted in a wireless way to a fixed base station. On the other hand, the fixed base station needs to send back data to the wearable system, for example alarms or other useful information. A wearable textile system can be used in applications such as health monitoring of patients [1], coordination of interventions

Manuscript received May 25, 2009; revised August 31, 2009; accepted September 25, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. L. Vallozzi and H. Rogier are with the Information Technology Department (INTEC), Ghent University, 9000 Ghent, Belgium (e-mail: Luigi. [email protected]; [email protected]; [email protected]). P. Van Torre is with the Information Technology Department (INTEC), Ghent University, 9000 Ghent, Belgium and also with the Hogeschool Gent, INWE department, 9000 Ghent, Belgium (e-mail: [email protected]). C. Hertleer is with the Department of Textiles, Ghent University, 9052 Zwijnaarde, Belgium (e-mail: [email protected]). M. Moeneclaey is with the Department of Telecommunications and Information Processing (TELIN), Ghent University, 9000 Ghent, Belgium (e-mail: [email protected]). J. Verhaevert is with Hogeschool Gent, INWE Department, 9000 Ghent, Belgium (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.2010.2041168

by rescue workers during emergencies [2], sports, entertainment and so on. In this article the particular application of data communication between a firefighter and a base station in an indoor environment is considered. More specifically, it focuses on the textile antenna system, which is that part of a wearable textile system that enables the transmission/reception of the RF-modulated data. Wearable and textile antennas for body-centric communication have been the object of extensive research in the last years [3]–[6]. More recently, systems composed of several wearable or textile antennas have been investigated, proving the effectiveness of diversity techniques, both for on-body communication [7], [8] and for off-body communication [9], [10]. In this article, the specific case of communication between firefighter and base station is treated. In contrast to the existing literature, the proposed antenna system was designed in such a way in order to combat the following specific disturbances, that can make communication unreliable. 1) Firefighters often operate in indoor environments, characterized by rich multipath scattering of the signal, resulting in the received signal fading and consequent increase of the bit error rate; 2) The harsh environmental conditions in which firefighters often operate (i.e., high temperatures and/or high humidity) pose extra challenges in realizing the antennas with appropriate textile materials [4], [11], whose performance must not be affected by the severe environment; 3) The firefighter can optionally carry a metal oxygen bottle which may partially shield the antennas and restrict the suitable places where the antenna can be integrated into the jacket; 4) The close proximity of the human body to the antennas and the continuous deformation which they are subjected to, affect the designed radiation characteristics such as gain, resonance frequency and polarization, with a potential increase of the bit error rate. The proposed wearable antenna system provides answers to the above needs by means of the following features. 1) To mitigate the severe fading of the indoor multipath environment, the antenna system uses pattern and polarization diversity, known to be effective diversity techniques [9], [12], [13]. This was realized by using two dual-polarized textile patch antennas [14], providing a total diversity . Each antenna provides polarization diverorder of sity, while pattern diversity is added by placing one antenna

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at the front and the other at the rear of the body, thus creating two complementary hemispheres covering the complete 360 azimuth range; 2) To prevent the antennas from being damaged by the extreme environmental conditions, the antenna substrate was realized in a protective, shock absorbing, fire retardant and water repellent foam, together with breathable conductive textile materials for patch and ground plane; 3) The two antennas were positioned in such a way in order to obtain minimum shielding or disturbance from the firefighter equipment, as it will be explained more in detail in the next sections; 4) Moreover, the antennas were placed at well-chosen locations in the jacket, where they are almost flat and not much subjected to large movements or bending due to the movements of the firefighter. The performance and reliability of the realized system were tested during a measurement campaign, by means of a Signalion-HaLo 420 wireless testbed, which performs a real transmission of data allowing to determine the performance of the wireless link containing the textile antenna system. The article is organized as follows: first, in Section II the textile antenna system is described, together with a complete description of the dual-polarized textile patch antenna, including the topology, design considerations and measured radiation characteristics such as gain patterns and polarization, in open space and on body. Section III-A describes the diversity settings and the combining techniques, while Section III-B describes the structure of the indoor environment. After that, in Sections III-C and III-D, the measurements conducted on the 2nd-order receiver polarization diversity and 4th-order receiver pattern-polarization diversity system are presented, together with a performance evaluation in terms of bit error rate and signal-to-noise ratio. Next, channel estimation and received signal correlation analysis are provided respectively in Sections III-E and III-F. After that, a more detailed analysis of bit error rate, including BERs after data detection and BER characteristics, is described in Sections III-G and III-H, respectively. Finally, in Section IV the conclusions are drawn. II. TEXTILE ANTENNA SYSTEM A. Measurement Setup The measurement setup is composed of a transmitting vertically-polarized dipole antenna and the receiving wearable antenna system under test, which consists of two dual-polarized textile antennas, resulting in a total of four received diversity signals (two diversity signals per antenna). The proposed wearable antenna system is realized by integrating the two antennas into the front and back side of a firefighter jacket, worn by a test person, as shown in Fig. 1. The placement of the antennas inside the firefighter garment is shown in Fig. 2. All antennas are then connected to a Signalion-HaLo 420 measurement testbed, interfacing to Matlab by means of PCs. The measurement setup scheme is depicted in Fig. 3. The transmitting unit of the testbed transmits bursts of QPSK data symbols modulating a carrier wave at 2.45 GHz. Note that in this measurement campaign we focus on QPSK modulation, in order to fix ideas. However, the measurement system allows

Fig. 1. Positions of front and rear antenna inside the jacket, covering two complementary azimuth hemispheres and ensuring minimal deformation of the antennas due to movement of the body.

Fig. 2. Textile antenna inside the fireman’s vest, behind the combined moisture and thermal barrier, protecting the antenna against harsh environmental conditions.

the use of any modulation schemes, so the obtained results are easily generalized to other modulation schemes. A total of 244 bursts are transmitted, concatenating all measurement cases, as explained in Section III. Symbols are generated and modulated in Matlab and then upconverted to RF by the testbed transmitter. The testbed receiver downconverts and samples the signals received by the textile antenna system. Demodulation, channel estimation, measurement and storage of the signal amplitude levels, diversity combining, detection and calculation of BER and SNR are performed in post processing on a PC by means of Matlab. During the measurements, particular care was taken in placing the testbed at positions for which its effects on the propagation channel were reduced as much as possible.

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Fig. 3. Measurement setup scheme: communication link including transmitter, propagation channel and receiver implementing pattern-polarization diversity.

Fig. 5. Textile antenna in open space, measured radiation pattern.

Fig. 4. Layout of the dual-polarized rectangular ring textile patch antenna. Feed points positioned symmetrically on the two diagonals, ensuring two orthogonal polarized waves. L : ,W : ,L : ,W : . Feed points positions: x; y : ; : . : : ,  : . The conSubstrate parameters: h , ductive materials are ShieldIt (patch), FlecTron (ground plane) and the substrate material is a protective closed-cell foam.

5 94 mm

= 49 59mm = 50 27 mm = 14 34mm = (6 6 ) = (67 7 mm 66 08 mm) = 5 55 mm = 1 12 tan = 0 003

B. Structure and Radiation Characteristics of the Antennas The dual-polarized wearable antenna is a patch antenna completely made out of textile materials, suitable for integration into protective clothing such as firefighter suits. The substrate material is a protective, water-repellent, fire-retardant foam, commonly used in firefighter garments, whereas the ground plane and patch are made out of FlecTron and ShieldIt respectively, two breathable and highly conductive textile materials. The layout of the dual-polarized patch antenna consists of a rectangular patch with a slot. The antenna possesses two feed points, each one corresponding to an antenna terminal or port, located on the patch diagonals, symmetrically with respect to the Y-axis, as shown in the scheme in Fig. 4. The topology and feeding structure ensure the excitation of two signals with different polarizations, enabling the implementation of 2nd-order diversity in a compact single antenna system. The dimensions for this layout and the parameters for the used materials are listed in Fig. 4.

Fig. 6. Textile antenna in open space, measured radiation pattern.

The measured radiation patterns of the antenna in open space, for both ports, are shown in Figs. 5 and 6. The radiation patterns were also measured while the antenna is embedded in the jacket, which is worn by a test person. Measurements of the XZ-plane gain pattern with and without the oxygen cylinder carried on the back are compared in Figs. 7–10, for both ports of rear and front antenna, respectively. The oxygen bottle has a moderate influence on the gain patterns of rear and front antennas, deforming

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Fig. 7. Rear antenna on firefighter, port 1, measured XZ-plane gain pattern. The presence of the metal oxygen bottle, carried by the firefighter, produces deformation of the gain pattern and lowering of the maximum gain.

Fig. 9. Front antenna on firefighter, port 1, measured XZ-plane gain pattern. The presence of the metal oxygen bottle, carried by the firefighter, produces deformation of the gain pattern and lowering of the maximum gain.

Fig. 8. Rear antenna on firefighter, port 2, measured XZ-plane gain pattern. The presence of the metal oxygen bottle, carried by the firefighter, produces deformation of the gain pattern.

Fig. 10. Front antenna on firefighter, port 2, measured XZ-plane gain pattern. The presence of the metal oxygen bottle, carried by the firefighter, produces deformation of the gain pattern.

them and in some cases lowering the maximum gain. Very little power is radiated towards the body. The antenna, in flat state, along broadside and at center fre, was designed to transmit/receive two quency quasi-linearly polarized waves, which are almost orthogonal in space, with the two polarizations oriented at tilt angles of about . Each port signal radiates a field characterized by a radiation vector, expressed by the phasor vector . This vector rotates in time, when the signal is propagating, tracing a polarization ellipse, which completely describes the polarization of the signal on the XY-plane, which is characterized by the following polarization parameters.

• Tilt angle , that is the angle between the X-axis of the antenna (as shown in Fig. 4) and the major axis of the polarization ellipse; • Eccentricity , that is the ratio between the minor and the major axis of the polarization ellipse; • Axial ratio for linear polarization (which provides the same information of ), defined as . The polarization parameters for both port signals were measured in open space and with the antenna worn by the test person at the front and back, with and without the oxygen cylinder present. Measurements of the tilt angle, eccentricity

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TABLE I ORTHOGONALITY OF THE ANTENNA PORTS IN OPEN SPACE

TABLE II ORTHOGONALITY OF THE ANTENNA PORTS OF THE FRONT ANTENNA ON THE BODY

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signals, increasing their correlation. This effect is further studied in Section III. Additional effects of bending and deformation of the textile antennas are not discussed in this article, as these were already investigated in previous publications. In [14] is shown that for a dual-polarized textile patch antenna, subjected to bending, the return loss remains acceptable over the frequency band of interest. Moreover, in [4] is proved that, for a single circularly polarized antenna, the return loss, gain and axial ratio are still satisfactory under bending. III. RECEIVER DIVERSITY USING DUAL POLARIZED TEXTILE ANTENNAS

TABLE III ORTHOGONALITY OF THE ANTENNA PORTS OF THE BACK ANTENNA ON THE BODY

and axial ratio for linear polarization, for the broadside direction, are listed in Tables I–III. For each listed case, a coefficient describing the orthogonality of the polarization ellipses of the , is also given. is defined by two ports, being

A. Diversity Schemes and Combining Techniques In this paper, we focus on receiver diversity employed by the off-body antenna system. We do not consider diversity at the base station/access point side, as this is easier to implement. We considered two different receiver-diversity schemes: • 2nd-order receiver polarization diversity, i.e., with one transmitting dipole on the test bed and one receiving dual-polarized antenna (front or back) on the firefighter; • 4th-order receiver combined pattern-polarization diversity, i.e., with one transmitting dipole on the testbed and two receiving dual-polarized antennas (front and back) on the firefighter. Moreover, for each diversity scheme, two different combining schemes were employed: Maximal ratio combining (MRC) and selection combining (SC). In all considered cases, performance analysis demonstrated the advantage of using diversity for the wearable antenna system. B. The Indoor Environment

with . On the one hand, if the two polarization ellipses are orthogonal, thus independent. On the the two ellipses are coincident. For a other hand, if , in order to dual-polarized antenna the desired value is have minimal correlation between the two port signals. From the measurements, we conclude the following. • Compared to the open-space case, when the antenna is worn inside the jacket on a human body (with and without bottle), the polarization departs from being linear to become elliptical. This is seen from the values for or ; • In some cases the presence of the oxygen bottle also produces a slight rotation of the polarization ellipses and changes the axial ratios, having an influence on the signal correlation; • When the antenna is worn, for both front and back antenna and with and without oxygen bottle, the orthogonality coincreases with respect to the open-space sitefficient uation, ranging from a minimum of 0.07 to a maximum of 0.41. This clearly indicates that the integration of the antennas into the jacket and the presence of the bottle has a negative effect on the orthogonality of the two ports

A floor plan of the indoor environment where the measurements were performed is displayed in Fig. 11. The path followed by the test person during the measurements is marked, as well as the position of the transmitter. The considered cases are listed here as a function of the labels shown in Figs. 12, 13 and 14, in Sections III-C and III-D. 1) Path : the test person walking towards the transmitter from a distance of 15 m and ending at 3 m from the transmitter; : walking away from the transmitter, in the 2) Path opposite direction of the first path; : walking towards the transmitter 3) Path and returning, i.e., the two previous paths combined in one measurement series; : walking sideways, along a path 4) Path perpendicular to the transmitter, at up to 18 m of distance. This is the hardest receiving path, since the distance is large and there are many obstacles in the signal path. The firefighter can optionally carry a metal oxygen bottle on the back. Measurements presented in Section II-B suggest a minor performance degradation due to the presence of the bottle. For that reason, to test the worst-case scenario, the following measurements were performed with the oxygen bottle present.

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symbol rate was 1 MSymbol per second, corresponding to a bit rate of 2 MBit/s. Each burst is composed of 300 pilot symbols for channel estimation, followed by 396 data symbols and it has a total duration of 696 , which is short enough to consider the channel invariant during one burst. At the receiver side, MRC and SC were applied to the two output signals coming from a single patch antenna. In Figs. 12 and 13 the signal-to-noise ratios corresponding to each of the two polarizations are displayed, together with the bit error rates obtained using MRC and SC. The labels on top of the graphs correspond to the paths walked, as described for the floor plan. One notices that, for both front and rear antennas, the signal-to-noise ratios increase as the test person is approaching the transmitter and decrease when walking away from the transmitter (cases 1,2,3). In the sidewalk path (case 4), when there is no line of sight, the SNR has a constant mean value, since the scattered rays are dominant compared to the direct ray. As far as the bit errors are concerned, by using MRC, the BER decreases significantly compared to the BER for each of the original signals, the average BER being reduced from an with a single polarization, to about order of about with dual polarization, using SC or MRC. Thus, application of polarization diversity with a dual polarized antenna clearly results in improvement of the communication quality. A more detailed analysis of the BER, together with the achieved diversity gain, is provided in Section III-H. Moreover, the measurements using polarization diversity indicate that both antenna signals fade independently as bit errors often occur at different times for each signal. This reveals low correlation between the two diversity signals received by the antenna, as confirmed by the correlation analysis, presented in Section III-F. D. 4th-Order Receiver Diversity Using Two Dual-Polarized Textile Antennas

Fig. 11. Floor plan of the indoor environment. The position of the fixed TX unit is shown together with the paths walked by the test person: walking to and/or from TX (A-B) and walking sideways (A-C).

C. 2nd-Order Receiver Diversity Using One Dual-Polarized Textile Antenna The indoor environment where the measurements were performed is characterized by a high amount of multipath propagation, resulting in small-scale fading. For the walking firefighter the signal levels are also influenced by shadowing and changing path loss. In order to investigate the use of polarization diversity to mitigate fading effects, 2nd-order receiver diversity based on a single dual-polarized antenna was first implemented. At the transmitting unit a transmit power of 10 dBm was applied. Symbols were transmitted using quadrature phase shift keyed (QPSK) modulation and organized in sequential bursts. The

Compared to the 2nd-order polarization diversity with only one receiving antenna (front or back), additional improvement is expected by exploiting 4th-order pattern-polarization diversity using two dual-polarized antennas, i.e., using the four signals coming from both the front and rear antennas. Also exploiting pattern diversity is advantageous for several reasons. First, because the firefighter can block the path between the transmitter and receiver antenna, i.e., his body shields the antenna that is oriented away from the transmitter. Moreover, the directional radiation pattern of the patch antennas increases the influence of the orientation. Because of these effects, the signal is stronger for the front antenna (compared to the rear antenna) when approaching the transmitter and vice versa when walking away from the transmitter. However the antenna oriented away from the transmitter still receives an acceptable signal caused by the multipath propagation of the signals. The hardest path is the sideways track, at a large distance from the transmitter (for the 0.1 mW transmitted power) and with many obstacles in between. In order to have significant advantage from the 4th-order diversity, sufficiently low correlations are necessary between the

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Fig. 12. Received SNR, BER for each polarization, as well as BER after SC and MRC, for the front antenna on the walking firefighter equipped with a metal oxygen bottle strapped on the back. Application of polarization diversity with a single dual-polarized textile antenna, with SC or MRC, results in a drastic reduction of BER.

Fig. 13. Received SNR, BER for each polarization, as well as BER after SC and MRC, for the back antenna on the walking firefighter equipped with a metal oxygen bottle strapped on the back. Application of polarization diversity with a single dual-polarized textile antenna, with SC or MRC, results in a drastic reduction of BER.

four signals. In Fig. 14 the different SNR curves show that the fading patterns are different for the four signals, suggesting low correlations. Especially the front and rear signals fade independently over a large period of time.

The performance investigation was repeated for the 4th-order diversity case in the same way as for the 2nd-order diversity case. The resulting measured SNR and computed BER, using MRC and SC, are shown in Fig. 14. Combining the four signals

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Fig. 14. Received SNR, BER for each antenna, as well as BER after SC and MRC, for a walking firefighter walking with a metal oxygen bottle strapped on the back. Application of pattern-polarization diversity with two dual-polarized textile antennas results in a large improvement of the quality. No errors are left after detection, with both SC or MRC.

with SC or MRC shows a significant improvement in performance, here pattern and polarization diversity are combined. In our case no errors are left after applying MRC to the signals and detecting the data.

TABLE IV ON A 50 LOAD) AVERAGE CHANNEL ESTIMATES (SIGNAL LEVEL IN dB FOR DIFFERENT DIVERSITY SCENARIOS

E. Channel Estimations The average channel estimations for both antennas and for the combined signals also allow to evaluate the effectiveness of receiver diversity. To estimate the gain obtained by using polarization diversity, MRC is considered for the two signals received by the same patch antenna. The gain obtained by combining 4 signals (pattern and polarization diversity together) is also listed, for MRC and SC. to being the random variables representing Given the the channel estimates for each complete set of measurements corresponding to the different cases, the average channel estimations were calculated as follows: Back antenna, channel 1; • Back antenna, channel 2; • MRC 1,2; • Front antenna, channel 3; • • Front antenna, channel 4; MRC 3,4; • MRC 1–4; • SC 1–4. • In the above expressions indicates the statistical expectation, calculated as an arithmetical average over the total samples, representing realizations of the correavailable sponding random variable.

The average is calculated for each scenario (cases 1–4, as defined in Section III-B), together with the average signal levels resulting from different combining schemes. The results for the on a 50 load, different measurement paths, converted to are listed in Table IV. For the interpretation of the values in this table, the combined signals should be compared to all corresponding input signals, not only to the best input signal involved in the calculation. When using only one antenna, in a real-life situation, the received signal will sometimes correspond to the worst input signal. From this table it is clear that polarization diversity by itself provides a significant gain in average signal level. By MRC combining 4 signals, adding pattern diversity, the gain is always significantly larger. Comparing the difference in average signal levels for the four inputs reveals that the difference between front and back signals can be much larger than the difference between signals received on the same patch antenna, differing only in polarization.

VALLOZZI et al.: WIRELESS COMMUNICATION FOR FIREFIGHTERS USING DUAL-POLARIZED TEXTILE ANTENNAS

This behavior is expected, taking into account the shadowing by the body and the directional radiation pattern of the antennas. The use of pattern diversity combined with polarization diversity provides the largest gain when a direct signal path is present, in that case the antenna oriented to the transmitter will consistently provide stronger signals. However, also in the absence of a direct path the use of all four signals provides a significant performance gain. Bit-error-rate graphs presented in Section III-H will confirm that an effective fourth order diversity is obtained. From a hardware point of view, SC of the signals is a simpler way to obtain diversity but performs up to 2 dB worse than MRC in our measurements. The average values only partially display the potential performance gain resulting from diversity. Because the variance for the combined signals will be smaller than for the individual signals, a lower bit error rate will be obtained for the same average signal level. The bit-error-rate characteristics in Sections III-G and III-H further demonstrate this additional performance improvement.

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TABLE V CORRELATION OF THE RECEIVED SIGNAL POWER BETWEEN THE ANTENNAS FOR THE WALK TO THE TRANSMITTER

TABLE VI CORRELATION OF THE RECEIVED SIGNAL POWER BETWEEN THE ANTENNAS FOR THE WALK AWAY FROM THE TRANSMITTER

TABLE VII CORRELATION OF THE RECEIVED SIGNAL POWER BETWEEN THE ANTENNAS FOR THE WALK TO AND RETURN FROM THE TRANSMITTER WITHOUT OXYGEN BOTTLE

F. Correlation Between the Signals The correlation coefficient matrices were calculated based on the channel estimations corresponding to each signal. The correlation coefficients are determined for a sliding window of 10 measurements at a time. The reason why this method was adopted, is to account for the fading only and not for shadowing or path loss, as the latter two parameters are assumed to be constant within the used window of 10 channel estimates. and , scalar random variables Given representing the squared channel gains, and indicating the statistical expectation, the correlation coefficient is given by

TABLE VIII CORRELATION OF THE RECEIVED SIGNAL POWER BETWEEN THE ANTENNAS FOR THE WALK SIDEWAYS TO THE TRANSMITTER

(1)

TABLE IX CORRELATION OF THE RECEIVED SIGNAL POWER BETWEEN THE ANTENNAS FOR ALL THE PREVIOUS SCENARIOS TOGETHER

In practice, the statistical expectations in the right hand side of (1) were calculated as arithmetical averages over samples contained in a window, representing 10 realizations of the random variables. The correlation coefficients obtained in this way are then averaged to provide the final correlation value. A direct calculation of the correlation coefficient based on the complete measurement series often results in low correlation values that are not necessarily related to the performance of the communication system. Especially the shadowing of the body, combined with changes in orientation of the firefighter, can result in altering statistics for both antennas during the same measurement series. Within the window of 10 measurements used, the statistics of the signals do not vary significantly. The results are displayed in Tables V to IX. Antennas F1 and F2 correspond to the two polarizations for the front antenna, likewise B1 and B2 are the signals for the back antenna. For front and back signals the correlation is very low, hence using two antennas improves the performance of the system a lot. The

gain obtained by using front and back antennas is not only related to fading, as the radiation pattern of the antennas is directional and the front and back antennas each cover approximately a complementary hemisphere around the body. The correlation is higher for signals differing only in polarization compared to the values for front and back antennas. However polarization diversity is still present to a varying degree depending on the situation. The bit-error-rate measurements in the next section will confirm that a significant performance gain can be obtained by adding polarization diversity. G. Bit Error Rates After Data Detection The bit error rates for all measurements combined are shown in Table X. These values have been determined by demodulating

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TABLE X BIT ERROR RATES FOR DIFFERENT RECEIVED AND COMBINED SIGNALS

the received signal, detecting the data and counting the number of errors. The values clearly illustrate the performance gain obtained by using maximal ratio combining of the signals. The values obtained by combining the two orthogonally polarized signals from the same antenna already show a significant diversity gain. In this experiment and for the limited number of symbols transmitted, using maximal ratio combining with all four signals results in error-free data after combination, demodulation and detection. Using selection combining with four signals, some errors remain but the bit error rate is still much lower by using all four signals.

Fig. 15. BER characteristics as a function of average E =N for the “average antenna.” Both curves for selection combining (SC) and maximal ratio combining (MRC) of four antenna signals, result in a diversity order of four.

This bit error rate is expressed as a function of the average per receive antenna, where denotes the average received bit energy per antenna. Assuming a QPSK constellation is computed as with variance equal to 1,

H. Bit-Error-Rate Characteristics

where denotes the ratio of useful signal power to noise power at the detector during the i-th burst. The quantities are computed from the one-sided noise power specat the receiver, the measured squared channel tral density to related to the different antennas/polargains izations, and the considered signal combining strategy. Calculating for all measurement points and averaging over the transmitted bursts yields

(4) corresponds to the average energy per bit reThe above ceived by the “average antenna,” and results from the concatenation of all measurement series with the two dual-polarized antennas, involving all received signals, from all inputs. Thus, it represents the energy per bit averaged over the transmitted bursts and over the receiver branches, before combining. This method was adopted because the average signal level is different for each of the four antennas, which implies that selecting the of only one of the antennas would not be appropriate. Based on the measured set of signal levels, the BER for values is calculated by scaling the different average and values by a common factor, folrecorded lowed by repeating (2), (3) and (4). The BER values obtained correspond to signals with the same distribution but different arithmetic means. Fig. 15 shows the average BER as a function of the average . The curve “average antenna” refers to the BER for a single antenna, averaged over the four antennas. The four original antenna signals are combined using SC and MRC. The resulting graph in Fig. 15 shows array gain and diversity gain combined. The largest possible array gain for four antennas is 6 dB, occurring when all four average signal strengths are equal. Also shown in Fig. 15 are the theoretical BER curves for a single antenna and for MRC using four antennas, under the assumption of Rayleigh fading. The theoretical curves, displayed for 1st- and 4th-order diversity, are calculated as [15, p. 825]

(3)

(5)

The bit-error-rate characteristics can be calculated based on the received signal-to-noise ratios. For these calculations only measurement data recorded along the sideways track were used. In this measurement series the path loss is nearly constant. Inevitably some shadowing will be present, making the signal worse than Rayleigh distributed. To be able to calculate relevant BERs, a new measurement series was recorded along the sideways track. The transmitted power was increased to 4 mW and the number of measurements was incremented to 300 data bursts, received along the sideways track. The other conditions for the measurement are comparable to the measurements discussed earlier in this text (Section III-B, case 4). Assuming the channel invariant during the time of one received burst, the bit error rates have been calculated for each measurement point, i.e., for each symbol burst, as

(2)

VALLOZZI et al.: WIRELESS COMMUNICATION FOR FIREFIGHTERS USING DUAL-POLARIZED TEXTILE ANTENNAS

with

the diversity order and

(6) The curve for the average antenna reveals a higher bit error rate compared to the Rayleigh distributed signal. This is probably due to shadowing, both by the indoor environment and by the human body, resulting in a worse than Rayleigh distributed signal. The bit error rates for the combined signals show a significant improvement, which is clearly best for MRC. For bit error , the curve for MRC approximates the theorates down to retical characteristic for 4th-order diversity in a Rayleigh fading environment. Since the bit error rate for the original signals is a little higher than for Rayleigh fading, consequently, the bit error rate for the MRC combined signal is also slightly higher compared to the theoretical curve for Rayleigh fading with 4th-order diversity. the curve for the measured For bit error rates lower than signals deviates increasingly from the theoretical characteristic. This can be attributed to the absence of a sufficient number of very low signal levels after Maximal Ratio Combining, resulting in an excessively low bit error rate. More measurements are needed to obtain more accurate values for the low error probabilities involved.

IV. CONCLUSION The measurement data clearly indicate the advantage of using two dual-polarized textile patch antennas for off-body communication. The correlation values show that the signal levels for both feeds coming from the same patch antenna and corresponding to orthogonal polarizations fluctuate independently in our indoor test environment. Examining the data after demodulation and detection confirmed that the number of bit errors left after combining is lower than what would be obtained for each polarization alone. Hence the dual-polarized antenna, which produces two signals which are then MRC combined, provides a more reliable communication link compared to an antenna producing only one signal. Using two dual-polarized antennas, on the front and rear of the body, is recommended, as the body always shields the radiation from the antenna such that no omnidirectional coverage around the body is possible with a single antenna. Using two antennas permits each antenna to cover a hemisphere around the body. The pattern and polarization diversity complement each other to achieve a significant additional performance gain. The bit error rates and channel estimates after combining clearly demonstrate the performance increase by using all four signals. The bit-error-rate curves prove that an effective 4th-order diversity is realized in our application: the performance of the combined signal approximates the theoretical characteristics

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for 4th-order diversity in a Rayleigh fading environment when using MRC. Selection Combining results in diversity of the same order but performs down to 2 dB worse compared to MRC. The proposed antenna system is therefore a suitable candidate for future use in wearable textile systems for firefighters. The integrated research project “PROeTEX” [16] already developed some initial prototypes of the discussed wearable textile system including textile antennas and implementing diversity. Though, in the current prototypes, the antennas are fed by a non-textile portable transceiver, available on the market, implementing dual diversity [17]. The connection with the antenna terminals is realized with flexible coaxial cables [2]. Future research will aim to develop transceivers, connection cables and in general electronics, designed on or even completely made out of textile materials and therefore fully integrable into garments. REFERENCES [1] L. Van Langenhove, Smart Textiles for Medicine and Healthcare. Cambridge, U.K.: Woodhead Publishing, 2007. [2] D. Curone, G. Dudnik, G. Loriga, J. Luprano, G. Magenes, R. Paradiso, A. Tognetti, and A. Bonfiglio, “Smart garments for safety improvement of emergency/disaster operators,” in Proc. 29th Annu. Int. Conf. IEEE EMBS, 2007, pp. 3962–3965. [3] P. S. Hall and Y. Hao, Antennas and Propagation for Body-Centric Wireless Communications. Boston/London: Artech House, 2006. [4] C. Hertleer, H. Rogier, L. Vallozzi, and L. Van Langenhove, “A textile antenna for off-body communication integrated into protective clothing for firefighters,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 919–925, Apr. 2009. [5] T. F. Kennedy, P. W. Fink, A. W. Chu, N. J. Champagne, G. Y. Lin, and M. A. Khayat, “Body-worn E-textile antennas: The good, the lowmass, and the conformal,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 910–918, Apr. 2009. [6] Y. Rahmat-Samii, “Wearable and implantable antennas in body-centric communications,” in Proc. 2nd European Conf. Antennas and Propag., EuCAP 2007, Edinburgh, U.K., 2007, pp. 1–5. [7] I. Khan, P. S. Hall, A. A. Serra, A. R. Guraliuc, and P. Nepa, “Diversity performance analysis for on-body communication channels at 2.45 GHz,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 956–963, Apr. 2009. [8] I. Khan and P. S. Hall, “Multiple antenna reception at 5.8 and 10 GHz for body-centric wireless communication channels,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 248–255, Jan. 2009. [9] Y. Ouyang, D. J. Love, and W. J. Chappell, “Body-worn distributed MIMO system,” IEEE Trans. Veh. Technol., vol. 58, no. 4, pp. 1752–1765, May 2009. [10] S. L. Cotton and W. G. Scanlon, “Spatial diversity and correlation for off-body communications in indoor environments at 868 MHz,” in Proc. IEEE 65th Vehicular Technology Conf., VTC2007-Spring, Dublin, Ireland, 2007, pp. 372–376. [11] C. Hertleer, A. Tronquo, H. Rogier, and L. Van Langenhove, “The use of textile materials to design wearable microstrip patch antennas,” Textile Res. J., vol. 78, no. 8, pp. 651–658, Aug. 2008. [12] R. G. Vaughan, “Polarization diversity in mobile communications,” IEEE Trans. Veh. Technol., vol. 39, no. 3, pp. 177–186, Aug. 1990. [13] R. U. Nabar, H. Bölcskei, V. Erceg, D. Gesbert, and A. J. Paulraj, “Performance of multiantenna signaling techniques in the presence of polarization diversity,” IEEE Trans. Signal Processing, vol. 50, no. 10, pp. 2553–2562, Oct. 2002. [14] L. Vallozzi, H. Rogier, and C. Hertleer, “Dual polarized textile patch antenna for integration into protective garments,” IEEE Antennas Wireless Prop. Lett., vol. 7, pp. 440–443, 2008. [15] J. G. Proakis, Digital Communication. New York: McGraw-Hill, 2001. [16] PROeTEX [Online]. Available: http://www.proetex.org/index.htm [17] WLAN Reference Design With the MAX2830 Maxim Integrated Products [Online]. Available: http://www.maxim-ic.com/appnotes.cfm/ an_pk/4276

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Luigi Vallozzi was born in Ortona, Italy, in 1980. He received the Laurea degree in electronic engineering from the Università Politecnica delle Marche, Ancona, Italy, in 2005 and is currently pursuing the Ph.D. degree in electrical engineering at Ghent University, Ghent, Belgium. His research focuses on design and prototyping of antennas for wearable textile systems, and the modeling and characterization of multiple-input multipleoutput wireless communication systems.

Patrick Van Torre was born in 1971. He received the Master’s degree in electrical engineering from Hogeschool Gent, Belgium, in 1995. From August 1995 to October 1998, he was working as a development Engineer in the private sector. Since November 1998, he has been active as an educator in electronics and researcher in the field of ultrasound technology. He is currently employed by Hogeschool Gent where he teaches theory courses in analog electronics, organizes project oriented lab sessions and is involved in public relations activities and hardware development projects for third parties. He is also a part-time researcher, affiliated with the Department of Information Technology, Ghent University.

Carla Hertleer received the M.Sc. degree in textile engineering and the Ph.D. degree in engineering from Ghent University, Ghent, Belgium, in 1990 and 2009, respectively. Her dissertation was on the research topic of textile-based antennas. For three years, she worked in a vertically integrated textile company that produces terry cloth. Since June 2000, she has worked as a Researcher in the Textile Department, Ghent University. She has given classes in weaving and Jacquard technology, but her recent activities are mainly concentrated on smart textiles, more specifically the research and development of textile sensors for integration in biomedical clothing and textile antennas. The latter research is carried out in collaboration with the Department of Information Technology, Ghent University. Her research is carried out in the framework of national and European projects.

Hendrik Rogier (SM’06) was born in 1971. He received the Electrical Engineering and the Ph.D. degrees from Ghent University, Gent, Belgium, in 1994 and in 1999, respectively. He is currently a Postdoctoral Research Fellow of the Fund for Scientific Research Flanders (FWO-V), Department of Information Technology, Ghent University, where he is also an Associate Professor with the Department of Information Technology. From October 2003 to April 2004, he was a Visiting Scientist at the Mobile Communications Group, Vienna University of Technology. He authored and coauthored about 50 papers in international journals and about 70 contributions in conference proceedings. His current research interests are the analysis of electromagnetic waveguides, electromagnetic simulation techniques applied to electromagnetic compatibility (EMC) and signal integrity (SI) problems, as well as to indoor propagation and antenna design, and in smart antenna systems for wireless networks. Dr. Rogier was awarded the URSI Young Scientist Award at the 2001 URSI Symposium on Electromagnetic Theory and at the 2002 URSI General Assembly. He is serving as a member of the Editorial Board of IET Science, Measurement Technology and acts as the URSI Commission B representative for Belgium.

Marc Moeneclaey (F’02) received the Diploma of Electrical Engineering and the Ph.D. degree in electrical engineering from Ghent University, Gent, Belgium, in 1978 and 1983, respectively. He is Professor in the Department of Telecommunications and Information Processing (TELIN), Gent University. His main research interests are in statistical communication theory, (iterative) estimation an detection, carrier and symbol synchronization, bandwidth-efficient modulation and coding, spread-spectrum, satellite and mobile communication. He is the author of more than 400 scientific papers in international journals and conference proceedings. He is a coauthor of the book Digital communication receivers—Synchronization, channel estimation, and signal processing. (Wiley, 1998). Dr. Moeneclaey is co-recipient of the Mannesmann Innovations Prize 2000. During the period 1992–1994, he was Editor for Synchronization, for the IEEE TRANSACTIONS ON COMMUNICATIONS. He served as co-Guest Editor for special issues of the Wireless Personal Communications Journal (on Equalization and Synchronization in Wireless Communications) and the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS (on Signal Synchronization in Digital Transmission Systems) in 1998 and 2001, respectively.

Jo Verhaevert received the Engineering degree and the Ph.D. degree in electronic engineering from the Katholieke Universiteit Leuven, Belgium, in 1999 and 2005, respectively. He currently teaches courses on telecommunication at the Department of Applied Engineering Sciences, University College Ghent, Ghent, Belgium, where he also performs research. His research interests include indoor wireless applications (such as wireless sensor networks), indoor propagation mechanisms and smart antenna systems for wireless systems.

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A Compact Six-Port Dielectric Resonator Antenna Array: MIMO Channel Measurements and Performance Analysis Ruiyuan Tian, Student Member, IEEE, Vanja Plicanic, Member, IEEE, Buon Kiong Lau, Senior Member, IEEE, and Zhinong Ying, Senior Member, IEEE

Abstract—MIMO systems ideally achieve linear capacity gain proportional to the number of antennas. However, the compactness of terminal devices limits the number of spatial degrees of freedom (DOFs) in such systems, which motivates efficient antenna design techniques to exploit all available DOFs. In this contribution, we present a compact six-port dielectric resonator antenna (DRA) array which utilizes spatial, polarization and angle diversities. To evaluate the proposed DRA array, a measurement campaign was conducted at 2.65 GHz in indoor office scenarios for 6 multiple antenna systems. Compared to the reference four 6 system of monopole arrays which only exploit spatial diversity, the use of dual-polarized patch antennas at the transmitter enriches the channel’s DOF in the non-line-of-sight scenario. Replacing the monopole array at the receiver with the DRA array that has a 95% smaller ground plane, the 10% outage capacity evaluated at 10 dB reference signal-to-noise ratio becomes equivalent to that of the reference system, due to the DRA’s rich diversity characteristics. In the line-of-sight scenario, the DRA array gives a higher DOF than the monopole array as the receive counterpart to the transmit patch array. However, the outage capacity is 1.5 bits/s/Hz lower, due to the DRA array’s lower channel gain. Index Terms—Antenna diversity, dielectric resonator antennas (DRAs), MIMO systems, polarization.

I. INTRODUCTION

M

IMO systems can achieve high spectrum efficiency in wireless communications by employing multiple antennas at both the transmit (TX) and receive (RX) sides. Such systems perform best when the spatial correlation among signals on different antenna branches is low [1]. However, the compactness of today’s terminal devices limits the degrees of freedom (DOFs), and consequently the correlation performance Manuscript received August 02, 2009; revised September 30, 2009; accepted October 12, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. This work was supported in part by VINNOVA under Grants 2007-01377 and 2008-00970, Vetenskapsrådet under Grant 2006-3012, and in part by Sony Ericsson Mobile Communications AB. This paper was presented in part at the 2nd COST2100 Workshop, Valencia, Spain, May 20, 2009. R. Tian and B. K. Lau are with the Department of Electrical and Information Technology, Lund University, SE-221 00 Lund, Sweden (e-mail: Ruiyuan. [email protected]; [email protected]). V. Plicanic is with Department of Electrical and Information Technology, Lund University, SE-221 00 Lund, Sweden and also with Sony Ericsson Mobile Communications AB, SE-221 88 Lund, Sweden (e-mail: Vanja.Plicanic@eit. lth.se; [email protected]). Z. Ying is with Sony Ericsson Mobile Communications AB, SE-221 88 Lund, Sweden (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2041174

in such systems. This motivates the need for efficient antenna design techniques to exploit all available DOFs. Recent examples on the design and performance evaluation of compact multiple antenna terminals include [2]–[4]. In 1938, Richtmyer showed that a suitably shaped dielectric material can function as electrical resonators for high frequency oscillations [5]. The characteristics of such dielectric resonators have been the subject of many early studies, e.g., [5]–[7]. More recently, their application as antenna elements has been demonstrated [8]–[10]. One interesting feature of dielectric resonator antennas (DRAs) is that the antenna can be electrically small (at the expense of efficiency bandwidth) when high permittivity material is used. This makes it attractive for compact implementations in wireless communications. In [11], a low profile single-port L-shape DRA fed by a planar inverted-F antenna (PIFA) is designed for laptop in Wireless LAN (WLAN) applications. In [12], a compact cylindrical DRA is designed for triple-mode operation, where two modes are excited for radiation in two different frequency bands, and the third mode is used as a filter. In [13]–[16], a single three-port rectangular DRA element is developed and evaluated for diversity and MIMO antenna systems in WLAN-type applications. Most of the existing studies focus on the antenna performance of scattering parameters and radiation patterns, and they do not consider the effect of the propagation channel. In [17], the single three-port DRA proposed in [13] is evaluated with ray tracing simulations of an indoor environment, and it is shown to achieve comparable capacity performance as a conventional uniform linear array of ideal dipoles despite its significantly more compact size. Even though the ray tracing simulations in [17] can give an initial indication of the DRA’s performance in its usage environment, they utilize simplifying assumptions of the modeled 3D environment and the propagation mechanisms. Therefore, measurements in real environments are crucial to fully substantiate the DRA’s practicality. In this contribution, we propose a diversity-rich yet compact six-port antenna array. The proposed array consists of two three-port DRA elements, which jointly utilizes spatial, polarization and angle diversities. In order to evaluate its performance for WLAN-type applications, a 6 6 MIMO channel measurement campaign was conducted at 2.65 GHz in indoor scenarios. Two common types of six-port antenna arrays were also measured in the campaign for the purpose of comparison: a single-polarized monopole array exploiting only spatial diversity, and a dual-polarized patch array exploiting spatial and

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TABLE I LIST OF ANTENNA SYSTEMS UNDER EVALUATION

Fig. 1. Floor map of the measurement campaign. (a) TX antenna array positions marked by black stars. (b) Room E:2521 with RX antenna array positions marked by black dots. The light gray rectangles are the office desks.

polarization diversities. The measurement results are analyzed in order to demonstrate the potential use of the compact DRA array for MIMO communications, in comparison to the larger monopole and patch arrays. The reminder of paper is structured as follows. Section II describes the measurement campaign in detail. Section III presents the characteristics of the three types of six-port antenna arrays used in this study. In Section IV, the performance of array dependent measured MIMO channels are evaluated and analyzed. Section V concludes the discussion. II. MEASUREMENT CAMPAIGN A. Setup The channel transfer functions between the TX/RX antenna pairs were measured using the RUSK LUND wideband channel sounder, which performs MIMO channel measurements based on the “switched array” principle [18]. The measurements were performed using 321 subcarrier signals over 200 MHz bandwidth with a center frequency at 2.65 GHz. However, we only used the measured data over 100 MHz bandwidth (i.e., 2.6 GHz–2.7 GHz) for this study. The output power of the channel sounder was 0.5 W (27 dBm). The length of the test signal to obtain one snapshot (in time) for one TX-RX channel branch was set to be 1.6 , which ensured long enough “excess runlength” of multipath components to avoid overlap of subsequent impulse responses in the considered environment [19]. A block of 20 consecutive snapshots was obtained for each RX measurement position (of a given array orientation). The measured channel matrices obtained from the consecutive snapshots are used in post processing to enhance the signal-to-noise ratio (SNR) of the measurement and to estimate the noise power. B. Scenario The channel measurements were performed in a corridor and office room E:2521 on the second floor of the E-building at LTH, Lund University, Sweden, as shown in Fig. 1. The dimensions of the room are . There were desks, chairs, a large white board, and other typical office furniture in the room. The RX unit of the channel sounder was stationed at the end of the corridor, which is just outside room E:2521 during the entire measurement campaign. The RX antenna array, placed on top of a trolley at the height of an office desk (i.e., 0.7 m),

was stationed at different measurement locations inside the office [see Fig. 1(b)]. In total, 5 rectangular grids of measurement positions (A–E) were chosen in proximity of the desks. Within each grid, 12 positions were measured in order to obtain good fading statistics. Adjacent measurement positions within each grid were two wavelengths (226 mm at 2.65 GHz) apart from each other. At each measurement position, two orientations of the RX array were performed. The array was rotated 90 horizontally with respect to the orientation in the first measurement to obtain the second measurement set. For both orientations, the rectangular ground plane of the RX array was aligned in parallel with the office walls, i.e., in the first (second) orientation, the longer side of the array’s ground plane was aligned in parallel with the longer (shorter) side of the office room. Two propagation scenarios were measured: line-of-sight (LOS) and non-line-of-sight (NLOS). In the NLOS scenario, the TX antenna array was located 9.7 m away from the end of corridor [see Fig. 1(a)]. The TX unit of the channel sounder was stationed in the hall area behind the TX antenna array. In the LOS scenario, the TX and RX array structures are in LOS of each other. The TX antenna was located next to the door inside the office, whereas the TX unit of the channel sounder was stationed outside the office. It is further noted that even though the TX and RX array structures are in LOS, the LOS path may not necessarily exist between the TX and RX array elements, depending on the orientation of the TX array. The effect of LOS obstruction will be examined in Section IV. In both scenarios, the TX antenna array was placed at a height of 1.8 m, corresponding to the height of an elevated WLAN Access Point (AP). The following steps were taken in order to minimize disturbances and assure a static measurement environment: 1) The MIMO channel was measured for the center frequency of 2.65 GHz, instead of 2.45 GHz, in order to avoid interference from the existing WLAN systems while maintaining similar propagation characteristics. Furthermore, a spectrum analyzer was used to ensure no detectable interfering sources in the measured environment. WLAN APs in proximity of the measured site were disabled during the measurement. 2) The measurement campaign was conducted during one occasion, from late one afternoon to early next morning. No significant movement within the measurement site during the campaign was ensured. After each measurement run, a person either rotated the RX antenna array or moved it to the next position for the next measurement. 3) The doors of other offices along the corridor were closed. 4) Absorber units were used to cover the body of the TX/RX unit of the channel sounder. III. ANTENNA CONFIGURATIONS The measurement campaign comprises the evaluation of four TX/RX multiple antenna systems, listed in Table I.

TIAN et al.: A COMPACT SIX-PORT DRA ARRAY: MIMO CHANNEL MEASUREMENTS AND PERFORMANCE ANALYSIS

Fig. 2. Photos of (a) the TX patch antenna array on a tripod in the NLOS scenario; (b) the RX monopole antenna array; (c) the RX DRA array.

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Fig. 4. Measured realized gain patterns [dBi] of the six-port monopole ( = 90 )[dBi], = 1 . . . 6; antenna array at 2.65 GHz. (a) (b) ( = 0 )[dBi], = 1 . . . 6.

G ;

i

G  ; ;

;

i

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Fig. 3. Sketches of (a) the monopole and (b) the patch antenna arrays.

Using sparsely separated single-polarized monopoles at both TX and RX antenna systems, Case I represents a reference system for WLAN-type applications. This is due to the well-known characteristics of monopole antennas. In addition, the antenna spacing between monopoles is designed to be one wavelength to minimize coupling and spatial correlation. In Case II, dual-polarized patch antennas are used at the TX subsystem to exploit polarization diversity and to characterize polarized propagation. In Case III, the proposed DRA array is investigated at the RX subsystem, such that the RX antenna system is physically compact (e.g., wireless terminals). In this system, the patch array is chosen as the TX counterpart since it is able to excite propagation in two orthogonal polarizations, and is thus a suitable match for the multi-polarized DRA array. In addition, the compact DRA array is evaluated against the monopole array at the RX end by comparing Cases II and III. In Case IV, the compact DRA array is used at both TX and RX antenna systems. This case corresponds to a more compact design of the TX antenna system, i.e., the WLAN AP. Details of the antenna arrays are given below. The monopole and DRA arrays were evaluated with a vector network analyzer and in a Satimo Stargate measurement system [20]. The DRA was also simulated with the CST software [21]. The specifications of the patch array are available in [22]. A. Monopole Array The monopole array consists of six vertical quarter-wavelength monopole antennas spaced one wavelength (113 mm at 2.65 GHz) apart from one another in a rectangular grid on a ground plane [see Figs. 2(b) and 3(a)]. The ground . plane size is The monopole array covers the evaluated frequency band of 2.6–2.7 GHz. The reflection coefficient is less than 19 dB, and the coupling between the neighboring elements is less than 20 dB. The antenna total efficiency, taking into account mismatch, dielectric and conductive losses, is 82% on average. The

Fig. 5. (a) Simulation model of the compact DRA array. (b) Drawing of the single DRA prototype.

measured realized gain patterns are given in Fig. 4. Approximately uniform patterns are observed in the azimuth plane for all elements of the array. In the elevation plane, however, the maximum gain of 5 dBi is obtained at 30 above the azimuth plane, due to the finite ground plane size [23]. The impact of the elevated radiation patterns on measured channel characteristics is examined in Section IV. During the measurement of Cases I and II, the ground planes of the monopole arrays were placed horizontally such that the monopole elements were vertically polarized. B. Patch Array The patch array [22] is a uniform planar array with 4 8 dualpolarized radiating square patch elements (64 ports in total) spaced half a wavelength (56.6 mm at 2.65 GHz) apart in a rectangular grid [see Fig. 2(a)]. For the 6 6 MIMO channel measurements, six patch elements in a row were chosen, such that the {1, 3, 5}-th elements were vertically polarized and the {2, 4, 6}-th elements were horizontally polarized. All other ports were terminated with 50 loads. A sketch of the antenna array is shown in Fig. 3(b). The patch antenna array has a reflection coefficient of less than 12 dB within the 2.6–2.7 GHz frequency band. The coupling between adjacent co-polarized elements is less than 11 dB. For the cross-polarized ports on the same patch element, the polarization isolation is more than 23 dB. The antenna total efficiency is approximately 83%, with a maximum gain of 6.6 dBi. To take into consideration the patch antennas’ radiation characteristics in Cases II and III, the patch elements were facing the

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Fig. 6. Measured and simulated S-parameters of the compact DRA array. (a) Measured reflection coefficient; (b) measured isolation; (c) simulated reflection coefficient; (d) simulated isolation.

end of the corridor in the NLOS scenario and into room E:2521 in the LOS scenario (see Fig. 1). C. Compact DRA Array The compact DRA array consists of two three-port DRA elements placed with reflection symmetry (see Figs. 2(c) and 5). Each DRA element comprises a cube of open dielectric material on a ground plane. Two microstrip excitation ports are on the sides and a monopole port is in the middle of the structure. A rectangular dielectric resonator theoretically supports two fundamental TE-modes which radiate like magnetic dipoles [10]. The two silver microstrips on two perpendicular faces of the cube are used to excite these modes. A monopole antenna is inserted into the center of the dielectric resonator to create a third port without disturbing the radiating modes of the dielectric resonator. Fig. 5(a) shows the simulation model of the compact six-port DRA array that is tuned to the center frequency of 2.65 GHz. Ports {1, 2} and {4, 5} denote the microstrip excitation ports, and ports 3 and 6 denote the monopole ports on the two DRA elements, respectively. The dimensions of the single DRA element are given in Fig. 5(b). The dielectric resonator cube, which of 19.6 is obtained from TDK, has relative permittivity of 0.0001. At the center frequency and a loss tangent of 2.65 GHz, the electrical dimensions of each dielectric resand the monopole has a onator cube are . The separation distance between the center of length of . The ground plane size the two DRA elements is 50 mm of the DRA array is , i.e., 95% smaller compared to the monopole array. The compactness makes the proposed DRA array an attractive candidate for use in wireless terminals. It should be noted that the sizes of both the DRA element and the

DRA array can be further reduced for specific terminal devices, if required. However, the prototype used suffices as a proof of concept for the suitability of DRA arrays for MIMO communications in an indoor environment. Fig. 6 shows the S-parameters of the DRA array obtained from both measurements and simulations (using CST software [21]). As can be seen, the DRA array covers the 100 MHz frequency band at 2.65 GHz with a reflection coefficient of less than 10 dB. The measured and simulated results agree well, except for some minor detuning observed in the measured port 5. The worst isolation of 10 dB is observed between a given microstrip excitation port and the monopole port on the same DRA element (e.g., between ports {1, 2} and 3). The isolation between all other ports is more than 15 dB. The S-parameters that are not shown exhibit similar behavior due to the symmetrical structure of the DRA array. The measured radiation patterns of the compact DRA array in are shown in three different cuts: one -plane , 135 ) in Figs. 8 and 9. The Fig. 7, and two -planes ( two -planes are chosen to be aligned with the vertical faces of the DRA elements [see Fig. 5(a)]. First, reflection symmetry is observed between the patterns of the two antenna elements. The six radiation patterns of the two DRAs are directed towards a broad range of distinct directions. For example, the two monopole patterns (ports 3 and 6) provide coverage in opposite directions (i.e., compare subplots (e) and (f) in Figs. 7–9). Since different antenna ports primarily see different directions, angle diversity can be achieved. Second, polarization diversity is also exploited. In Fig. 7, the monopole patterns (ports 3 and 6) have a stronger contribution in the -component, whereas the two polarizations ( - and -components) are orthogonal across the patterns of the microstrip excitation ports (ports {1, 2, 4,

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G ;

Fig. 8. Measured realized gain patterns ( = 45 )[dBi] of the six-port compact DRA array at 2.65 GHz. Black solid line: total gain; Light gray dashed line: -component; Black dashed line: -component. (a) Port 1; (b) port 4; (c) port 2; (d) port 5; (e) port 3; (f) port 6.



G 

;

Fig. 7. Measured realized gain patterns ( = 90 )[dBi] of the six-port compact DRA array at 2.65 GHz. Black solid line: total gain; Light gray dashed line: -component; Black dashed line: -component. (a) Port 1; (b) port 4; (c) port 2; (d) port 5; (e) port 3; (f) port 6.





5}). Moreover, as shown in Fig. 8 for the -cut, the patterns of ports 1 and 5 are dominated by the -component, whereas the patterns of ports 2 and 4 are dominated by the -component. The diverse radiation patterns make the compact DRA array robust to incoming waves with arbitrary directions and polarizations, thus making good use of the available DOF. The simulated radiation patterns are in good agreement with the measured ones, and are not included here due to space constraint. Nevertheless, the simulations gave the insight that the strong currents are concentrated under the DRA elements and around the microstrip excitation ports when the TE-modes are excited. When the monopole is excited, the strong currents mainly reside on the monopole element. Thus, no significant radiation is associated with the ground plane. The DRA array has an antenna total efficiency of 68% on average, with a maximum gain of 5.1 dBi. During the measurement of Cases III and IV, the ground plane of the RX DRA array was placed

horizontally. However, the ground plane of the TX DRA array in Case IV was placed vertically, in order to account for the array’s radiation characteristics. IV. ANALYSIS In this section, the measured MIMO channels for the four antenna systems are evaluated and analyzed. The investigated parameters include RX power, SNR, branch power ratio (BPR), channel envelope distribution and MIMO performance in terms of channel capacity. A. RX Power The RX power is calculated from the measured MIMO channel matrices of each evaluated antenna system , , as (1) denotes the Frobenius norm operator, dewhere notes the number of measured antenna systems (Cases I–IV),

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Fig. 10. Averaged RX power relative to Case I of each antenna system (Cases II–IV) at each measurement grid position (A–E). (a) LOS; (b) NLOS.

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Fig. 9. Measured realized gain patterns ( = 135 )[dBi] of the six-port compact DRA array at 2.65 GHz. Black solid line: total gain; Light gray dashed line: -component; Black dashed line: -component. (a) Port 1; (b) port 4; (c) port 2; (d) port 5; (e) port 3; (f) port 6.



TABLE II AVERAGED RX POWER RELATIVE TO CASE I

is the number of measured grid positions (A–E), and is the number of narrowband channel realizations obtained from 161 frequency subcarriers (i.e., within 2.6–2.7 GHz) at 12 measured points within each grid and with two array orientations. The RX power of all other antenna systems under evaluation (Cases II–IV) are compared to that of the reference system (Case I). Table II summarizes the relative RX power averaged among the five measured grids. Case II collects significantly more power on average (4.1 dB in LOS and 1.6 dB in NLOS) than other cases. The strong channel gain is due to the patch and the monopole array having the highest antenna efficiency and gain at the TX and RX subsystems, respectively. In addition, the orientation of the TX patch array makes Case II favorable for a higher RX power, since the maximum gain is directed

Fig. 11. Path loss of each antenna system (Cases I–IV) at each measurement grid position in the LOS scenario. {A, B, C, D, E} denote the distance (in linear scale) from the TX array to the center of each measured grid. Grid D is used as a reference position. Gray diamonds: path loss model; Black circles and stars: measured channel branches. (a) Case I; (b) Case II; (c) Case III; (d) Case IV.

towards the general direction of the RX array. In Case I, however, the finite ground plane of the TX monopole array makes the radiation pattern elevated such that the maximum gain is directed away from the RX antennas. Cases III and IV exhibit channel gains of within 1 dB relative to that of the reference system. Fig. 10 illustrates the relative RX power of Cases II–IV with respect to Case I measured at each grid position (A–E), in LOS and NLOS scenarios, respectively. In LOS, see Fig. 10(a), a trend of increasing RX power from grid positions A to E is observed. In order to further study this effect, a simple path loss model [24] is used (2) where denotes the path loss, denotes the TX-RX separation distance, and is the path loss exponent. Denoting grid position D, which is the closest to the TX array position [see Fig. 1(b)], as a reference distance , the path loss difference is found for other measured grid positions, as (3) Fig. 11 compares the path loss difference obtained from the measured RX power of each channel branch with the calcula. While the increase of path loss is tion using (3) with

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TABLE III GROUPS OF CHANNEL BRANCHES

TABLE IV BPR OF MEASURED ANTENNA SYSTEMS

C. Branch Power Ratio

Fig. 12. Averaged RX power for each TX/RX antenna pair, in total of 36 channel branches. Note the negative sign on y-axis, i.e., the higher the box the less the power. (a) Case I, LOS; (b) Case I, NLOS; (c) Case II, LOS; (d) Case II, NLOS; (e) Case III, LOS; (f) Case III, NLOS; (g) Case IV, LOS; (h) Case IV, NLOS.

partly due to the increase of the TX-RX separation, the measured RX power is also strongly influenced by the choice of antenna systems. Specifically, the RX power in Case I is approximately invariant to the change in TX-RX separation distance [see Fig. 11(a)]. For Case II, a closer examination reveals that the RX power of the co-polarized channel branches [indicated by stars in Fig. 11(b)] agree with the path loss model, whereas this is not the case for the cross-polarized channel branches (indicated by circles). These effects are further investigated in the following sections. B. SNR The SNRs of the measured channels are estimated using the 20 consecutive snapshots obtained at each measurement location and array orientation. Since the measured channel at each location is supposed to be static as described in Section II, the differences between the measured channel matrices obtained from the consecutive snapshots are used to estimate the noise power (or variance). On average, the SNR is found to be 27.4 dB in LOS and 23.1 dB in NLOS.

In order to investigate the branch power of each TX/RX antenna pair, Fig. 12 illustrates the averaged RX power of each TX/RX channel (i.e., 36 channel branches in total) for all antenna systems (Cases I–IV) in LOS and NLOS. 1) Case I: As shown in Fig. 12(a) and (b), the RX power in the reference system (Case I) is approximately uniformly distributed among all channel branches. However, in other antenna systems (Cases II–IV), branch power imbalance is observed. Channel branches with distinct characteristics can be identified in groups, see Table III. The power ratios between different groups of channel branches are given in Table IV. 2) Case II: Fig. 12(c) and (d) show that significant branch power imbalance is observed between vertically excited channels (TX ports {1, 3, 5}) and horizontally excited channels (TX ports {2, 4, 6}). The BPR is defined as the power ratio between and cross-polarized channels . co-polarized channels In LOS, the direct propagation dominates so that the cross-polarization ratio (XPR) is inherently high. The co-polarized channels are found to be 11.3 dB stronger than the cross-polarized channels. In NLOS, the multipath propagation induces significant cross-polarization response. Thus, the BPR is reduced to 4.2 dB. 3) Case III: Three groups of channel branches can be identified from Fig. 12(e) and (f), according to the characteristics of the microstrip excitation and monopole ports on the RX DRA array. In LOS, the channel branches with co-polarized achieves about 8 dB higher power than the monopoles , whereas the BPR of the cross-polarized monopoles to the channel branches with microstrip excitation ports cross-polarized monopoles on the RX DRA array is about 5.9 dB. In NLOS, branch power imbalance is mitigated to and to are reduced to about and BPRs of 5 dB and 2 dB, respectively. Compared to Case II, the BPR is reduced due to the RX DRA array exploiting polarization

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diversity. Thus, the compact DRA array is shown to be more robust in polarized propagation channels, but with 1.7–3.1 dB less of RX power (compare Cases II and III in Table II). 4) Case IV: Two groups of channel branches can be identified from Figs. 12(g) and (h), according to the characteristics of the microstrip excitation and monopole ports on the TX DRA array. On the one hand, the TX DRA array is placed perpendicular with respect to the RX DRA array. This results in cross-polarized monopole elements in the corresponding TX/RX pairs. On the other hand, the microstrip excitation ports achieve higher gain than the monopole ports discussed in Section III-C. Consequently, the BPR is about 3.6 dB and 5.5 dB in LOS and NLOS, respectively. This can be understood by the rich angle diversity characteristics of the compact DRA array. The radiation patterns cover a broad range of directions such that branch power imbalance is mitigated in LOS. D. Power Normalization The above discussion on the RX power reveals that it is important to account for the impact of path loss in the normalization of the channel realizations. In this work, the measured channel matrices of each antenna system are normalized lo, cally within each measurement grid. For channel matrices , the normalized MIMO channel matrices are obtained as [1] (4) and are the numbers of TX and RX antenna ports, respectively. With this approach, the small-scale power variation among the measured points within each grid is preserved, whereas the large-scale power variations and the differences in path loss between different grids are neglected. The following two normalization principles are considered: 1) Normalized RX SNR: (assuming power control) The measured channel matrices of each antenna system (Cases I–IV) are normalized independently, such that all antenna systems have the same average evaluation SNR at the RX side. Thus the channel’s DOF can be investigated regardless of the relative power difference among the four antenna systems. 2) Case I as Reference: In order to account for the relative channel gain discussed in Section IV-A, the measured channel matrices of Cases II–IV are normalized with respect to the refinside erence system (Case I). This is achieved by setting of (4) for all the measured antenna systems. E. Envelope Distribution In order to obtain channel envelope distributions for all measured antenna systems, channel matrices are normalized according to the principle of Normalized RX SNR. The envelope distribution is found for each group of channel branches defined in Table III, respectively. The BPR is further removed by normalizing each group of channel branches independently. Each of the measured envelope distributions is fitted to a theoretical cumulative distribution function (CDF) of Rician distribution using maximum likelihood estimation. The -factor can

TABLE V

ESTIMATED RICIAN

K -FACTOR OF MEASURED CHANNELS IN LOS

be calculated as the power ratio between the dominant and the Rayleigh components [23]. In the measured NLOS scenario of all antenna systems, the Rician distribution reduces to Rayleigh, as the obtained -factors approach 0. For LOS, Table V summarizes the -factor of each antenna system obtained at each grid position in detail. 1) Case I: The Rician -factors are negligibly small in the measured LOS scenario. This is mainly due to the elevated radiation patterns of the monopoles discussed in Section III-A. Moreover, since the RX and TX antennas are placed at different heights, the ground plane of the TX array obstructs the LOS path between the TX/RX array elements, which further restricts the number of dominant components and results in small -factors in this particular LOS scenario. 2) Case II: A clear distinction of the envelope distribution between co- and cross-polarized channels is observed in the is described by Rician distrimeasured LOS scenario. butions with larger -factors, which indicate the presence of dominant components. The obtained -factor is found to be increasing from grid position A to E with decreasing TX/RX separation distance (see Table V), which implies that the corresponding increases in RX power observed in Section IV-A is mainly attributed to the increase in power of the dominant component. On the other hand, as indicated by the high BPR is described by negligibly small and XPR in Section IV-C, -factors, suggesting a Rayleigh distribution. 3) Case III: In this case, although channel branches with is described by Rician districross-polarized monopoles bution with negligibly small -factors, other channel branches show the presence of slightly more dominant components. In particular, in the extreme LOS scenario at measured grid position E, channel branches with microstrip excitation ports as well as co-polarized monopoles on the RX DRA array exhibit relatively strong -factors. Nevertheless, the dominant components are much less significant than those in Case II, as the obtained -factors in Case II are 5 times greater than the corresponding ones in Case III. Recall that the BPR is also mitigated in Case III relative to Case II, which indicates that the tri-polarized DRA array is more robust in polarized channels than the monopole array. 4) Case IV: The -factors of the two groups of channel branches are found to be small in the measured LOS scenario, which suggest that there is no significant dominant component. This can be understood by the mismatch in the array orientation of the TX/RX antenna pair, where the TX DRA array is oriented perpendicular to its RX counterpart. Moreover, the diverse radiation patterns of the DRA array cover a broad range of

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Fig. 13. Cumulative distribution function (CDF) of measured MIMO channel capacity. (a) NLOS, Normalized RX SNR; (b) NLOS, Case I as Reference; (c) LOS, Normalized RX SNR; (d) LOS, Case I as Reference.

directions with different polarization contributions, which further reduce the number of direct propagation paths in LOS. F. MIMO Capacity The channel capacity is evaluated for the measured MIMO channels, assuming no channel knowledge at the TX end, i.e., with equal TX power allocation. The capacity is evaluated at the SNR of 10 dB (recall that the SNRs obtained from the measurements were higher than 20 dB). The measured channel matrices are normalized according to the two normalization principles discussed in Section IV-D. Fig. 13 shows CDF of the measured channel capacity of all four antenna systems, applying the two normalization principles, in NLOS and LOS, respectively. In subplots (a) and (c), the channel’s DOF is compared with the 6 6 and 5 5 i.i.d. Rayleigh fading channels. Eigenvalue dispersion is also studied as a scale-invariant metric to describes the multipath richness and the channel’s DOF [25], and similar findings are obtained. In subplots (b) and (d), the channel capacity of Cases II-IV are compared to the reference system. 1) Capacity I—Normalized RX SNR: a) NLOS: Fig. 13(a) shows that none of the evaluated antenna systems can achieve the performance of the 6 6 i.i.d. Rayleigh fading channel in NLOS. The reference system (Case I) that only exploits spatial diversity achieves a similar 10% 5 i.i.d. Rayleigh channel. Case II outage capacity as the 5 achieves the best performance since both polarization and spatial diversities are exploited to enrich the channel’s DOF. In Case III, the ground plane of the compact RX DRA array is 95% smaller than that of the RX monopole array in Case II, which restrics the available spatial diversity. However, the diversity-rich design of the DRA array employs polarization and angle diversities to achieve a similar DOF performance as that in Case II.

In Case IV, the performance is worse than that of the 5 5 i.i.d. Rayleigh channel. This indicates that the channel’s DOF is limited by the compact DRA array when it is used at both ends of the communication links. b) LOS: In Fig. 13(c), the reference system (Case I) and Case IV maintain similar performances as those in NLOS. However, for Case II, the performance decreases significantly compared to that in NLOS, and is the worst performance among all cases. This is attributed to the strong branch power imbalance observed in Section IV-C, where distinct channel envelope distributions are identified for the co- and cross-polarized channels. The Rician -factor is found to be 14 for the co-polarization. Given the normalized RX SNR, the significant branch power imbalance is detrimental to the capacity and DOF performance. On the other hand, the high BPR and Rician -factor is mitigated in Case III when the compact DRA array is employed at the RX subsystem. Thus, the robust performance of the compact DRA array also provides a better DOF performance than the RX monopole array in Case II. 2) Capacity II—Case I as Reference: a) NLOS: When taking into account the relative power differences among the four antenna systems [see Fig. 13(b)], Case II achieves significantly better capacity performance, where the 10% outage capacity is 2.6 bits/s/Hz higher than that of the reference system. The improved capacity relative to Fig. 13(a) is due to its 1.6 dB higher channel gain as discussed in Section IV-A. Case III achieves the same outage capacity as the reference system, since the DOF [Fig. 13(a)] and the channel gain (Table II) is similar between Cases I and III in NLOS. Case IV has the worst outage capacity, which is 1.3 bits/s/Hz lower than that of the reference system. b) LOS: Fig. 13(d) shows that Case II achieves a slightly lower outage capacity as that of the reference system. Although

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TABLE VI SUMMARY OF PERFORMANCE

ACKNOWLEDGMENT Helpful discussions with Dr. S. Wyne and Assoc. Prof. F. Tufvesson of Lund University, Prof. J. B. Andersen of Aalborg University, Prof. M. A. Jensen of Brigham Young University and Mr. T. Bolin of Sony Ericsson Mobile Communications AB are gratefully acknowledged. The authors thank Mr. J. Långbacka, Mr. L. Hedenstjerna and Mr. M. Nilsson for their support in the measurement campaign. They also thank anonymous reviewers for valuable comments which have helped to improve the quality of this paper. REFERENCES

Case II has poorer DOF as shown in Fig. 13(c), its performance is improved due to the 4.1 dB higher channel gain. Case III gives 1.5 bits/s/Hz lower 10% outage capacity compared to Case II, although the use of RX DRA array achieves higher DOF performance. This is due to its 3.1 dB lower channel gain than Case II. In Case IV, the 10% outage capacity is 3.4 bits/s/Hz lower than the reference system, due to both limited DOF and lower channel gain. V. CONCLUSIONS In this work, a compact six-port DRA array is proposed. In order to demonstrate its suitability for WLAN-type applications relative to common (but larger) array types, its performance is evaluated with measured MIMO channels of indoor office scenarios. Table VI summarizes the performance of the measured MIMO channels for the four different 6 6 multiple antenna systems. Compared to the reference system using sparsely separated monopoles at the TX/RX subsystems, the use of dual-polarized patch antennas at the TX end and the proposed DRA array at the RX end shows rich characteristics of spatial, polarization and angle diversities. The channel’s DOF (Capacity I in Table VI) is found to be higher than that of the reference system in the measured NLOS scenario, which together with its slightly lower channel gain, results in similar channel capacity for the two cases (Capacity II in Table VI). In addition, the proposed DRA array is shown to be more robust than the monopole array as the RX counterpart to the TX patch array. It achieves a higher DOF than the RX monopole array in the measured LOS scenario, whose performance is shown to be limited due to the high BPR of the cross-polarized channels, as well as the strong Rician -factor of the co-polarized channels. However, the achievable 10% outage capacity at 10 dB reference SNR is 1.5 bits/s/Hz lower. This is attributed to the 3.1 dB lower channel gain of the RX DRA array. Furthermore, the antenna system with the proposed DRA array at both ends of the communication link is also evaluated. This corresponds to implementing compact multiple antenna solutions at both link ends. The penalty for implementing the compact TX array is the reduction in DOF, which results in 1.3 bits/s/Hz and 3.4 bits/s/Hz lower outage capacity than the reference system in NLOS and LOS, respectively.

[1] M. Jensen and J. Wallace, “A review of antennas and propagation for MIMO wireless communications,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 2810–2824, Nov. 2004. [2] Y. Gao, X. Chen, Z. Ying, and C. Parini, “Design and performance investigation of a dual-element PIFA array at 2.5 GHz for MIMO terminal,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3433–3441, Dec. 2007. [3] J. Villanen, P. Suvikunnas, C. Icheln, J. Ollikainen, and P. Vainikainen, “Performance analysis and design aspects of mobile-terminal multiantenna configurations,” IEEE Trans. Veh. Technol., vol. 57, no. 3, pp. 1664–1674, May 2008. [4] A. Diallo, C. Luxey, P. Le Thuc, R. Staraj, and G. Kossiavas, “Diversity performance of multiantenna systems for UMTS cellular phones in different propagation environments,” Int. J. Antennas Propag., 2008. [5] R. D. Richtmyer, “Dielectric resonators,” J. of Appl. Phys., vol. 10, no. 6, pp. 391–398, 1939. [6] J. Van Bladel, “The excitation of dielectric resonators of very high permittivity,” IEEE Trans. Microw. Theory Tech., vol. 23, no. 2, pp. 208–217, Feb. 1975. [7] J. Van Bladel, “On the resonances of a dielectric resonator of very high permittivity,” IEEE Trans. Microw. Theory Tech., vol. 23, no. 2, pp. 199–208, Feb. 1975. [8] M. McAllister, S. Long, and G. Conway, “Rectangular dielectric resonator antenna,” Electron. Lett., vol. 19, no. 6, pp. 218–219, Mar. 1983. [9] A. Kishk, H. Auda, and B. Ahn, “Accurate prediction of radiation patterns of dielectric resonator antennas,” Electron. Lett., vol. 23, no. 25, pp. 1374–1375, Dec. 1987. [10] R. K. Mongia and A. Ittipiboon, “Theoretical and experimental investigations on rectangular dielectric resonator antennas,” IEEE Trans. Antennas Propag., vol. 45, no. 9, pp. 1348–1356, Sep. 1997. [11] W. Huang and A. Kishk, “A DRA fed by PIFA for laptop WLAN application,” presented at the IEEE Antennas Propag. Soc. Int. Symp., San Diego, CA, Jul. 2008. [12] L. Hady, D. Kajfez, and A. Kishk, “Triple mode use of a single dielectric resonator,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1328–1335, May. 2009. [13] Z. Ying, “Compact Dielectric Resonant Antenna,” U.S. Patent Application 20080122703, Sep. 22, 2006. [14] K. Ishimiya, J. Långbacka, Z. Ying, and J.-I. Takada, “A compact MIMO DRA antenna,” in Proc. Int. Workshop Antenna Technol. (IWAT 2008), Chiba, Japan, Mar. 4–6, 2008, pp. 286–289. [15] K. Ishimiya, Z. Ying, and J.-I. Takada, “A compact MIMO DRA for 802.11n application,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., San Diego, CA, Jul. 2008. [16] N. Oland, “WLAN MIMO terminal test in reverberation chamber” Master’s thesis, Chalmers Univ. Technol., Gothenburg, Sweden, 2008 [Online]. Available: http://publications.lib.chalmers.se/records/fulltext/70869.pdf [17] I. Shoaib, Y. Gao, K. Ishimiya, X. Chen, and Z. Ying, “Performance evaluation of the 802.11n compact MIMO DRA in an indoor environment,” presented at the Proc. 3rd Eur. Conf. Antennas Propag. Berlin, Germany, Mar. 23–27, 2009. [18] R. Thoma, D. Hampicke, A. Richter, G. Sommerkorn, A. Schneider, U. Trautwein, and W. Wirnitzer, “Identification of time-variant directional mobile radio channels,” IEEE Trans. Instrum. Meas., vol. 49, no. 2, pp. 357–364, Apr. 2000. [19] J. Kåredal, A. Johansson, F. Tufvesson, and A. Molisch, “A measurement-based fading model for wireless personal area networks,” IEEE Trans. Wireless Commun., vol. 7, no. 11, pp. 4575–4585, Nov. 2008. [20] SATIMO Homepage [Online]. Available: http://www.satimo.fr

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[21] CST Computer Simulation Technology AG Homepage [Online]. Available: http://www.cst.com [22] Ö. Isik, “Planar and Cylindrical microstrip array antennas for MIMO-Channel sounder applications,” Master’s thesis, Chalmers Univ. Technol., Gothenburg, Sweden, 2004. [23] R. Vaughan and J. B. Andersen, Channels, Propagation and Antennas for Mobile Communications. London, U.K.: Inst. Elect. Eng., 2003. [24] A. F. Molisch, Wireless Communications. Piscataway, NJ: IEEE Press, 2005. [25] J. Salo, P. Suvikunnas, H. El-Sallabi, and P. Vainikainen, “Ellipticity statistic as measure of MIMO multipath richness,” Electron. Lett., vol. 42, no. 3, pp. 160–162, Feb. 2006.

Buon Kiong Lau (S’00–M’03–SM’07) received the B.E. degree (with honors) from the University of Western Australia, Crawley and the Ph.D. degree from Curtin University of Technology, Perth, Australia, in 1998 and 2003, respectively, both in electrical engineering. During 2000–2001, he took a year off from his Ph.D. studies to work as a Research Engineer with Ericsson Research, Kista, Sweden. From 2003 to 2004, he was a Guest Research Fellow at the Department of Signal Processing, Blekinge Institute of Technology, Sweden. In 2004, he was appointed a Research Fellow in the Department of Electrical and Information Technology, Lund University, Sweden, where he is now an Assistant Professor. During 2003, 2005 and 2007, he was also a Visiting Researcher at the Department of Applied Mathematics, Hong Kong Polytechnic University, China, the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, and Takada Laboratory, Tokyo Institute of Technology, Japan, respectively. His research interests include array signal processing, wireless communication systems, and antennas and propagation. Dr. Lau is an active participant of EU COST Action 2100, where he is the Co-Chair of Subworking Group 2.2 on “Compact Antenna Systems for Terminal.”

Ruiyuan Tian (S’07) received the Bachelor’s degree in electronic engineering from Beijing Institute of Technology (BIT), China, in 2005, and the M.S. degree in digital communications from Chalmers University of Technology, Sweden, in 2007. He is currently working towards a Ph.D. degree at Lund University, Sweden. During 2006 to 2007, he worked in the Telecommunications Research Center Vienna (FTW), Austria. His current research interests include compact antennas in MIMO systems and its fundamental limitations, antenna channel interaction in wireless communications.

Zhinong Ying (SM’05) is an Expert in Antenna Technology at Sony Ericsson Mobile Communication AB, Lund, Sweden. He joined Ericsson AB in 1995 where he became Senior Specialist in 1997 and in 2003 an Expert in engineering. His main research interests are small antennas, broad and multiband antenna, multichannel antenna system (MIMO), near-field and human body effects and measurement techniques. He has authored and coauthored over 60 papers in various journal, conference and industry publications. He holds more than 60 patents and patents pending in the antenna and mobile terminal areas. He contributed a chapter to the well-known book Mobile Antenna Systems Handbook (Artech House, 200). He has invented and designed various types of multiband antennas and integrated antennas for the mobile industry. His most significant contributions in the 1990s are the development of nonuniform helical antenna and multiband integrated antenna. The innovative designs are not only used in Ericsson products, but also in mobile industry worldwide. His patented designs have reached a commercial penetration of more than several hundreds million products worldwide. He was also involved in the evaluation of Bluetooth Technology which was invented by Ericsson. Mr. Ying received the Best Invention Award at Ericsson Mobile in 1996 and the Key Performer Award at Sony Ericsson in 2002. He was nominated for the President Award at Sony Ericsson in 2004 for his innovative contributions. He served as TPC Co-Chairmen in the International Symposium on Antenna Technology (iWAT), 2007, and serves as a Reviewer for several academic journals. He was a member of the scientific board of the Antenna Centre of Excellent European 6th frame from 2004 to 2007.

Vanja Plicanic (M’07) received the M.S. degree in electrical engineering and technology management from Lund University, Lund, Sweden, in 2004, where she is currently working toward the Ph.D. degree. From 2004 to 2005, she was a Young Graduate Trainee at the Antenna and Submillimetre Wave Group, European Space Research and Technology Centre, ESTEC, The Netherlands. In 2005, she joined Sony Ericsson Mobile Communications and is currently working in the Communications and Networking Group. Her research interest comprises multiband multi-antenna systems and their implementation in compact mobile terminals.

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Communications Dual-Band Multiple Beam Antenna System Using Hybrid-Cell Reuse Scheme for Non-Uniform Satellite Communications Traffic Jim Wang, Sudhakar K. Rao, Minh Tang, and Chih-Chien Hsu

Abstract—An advanced dual-band antenna system suitable for broadband satellites that is capable of providing higher EIRP, higher G/T, and improved copolar isolation among frequency reuse beams is presented. The antenna system employs a hybrid-cell frequency reuse scheme instead of conventional fixed cell reuse (4-cell or 7-cell) in order to efficiently use spectrum based on traffic demands. It is shown that the advanced antenna employing high-efficiency horns and shaped reflectors provides about 1.0 dB EIRP improvement, 2.0 dB G/T improvement, and 3.0 dB improvement in C/I when compared to conventional antennas.

Fig. 1. Typical MBA spacecraft layout.

Index Terms—Dual-band horns, high efficiency horns, reflector antennas, satellite antennas.

I. INTRODUCTION There has been a significant growth in the broadband communications satellite market in the recent past. These systems provide communications among several users that are spread over a given coverage region by using multiple overlapping spot beams. Communications among users is established through the hubs and the satellite. The users within the coverage region establish two-way communications links with other users via the satellites and the hubs through forward (hub uplink and user downlink) and return (user uplink and hub downlink) links. This requires that each spot beam on-board the satellite needs to support simultaneously Ka-band uplink signals and K-band downlink signals that are spread over large bandwidth ratio of about 1.64. This communication discusses advanced dual-band antenna system employing dual-band high efficiency horns (DBHEH) [1], shaped reflectors, and use of hybrid-cell reuse scheme [2] that provides significant improvements in RF performance and efficient traffic distribution within the coverage region.

Fig. 2. Conventional MBA using 4-Cell reuse.

II. HYBRID-CELL REUSE SCHEME Fig. 1 shows typical layout of the multiple beam antenna (MBA) in the deployed view of the spacecraft. It employs four multiple reflector apertures, where by each reflector is fed with multiple horns producing about one fourth of the total number of spot beams. The beams from the four reflectors are interleaved on ground with hexagonal grid layout and providing contiguous spot beam coverage from a given orbital location of the satellite. Conventional layout of the beams using a 4-cell fixed reuse scheme is shown in Fig. 2. There are 45 spot beams covering CONUS from 105 W orbital slot, each with a diameter of 0.7 and with adjacent beam Manuscript received June 25, 2009; revised July 31, 2009; accepted October 02, 2009. Date of publication January 26, 2010; date of current version April 07, 2010. The authors are with Lockheed Martin Space Systems Company, Newtown, PA 18966 USA (e-mail: [email protected]; [email protected]; minh. [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.2010.2041172

Fig. 3. MBA using proposed “hybrid-cell” reuse.

spacing of 0.6 . RF performance of each beam needs to be evaluated by expanding the beam diameter including satellite pointing error. Alternate beams are generated from the same reflector. A total of 16 channels are used with each beam carrying four channels on the four-cell re-use pattern using cells A, B, C, & D. The disadvantage with this approach is that each beam has the identical capacity independent of the population density and demand, carries a total of 180 channels that are too many to support for any spacecraft, and requires a large number (12 for this example) of hubs on the ground. The hybrid-cell reuse scheme, shown in Fig. 3, overcomes most of above limitations of the fixed-cell reuse scheme. Each beam has designations of the beam number, hub number, and specific channel numbers it is using out of the 16 available channels (8 channels are left-hand

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circularly polarized and the remaining 8 channels are right-hand circularly polarized). It employs a 4-cell reuse scheme for the densely populated areas such as east-coast and west-coast and an N-cell reuse 4) for the low population regions of the contiguous US scheme ( (CONUS). All the east-coast and west-coast beams with high population density carry 4 channels each, while the beams located in the mid-west, north, and south-west of CONUS with low population density carry 1 to 3 channels each depending on the population and traffic demand in these beams. This method provides capacity in each beam based on the traffic demand, requires only 107 total number of channels that can be supported by several spacecrafts in the industry, and demands only 8 hubs to support both forward and return channels. The hybrid-cell reuse scheme provides tremendous flexibility in the beam/channel layout, and provides better overall antenna carrier-to-interference ratio (C/I) for the beams. The C/I is the aggregate value where all the copolar and cross-polar interferers are added in power and compared to the carrier within the beam.

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Fig. 4. Typical geometry of DBHEH using “slope-discontinuities”.

III. DUAL-BAND ANTENNA The MBA shown in Fig. 1 employs four graphite reflectors, each being fed with about 12 horns to generate about 12 beams. The antenna has to support downlink beams at K-band (20 GHz) and uplink beams at Ka-band (30 GHz). Conventional antennas are designed with reflector size meeting the desired beam size at the downlink K-band. However, the uplink beams at Ka-band are 50% smaller than the downlink beams due to larger electrical size of the reflector and suffer from lower edge-of-coverage (EOC) gain, increased gain loss due to pointing error, and suffer from large peak-to-edge gain variation and lower C/I. Advanced MBA design for the dual-band antenna employs the following design features to achieve significant performance advantages at both bands. • High-efficiency dual-band horns [1], [3] with more than 82% efficiency at both bands (conventional corrugated horn has about 54% efficiency) to provide more illumination taper on the reflector resulting in improved EOC gain and lower sidelobes that improve C/I on the downlink; • Shaped reflector surface to broaden the receive beams to improve EOC gain, reduce peak-to-edge variation, and improve C/I on the uplink beams. The shaping will have minimal impact on downlink beams EOC gain; • Stepped-reflector to generate a flat-top receive beam and improve G/T performance; • Use of hybrid-cell reuse scheme to provide non-uniform traffic based on the population density, to reduce the number of required hubs, and to improve C/I. The high efficiency horn (HEH) employs TE1m type modes (TE12, TE13, TE14, TE15 etc.) in addition to the dominant TE11 mode to make the aperture illumination more uniform resulting in higher efficiency. The DBHEH typical geometry is shown in Fig. 4 and it employs five “slope-discontinuities” to generate the desired higher order modes in order to achieve high efficiency of 82% at both bands. The horn geometry is synthesized using the mode-matching analysis of discrete step-junctions combined with a generalized scattering matrix (GSM) to evaluate the horn performance. Fig. 5 shows performance of DBHEH compared to other conventional horns, such as corrugated and Potter type horns. Note that the Potter horn model is ideal and can not be realized in practice due to its bandwidth limitations. These primary patterns are used to evaluate the secondary patterns of an 80 in. offset reflector antenna. The reflector surface is shaped mainly to broaden the receive beams and thereby improving the radiation patterns at Rx frequencies. The edge of coverage gain and worst case C/I of the antenna beams are evaluated by enlarging the beam size to account for the satellite pointing error of + 0 0 05 i.e., using 0.8 beam instead

Fig. 5. Aperture efficiency and edge taper comparison of various dual-band MBA horns. TABLE I RF PERFORMANCE COMPARISON OF PRIMARY AND SECONDARY PATTERNS OF THE DBHEH WITH CONVENTIONAL CORRUGATED HORN

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of the nominal 0.7 beam. The carrier-to-interference ratio (C/I) is calculated in an aggregate manner that includes power addition of all the copolar interferers coming from several adjacent beams that re-use the same frequency as the carrier and also cross-polar interferences. Table I summarizes performance comparison of the MBA with DBHEH and conventional corrugated horn. The DBHEH improves the EOC gain by about 0.9 dB at Tx and by about 2.0 dB at Rx, and improves the Tx C/I by about 3.0 dB.

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Fig. 9. Computed EIRP contours of MBA. Fig. 6. Concept of “Stepped Reflector Antenna”.

Fig. 10. Computed C/I contours of Tx MBA. Fig. 7. Near-field phase distribution of the SRA at Rx frequencies.

Fig. 8. Computed DMBA patterns of the SRA showing “flat-topped” beam with increased EOC gain.

The DMBA performance can be further improved by the use of Lockheed Martin’s patented “stepped-reflector antenna” (SRA) technology [4]. The SRA concept is illustrated in Fig. 6 where it employs an outer annular region that is stepped relative to the central region. The height of the stepped region is designed in conjunction with the DBHEH feed phase characteristics at the Rx frequencies in order to provide a 180 degree phase reversal in the stepped region resulting in a “flat-topped” receive beam with improved EOC gain. Both central and outer annular regions can be shaped to improve the overall RF performance and the transition region can be blended smoothly near the stepped region to avoid abrupt discontinuities. The computed near-field phase patterns of the SRA plotted in Fig. 7 show the 180-degree phase reversal at Rx frequencies near the transition region of the step. As a result, the Rx beam patterns computed in Fig. 8 show “flat-topped” radiation patterns with increased

EOC gain. EOC gain improvement at Rx band is about 1.2 dB relative to the conventional reflector. The feed horn will have a quadratic error across the reflector aperture due to the fact that it is typically aligned with the reflector such that the Tx phase center coincides with the focal-plane of the reflector. The Rx phase center will be away from the focal-plane and inside the horn which results in a quadratic phase front across the reflector aperture. This property could be utilized to reduce the step height from quarter wavelength to less than quarter length depending on the DBHEH design. The reduced step size eases the fabrication of the shaped reflector blending the smaller step smoothly into its surface. By combining the fed quadratic phase variation and the phase variation due to the stepped region, the height of the step can be minimized allowing the stepped region to be blended with the reflector shape. The SRA concept works well for wide angle coverage regions like CONUS and achieves significant improvements in Rx gain, Rx C/I, Tx C/I, and moderate improvement of Tx gain when compared to a reflector without the step region. Typical computed EIRP and C/I patterns of the spot beams over the CONUS coverage are shown in Figs. 9 & 10, respectively. The EIRP is weighted to give higher values over the rain regions and the C/I values are better than 18 dB on the average. The coverage region is not shown in Fig. 9 for those beams with EIRP computations for better clarity. IV. SUMMARY AND CONCLUSION An advanced multiple beam antenna that provides communications traffic based on the population density and demand is presented. It employs a hybrid-cell frequency reuse scheme to provide non-uniform traffic distribution. By employing dual-band high efficiency horns, shaped reflectors, and stepped-reflector with annular region, it is shown that significant improvement of the order of 1.0 dB for EIRP, 3.0 dB for G/T, and 3.0 dB for C/I can be achieved over conventional

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MBA designs. It is feasible to provide limited on-orbit reconfigurability for future satellites in terms of number of channels per beam by using switch matrices at the payload repeater level. The trades are the flexibility versus added complexity of the switch matrices and its associated losses. ACKNOWLEDGMENT The authors thank the anonymous reviewers for their valuable suggestions that have helped to improve the quality of the communication.

REFERENCES [1] S. Rao and M. Tang, “High-efficiency horns for an antenna system,” U.S. Patent 7,463,207, Dec. 09, 2008, Lockheed Martin. [2] S. Rao, M. Sheshadri, and J. Wang, “Multiple-beam antenna system using hybrid frequency-reuse scheme,” U.S. Patent 7,382,743, Jun. 03, 2008, Lockheed Martin. [3] S. Rao, K. K. Chan, and M. Tang, “Design of high efficiency circular horn feeds for multibeam reflector applications,” in Proc. IEEE AP-S Symp., Washington, DC, Jul. 2005, pp. 359–362. [4] S. Rao and M. Tang, “Stepped-reflector antenna for dual-band multiple beam satellite communications payloads,” IEEE Trans. Antennas Propag., vol. 54, pp. 801–811, Mar. 2006.

A Low Cross-Polarization Smooth-Walled Horn With Improved Bandwidth Lingzhen Zeng, Charles L. Bennett, David T. Chuss, and Edward J. Wollack

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angle. Corrugated feeds [1] approximate this idealization by providing the appropriate boundary conditions for the HE11 hybrid mode at the feed aperture. Alternatively, an approximation to a scalar feed can be obtained with a multimode feed design. One such “dual-mode” horn is the Potter horn [2]. In this implementation, an appropriate admixture of TM11 is generated from the initial TE11 mode using a concentric step discontinuity in the waveguide. The two modes are then phased to achieve the proper field distribution at the feed aperture using a length of waveguide. The length of the phasing section limits the bandwidth due to the dispersion between the modes. Lier [3] has reviewed the cross-polarization properties of dual-mode horn antennas for selected geometries. Other authors have produced variations on this basic design concept [4], [5]. Improvements in the bandwidth have been realized by decreasing the phase difference between the two modes by 2 [6], [7]. To increase the bandwidth, it is possible to add multiple concentric step continuities with the appropriate modal phasing [8], [9]. A variation on this technique is to use several distinct linear tapers to generate the proper modal content and phasing [10], [11]. Operational bandwidths of 15–20% have been reported using such techniques. A related class of devices is realized by allowing the feedhorn profile to vary smoothly rather than in discrete steps. Examples of such smoothwalled feedhorns with 15% fractional bandwidths exist in the literature [12], [13]. In this work, we describe the design and optimization of a smoothwalled feed that has a 30% operational bandwidth, over which the cross-polarization response is better than 030 dB. The optimization technique is described, and the performance of the feed is compared with other published dual-mode feedhorns. The feedhorn described here has a monotonic profile that allows it to be manufactured by progressively milling the profile using a set of custom tools. II. SMOOTH-WALLED FEEDHORN OPTIMIZATION

Abstract—Corrugated feed horns offer excellent beam symmetry, main beam efficiency, and cross-polar response over wide bandwidths, but can be challenging to fabricate. An easier-to-manufacture smooth-walled feed is explored that approximates these properties over a finite bandwidth. The design, optimization and measurement of a monotonically-profiled, 14 FWHM smooth-walled scalar feedhorn with a diffraction-limited beam is presented. The feed was demonstrated to have low cross polariza30 dB) across the frequency range 33–45 GHz (30% fractional tion ( bandwidth). A power reflection below 28 dB was measured across the band. Index Terms—Feeds, horn antennas, millimeter-wave antennas.

The performance of a feedhorn can be characterized by angle- and frequency-dependent quantities that include beam width, sidelobe response and cross-polarization. Quantities such as reflection coefficient and polarization isolation that only depend on frequency are also important considerations. All of these functions are dependent upon the shape of the feed profile. In the optimization approach described, a weighted penalty function is used to explore and optimize the relationship between the feed profile and the electromagnetic response. A. Beam Response Calculation

I. INTRODUCTION Many precision microwave applications, including those associated with radio astronomy, require feedhorns with symmetric E - and H -plane beam patterns that possess low sidelobes and cross-polarization control. A common approach to achieving these goals is a “scalar” feed, which has a beam response that is independent of azimuthal Manuscript received July 10, 2009; revised October 16, 2009; accepted November 08, 2009. Date of publication January 26, 2010. Date of current version April 07, 2010. This work was supported in part by a NASA ROSES APRA Grant and a Johns Hopkins University/APL partnership research grant. L. Zeng and C. L. Bennett are with the Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 20723 USA (e-mail: [email protected]; [email protected]). D. T. Chuss and E. J. Wollack are with the Observational Cosmology Laboratory, NASA GSFC, Greenbelt, MD 20771 USA (e-mail: David.T.Chuss@nasa. gov; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2041318

The smooth-walled horn was approximated by a profile that consists of discrete waveguide sections, each of constant radius. With this approach, it was important to verify that each section is thin enough that the model is a valid approximation of the continuous profile. For profiles relevant to our design parameters, section lengths of 1l  c =20 were found to be sufficient by trial and error, where c is the cutoff wavelength of the input waveguide section. It is possible in principle to dynamically set the length of each section to optimize the approximation to the local curvature of the horn. This would increase the speed of the optimization; however, for simplicity, this detail was not implemented in our study. For each trial feedhorn the angular response was calculated directly from the modal content at the feed aperture. This in turn was calculated as follows. The throat of the feedhorn was assumed to be excited by the circular waveguide TE11 mode. The modal content of each successive section was then determined by matching the boundary conditions at each interface using the method of James

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[14]. The cylindrical symmetry of the feed limits the possible propagating modes to those with the same azimuthal functional form as T E11 [15]. This azimuthal-dependence extends to the resulting beam patterns, allowing the full beam pattern to be calculated from the E and H -plane angular response. Ludwig’s third definition [16] is employed in calculation and measurement of cross-polar response. We note that an additional consequence of the feedhorn symmetry is that to the extent that the E - and H -planes are equal in both phase and amplitude, the cross-polarization is zero [17]. Changes in curvature in the feed profile can excite higher order modes (e.g., T E12 and T M12 ), the presence of which can potentially degrade the cross-polarization response of the horn.

Fig. 1. The initial, intermediate and final profiles are shown. All dimensions are given in units of the cuttoff wavelength of the input circular waveguide.

B. Penalty Function We constructed a penalty function to optimize the antenna profile. The penalty function with normalized weights, j , is written as

2 =

N

M

i=1 j =1

j 1j (fi )2

(1)

where i sums over a discrete set of (N ) frequencies in the optimization frequency band, and j sums over the number (M ) of discrete parameters one wishes to take into account for the optimization. In the parameter space considered, this function was minimized over the frequency range 1:25fc < f < 1:71fc (1f=f0 = 0:3) to find the desired solution. Results reported here were obtained by restricting this penalty function to include only the cross-polarization and reflection (jS11j2 ) with uniform weights (M = 2). Additional parameters were explored; however, they were found to be subdominant in producing the target result. These functions were evaluated at 13 equally-spaced frequency points in (1). The explicit forms used for 11 (f ) and 12 (f ) are

(f ) 11 (f ) = XP 0 (f ) 12 (f ) = RP 0

XP0 if XP (f ) > XP0 if XP (f )  XP0 0 RP0 if RP (f ) > RP0 , if RP (f )  RP0 0

(2) (3)

where XP (f ) and RP (f ) are the maximum of the cross-polarization XP (f ) = M ax[XP (f; )] and reflected power at frequency f , respectively. XP0 and RP0 are the threshold cross-polarization and reflection. If either the cross-polarization or reflection at a sampling frequency were less than its critical value, it was omitted from the penalty function. Otherwise, its squared difference was included in the sum in (1). C. Feedhorn Optimization The feedhorn was optimized in a two-stage process that employed a variant of Powell’s method [18]. Generically, this algorithm can produce an arbitrary profile. To produce a feed that is easily machinable, we restricted the optimization to the subset of profiles for which the radius increases monotonically along the length of the horn. Without this constraint, this method was observed to explore solutions with corrugated features and the serpentine profiles explored in [19]. The aperture diameter of the feedhorn was initially set to  4c , but was allowed to vary slightly to achieve the desired beam size. A single discontinuity exists between the circular waveguide and the feed throat. The remainder of the horn profile adiabatically transitions to the feed aperture. The total length of the feedhorn from the aperture to the single

mode waveguide was fixed at 12:3c during optimization. This length is somewhat arbitrary, but chosen to produce a stationary phase center and a diffraction-limited beam in a practical volume. The approach of [12] was followed as an initial input to the Powell method. Specifically, the feed radius, r , is written analytically as a function of the distance along the length of the horn, z , as:

r(z ) =

0:293 + 0:703sin0:75 (0:255z) 0:293+0:703 1+[0:282(z 6:15)]2 0

0 z 6:15 6:15 50 dB dynamic range from the peak response over  2 steradians with an absolute accuracy of