IEEE Transactions on Antennas and Propagation [volume 59 number 12]

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DECEMBER 2011

VOLUME 59

NUMBER 12

IETPAK

(ISSN 0018-926X)

2011 ANTENNAS AND PROPAGATION SOCIETY AWARDS

2011 Distinguished Achievement Award . ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... . 2011 Chen-To Tai Distinguished Educator Award ....... ......... ......... ........ ......... ......... ........ ......... ......... . 2011 John Kraus Antenna Award ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... . 2011 Donald G. Dudley Jr. Undergraduate Teaching Award ..... ......... ........ ......... .... ...... ........ ......... ......... . 2011 R. W. P. King Award ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... . 2011 Harold A. Wheeler Applications Prize Paper Award ....... ......... ........ ......... ... ....... ........ ......... ......... . 2011 Sergei A. Schelkunoff Transactions Prize Paper Award .... ......... ........ ......... ......... ........ ......... ......... . 2011 AWPL Piergiorgio L. E. Uslenghi Prize Paper Award ...... ......... ........ ......... ......... ........ ......... ......... . 2011 IEEE Antennas and Propagation Edward E. Altshuler Prize Paper . ........ ......... ......... . ....... ......... ......... . 2011 IEEE Fellow Awards From the Antennas and Propagation Sociey . ........ ......... ......... ........ ......... ......... .

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PAPERS

Antennas A Reconfigurable Patch Antenna Using Liquid Metal Embedded in a Silicone Substrate ........ ........ ......... ......... .. .. ........ ......... ......... ........ ......... .... S. J. Mazlouman, X. J. Jiang, A. Mahanfar, C. Menon, and R. G. Vaughan A Multifunctional Reconfigurable Pixeled Antenna Using MEMS Technology on Printed Circuit Board ....... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ........ A. Grau Besoli and F. De Flaviis Design, Simulation, Fabrication and Testing of Flexible Bow-Tie Antennas .... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ... A. C. Durgun, C. A. Balanis, C. R. Birtcher, and D. R. Allee Spiral Leaky-Wave Antennas Based on Modulated Surface Impedance . ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ..... G. Minatti, F. Caminita, M. Casaletti, and S. Maci Radiation Bandwidth Enhancement of Aperture Stacked Microstrip Antennas . .......... ......... H. Oraizi and R. Pazoki Compact Metallic RFID Tag Antennas With a Loop-Fed Method .... P. H. Yang, Y. Li, L. Jiang, W. C. Chew, and T. T. Ye Mode Excitation in the Coaxial Probe Coupled Three-Layer Hemispherical Dielectric Resonator Antenna .... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ..... A. B. Kakade and B. Ghosh Compact Disc Monopole Antennas for Current and Future Ultrawideband (UWB) Applications ....... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ .. M. N. Srifi, S. K. Podilchak, M. Essaaidi, and Y. M. M. Antar A New Multimode Antenna for MIMO Systems Using a Mode Frequency Convergence Concept ..... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ........ J. Sarrazin, Y. Mahé, S. Avrillon, and S. Toutain Wearable Circularly Polarized Antenna for Personal Satellite Communication and Navigation . ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... . E. K. Kaivanto, M. Berg, E. Salonen, and P. de Maagt 24 GHz Balanced Doppler Radar Front-End With Tx Leakage Canceller for Antenna Impedance Variation and Mutual Coupling ....... ......... ........ ......... ......... ........ ......... ......... ... H. L. Lee, W.-G. Lim, K.-S. Oh, and J.-W. Yu

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(Contents Continued from Front Cover) Arrays Sum, Difference and Shaped Beam Pattern Synthesis by Non-Uniform Spacing and Phase Control .... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ... H. Oraizi and M. Fallahpour Optimal Polarization Synthesis of Arbitrary Arrays With Focused Power Pattern ....... . ....... B. Fuchs and J. J. Fuchs Low Sidelobe Phased Array Pattern Synthesis With Compensation for Errors Due to Quantized Tapering ..... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... W. P. M. N. Keizer Interference Suppression in Uniform Linear Arrays Through a Dynamic Thinning Strategy .... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .... P. Rocca, R. L. Haupt, and A. Massa Connecting Spirals for Wideband Dual Polarization Phased Array ....... ........ ......... . R. Guinvarc’h and R. L. Haupt Wideband Planar Microwave Lenses Using Sub-Wavelength Spatial Phase Shifters .... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ...... M. A. Al-Joumayly and N. Behdad Light Weight and Conformal 2-Bit, 1 4 Phased-Array Antenna With CNT-TFT-Based Phase Shifter on a Flexible Substrate ....... ......... ........ ......... ......... ....... D. T. Pham, H. Subbaraman, M. Y. Chen, X. Xu, and R. T. Chen High Gain Metal-Only Reflectarray Antenna Composed of Multiple Rectangular Grooves ..... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .. Y. H. Cho, W. J. Byun, and M. S. Song Numerical Techniques Integral-Equation Analysis of Scattering From Doubly Periodic Array of 3-D Conducting Objects .... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ... F.-G. Hu and J. Song A Calderón Multiplicative Preconditioner for the PMCHWT Integral Equation ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ... K. Cools, F. P. Andriulli, and E. Michielssen Fast-Factorization Acceleration of MoM Compressive Domain-Decomposition ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ .. A. Freni, P. De Vita, P. Pirinoli, L. Matekovits, and G. Vecchi Multilevel Adaptive Cross Approximation (MLACA) ... ......... ......... ..... J. M. Tamayo, A. Heldring, and J. M. Rius Efficient Analyzing EM Scattering of Objects Above a Lossy Half-Space by the Combined MLQR/MLSSM . ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ... Z. Jiang, Y. Xu, Y. Sheng, and M. Zhu Method-of-Moments Analysis of Resonant Circular Arrays of Cylindrical Dipoles ..... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ....... G. Fikioris, S. Lygkouris, and P. J. Papakanellos Fast and Accurate Wide-Band Analysis of Antennas Mounted on Conducting Platform Using AIM and Asymptotic Waveform Evaluation Technique ....... ......... ........ ........ X. Wang, S.-X. Gong, J.-L. Guo, Y. Liu, and P.-F. Zhang A Practical Implementation and Comparative Assessment of the Radial-Angular-Transform Singularity Cancellation Method ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ....... G. Kaur and A. E. Yılmaz An Improved Marching-on-in-Degree Method Using a New Temporal Basis ... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ...... Z. Mei, Y. Zhang, T. K. Sarkar, B. H. Jung, A. García-Lampérez, and M. Salazar-Palma Implicit Runge-Kutta Methods for the Discretization of Time Domain Integral Equations .... .. X. Wang and D. S. Weile Coarse-Grid FDTD Approach for Capacitively Loaded Cylindrical Tube Antennas ..... ... ...... S.-Y. Hyun and S.-Y. Kim Calculation of the SAR Induced in Head Tissues Using a High-Order DGTD Method and Triangulated Geometrical Models . ......... ......... ........ ......... ......... ........ ...... H. Fahs, A. Hadjem, S. Lanteri, J. Wiart, and M.-F. Wong Gaussian Beam-Based Hybrid Method for Quasi-Optical Systems .... A. Rohani, S. K. Chaudhuri, and S. Safavi-Naeini Wave Phenomena and Imaging A Theory of Antenna Electromagnetic Near Field—Part I ....... ......... ........ ......... . S. M. Mikki and Y. M. M. Antar A Theory of Antenna Electromagnetic Near Field—Part II ...... ......... ........ ......... . S. M. Mikki and Y. M. M. Antar Bridging the Gap Between the Babinet Principle and the Physical Optics Approximation: Scalar Problem ..... ......... .. .. ........ ......... ......... ........ ......... ......... ....... G. Kubické, Y. A. Yahia, C. Bourlier, N. Pinel, and P. Pouliguen Pulsed Radiation From a Line Electric Current Near a Planar Interface: A Novel Technique .... ........ .... L. A. Pazynin Off-Axis Gaussian Beam Scattering by an Anisotropic Coated Sphere .. ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ...... Z.-S. Wu, Z.-J. Li, H. Li, Q.-K. Yuan, and H.-Y. Li Analytic Propagation Model for Wireless Body-Area Networks . ......... ........ . ........ ......... D. Ma and W. X. Zhang Experimental Characterization of Microwave Radio Propagation in ICT Equipment for Wireless Harness Communications ........ ........ ......... ......... ........ ....... M. Ohira, T. Umaba, S. Kitazawa, H. Ban, and M. Ueba Sub-Wavelength Focusing at the Multi-Wavelength Range Using Superoscillations: An Experimental Demonstration . .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ . A. M. H. Wong and G. V. Eleftheriades Three-Dimensional Near-Field Microwave Holography Using Reflected and Transmitted Signals ..... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ... R. K. Amineh, M. Ravan, A. Khalatpour, and N. K. Nikolova A Multiplicative Regularized Gauss–Newton Inversion for Shape and Location Reconstruction ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ..... P. Mojabi, J. LoVetri, and L. Shafai Application of DCIM on Marine Controlled-Source Electromagnetic Survey ... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... .. H. Ju, G. Fang, Z. Lin, F. Zhang, L. Huang, and Y. Ji

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(Contents Continued from Page 4393) COMMUNICATIONS

RF MEMS Switchable Slot Patch Antenna Integrated With Bias Network ...... ..... ..... .... I. Kim and Y. Rahmat-Samii Reconfigurable Small-Aperture Evanescent Waveguide Antenna ......... ........ ......... ......... P. Ludlow and V. Fusco Plastic-Based Supershaped Dielectric Resonator Antennas for Wide-Band Applications ........ ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... . M. Simeoni, R. Cicchetti, A. Yarovoy, and D. Caratelli CPW-Fed Wideband Printed Dipole Antenna for Digital TV Applications ...... ......... ... O. T.-C. Chen and C.-Y. Tsai Measurements and Analysis of a Helical Antenna Printed on a Layered Dielectric Hemisphere ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ... T. A. Latef and S. K. Khamas Passive RFID Strain-Sensor Based on Meander-Line Antennas . ......... ...... C. Occhiuzzi, C. Paggi, and G. Marrocco Millimeter-Wave Miniaturized Substrate Integrated Multibeam Antenna ........ ......... ......... .. Y. J. Cheng and Y. Fan Polarization, Gain, and Q for Small Antennas .... ........ ......... ......... ........ ......... ......... ........ ........ H. L. Thal On the Robustness to Element Failures of Linear ADS-Thinned Arrays . ....... .. .... M. Carlin, G. Oliveri, and A. Massa A Novel Planar Slot Array Antenna With Omnidirectional Pattern ....... ...... ... ....... X. Chen, K. Huang, and X.-B. Xu A Standing-Wave Microstrip Array Antenna ..... ........ ......... ......... ........ ......... ... A. Lakshmanan and C. S. Lee Application of Spatial Bandwidth Concepts to MAS Pole Location for Dielectric Cylinders .... ........ ...... J. E. Richie Seasonal Additional Attenuation in Woodlands for Satellite Services at L-, S- and C-Bands .... . P. Horak and P. Pechac High-Sensitivity Millimeter-Wave Imaging Front-End Using a Low-Impedance Tapered Slot Antenna ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... D. Radenamad, T. Aoyagi, and A. Hirose

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CORRECTIONS

Corrections to “Creation of a Magnetic Boundary Condition in a Radiating Ground Plane to Excite Antenna Modes” . .. .. ........ ......... ......... M. Sonkki, M. Cabedo-Fabrés, E. Antonino-Daviu, M. Ferrando-Bataller, and E. T. Salonen

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CALL FOR PAPERS

Special Issue on Antennas and Propagation at mm- and sub mm-waves . ........ ......... ......... ........ ......... ......... .

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2011 INDEX ....... ......... ........ ......... ......... ........ ......... ... ....... ........ ......... ......... ........ ......... ......... .

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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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2011 Chen-To Tai Distinguished Educator Award HE Chen-To Tai Distinguished Educator Award was established in 2000 by the IEEE Antennas and Propagation Society to recognize outstanding career achievements by a distinguished educator in the field of antennas and propagation. The 2011 Award was presented to John L. Volakis “For exemplary contribution as an inspiring teacher and mentor, and for advancing electromagnetic technology.”

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John L. Volakis was born in Chios, Greece on May 13, 1956 and immigrated to the U.S.A. in 1973. He received the Ph.D. degree from the Ohio State University in 1982. After two years at Rockwell International (now Boeing), he was appointed (1984) Assistant Professor at The University of Michigan, becoming a full Professor in 1994. Since January 2003, he has been the Chope Chair Professor at The Ohio State University, Electrical and Computer Engineering Department. He also serves as the Director of the ElectroScience Laboratory with $7.5M in external research funding. Over the years he has carried out research on diffraction theory and scattering, antennas, computational methods, electromagnetic compatibility and interference, propagation, design optimization, RF materials, multi-physics engineering and bioelectromagnetics. His publications include seven books (among them: Approximate Boundary Conditions in Electromagnetics, 1995; Finite Element Methods for Electromagnetics, 1998; the classic 4th ed. Antenna Engineering Handbook, 2007 and Small Antennas, 2010), over 290 journal papers and over 500 conference papers. He has graduated/mentored over 65 doctoral students/post-docs with 14 of them receiving best paper awards at international conferences. Prof. Volakis is a Fellow of IEEE and a Fellow of ACES. His service to professional societies include: 2004 President of the IEEE Antennas and Propagation Society, twice the General Chair of the IEEE Antennas and Propagation Symposium, IEEE APS Fellows Committee Chair, IEEE-wide Fellows Committee member and Associate Editor of several journals. He is listed by ISI among the top 250 most referenced authors.

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2011 John Kraus Antenna Award The John Kraus Antenna Award honors an individual or team that has made a significant advance in antenna technology. In 2011, this was awarded to Daniel Filipovic and Gabriel M. Rebeiz “For the analysis and design of extended dielectric-lens antennas and imaging arrays for millimeter-wave and terahertz applications.”

Daniel F. Filipovic received the B.S.E, M.S. and Ph.D. degrees in electrical engineering from the University of Michigan, Ann Arbor, in 1990, 1991, and 1995, respectively. He also received an MBA in Finance from San Diego State University, CA, in 2008. Previously, his research included millimeter-wave antennas and imaging arrays, integrated receivers and multipliers, and solid-state fabrication. Since 1995, he has worked at Qualcomm, Inc., where he is a Principal Engineer. His work at Qualcomm has spanned antennas, RF circuits, modems, and precision frequency compensation techniques. He holds 20 patents in these various areas of wireless communications with 20 additional patents pending.

Gabriel M. Rebeiz (F’97) received the Ph.D. degree from the California Institute of Technology. He is a Professor of electrical and computer engineering at the University of California, San Diego (UCSD). His research effort is now in silicon RFICs for phased array applications, high efficiency millimeter-wave and THz planar antennas, and RF MEMS components and reconfigurable systems. He has graduated over 40 Ph.D. students, and currently leads a group of 20 Ph.D. and Postdoctoral Fellows in the area of mm-wave RFIC, microwaves circuits, RF MEMS, planar mm-wave antennas and terahertz systems, and is the Director of the UCSD/DARPA Center on RF MEMS Reliability and Design Fundamentals. He is the author of the book, RF MEMS: Theory, Design and Technology (Wiley, 2003). Prof. Rebeiz is an IEEE Fellow, an NSF Presidential Young Investigator, an URSI Koga Gold Medal Recipient, the IEEE MTT 2003 Distinguished Young Engineer, and is the recipient of the IEEE MTT 2000 Microwave Prize, and the IEEE MTT 2010 Distinguished Educator Award. He also received the 1998 Eta-Kappa-Nu Professor of the Year Award and the 1998 Amoco Teaching Award given to the best undergraduate teacher at the University of Michigan, and the 2008 Teacher of the Year Award at the Jacobs School of Engineering, UCSD. His students have won a total of 19 best paper awards at IEEE MTT, RFIC and AP-S conferences.

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2011 Donald G. Dudley Jr. Undergraduate Teaching Award HE Donald G. Dudley, Jr. Undergraduate Teaching Award was established in 2009, and it recognizes outstanding and original contributions to undergraduate education by an individual at that early stage of his/her career as an educator in the general field of electromagnetics including theory, analytical solutions, numerical methods, antennas, propagation, phenomena visualization, and measurements. The 2011 Award was presented Vitaliy Lomakin “For excellence in developing and implementing research inspired applied electromagnetics curriculum, and mentoring students.”

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

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2011 R. W. P. King Award HE R. W. P. King Award is presented to an author under 36 years of age for the best paper published in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION during the previous year. The 2011 prize was awarded to Simon Chu and David Michelson for “Effect of Human Presence on UVB Radio Wave Propagation Within the Passenger Cabin of a Midsize Airliner,” IEEE Trans. Antennas Propag. vol. 58, no. 3, pp. 917–926, March 2010.

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Simon Chiu was born in Hong Kong, China, in 1984. He received the B.A.Sc. and M.A.Sc. degrees in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 2006 and 2009, respectively. During his graduate studies, his main research interests focused on UWB propagation in passenger aircraft cabins and outdoor industrial environments, including the effects of human presence on the wireless channel. He has since moved into the financial services sector and currently serves as Senior Training Manager at Inslink Financial Group, Richmond, BC.

David G. Michelson (S’80–M’89–SM’99) received the B.A.Sc., M.A.Sc., and Ph.D. degrees in electrical engineering from the University of British Columbia (UBC), Vancouver, BC, Canada. From 1996 to 2001, he served as a member of a joint team from AT&T Wireless Services, Redmond, WA, and AT&T Labs-Research, Red Bank, NJ, where he was concerned with the development of propagation and channel models for next-generation and fixed wireless systems that were later adopted by IEEE 802.16. From 2001 to 2002, he helped to oversee the deployment of one of the world’s largest campus wireless local area networks at UBC while also serving as an Adjunct Professor with the Department of Electrical and Computer Engineering. Since 2003, he has led the Radio Science Laboratory, Department of Electrical and Computer Engineering, UBC, where his current research interests include propagation and channel modeling for fixed wireless, ultra wideband, and satellite communications. Prof. Michelson currently serves as Chair of the IEEE Vehicular Technology Society’s Propagation Committee, as a member of the IEEE Antenna and Propagation Society’s Wave Propagation Standards Committee, and as an editor for Antenna Systems and Channel Characterization for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. He received the E. F. Glass Award from IEEE Canada in 2009 and currently serves as Chair of IEEE Canada’s Industry Relations Committee.

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2011 Harold A. Wheeler Applications Prize Paper Award HE Harold A. Wheeler Applications Prize Paper Award is presented to the authors of the best applications paper published in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION during the previous year. The 2011 Award was presented to Mark S. Mirotznik, Brandon L. Good, Paul Ransom, David Wikner, and Joseph N. Mait for the paper, “Broadband Antireflective Properties of Inverse Motheye Surfaces,” IEEE Trans. Antennas Propag., vol. 58, no. 9, pp. 2969–2980, Sept. 2010.”

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Mark S. Mirotznik (S’87-M’92) received the B.S.E.E. degree from Bradley University, Peoria, IL, in 1988 and the M.S.E.E. and Ph.D. degrees from the University of Pennsylvania, Philadelphia, in 1991 and 1992, respectively. From 1992 to 2009, he was a faculty member in the Department of Electrical Engineering, The Catholic University of America, Washington, DC. Since 2009, he is an Associate Professor and Director of Educational Outreach in the Department of Electrical and Computer Engineering, University of Delaware, Newark. In addition to his academic positions he an Associate Editor of the Journal of Optical Engineering and also holds the position of Senior Research Engineer for the Naval Surface Warfare Center (NSWC), Carderock Division. His research interests include applied electromagnetics and photonics, computational electromagnetics and bioelectromagnetics.

Brandon L. Good (M’09) received the B.A. and M.S. degrees in electrical engineering from The Catholic University of America, Washington, DC, in 2009. He is currently working toward the Ph.D. degree at the University of Delaware, Newark. Currently, he is a Junior Engineer at the Carderock Division, Naval Surface Warfare Center, West Bethesda, MD. His research interests include meta-material design and fabrication and advanced material measurement techniques.

Paul Ransom (M’09) received the B.S.E.E. degree from Southern University, Baton Rouge, LA and the M.S.E.C.E. degree from the Georgia Institute of Technology, Atlanta. He is currently working toward the Ph.D. degree at the Catholic University of America, Washington, DC. He is currently a Senior Research Engineer a the Carderock Division, Naval Surface Warfare Center, West Bethesda, MD. His research interest include computational electromagnetics and electromagnetically engineered materials.

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David Wikner received the B.A. degree in physics and mathematics from Ohio Wesleyan University, Delaware, in 1986 and the M.S. degree in physics from Stanford University, Stanford, CA, in 1988. In 1988, he joined the U.S. Army Research Laboratory, Adelphi, MD, where he currently leads the Millimeter Wave (MMW) Sensor Technology Team. He and his group explore MMW imaging phenomenology, system designs, and novel device technology. He has worked extensively with 94-GHz radar systems for vehicle and ground clutter measurements. For over 15 years he has been involved in the study of MMW imaging technology. The last several years his work has focused on efforts to create affordable MMW imaging systems for Army applications using new device technology and quasi-optical signal processing techniques. Mr. Wikner and has been a Chair of the SPIE Passive Millimeter-Wave Imaging Technology Conference since 2002.

Joseph N. Mait (S’78-M’84-SM’03) received the B.S.E.E. degree from the University of Virginia, in 1979 and the M.S.E.E. and Ph.D. degrees from the Georgia Institute of Technology, in 1980 and 1985, respectively. Since 1989, he has been with the U.S. Army Research Laboratory, where he is currently Senior Technical Researcher (ST) for Electromagnetics. He was an Assistant Professor of Electrical Engineering at the University of Virginia from 1984–1989 and an adjunct associate professor at the University of Maryland, College Park from 1997 to 2005. He has also held visiting positions at the Lehrstuhl für Angewandte Optik, Universität Erlangen-Nürnberg, Germany and the Center for Technology and National Security Policy, National Defense University, Washington DC. His research interests include the application of optics, photonics, and electro-magnetics to sensing and sensor signal processing. Dr. Mait is a Fellow of SPIE and OSA, and is a member of the Raven Society, the University of Virginia’s oldest honorary society. He is currently Editor-in-Chief of Applied Optics.

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2011 Sergei A. Schelkunoff Transactions Prize Paper Award

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HE Sergeo A. Schelkunoff Transactions Prize Paper Award is presented to the authors of the best paper published in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION during the previous year. The 2011 prize was awarded to Tengmeng Tan and Mike Potter for “FDTD discrete planewave (FDTD-DPW) formulation for a perfectly matched source in TFSF simulations,” IEEE Trans. Antennas Propag., vol. 58, no. 9, pp. 2641–2648, August 2010.

Tengmeng Tan (M’10) received the B.Sc. degree in electrical and computing engineering (with a minor equivalence in applied mathematics and physics) and the Ph.D. degree from University of Calgary, Canada. Prior to the graduate program, he was with IDENTEC Solution Inc., working as a Functional Test Engineer for the active RFID tags and latter with Agilent Technology Inc., as a Board Level Design Engineer for the router tester project. His current research interests include both theoretical and practical aspects of wave propagation problems, particularly in the applications of biophotonics. Currently, he holds an NSERC (Natural Sciences and Engineering Research Council of Canada) Postdoctoral Fellowship working as a Research Associate in the Biomedical Engineering Department at Northwestern University, IL. Dr. Tan was awarded an Excellent Teaching Assistant Award in 2005, was twice a recipient of the Student Paper Competition at the AP-S/URSI, in 2008 and 2009, and received the University of Calgary Graduate Student High Productivity Achievement Award in 2010.

Mike Potter (M’94) received the B.Eng. degree in engineering physics (electrical) from the Royal Military College of Canada, Kingston, ON, Canada, in 1992 and the Ph.D. degree in electrical engineering from the University of Victoria, Victoria, BC, Canada, in 2001. From 1992 to 1997, he served as an officer in the Canadian Navy as a Combat Systems Engineer. After completing his service and attaining the rank of Lieutenant (Navy), he completed his doctoral work in Victoria, British Columbia, Canada. He was then a Postdoctoral Fellow at the University of Arizona, Tucson, from 2001 to 2002. He currently holds the position of Associate Professor in the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada. His research interests include computational electromagnetics and the FDTD method. Dr. Potter is also a member of the Optical Society of America, and serves as a member of the APS Education Committee.

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2011 AWPL Piergiorgio L. E. Uslenghi Prize Paper Award

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HE 2011 AWPL Piergiorgio L. E. Uslenghi Prize Paper Award is presented for the best paper in the IEEE Antennas and Wireless Propagation Letters during the previous year. The 2011 prize was awarded to Dan Sun, Wenbin Dou, Lizhi You for, “Application of Novel Cavity-Backed ProximityCoupled Microstrip Patch Antenna to Design Broadband Conformal Phase Array,” IEEE Antennas Wireless Propag. Lett, vol. 9, pp. 1010–1013, 2010. Dan Sun was born in Wuxi, Jiangsu Province, China. He received the B.S. degree in electronic information engineering from Hohai University, Changzhou Campus, China, in 2003 and the M.S. degree in communication and information system from Nanjing University of Science and Technology, Nanjing, China, in 2005, respectively. Currently he is working toward the Ph.D. degree (under Prof. Wenbin Dou’s guidance) at Southeast University, Najing, China. In 2005, he joined AVIC Radar and Avionics Institute, Wuxi, China, where he is currently an Engineer. His research interests include wideband microstrip antennas, three-dimensional interconnection circuits, conformal phased array antennas and development of smart skin antennas, and millimeter wave antennas. He has authored and coauthored 8 papers in refereed journals and conference proceedings.

Wenbin Dou (M’96) graduated in microwave technology from the University of Science and Technology of China, Hefei, China, in 1978. He received the M.S. and Ph.D. degrees in electronics and communications from University of Electronic Science and Technology of China, Chengdu, in 1983 and 1987, respectively. From 1987 to 1989, he was a Postdoctoral Fellow at Southeast University, Nanjing, China. Since 1989, he has been with the Department of Radio Engineering and in 1994 he was promoted to Professor. He is currently the Vice Director of the State Key Laboratory of Millimeter Waves, Southeast University. His research interesting include millimeter wave quasi-optics, millimeter wave focal imaging, antennas and scattering, millimeter wave binary optics, ferrite devices, and so on. He has completed many projects on millimeter waves from the State Ministries and Foundation and now is in charge of key projects. He has published over 100 papers in journals and two books on millimeter wave ferrite devices and quasi-optical techniques (in 1996 and 2000, respectively). The book on millimeter wave quasi-optics was republished in 2006 after some revision and supplements. Prof. Dou is a member of the State Ministry Expert Committee. He is senior member of CIE and a member of the Microwave Institute of CIE. He is an Editor of Progress In Electromagnetics Research (PIER) and is an invited Reviewer for journals such as Applied Optics, Journal of Optical Society of America (A), Optics Express, and other international journals. He is Co-Chairman of the Program Committee of IRMMW-THz 2006 and a member of the International Advisory Committee of IRMMW-THz 2009.

Lizhi You received the M.S. degree in electromagnetic field and microwave technology from the University of Electronic Science and Technology of China, Chengdu, in 2003 and the Ph.D. degree in electromagnetic field and microwave technology from the State Key Laboratory of Millimeter Waves, Southeast University, Nanjing, China, in 2008. After completing his Ph.D. work, he was a Postdoctoral Fellow (with Professor Jungang Miao) at Beihang University, Beijing, China, and a Research Fellow (with Xuequan Yan) at Leihua Electronic Technology Institute of China (LETIC), Wuxi, from 2008 to 2011. In 2008, he joined AVIC Radar and Avionics Institute, Wuxi, China, where he is currently a Senior Engineer. In 2010, he joined Aviation Key Laboratory of Science and Technology on AISSS as a Senior Engineer. His research interests focus on the development of active electronic scan antennas, millimeter wave antennas and co-aperture multiband antennas. He has authored and coauthored more than 18 papers in peer-reviewed journals and conference proceedings. Prof. You was awarded the Outstanding Postdoctoral Prize for his excellent postdoctoral research in 2011 at LETIC.

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2011 IEEE Antennas and Propagation Edward E. Altshuler Prize Paper HE 2011 IEEE Antennas and Propagation Edward E. Altshuler Prize Paper was established in 2010 to recognize the best contribution published in the IEEE Antennas and Propagation Magazine in the preceding year. The 2011 prize was awarded to Do-Hoon Kwon and Douglas H. Werner for “Transformation Electromagnetics: An Overview of the Theory and Applications,” IEEE Antennas Propag. Mag., vol. 52, no. 1, pp. 24–26, Feb., 2010.

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Do-Hoon Kwon (SM’08) received the B.S. degree from Korea Advanced Institute of Science and Technology (KAIST), Korea in 1994, and the M.S. and Ph.D. degrees from the Ohio State University, Columbus, OH in 1995 and 2000, respectively, all in electrical engineering. He was a Senior Engineer at the Central R&D Center and Samsung Advanced Institute of Technology, Samsung Electronics, Co., Korea, from 2000 to 2006. During 2006 to 2008, he was a Postdoctoral Researcher with the Material Research Science and Engineering Center and the Department of Electrical Engineering, Pennsylvania State University. In August 2008, he joined the Department of Electrical and Computer Engineering, University of Massachusetts Amherst, as an Associate Professor. He is affiliated with the Antennas and Propagation Laboratory and the Center for Advanced Sensor and Communication Antennas of the ECE Department. He is the inventor/co-inventor of 17 U.S. patents in antenna and wireless communication areas. His main research interests include antenna scattering theory, small/wideband antennas, frequency selective surfaces, metamaterials, cloaking, and transformation electromagnetic/optical device designs. Prof. Kwon served as the Student Program Co-Chair at the 2010 IEEE International Symposium on Phased Array Systems & Technology. Douglas H. Werner (F’05) received the B.S., M.S., and Ph.D. degrees in electrical engineering and the M.A. degree in mathematics from The Pennsylvania State University (Penn State), University Park, in 1983, 1985, 1989, and 1986, respectively. He is a Professor in the Department of Electrical Engineering, Pennsylvania State University and the Director of the Computational Electromagnetics and Antennas Research Lab (http://cearl.ee. psu.edu/) as well as a member of the Communications and Space Sciences Lab. He is also a Senior Scientist in the Computational Electromagnetics Department of the Applied Research Laboratory and a faculty member of the Materials Research Institute, Penn State. He has published over 450 technical papers and proceedings articles and is the author of eight book chapters. He edited the book Frontiers in Electromagnetics (IEEE Press, 2000). He contributed a chapter for Electromagnetic Optimization by Genetic Algorithms (Wiley Interscience, 1999) as well as for the book Soft Computing in Communications (Springer, 2004). He is the coauthor of Genetic Algorithms in Electromagnetics (Wiley/IEEE, 2007). He contributed an invited chapter on “Fractal Antennas” for the popular Antenna Engineering Handbook (McGraw-Hill, 2007) as well as a chapter on “Ultra-Wideband Antenna Arrays” for Frontiers in Antennas (McGraw-Hill, 2011). His research interests include theoretical and computational electromagnetics with applications to antenna theory and design, phased arrays, microwave devices, wireless and personal communication systems, wearable and e-textile antennas, conformal antennas, frequency selective surfaces, electromagnetic wave interactions with complex media, metamaterials, electromagnetic bandgap materials, zero and negative index materials, fractal and knot electrodynamics, tiling theory, neural networks, genetic algorithms and particle swarm optimization. Dr. Werner is a member of the American Geophysical Union (AGU), URSI Commissions B and G, Eta Kappa Nu, Tau Beta Pi and Sigma Xi. He is a Fellow of the IEEE, the IET, and ACES. He was presented with the 1993 Applied Computational Electromagnetics Society (ACES) Best Paper Award and the 1993 International Union of Radio Science (URSI) Young Scientist Award. In 1994, he received the Pennsylvania State University Applied Research Laboratory Outstanding Publication Award. He was co-recipient of the 2006 R. W. P. King Award. He has also received several Letters of Commendation from the Pennsylvania State University Department of Electrical Engineering for outstanding teaching and research. He was the recipient of a College of Engineering PSES Outstanding Research Award and Outstanding Teaching Award in March 2000 and March 2002 respectively. He was also presented with an IEEE Central Pennsylvania Section Millennium Medal. In March 2009, he received the PSES Premier Research Award. He was an Associate Editor of Radio Science and an Editor of the IEEE Antennas and Propagation Magazine.

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2011 IEEE Fellow Awards From the Antennas and Propagation Society HE following is a list of 2011 IEEE Fellows who are members of the Antennas and Propagation Society. Those with an asterisk after their names were evaluated by the AP-S Fellow Committee. Donald Barrick—for development of high frequency radars and applications. Goutam Chattopadhyay—for contributions to development of sources, sensors, and coupling structures at terahertz frequencies. Debabani Choudhury—for contributions to millimeter wave enabling technologies. Steve Cripps—for contributions to broadband and high-efficiency radio frequency power amplifiers. Steven Cummer*—for contributions to lightning remote sensing and artificial electromagnetic materials. Luis Jofre-Roca*—for contributions to antenna near-field characterization and imaging. Niels Kuster—for contributions to the area of near-field exposures and dosimetry for radiofrequency fields in biomedical research. Kwok Leung*—for contributions to the development of the dielectric resonator antenna. Eric Mokole—for leadership and contributions to ultra-wideband radar, waveform diversity, and transionospheric space radar. Michel Ney—for contributions to modeling in electromagnetics. Natalia Nikolova-Zimmerman—for contributions to computer-aided analysis of microwave systems. Ken O—for contributions to ultra-high frequency complementary metal-oxide semiconductor circuits. Yasutaka Ogawa*—for contributions to estimation techniques and antenna signal processing. Ioannis (John) Papapolymerou—for contributions to flexible, microwave and wireless components and systems. Stephen Schneider*—for leadership in integrated wide bandwidth array technology. Rodney Waterhouse*—for contributions to microwave photonic systems and printed antennas. Ke-Li Wu—for contributions to non-planar microwave filters and embedded radio frequency passive circuits. Quan Xue—for contributions to microwave transmission line structures and integrated circuits.

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A Reconfigurable Patch Antenna Using Liquid Metal Embedded in a Silicone Substrate Shahrzad Jalali Mazlouman, Member, IEEE, Xing Jie Jiang, Alireza (Nima) Mahanfar, Member, IEEE, Carlo Menon, Member, IEEE, and Rodney G. Vaughan, Fellow, IEEE

Abstract—A frequency-reconfigurable microstrip patch antenna is implemented using a stretchable silicone TC5005 substrate. The stretchable patch is fabricated by injecting liquid metal alloy Galinstan into a square reservoir fabricated in the silicone elastomer substrate. An aperture-coupled feeding method is presented to enhance the strain by separating the fixed feed element from the stretched radiating element. The implemented prototype is demonstrated to be stretchable by a strain of up to 300% (and the material has the potential for more). The electrical length of the patch antenna varies with the stretching, and we demonstrate frequency tuning from 1.3 to 3 GHz. A maximum radiation efficiency of 80% is measured for the antenna. The fabrication process of the antenna system and experimental results for impedance and radiation pattern are presented. Index Terms—Aperture-coupled patch antenna, reconfigurable antennas, stretchable antenna.

I. INTRODUCTION

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MERGING wireless communication devices call for antennas that can dynamically reconfigure their radiation parameters such as the resonance frequency or the pattern, e.g., [1]. Such antennas can potentially mitigate interference, increase signal-to-noise ratio, and help to provide spectrum reuse and multistandard wireless solutions. Many reconfigurable antennas incorporate switches, e.g., p-i-n diodes or RF/MEMS switches, to connect/disconnect part of the structure of the antenna. The switchable structures include matching or feeding networks or parasitic elements [1]–[4]. Reconfiguration of these antennas is however limited by the binary nature of the switch. In addition, these switches are placed in the RF signal path and limit the linearity and power handling of the antenna system [5]. Varactors, for example, offer a changeable impedance termination, but also suffer from nonlinearity issues (see [6], [7], and the references therein.) Another method for reconfiguring the antennas is to incorporate mechanical actuators to reconfigure the physical structure of the antenna [1], [8], [9]. One advantage of mechanically reconfiguring the antenna is that many configurations can be obtained. With advances in smart materials and very large Manuscript received December 13, 2010; revised February 25, 2011; accepted April 30, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. S. Jalali Mazlouman, X. J. Jiang, C. Menon, and R. G. Vaughan are with the School of Engineering Science, Simon Fraser University, Burnaby, BC V5A 1S6, Canada (e-mail: [email protected]). A. Mahanfar is with Microsoft Corporation, Redmond, WA 98052 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165501

scale integration (VLSI), low-cost actuators for mechanical reconfiguration are becoming increasingly viable. Mechanical reconfiguration offers a complexity reduction for multiple antenna systems whose multiple front ends are proving too costly for widespread uptake. For example, electroactive polymer (EAP) actuators have been shown as potential candidates for mechanically reconfigurable antennas [8], [9]. Several recent works are focused on lightweight, flexible, and stretchable antennas that can be implemented at low cost [10]–[13]. This has put a spotlight on the use of new materials. Flexible materials can be used to implement a new generation of mechanically reconfigurable antennas. Although copper, and other solid conductors, can provide highly efficient antennas, the fatigue of copper even in the form of thin foil does not allow antennas to be continually deformed for general adaptive antenna applications. Nevertheless, flexible structures are possible and have less-demanding applications. Flexible antennas can also be used in wearable health-monitoring devices or for sensors and security applications such as RFIDs. Such antennas can be integrated with the electronics on a flexible substrate [14] and can be embedded in the clothing or accessories such as a watch or bracelet. In many of these applications, flexibility is desired, but not a change of the antenna parameters, and here the design goals would be different from other reconfigurable antennas. A bendable, frequency-reconfigurable monopole antenna was implemented by embedding a liquid metal alloy (Galinstan) micro-channel in a polydimethylsiloxane (PDMS) substrate [10]. It was shown that the resonant frequency of the liquid metal wire antenna can be tuned by stretching the substrate, thereby altering the effective length of the antenna. Similarly, flexible loop and cone antennas were implemented [11], [12]. Although not designed for a reconfigurable antenna, some frequency and pattern variations due to stretching were reported for the loop antenna [12]. The measured mechanical strain for these antennas is limited to 40% due to their rigid feeding mechanism that can easily damage the antenna when stretched beyond this limit [11]. In addition, the feed cable was detached while stretching the antenna and reattached after each stretch [12]. A hybrid substrate was later presented to increase the strain by up to 120% by incorporating a more elastic substrate around the rigid components while maintaining the PDMS substrate [13]. Fabrication of the hybrid substrate is more complicated and requires extra steps. The rigid feed connections are again reported to decrease the durability of the flexible antennas [11], [12]. When the antenna is in a twisted or folded state, the rigid connections make measurements difficult or impossible [11], [12]. In addition, the

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MAZLOUMAN et al.: RECONFIGURABLE PATCH ANTENNA USING LIQUID METAL EMBEDDED IN SILICONE SUBSTRATE

PDMS substrate used in the previous works can potentially provide a maximum potential strain of 140% [15]. In this paper, implementation and experimental results for a frequency-reconfigurable microstrip patch antenna using new materials is reported. The presented design can be stretched by up to 300%, and with our choice of prototype dimensions, the narrowband antenna can operate at a frequency located from 1.3 to 3 GHz. To enhance the strain, a new TC5005 silicone substrate structure and a slot-aperture-coupled feeding technique are incorporated. The patch antenna is implemented using a planar reservoir of Galinstan that is inserted in the substrate. It is shown that by stretching the substrate along one axis of the antenna, the effective length of the antenna, and therefore its resonant frequency (i.e., lowest dominant mode), can be reconfigured as a smooth transition. The TC5005 silicone substrate can in principle provide much larger strains compared to PDMS, viz., up to 700% [16]. Fabrication of this antenna is straightforward. The problem of the rigid feeding in previously implemented antennas is solved by using a coupled feeding mechanism [17], [18]. Using this feeding mechanism, the feed system remains fixed while the rest of the antenna stretches. This, combined with the use of a softer substrate material, allows the patch antenna to realize its 300% measured strain without requiring the extra material in the hybrid substrate [13]. Planar patch antennas are very common in wireless communication devices such as cell phones because of their low profile and ease of fabrication, but to date, they do not use mechanical reconfiguration. The rest of the paper is as follows. The aperture-coupled microstrip patch antenna design, the feeding mechanism, and the prototype fabrication method are discussed in Section II. In Section III, experimental results on the fabricated prototype, including frequency reconfigurability of the flexible liquid patch antenna as well as pattern measurements and efficiency measurement results are presented. Section IV is the conclusion. II. ANTENNA DESIGN AND FABRICATION A. Slot-Aperture-Coupled Antenna Design Fig. 1 depicts the building blocks of the implemented patch antenna. The antenna is in two parts: the stretchable substrate with patch and the rigid groundplane with slot feed. The stretchable substrate consists of two, 40 150 mm TC5005 silicone elastomer layers enveloping a rectangular patch Galinstan reservoir of size 31 31 0.5 mm . The fabrication process is reviewed in Section II-B. Galinstan is an alloy of Ga, In, and Sn and is liquid between 19 and 1300 C [12]. Its conductivity is reported about S/m, and it is nontoxic [12]. The groundplane is a square copper plane with a slot used for feeding in the middle. The size of the groundplane is 300 300 mm to cater for the large stretch, but it could be smaller if the patch antenna is to be stretched less. The gap between the patch and the groundplane is 5 mm and can be provided using a Styrofoam or plastic support, as shown in Fig. 1. This separation gap is chosen for reasonable radiation efficiency feed coupling. The coaxial feed line is fixed and located

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Fig. 1. (a) Two-layer slot-aperture-coupled microstrip patch antenna with the enlarged view of the coaxial feed. (b) Side view of the antenna.

across the middle of the slot below the groundplane. The location of the feed on the slot affects the impedance matching of the antenna at the resonance frequency. The slot itself is smaller than half a wavelength (i.e., below slot resonance) to reduce radiation from the groundplane side while maintaining good coupling to the patch [19]. Various shapes of slots are possible, e.g., rectangular, square, or U-shaped. The prototype requires a symmetric Z-shaped slot (as shown in Fig. 1) of size 23 1 mm with two symmetric arms of 3 1 mm . The width of the slot also affects the coupling, but to a much less degree than the slot length [19]. Other design methods may be used to feed the slot in order to attain impedance matching for other patch lengths. For example, an open-circuited transmission line is often added to the feeding structure to attain impedance matching at a desired operating frequency [17], [19]. The Galinstan-filled reservoir is the patch element. For the dominant mode, the resonant frequency of the patch antenna is a function of its length [19], (1) where is the speed of light in free space and is the effective permittivity that is a function of the permittivity of the thin TC5005 silicone layer 2.8–3.1 and the air gap underneath it. Fringing fields are neglected in (1). In practice, other factors such as fringing capacitance and the aspect ratio of the patch affect the resonance frequency [19]. The original size of the patch is chosen to be 31 31 mm , and the resonance frequency of the patch with the slot feed turns out to be 2.8 GHz. Stretching the substrate along the depicted direction in Fig. 1 (perpendicular to the slot axis) will result in increasing the patch length . Therefore, the resonant frequency of the patch is decreased. In addition, the is varied since the thickness of the substrate is decreased with stretching. This is demonstrated in the measurements in the following section.

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Fig. 4. Top elastomer layer collapsing on the bottom elastomer layer due to the glue layer sprayed on the reservoir.

Fig. 2. Top and side views of the PMMA square mold for fabrication of the reservoir.

Fig. 3. Complete fabrication steps of the prototype: (a) The patch pattern is cut on the PMMA mold using a laser cutter. (b) The TC5005 substrate elastomer is filled into the mold and cured. (c) After curing, the reservoir appears on the resulting substrate elastomer. (d) The isolation plastic layer is inserted on the reservoir to avoid blocking the channel. (e) A flat thin layer of uncured TC5005 is sprayed on the sealing elastomer to act as the glue. (f) The substrate elastomer (that includes the reservoir) is inverted and placed on top of the sealing elastomer. (g) Finally, Galinstan is injected into the reservoir using a syringe.

B. Fabrication The prototype silicone TC5005 elastomer layers with a square-shaped Galinstan reservoir is handmade. The liquid metal is injected into the reservoir using a syringe. The bonding of the elastomer layers is provided by spraying on an uncured TC5005 layer. To fabricate the reservoir, first a mold is made out of polymethylmethacrylate (PMMA), as shown in Fig. 2. To fabricate this mold, a 60-W CO laser cutter is used to cut a square ring with an exterior side of 50 mm, interior side of 31 mm, and depth of 0.5 mm out of the PMMA. A 31 31 0.5-mm square island is left in the middle of the mold [shown in Figs. 2 and 3(a)]. Then, TC5005 is poured into the mold and cured [shown in Fig. 3(b)]. We refer to the resulting layer as the substrate elastomer. After curing, the substrate elastomer becomes a 50 50 mm sheet with a 31 31 0.5 mm reservoir for containing the liquid metal [shown in Fig. 3(c)]. To seal it, a uniform thin layer of uncured TC5005 is sprayed over a flat sheet of TC5005 using a spinner [shown in Fig. 3(e)]. This uncured layer acts as a glue

to bond the two elastomer layers. We refer to the second elastomer as the sealing elastomer. The substrate elastomer is inverted and placed on top of the glue layer (and the sealing elastomer), [shown in Fig. 3(f)]. However, the TC5005 is softer and more stretchable than the PDMS, and the planar square patch has a wider area compared to the narrow channel structures reported in [10]–[12]. Therefore, when the substrate elastomer is placed on the flat glue and sealing elastomer, the top layer collapses and touches the bottom layer, as shown in Fig. 4. Since the glue layer has been sprayed uniformly, the touching parts hold after curing time, therefore blocking the reservoir. One way to avoid blocking the reservoir is to hand-spray the glue on selected areas, but this results in a nonuniform glue layer, which in turn results in a distorted or irregular shaped sample that tears apart easily. In addition, controlling uncured TC5005 is difficult due to its liquid-like character. TC5005 silicone does not bond well with plastic, and therefore a plastic isolation layer should be placed on the reservoir to isolate it from the glue and therefore avoid blocking. This is shown in Fig. 3(d). The rest of the fabrication is as explained above. Finally the Galinstan is injected using a syringe, as shown in Fig. 3(g). The complete fabrication steps are shown in Fig. 3. Another advantage for using the plastic isolation layer on the reservoir is that it produces a capillary force to help expand the liquid metal uniformly despite the high surface tension of liquid metal. After injecting the liquid metal, the sample is ready to use. III. EXPERIMENTAL RESULTS The prototype is shown in Fig. 5(a) and (b) for top and bottom views, respectively, with the original length (not stretched). The Z-shaped slot on the groundplane and the coaxial feed connector are shown in Fig. 5(b). Fig. 6(a) and (b) depicts the patch as unstretched and stretched by 300%, respectively. As can be seen in Fig. 6(a) and (b), a (white) plastic spacer is used between the elastomer and the copper groundplane. This hollow plastic spacer has a thickness of 5 mm to support the gap (larger cavity volume below the patch) required to improve radiation efficiency. As seen in Figs. 5(a) and 6(a), and 6(b), there is extended elastomer area on the sides of the patch (not shown in the mold in Fig. 2). This facilitates the stretching of the prototype. The elastomer is fixed between a pair of plastic clamps holding the sample so that there is a uniform stretching force across the width of the patch. A caliper is used to measure the length of the patch antenna at each measurement point. The polymer substrate can be stretched by as much as 700% according to the manufacturer specifications, which is much higher than the stretching limit of 140% for PDMS. Our handmade prototypes are stretched manually, therefore a lower strain is measured. In an integrated product, the stretching can be implemented by incorporating, for example, EAP actuators on the same

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Fig. 7. Variations of the of the Galinstan patch antenna when altering its effective length by stretching.

Fig. 5. Prototype under test: (a) top and (b) bottom view, in relaxed state (nonstretched).

Fig. 8. Variations of the resonance frequency of the patch antenna by stretching.

Fig. 6. Complete implemented prototype under test: (a) In relaxed state (nonstretched). (b) The antenna can be stretched to 300% of its original length at maximum measured stretch.

substrate [20]. To the authors’ knowledge, the measured 300% strain is the largest stretching range reported to date for similar antenna substrates. However, a higher strain can be expected if the sample fabrication and the stretching mechanism are further optimized.

Fig. 7 depicts the measured for length variations of the patch. It is shown that by stretching the patch antenna, its resonant frequency can be varied continuously. As shown in this figure, the resonance frequency and the bandwidth of the antenna decrease as the antenna length is increased. A reasonable impedance matching is preserved below 6 dB) with different patch lengths. The measured resonance frequencies are also shown in Fig. 8 versus the lengths of the patch and compared to the results obtained from finite integration technique (FIT) modeling using CST simulation results. The measured and simulated results show very good agreement. In addition to variations of the length of the patch, other parameters such as the thickness of the patch and the substrate (and as a result the ) and the aspect ratio of the patch are varied by stretching. As stated before, these parameters affect the patch resonance frequency. Since the samples are made in the lab by hand, the dimensions are approximate, and we do not have a means of measuring the parameters (e.g., thickness of the embedded patch while stretching) with higher precision. Therefore, these effects are taken into account in the simulations by modeling the elastomer and the patch with a constant volume while stretching. With

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Fig. 10. Experimental results for the total radiation efficiency of the slot-aperture-coupled square Galinstan patch antenna in various stretching steps.

Fig. 9. Measured patterns of realized gain for (a) , (b) , and pattern cuts of the slot-aperture-coupled Galinstan patch antenna in (c) its nonstretched state (resonance frequency of 2800 MHz) and several stretched cases. The spherical coordinates are defined, with the patch on the plane.

the available tools, we tried our best to keep the width of the patch constant while stretching. Assuming a constant width, the thickness and, as a result, other parameter variations can be calculated accordingly. Simulation results agree well with the measurements, suggesting that these assumptions are valid. Measured radiation patterns for the principal cuts [(a) , (b) , and (c) ] of the antenna in its nonstretched state and several stretched states are shown in Fig. 9, and the total efficiency for various stretching steps are shown

in Fig. 10. These measurements are conducted using a Satimo StarLab anechoic chamber. The radiation pattern in Fig. 9, as expected, has a main lobe for the radiation of the patch in both the and pattern cuts. This main lobe is evident at . Low gain in the direction below the groundplane confirms that the slot is not radiating strongly in this direction. As seen in this figure, variations in the radiation pattern of the patch due to stretching are minor. The presented pattern measurement results report the realized gain and therefore includes the effect of the substrate (the loss due to the substrate) as well as the effect of impedance matching. The radiation efficiency of the antenna, which is governed by the ohmic losses in the antenna structure, is difficult to simulate or measure accurately. However, because the total efficiency is the product of the radiation efficiency and the impedance mismatch efficiency, we can at least conclude that if the measured total efficiency is good, then the radiation efficiency is good, and so the material losses must be reasonably small. A maximum total efficiency of 80% is attained for this prototype in the nonstretched case, as shown in Fig. 10. The radiation efficiencies for different stretching states were measured in a Satimo anechoic chamber. It can be seen that the maximum efficiency is slightly decreasing when the patch is stretched. At maximum efficiency, the impedance mismatch loss is negligible (see the resonant example in Fig. 7), and so these values essentially represent the radiation efficiency. For the unstretched patch, the radiation efficiency is about 80%, and for the stretched cases, the efficiency is still kept above 65% for more than 100% stretch. It is not easy to identify a single mechanism for the decrease of the efficiency of the antenna while stretching. There are many things changing that will influence the efficiency. First, the frequency is changing with the stretching of the elastomer and the patch. As the frequency gets lower, the overall antenna electrical size is decreasing (the wavelength is increasing), and generally, more electrically compact antennas tend to have lower efficiencies. A particular mechanism could be as follows: The 5-mm gap between the patch antenna and the slot becomes much smaller—for example, for a 50% stretch of the antenna length from 30 to 60 mm, the gap changes roughly from to (the decrease is further if the substrate thickness decrease

MAZLOUMAN et al.: RECONFIGURABLE PATCH ANTENNA USING LIQUID METAL EMBEDDED IN SILICONE SUBSTRATE

by 50% is taken into account). On the other hand, as the elastomer stretches, the elastomer substrate is getting thinner, and the dielectric losses should decrease accordingly, which acts to increase the efficiency. Finally, the patch conductor is getting thinner with stretching, and this would act to decrease the efficiency. The combination of all such effects is seen in the figure. The trend matches simulation results (not in the script). The prototype demonstrates the feasibility of frequency reconfigurability using aperture-coupled slot feeding and a flexible substrate. To improve the bandwidth and performance of the system, the thickness of the substrate and the shape of the patch antenna can be optimized, but this does not change the basic mechanisms of its operation. The measured antenna parameters did not change after six months and many hundred stretching times, however, the Galinstan tends to start leaking after many stretchings at the extreme length, owing to failure of the bonding. As stated before, a mechanized method for bonding the elastomers would probably result in better bonding and less leakage of Galinstan, as well as better stretchability. Although not the application of this paper, the samples are deformable and stretchable in every dimension; they can be rolled around a hand or worn on clothing. However, the change of antenna parameters in these cases has not been measured. IV. SUMMARY AND CONCLUSION We demonstrated a new configuration of elastomer and liquid metal for implementing a reconfigurable antenna with wide frequency agility. It uses a patch antenna comprising a planar reservoir of injected liquid metal alloy Galinstan fabricated within a TC5005 silicone elastomer. This patch antenna is aperture-coupled from a slot in a copper groundplane. The flexible patch can be stretched resulting in a changed resonance frequency. Two factors combine to enhance the strain compared to previous works. First, the presented aperture-coupling feeding method allows the coaxial feed line to be removed from the flexible structure, which solves the restriction caused by a rigid feed interfacing with a flexible antenna. Second, the use of the presented TC5005 elastomer instead of the PDMS substrate used in previous works allows a higher strain and enables a higher range of frequency reconfigurability. The fabrication process is presented and includes a plastic isolation layer to facilitate the planar reservoir. A thin layer of uncured TC5005 is used to seal the reservoir. Experimental results confirm good impedance matching over a wide range of resonant frequencies. Good antenna efficiency is obtained even when it is stretched and the patch conductor is at its thinnest. The presented concept is promising for lowcost, flexible, reconfigurable antennas that can be integrated with electronic actuators such as EAPs, and other flexible or rigid electronics such as the communications signal processing blocks.

REFERENCES [1] J. T. Bernhard, Reconfigurable Antennas. Morgan & Claypool, 2007.

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[2] J. Sarrazin, Y. Mahé, S. Avrillon, and S. Toutain, “Pattern reconfigurable cubic antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 2, pp. 310–317, Feb. 2009. [3] S. Chen, J. Row, and K. Wong, “Reconfigurable square-ring patch antenna with pattern diversity,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 472–475, Feb. 2007. [4] J. Kiriazi, H. Ghali, H. Radaie, and H. Haddara, “Reconfigurable dualband dipole antenna on silicon using series MEMS switches,” in Proc. IEEE/URSI Int. Symp. Antennas Propag., 2003, pp. 403–406. [5] N. P. Cummings, “Active antenna bandwidth control using reconfigurable antenna elements,” Ph.D. dissertation, Virginia Polytechnic Institute & State University, Blacksburg, VA, 2003. [6] J. H. Mulligan and C. A. Paludi, “Varactor tuning diodes as a source of intermodulation in RF amplifiers,” IEEE Trans. Electromagn. Compat., vol. EMC-25, no. 4, pp. 412–421, Nov. 1983. [7] P. S. Hall, S. D. Kapoulas, R. Chauhan, and C. Kalialakis, “Microstrip patch antenna with integrated adaptive tuning,” in Proc. Int. Conf. Antennas Propag., Apr. 14–17, 1997, vol. 1, pp. 506–509, (Conf. Publ. No. 436). [8] A. Mahanfar, C. Menon, and R. G. Vaughan, “Smart antennas using electro-active polymers for deformable parasitic elements,” Electron. Lett., vol. 44, no. 19, pp. 1113–1114, 2008. [9] K. Daheshpor, S. Jalali Mazlouman, A. Mahanfar, X. Han, J. Yun, C. Menon, F. Carpi, and R. G. Vaughan, “Pattern reconfigurable antenna based on moving vee-shaped parasitic elements actuated by dielectric elastomer,” Electron. Lett., vol. 46, no. 13, pp. 886–888, June 2010. [10] J.-H. So, J. Thelen, A. Qusba, G. J. Hayes, G. Lazzi, and M. D. Dickey, “Reversibly deformable and mechanically tunable fluidic antennas,” Adv. Func. Mater., vol. 19, pp. 3632–3637, 2009. [11] S. Cheng, Z. Wu, P. Hallbjorner, K. Hjort, and A. Rydberg, “Foldable and stretchable liquid metal planar inverted cone antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3765–3771, Dec. 2009. [12] S. Cheng, A. Rydberg, K. Hjort, and Z. Wu, “Liquid metal stretchable unbalanced loop antenna,” Appl. Phys. Lett., vol. 94, no. 14, pp. 144103–144103-3, Apr. 2009. [13] M. Kubo, X. Li, C. Kim, M. Hashimoto, B. J. Wiley, D. Ham, and G. M. Whitesides, “Stretchable microfluidic radiofrequency antennas,” Adv. Mater., vol. 22, pp. 2749–2752, 2010. [14] D. H. Kim, J. Xiao, J. Song, Y. Huang, and J. A. Rogers, “Stretchable, curvilinear electronics based on inorganic materials,” Adv. Mater., vol. 22, pp. 2108–2124, 2010. [15] “Sylgard 184 silicone elastomer datasheet,” Dow Corning, Midland, MI, 2010 [Online]. Available: http://www.dowcorning.com [16] “TC 5005 A/B-C datasheet,” BJB Enterprises, Tustin, CA, 2010 [Online]. Available: http://www.bjbenterprises.com/pdf/TC-5005.pdf [17] D. M. Pozar, “Microstrip antenna aperture-coupled to a microstripline,” Electron. Lett., vol. 21, no. 2, pp. 49–50, Jan. 1985. [18] P. Sullivan and D. Schaubert, “Analysis of an aperture coupled microstrip antenna,” IEEE Trans. Antennas Propag., vol. AP-34, no. 8, pp. 977–984, Aug. 1986. [19] D. M. Pozar and D. H. Schaubert, Microstrip Antennas. New York: Wiley–IEEE Press, 1995. [20] Y. Bar-Cohen, Electroactive Polymer (EAP) Actuators as Artificial Muscles—Reality, Potential and Challenges. Bellingham, WA: SPIE Press, 2004.

Shahrzad Jalali Mazlouman (M’09) received the B.Sc. and M.Sc. degrees in electrical engineering (electronics) from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran, in 2001 and 2003, respectively, and the Ph.D. degree in electrical and computer engineering from the University of British Columbia (UBC), Vancouver, BC, Canada, in 2008. She has worked on several mixed-signal, RF, and antenna system designs. In 2007, she worked as a mixed-signal intern with PMC-Sierra, Burnaby, BC, Canada. Since 2009, she has been a Post-Doctoral Fellow with the School of Engineering Science, Simon Fraser University (SFU), Burnaby, BC, Canada, where she is working on reconfigurable RF and antenna systems for wireless communication devices, using smart material actuators.

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Xingjie Jiang was born in Chengde, China. She received the B.Sc. degree in physics from Jilin University, Changchun, China, in 2002, and the M.Sc. degree in physics from the University of New Brunswick, Fredericton, NB, Canada, in 2007, and is currently pursuing the M.A.Sc. degree in engineering science at Simon Fraser University, Burnaby, BC, Canada. Her current research interests include microelectromechanical systems; the design, control, and characterization of electroactive polymers (EAPs); and

both the School of Biomedical Physiology and Kinesiology and the Institute of Micromachine and Microfabrication Research at SFU. His research team is focusing on mechatronics, smart materials and structures, robotics, and bio-inspired systems with applications especially in the biomedical and space sectors. Dr. Menon is an AIAA, ASME, BIONIS, and IAF member. He is currently a reviewer for about 20 international journals and a member of the Editorial Board of the Journal of Bionic Engineering. He received the International IAF Luigi G. Napolitano Award, Spain, in 2006, and the International BIONIS Award on Biomimetics, U.K., in 2007.

smart structures.

Alireza (Nima) Mahanfar (S’99–M’05) received the B.S. (Honors) and M.S. degrees from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran, in 1997 and 1999, respectively, and the Ph.D. degree from XLIM (formerly IRCOM), Limoges, France, in 2005, all in electrical engineering. From 1998 to 2009, he was with a number of organizations, including the Mobile Communications Lab, Simon Fraser University, Burnaby, BC, Canada; Sierra Wireless, Richmond, BC, Canada; and Nokia Mobile Phones, Burnaby, BC, Canada. Since December 2009, he has been with Windows Phone Innovations, Microsoft Corporation, Redmond, WA, where he is involved in the research and development on wireless systems. His research interests are design of antennas and radio-frequency circuits. Dr. Mahanfar is a recipient of NSERC Postdoctoral Fellowship in 2005 and the URSI Young Scientist Award in 2007.

Carlo Menon (M’04) received the Laurea degree in mechanical engineering from the University of Padua, Padua, Italy, in 2001, and the Ph.D. degree in space sciences and technologies from the Centre of Studies and Activities for Space, “G. Colombo,” Padua, Italy, in 2005. He was Visiting Scholar with Carnegie Mellon University, Pittsburgh, PA, in 2004, and a Research Fellow with the European Space Agency, Noordwijk, The Netherlands, in 2005 and 2006. Since 2007, he has been an Assistant Professor with Simon Fraser University (SFU), Burnaby, BC, Canada, where he leads the MENRVA Research Group (http://menrva.ensc.sfu.ca). He is an Associate Member with

Rodney G. Vaughan (F’07) received the Bachelor’s and Master’s degrees from the University of Canterbury, Christchurch, New Zealand, in 1975 and 1976, respectively, and the Ph.D. degree from Aalborg University, Aalborg, Denmark, in 1985, all in electrical engineering. He worked with the New Zealand Post Office (now Telecom NZ Ltd.) and the New Zealand Department of Scientific and Industrial Research, and Industrial Research Limited (IRL). Here, he undertook a wide variety of practical mechanical and electrical projects including network analysis and traffic forecasting and developed microprocessor and DSP technology for equipment ranging from abattoir hardware to communications networks. He was an URSI Young Scientist in 1982 for Fields and Waves, and in 1983 for Electromagnetic Theory. He developed research programs and personnel working in communications technology for IRL, revolving around signal processing, multipath communications theory (electromagnetic, line, and acoustic media), diversity design, signal theory, and DSP. Industrial projects included the design and development of specialist antennas for personal, cellular, and satellite communications, large-N MIMO communications systems design; and also capacity theory and spatial field theory. In 2003, he became a Professor of electrical engineering and Sierra Wireless Chair in Communications with the School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada. His current research for mobile communications involves propagation theory, communications signal processing, and theory and design of antennas. Recent projects include compact mammalian bio-implantable antennas; multielement antenna design and evaluation; circularly polarized antennas, multifaceted structures for large arrays; microelectronic antenna structures, MIMO capacity realization; and blind-, precoding-, and interference mitigation techniques for OFDM. Dr. Vaughan is a Fellow of the BC Advanced System Institute and an URSI Correspondent, and he continues to serve as the New Zealand URSI Commission B (Fields and Waves) representative. He has guest-edited for several special issues including the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION Special Issue on Wireless Communications.

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A Multifunctional Reconfigurable Pixeled Antenna Using MEMS Technology on Printed Circuit Board Alfred Grau Besoli, Member, IEEE, and Franco De Flaviis, Senior Member, IEEE

Abstract—A multifunctional reconfigurable pixeled antenna that uses microelectromechanical switches (MEMS) and operates within the 4–7 GHz frequency band is presented. The proposed antenna is a highly reconfigurable planar surface radiator composed of an px px matrix of metallic pixels interconnected through MEMS switches. The antenna can be reconfigured to perform multiple functions such as: frequency tuning, pattern diversity, and polarization adaptation. A functional prototype using integrated MEMS switches was fabricated on a printed circuit board substrate, to show its feasibility and the impact of MEMS switches during operation. Simulated and measured data of the return loss and radiation characteristics of the antenna are presented. Index Terms—Frequency tunable, MEMS, multifunctional antenna, pattern diversity, pixel antenna, polarization agile, reconfigurable antenna.

I. INTRODUCTION N recent years, new technologies such as microelectromechanical systems (MEMS) [1]–[3] have begun to be applied in wireless communications, which create new opportunities for antenna designers. Using MEMS switching technology, an antenna can be designed which can be tuned to different frequency bands while maintaining a good radiation efficiency. Such an antenna would not cover all bands simultaneously, but would provide narrow individual bandwidths that are selectable by the user through a software interface. An antenna with these characteristics is typically referred in the literature as a frequency reconfigurable antenna [3]–[6]. In communication systems employing antenna diversity techniques, one could also be interested in being able to excite and select among a finite set of different radiation patterns and different polarization states from a single antenna. Traditionally, this would require the use of antenna arrays, beamforming networks (e.g., Butler Matrix), and switching networks, thus producing bulky systems. However, MEMS switches can be used to design antennas that can reconfigure their radiation pattern and/or polarization [2], [7]–[10].

I

Manuscript received May 03, 2010; revised April 30, 2011; accepted June 03, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. This work was supported in part by the National Science Foundation Award ECS-0424454, Project TEC-2007-66698, Balsells Fellowships, and in part by the California-Catalonia Engineering Innovation Program 2004–2005. A. Grau Besoli was with the University of California at Irvine, Irvine, CA 92697-7625 USA. He is now with Broadcom Corporation, Irvine, CA 92617 USA (e-mail: [email protected]). F. De Flaviis is with the University of California at Irvine, Irvine, CA 926977625 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165470

Reconfigurable antennas work based on the principle that, by altering the antenna’s physical geometry, the current distribution on the antenna can be controlled in a desirable manner (to some extent) and therefore its radiating and electrical characteristics can be changed. In order to change the antenna’s physical configuration, one can use switches that strategically interconnect different metallic parts of the antenna. It is desirable to have an antenna which integrates several of the aforementioned reconfigurable capabilities. Such antenna can result in a compact design which can perform multiple tasks. In [11], we introduced the concept of a multifunctional antenna which, through the use of MEMS switches, can reconfigure its radiation pattern, polarization, and operational frequency. Within the same timeline, in [12] a similar antenna structure was proposed and fabricated which used photo-diodes as active switching devices to reconfigure the antenna in distinct geometries. In [12], however, no specific requirements on the manner as to how the antenna should radiate were part of the design criteria. A generic pixeled structure was also proposed in [13], although its focus is towards reconfigurable electromagnetic mechanism in general and control architectures. In [14], a frequency tunable antenna utilizing reconfigurable patch modules was introduced which used an adaptive feed mechanism, but ideal interconnects where used to model the MEMS switches. In this work, we develop the concept introduced in [11] and we present a multifunctional MEMS-reconfigurable pixeled antenna intended to operate within the 4–7 GHz frequency band. Throughout this paper, we will refer to the proposed antenna as the PIXEL antenna. The PIXEL antenna is a significant step towards the realization of software defined antennas (SDA), which in combination with software defined radio (SDR) architectures have the potential to satisfy the requirements of future communications, such as programmable software and hardware radio architectures with multiband and multifunctional operation. Nevertheless, the current prototype operates at a higher frequency range than the one traditionally allocated for SDR systems. Here, MEMS switches are used to reconfigure the antenna because they offer superior performance than active switching devices such as diodes or field effect transistors (FETs), in terms of insertion loss, isolation and power consumption [1], [15], [16]. Most importantly, MEMS switches can be integrated monolithically within the antenna because they can be fabricated on low cost substrates such as printed circuit board (PCB) [16]. This paper is organized as follows. A description of the proposed antenna, its modes of operation, and the design guidelines are given in Section II. Section III presents a functional 9 9

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PIXEL antenna with integrated MEMS switches. Section IV shows the simulated and measured scattering parameters and radiation characteristics. II. THE PIXEL ANTENNA A. Description The starting point in the concept design of the PIXEL antenna is an arbitrarily shaped metallic surface acting as a radiating surface. Our first step is to enable this surface to radiate with different characteristics. In order to achieve this, we consider the possibility of decomposing the metallic surface into electrically small metallic pieces, interconnected with switching devices. The natural approach is to begin with a simple geometry for these metallic pieces, thus we select square shape and we name each one of these metallic pieces a pixel (in analogy with the smallest reconfigurable region of a digital monitor). The square shape may not necessarily be the most effective shape in minimizing the number of switches and maximizing the number of functionalities, but this investigation is beyond the scope of this work. The lateral dimension ( in Fig. 1) has to be much smaller at the operating frequency, that is, , than a wavelength thus enabling the metallic surface to be highly reconfigurable. As a result, the PIXEL antenna is proposed and in the following paragraph we describe it in detail. Fig. 1(a)–(d) illustrates the aforementioned conceptual steps. The PIXEL antenna is a microstrip-type radiator consisting of a reconfigurable planar radiating surface composed of an square matrix of metallic pixels interconnected through MEMS switches, as shown in Fig. 1(d). These pixels have some intrusions to allocate the MEMS switches and the DC biasing vias for control. By interconnecting specific sets of pixels within the matrix, radiating surfaces of different shape can be mapped, thus creating microstrip patches of different forms and sizes, such as approximately circular, rectangular, triangular, and comb patches [11], among others, as exemplified in Fig. 1(e)–(f). These distinct mapped patches radiate through distinct radiation patterns, polarizations, and/or operating frequencies, and thus the antenna can be used to perform multiple functionalities. Fig. 2(a)–(c) shows different views of the geometry of the antenna and the basic dimensions of the PIXEL antenna. The antenna is fed through a single coaxial cable on the back side of the substrate. The inner conductor of the coaxial connects to one of the metallic pixels, while the external conductor connects to the ground plane. Notice that the pixel connected to the inner connector is the same in all modes of operation, and it is offset from the center of the matrix of pixels (to facilitate the matching of mapped patches). Each metallic pixel has dimensions 1.43 mm 1.43 mm with a spacing of 2 mm from center to center, leaving a gap of 0.57 mm between them. The pixel dimensions are chosen to maximize the number of functionalities that the antenna can generate (i.e., the smaller possible pixel dimension), for a given surface area, while ensuring enough space to allocate the MEMS switches and control system. Fig. 3(a)–(b) shows the dimensions of the pixeled structures and DC biasing vias for the control system, in detail.

Fig. 1. Sequence of conceptual steps followed during the realization of the PIXEL antenna: (a) solid metallic radiating surface, (b) decomposed surface into irregular metallic pieces, (c) decomposition into square pixels, (d) reconfigurable radiating surface by interconnecting the pixels with switches, (e) interconnected pixels (in black) forming a square mapped patch, and (f) forming a circular mapped patch of radius a.

A desired input impedance (and matching level) in a mapped patch is achieved by adjusting its relative position (and sometimes also its shape) with respect to the pixel in which the feeding coaxial line is connected, and by connecting or disconnecting strategically chosen pixels from the interconnected parts of the radiating surface, or from the surrounding regions using the pixels as parasitic elements. B. Modes of Operation and Functionalities The proposed PIXEL antenna can be operated in three distinct ways: (1) as a multimode antenna for antenna diversity wireless communication systems, (2) as a multiband frequency tunable antenna, and (3) as a polarization-agile antenna in polarization-sensitive applications (e.g., analog phased arrays). These functionalities can be achieved because the PIXEL antenna can excite a wide variety of radiating surfaces with different shape and size. Some of these radiating surfaces are analyzed next by focusing on the mapping of circular arrangement of pixels, and in some cases rectangular and irregular arrangements in order to improve matching or to tune the resonant frequency.

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Fig. 2. Geometry model and dimensions of an n n PIXEL antenna: (a) perspective view with the reference coordinate system. DC control wires are not shown, (b) top view, and (c) side view (with the DC control wires connected to the biasing via pads on the back side of the substrate).

1) Multimode Operation: Multimode operation allows the PIXEL antenna to have different radiation patterns and polarizations while keeping the same operating frequency. This functionality is of great interest for antenna diversity systems, because in these systems one desires to excite and select amongst a finite set of radiation patterns and polarization states in order to improve the receive signal-to-noise ratio (SNR) of a wireless communication system [8], [17], [18]. As shown in [8], [19], to maximize the benefits produced by these systems, it is desirable that the signals received through the different radiation patterns of the antenna are uncorrelated. Assuming a rich scattering environment, this can be achieved by exciting or capturing the fields with orthogonal radiation patterns [19]. In the PIXEL antenna, the generation of orthogonal radiation patterns (modes) is done by exciting the fundamental and higher order modes of a circular patch (through the mapping of distinct circular arrangements of pixels). For the sequel, we will also refer to a set of orthogonal radiation patterns as modes. As shown in [20], these modes can be excited using circular patches of different radius.

Fig. 3. Basic dimensions of the pixeled structures and control system: (a) perspective view of one of the PIXEL antenna’s corners, and (b) top view. In (c), zoom in of the used MEMS switch with its dimensions (in mm).

Mode reconfiguration is then achieved by adjusting the size of the mapped patch, which in the case of a circular patch is done . by adjusting its radius To achieve this functionality, the PIXEL antenna is designed with the capability to generate a finite set of orthogonal radiation patterns (modes). These modes correspond to the fundahigher-order modes that, for a given mental mode and the surface area, can be generated using a microstrip circular patch. The total number of modes that can be excited depend on the size of the matrix of metallic pixels ( in Fig. 2) and the size of

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the pixels. The and components of the far-field pattern for mode of a circular patch the th transverse magnetic antenna with radius can be described as follows [20]:

(1) (2) where and represent the Bessel funcis the magnitude of the tion of the first kind and order is the free space magnetic ring current for the th mode, is the reference angle which corpropagation constant, and responds to the peak magnetic current . A definition of the mode is given in [21]. As shown in [8], [19], a communication system using reconfigurable antennas capable of exciting orthogonal radiation patterns, could improve its diversity order by -fold, when compared to its non-reconfigurable counterpart, under ideally rich scattering conditions. In reality, however, signals received with different modes are partially correlated and thus this -fold can only be approached. Therefore, the potential benefits of the PIXEL antenna for diversity system is of great significance. 2) Multiband Frequency Tunable Operation: As a frequency tunable antenna, the PIXEL antenna can be used to adapt its operating frequency (while keeping the same radiation pattern and polarization) into different bands to serve distinct wireless applications. Similarly as in multimode operation, frequency tuning is achieved by adjusting the size of the mapped patch, which in the case of a circular patch is done by adjusting its radius [20]. Notice that the tunable frequency range of the PIXEL antenna is limited by the length of the lateral dimensions of the matrix of metallic pixels ( in Fig. 2), because these dimensions directly relate to the area available to map the radiating surfaces. In addition, the PIXEL antenna can also be used in MIMO systems employing frequency diversity techniques, such as in OFDM-MIMO [22]. A -fold increase on the diversity order of such a reconfigurable system could also be attained (when compared to its non-reconfigurable counterpart), where in this case is the number of uncorrelated frequency bands in which the PIXEL antenna can operate. 3) Polarization Agile Operation: With this functionality, the PIXEL antenna can be used to excite a specific radiation pattern with distinct polarizations while maintaining the operating frequency unchanged. In particular, any of the TM modes can be in order to match the polarrotated at desired angular steps ization of linearly polarized incoming waves. This is achieved by rotating the mapped circular patches around the feeding location in the matrix of metallic pixels. Notice that polarization agility is not achieved by changing the antenna feeding point nor by mechanically rotating the antenna, but instead by changing the ON/OFF state of specific sets of MEMS switches. Circular polarization can also be generated with the PIXEL antenna by using conventional techniques and disconnecting strategically located pixels on the mapped patches.

C. Design Guidelines In this section, we describe the procedure that we use to create circular patches (as the one pictured in Fig. 1(f)) through the mapping of circular arrangements of pixels. We provide details on the selection of the number of pixels and their dimensions. The effects on the discrete nature of the PIXEL antenna and of the integrated MEMS switches are also investigated. We begin by revising the design rules of a microstrip circular patch. Two parameters are critical for its design: the radius of the circular patch and the relative position of the coaxial feed with respect to the center of the circular patch. In [23], an expression for the resonant frequency of the TM modes in a circular disk is given. Using this expression, one can predict the radius needed to excite a particular TM mode with an accuracy in the order of 2.5%. Therefore, for a fixed desired frequency mode, the radius of of operation and th resonant the circular patch is found numerically by solving the following equation for [23] (3)

is the th zero In (3), is the thickness of the substrate, is the speed of the derivative of the Bessel function of order of light in free space, and is the relative dielectric constant of the substrate. The relative position of the coaxial feed (with ) is found by numerically solving (4) where the desired input impedance value is set to represent the Bessel function of the first kind and order , and is the total resistance of the antenna given by the sum of the radiation resistance, dielectric, and ohmic losses [23]. The feeding point in the PIXEL antenna is lodirection (in Fig. 2). A visual cated along the radius in the representation of the and parameters is given in Fig. 10(e). We continue by investigating the effect of pixelization of a circular patch when mapped on the PIXEL antenna. For now we assume a hardwired connection between the pixels. To investigate this effect we use a feeding mechanism consisting of a microstrip feedline that couples the fields into the pixeled structure. This allows us to avoid discretization errors in the feeding , and extract only the pixelization effect in the ralocation diating surface. The microstrip feedline is located 0.762 mm above the ground plane. It has a characteristic impedance of 50 with a width of 1.78 mm, and mm (notice that in is not the radial feed location but the distance from this case the edge of the reconfigurable pixeled surface to the end of the microstrip line). Fig. 4(a)–(b) shows the structure used to investigate the pixelization effect on a circular radiating surface. The PCB substrate is RO-TMM3, with , and a 17 m thick copper metallization was used. The investigation is conducted on the mode, with the mapped rawhere mm. To condius being fixed at is fixed and we sweep duct this investigation, the value of the number of pixels per dimension, , from 3 to 17 (in steps

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Fig. 5. Impact of pixelization on the resonant frequency, directivity and radiation efficiency.

Fig. 4. Antenna structure used to investigate the pixelization effect on a circular radiating surface. As shown in (a) and (b), notice that the antenna is fed through a microstrip line. (c) shows the sequence of pixelization steps that we used to investigate this effect. 8 configurations have been simulated, from a pixelization of 3 pixels up to 17 pixels in steps of 2 pixels.

TABLE I SUMMARY OF THE PIXEL DIMENSIONS VS. NUMBER OF PIXELS

of 2), while the lateral dimension of each pixel is determined . Table I gives a sumusing mary of the pixel dimensions vs. number of pixels, and Fig. 4(c) shows the sequence of 8 mapped circular patches with different pixelization values from 3 to 17, used to investigate this effect. All the remaining dimensions, including those of the modeled switches, are the same as those described in Fig. 3, and do not change from one pixelization to another. Fig. 5 shows the pixelization impact on the simulated resonant frequency, directivity and radiation efficiency. All three quantities are affected by pixelization, and all of them increase as the number of pixels increases. All three quantities are compared against the solid case scenario (that is, against a non-pixeled circular patch with radius , whose performance is represented with dashed lines in Fig. 5). We first observed that for the mode, pixelization produces a reduction on the resonant frequency (with respect to the solid circular patch). At a pixelization of 13 pixels, the reduction on the resonant frequency is about 9.85%, because the pixelization increases the effective electrical length of the antenna due to the bendings that the surface currents suffer when passing through the pixels interconnects. As shown in Fig. 5, the larger the number of pixels, the closer the performance of the antenna to the ideal solid configu-

ration, because the surface currents follow a similar path to that in the non-pixeled case when the number of pixels is large. In the limit case, when only 3 pixels are allocated within the diameter, the efficiency is severely degraded. This is due to the fact that the coarse pixelization causes a large part of the surface currents (the ones perpendicular to the feedline) to cancel out. For all three quantities, there exists a pixelization number after which by increasing the number of pixels the performance of the antenna does not change significantly. In our case, this pixelization number is around 7, and thus any number above that should result in an efficient design. Notice that these pixelization trends depend on the radiating surface size and the overall of the matrix of metallic pixels, therefore this dimension estimation should be repeated if different values are used. Similar trends were observed for the other TM modes. A third aspect that we investigate is the effect of the DC biasing vias for the control system. In this case, we use a feeding coaxial mechanism as in the final prototypes. We observe that mode, the inclusion of biasing vias decreases for the its resonant frequency by 10% approximately. This can be explained from the fact that the introduction of vias increases the capacitance to the ground. Therefore, the combined effect of the biasing vias and pixelization produces a reduction on the resomode of about 18–20%. nant frequency for the At this point, the parameters and are compensated ( and ) to account for the biasing vias and pixelization effect. These parameters are adjusted through simulations. We and for typically use modes mapped using patches with 5 or more pixels in diameter. Finally, the compensated parameters are discretized by finding the optimal number of pixels for which the radius and feed point of the resultant mapped patch are as close as possible to and , respectively. The number of pixels for the and feeding location can be obtained by using radius the formula (5)

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where is used either for or is used either with or is the number of pixels per dimension, is the center to center distance between the pixels, and is the lateral dimension of a pixel, as shown in Fig. 3. Notice that in the limit, and , the PIXEL antenna becomes when a continuous patch. The above design methodology has been programmed in MATLAB. This MATLAB code can map any radiating surface on the matrix of pixels with a desired resonant frequency and radiation pattern/polarization, then automatically writes an HFSS script (with the specific set of MEMS switches in the ON and OFF state) and finally launches an HFSS full wave simulation. This code also generates the DC voltage mapping that one needs to apply to each one of the pixels in order to synthesize the desired radiating surface (see Fig. 10(c) and (d)), that is, to produced the specific set of MEMS switches in the ON and OFF state. Sometimes, during the discretization process the error committed with respect to the compensated values may be non-negligible, especially when the mapped patch’s area is small compared to the pixel dimensions. In these cases, the mapped circular patches need to be slightly reshaped to resonate at the desired frequency with a good matching level. The discretization errors are normally small for large mapped patches. The percent of total adjustments associated with the pixelization effect, biasing vias, and discretization errors, with respect to the theoretical values and of a solid circular patch, can be (in %) in the case calculated as is the actual radius while of the radius (notice that is the theoretical value), and (in %) is the actual in the case of the feeding location (notice that feeding location while is the theoretical value). Finally, we also investigate the antenna performance when 3D integrated MEMS switches are used and we compare it with the hardwired case using ideal open and short circuit interconnections. Overall, good isolation on the surface current levels between isolated pixels was observed, in both cases. In fact, the dB [16], isolation of the MEMS switches is in the order of in the 3–7 GHz frequency range. As shown in Fig. 5, in the low similar resonant frequency is obpixelization regime served either with hardwired connections or integrated MEMS , the switch. As the pixelization regime increases resonant frequency when using integrated MEMS switches is about 4.6% lower than in the hardwired case. Also the directivity and radiation efficiency are slightly lower. This is due to the fact that the surface currents follow a slightly electrically longer path, when going through an integrated MEMS switch than when passing through a hardwired interconnection, as a result of the larger capacitance (between the top and bottom electrodes) existing in the real MEMS switch. This effect becomes more apparent as the number of pixel interconnections increases. Finally, notice the impact of the biasing lines on the antenna performance was not taken into account in Fig. 5. III. MEMS INTEGRATED PIXEL ANTENNA We now present a functional MEMS integrated PIXEL antenna to demonstrate its principle and modes of operation (multimode, multifrequency, and polarization agile), its fabrication

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Fig. 6. Image composition with: (a) picture of the fabricated 9 9 PIXEL antenna with integrated MEMS switches, (b) zoom of the matrix of pixels, (c) integrated 3D MEMS switch, and (d) pixel with its biasing via in the center.

feasibility, and to evaluate the impact of real MEMS switches during operation. This antenna prototype uses a 9 9 matrix of metallic pixels and was also fabricated on a RO-TMM3 PCB substrate with dimensions 50.8 50.8 mm and thickness of 1.5875 mm. The lateral dimension of the pixels is equal and mm. fixed to Fig. 6(a) shows a picture of the fabricated 9 9 prototype, while in (c) and (d) a magnification of the MEMS switch and a pixel with its DC biasing via for control in the center are shown. The dimensions for this prototype are given in Fig. 3 with mm. For the current design, we have used a 0.5 m thick gold metallization, which we modeled with a sheet resistance of . 0.0488 When used in multimode operation, the 9 9 PIXEL antenna modes which correspond to the can excite a set of modes with , and with . The lateral length of the matrix of metallic pixels, , was chosen slightly larger than the diameter required to map the two modes (thus ) at 5.15 GHz. This is done in order to be able to tune this mode over a specific range of frequencies during multifrequency operation. When the 9 9 PIXEL antenna prototype is adopted in multifrequency operation, it can be used to excite the mode over a wide range of frequencies from 4.5 GHz up to 7 GHz. This frequency band is chosen not based on the interest to target specific wireless applications, but on the limitations of the fabrication process within our clean-room. Designing the antenna for a lower frequency range would have required using a larger substrate (to accommodate larger patches sizes),

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which was unavailable in our case due to the size limit of the boards in which a lithographic process can be employed. Also, the MEMS switches used in this work are of capacitive type which exhibit best performance at operating frequencies above 3–4 GHz, therefore the concept was illustrated above such frequencies. It is possible to use metal-contact type MEMS switches [24] in a very similar configuration as the one presented here to cover lower frequencies of operation. However, it has been shown that contact MEMS have problems as well such as stiction. Designing the antenna for a higher frequency range is also possible. However, in order to attain a high degree of reconfigurability the pixel sizes may need to be smaller than the current ones and thus their size may become comparable to that of the MEMS switches, thus creating a limitation. The antenna was controlled using a set of 80 wires (for DC control) soldered to each one of the biasing via pads in the back side of the antenna. Then, these wires were grouped and connected to either ground or 30 V on a DC voltage supply. Notice the fact that the MEMS switches are electrostatically actuated and that the pixels are DC isolated from each other. A. Structure of MEMS Switches The MEMS switches used are capacitive whose structure is based on a double support suspended membrane over a microstrip line, and was developed during previous research studies within our group [16], [25], [26]. Its basic dimensions are given in Fig. 3(c). The height of the metal bridge is 5 m, the membrane thickness is 0.5 m, and the membrane holes have a diameter of 10 m. The membrane is made of gold alloy while the two posts over which the membrane is suspended are made of nickel. Notice that a 0.2 m thick silicon nitride (SiN) layer is deposited on the bottom electrode to guarantee DC isolation when the membrane is down. For this switch, the capacitance in the ON state is around 4 pF and in the OFF state around 0.05 pF. These values are consistent with an air gap of 5 between membrane and signal line. Further information on the scattering matrix and equivalent circuit of the switch can be found in [26]. In the proposed antenna, each pixel (with exception of the sides and corner ones) has four switches connecting the four sides of the pixel with its neighbor. This results in a total number . Notice of MEMS switches given by that we would normally have switches if all the pixels were switchable. The feeding pixel has no switches and is always connected to the adjacent pixels, therefore we must subtract 4 from the calculated total. The actuation voltage of the used MEMS switches is 30 V and in practical systems can be implemented by voltage pump circuits such as the ones designed on CMOS that utilize a supply voltage of 3.3 V [27]–[29]. Notice that the MEMS switches are monolithically integrated within the antenna on a PCB substrate. Therefore, the total cost for adding MEMS switching technology is independent from the number of switches, because the price on this lithographic process is mainly determined by the number of masks being used and the minimum featured dimension. On the other hand, MEMS still have some disadvantages compared to solid state

Fig. 7. Cross section of the main parts of the DC biasing control system, including the pixels and MEMS switch structures: (a) view of the control system from the biasing pad (A) to the upper electrode, that is, the membrane (following path (i)), and (b) view from the biasing pad (B) to the bottom electrode of the MEMS switch (following path (ii)).

counterparts, in particular in the area of reliability and packaging. Currently, the estimated yield for the MEMS switch developed in our group is around 80%, when using our fabrication facility. Therefore, it took a significant amount of work to get a functional PIXEL antenna prototype capable of demonstrating the different modes of operation. Reliability aspects are not covered in this paper and the reader can refer to [16], [30] for details. B. Control System The bias mechanism for the control of the PIXEL antenna is shown in Fig. 7. The MEMS switches around each pixel are actuated ON and OFF depending on the DC voltage that is present between the two adjacent pixels to which it is attached to. Therefore, to interconnect two metallic pixels, the voltage difference between them has to be around 30 V. That is, the switch shown in Fig. 7 will be ON when 0 V and 30 V are applied to points (A) and (B) respectively, or vice-versa, while the switch will be OFF when both points (A) and (B) are at the same potential (0 V or 30 V). The DC connectivity is done through a system of biasing vias, that connects the bottom and upper electrodes

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of each MEMS switch through the metallic pixels and to the back side of the substrate through the biasing vias, as shown in Figs. 2(c), 3 and 7. A wire is soldered on the via pad on the backside of the substrate, which connects to the DC voltage supply. Path (i) in Fig. 7(a) and path (ii) in Fig. 7(b) represent the DC paths that are followed to bring a particular voltage level to the upper and bottom electrode of the MEMS switch, respectively, from the biasing pads (A) and (B). The requirements on the bias network for control are that it needs to be RF isolated from the matrix of pixels, in order to not alter the resonance and radiation pattern characteristics of the antenna, while being controlled. Therefore, notice that the metallic pixels and the biasing vias are connected through RF resistive high-impedance lines made of Ni-Chrome (NiCr) alloy. The combined effect of impedance mismatch (due to the small line width) and the high resistivity of the material allow us to effectively produce an RF choke within the frequency range of interest, which isolates the control system from the microstrip-type surface radiator. The . sheet resistance of the used NiCr alloy is around 12 K On the other hand, notice that each pixel is DC isolated from the other surrounding pixels because of the SiN layer, therefore each pixel can be biased independently. C. Fabrication Process The fabrication process starts by creating the feeding hole for the coaxial cable and the via holes into the PCB using a precision drill. Due to its complicated features and the small size of the holes, the drilling and electroplating of the vias were carried out by Cal-Tronics, a local company in Irvine, California. After the drilling, the fabrication process is similar but not the same as in [16]. The process steps are graphically shown in Fig. 8. Rogers’s standard TMM3 composite is used as substrate for our antenna. The substrate is made of hydrocarbon ceramic and has good rigidity which helps during the polishing process. The fabrication process continues by patterning the top and bottom sides of the copper cladding (Fig. 8(a)). The next steps consist of mechanical polishing of the substrate to obtain mirror like surfaces on the PCB with roughness below 50 nm. The polishing was conducted manually using sandpaper of different grit sizes. After the mechanical polishing, nickel posts are electroplated. Next, the fabrication process continues with the deposition and patterning of the Ni-Chrome resistive lines (Fig. 8(b)). Following this step is the deposition and patterning of titanium (Ti) and 0.5 m thick gold (Au) layer to form the matrix of metallic pixels of the PIXEL antenna (Fig. 8(c)). Titanium is used to adhere gold on a PCB substrate. In Fig. 8(d) a thin layer of SiN dielectric film is then deposited using highdensity inductively coupled plasma chemical vapor deposition (HDICP-CVD); the deposition is carried out at low tempera, which meets the temperature requirements of tures PCB (below 200 C). The dielectric film is then patterned with a thick photoresist layer, followed by reactive ion etching (RIE). Later, as shown in Fig. 8(e), a thick photoresist (5–6 m) is used to create the mold for nickel plating. After the nickel is plated a sacrificial photoresist layer is formed before deposition of the gold membrane using spin coating (Fig. 8(e)). Next, a layer of 5 m-thick photoresist is spincoated and compressive molding planarization (COMP) technique is employed to planarize the

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Fig. 8. Fabrication steps (a)–(g) of the 9 9 PIXEL antenna prototype with monolithically integrated RF-MEMS switches: (a) patterning of copper cladding, (b) deposition of Ni-Cr lines, (c) deposition and patterning of titanium and gold, (d) deposition and patterning of SiN dielectric, (e) nickel plating, (f) gold deposition and patterning, and (g) releasing of the sacrificial layer.

photoresist, and then the thin layer of photoresist left above the nickel post is photo-lithographically patterned. Next, a layer of 0.5 m-thick gold is deposited on the planarized photoresist, and later patterned (Fig. 8(f)). The last step is to release the switch membrane (Fig. 8(g)). Circular holes on the membrane are used to facilitate the releasing of the sacrificial layer under the MEMS switch membrane. An acetone based solution is used to perform the final release of the membrane. To minimize stiction, boiling methanol is applied right after acetone releasing. IV. SIMULATIONS AND MEASUREMENTS OF THE MEMS INTEGRATED PIXEL ANTENNA A. Impedance Matching Response for the different modes of The scattering parameters operation of the 9 9 antenna prototype are shown in Fig. 9. All the simulations have been conducted using HFSS(TM) [31]. Plots (a) and (b) show the simulated and measured reflection coefficients, respectively, of the antenna when used in multimode operation. Notice that the two modes of the antenna resonate at a center frequency near 5.15 GHz. Measured data shows good

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Fig. 10. Sequence of mapped configurations for the 9 9 PIXEL antenna prototype when used in multimode operation (a), (b) and multifrequency operation (e)–(h). Notice that (a) and (b), correspond to the mapped patches exciting the TM modes with  = 0 and  = 90 , respectively. In (a), (b), (e)–(h) black pixels denote disconnected pixels, and gray pixels represent connected ones. The white pixels show the location where the feed coaxial cable is connected. Sub-Figs. (c), (d) are the DC voltage maps used to generate the structures in (a), (b), respectively.

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Fig. 9. Scattering parameters (S ) of the 9 9 PIXEL antenna prototype: (a) simulation data when used in multimode operation, (b) measured data when used in multimode operation, (c) simulation data when used in multifrequency operation.

agreement with the simulated reflection coefficient characteriswith ) is off tics although one of the modes (the by 0.4 GHz with respect to the design frequency. We were able to confirm by visual inspection under the microscope that the membrane of one of the MEMS switches had peeled off, thus mode affecting the resonance of this mode. As for the with , the MEMS were also visually inspected and found to be working properly. Nevertheless, this shows the importance to develop a reliable protective package, even though this is outside the scope of this paper. The sequence of mapped patches in Fig. 10 from (a) to (b), correspond to those exciting modes with and , respectively. the Fig. 9(c) presents the simulated response of the antenna when mode at different used in multifrequency operation (i.e., frequencies). It shows that the mode can be excited at any frequency between 4.5 GHz and 7 GHz. The sequence of curves, from lower to higher resonant frequency corresponds to the sequence of mapped patches in Fig. 10 from (e) to (h), respectively. Notice that as the mapped patches become smaller, mode. The the higher the resonant frequency of the bandwidth of most resonances, defined at a return loss level of dB, ranges from 1.5–4% depending on the matching level, which is a typical value for the mode of a circular patch antenna [21]. In Fig. 10(a), (b), (e)–(h) black pixels denote disconnected pixels, and gray pixels represent connected ones, thus forming

a mapped patch. The white pixels show the location where the feed coaxial cable is connected. Sub-Figs. (c) and (d) represent the DC voltage maps used to excite the structures in (a) and (b), respectively, where black and white denote 0 V and 30 V, respectively. These DC voltage maps provide the required voltage difference among pixels to interconnect a specific set of pixels thus forming a mapped patch. The voltage maps are applied to the antenna through the DC biasing vias and wires described in Section III.B. Notice that some of the mapped circular patches shown in Fig. 10 are not perfectly circular. This has to do with the fact that, in small mapped patches, the discretization errors of the radius and feeding location are non negligible. For example, for the patch in Fig. 10(a), the calculated value for the radius is 6.6 mm and the actual mapped radius once discretized is 5.43 mm. As a result, the mapped patches are either not perfectly matched or resonate at slightly different frequencies. In these cases, the resonant frequency and matching can be adjusted by slightly reshaping the mapped patches (by disconnecting or connecting some of the pixels around the mapped area). As we can see on Fig. 10(e), as the mapped patches become larger, the discretization becomes negligible and thus there is no need to reshape the patches. Finally, discretization errors can only be corrected through carefully reshaping the mapped patches. B. Radiation Parameters Fig. 11 shows the simulated and components of the multimode far-field radiation patterns of the 9 9 antenna prototype, on a flattened 3D representation. In this representation, the radial axis of the polar plot represents the angle spanning from 0 to , while the polar angle represents the angle spanning from 0 to . The field components for the modes with with and are shown. Notice order modes with and are northat the mally referred in the literature as the fundamental modes of a

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Fig. 11. Simulated  and  components of the far-field radiation pattern on modes with order n = 1 with a flattened 3D representation, for the TM  = 0 and  = 90 of the 9 9 PIXEL antenna prototype. In the flattened 3D radiation pattern  2 [0;  ] and  2 [0; 2 ].

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Fig. 12. Simulated (gray) and measured (black)  and  components of the far-field radiation pattern, in the azimuth (x-y ) plane (a), (c) and in the  = 0 elevation plane or x-z plane (b), (d), associated with TM modes with order n = 1 with  = 0 (top row) and with  = 90 (bottom row) of the 9 2 9 PIXEL antenna prototype.

circular patch with vertical and horizontal polarization, respectively. The simulated radiation patterns also agree well with the ones predicted from (1) and (2) in Section II.B.1. These modes, in this flattened representation, resemble the modes that can be excited inside a circular waveguide, and thus it is a useful representation to identify the distinct radiation characteristics among them. As commented in Section II.B.1, antenna diversity systems perform optimally when the radiation patterns of the antennas are orthogonal. Following the approach in [32], it was verified that in rich scattering conditions, the radiation pattern of the two modes are practically orthogonal to each other.

Fig. 13. Experimental setup used in the measurement of the PIXEL antenna’s radiation patterns inside the anechoic chamber.

Fig. 12 shows the simulated (gray) and measured (black) and components of the far-field radiation pattern, in the azelevaimuth ( - ) plane (Fig. 12(a), (c)) and in the tion plane or - plane (Fig. 12(b), (d)), associated with modes with order with (top row) and with (bottom row) of the 9 9 antenna prototype. Good agreement exists between the simulated and measured patterns. It is worthwhile to notice that a polarization isolation level (between the copolar to crosspolar components) of more than 15 dB exists in the - plane patterns over a wide range of angles around the maximum radiation, which is typically expected mode on a microstrip patch antenna. Notice from the mode has been reconfigured that the polarization of the at angular steps of , although a much smaller step is possible. Fig. 13 shows a block diagram of the experimental setup used in the measurement of the PIXEL antenna’s radiation patterns inside the anechoic chamber. Notice that the MEMS switches are actuated as described in Section III.B, using the DC voltage maps given in Fig. 10(c) and (d). Table II summarizes the resonant frequency values, gain, directivity and radiation efficiency for each one of the simulated and measured modes of the 9 9 prototype (in multimode operation). Overall, simulated and measured values agree well, but some small differences exist on the absolute gain between simulated and measured data, of about 0–1 dB, which may be due to soldering imperfections of the coaxial cable on the feeding pixel. In this case, the measured gain is around 3.2 dBi for both modes, while the estimated radiation efficiency is of the order of 50%. These values are relatively low for a patch antenna, but the explanation can be found on the NiCr lines which absorb part of the energy that otherwise would be radiated. The fact that the metal thickness (0.5 m) is a fraction of the skin depth ( m at 5.15 GHz) also contributes to it but in a much smaller manner. Further investigation in novel RF choke materials is needed to improve the antenna performance. Table II

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TABLE II SUMMARY OF RADIOELECTRIC PROPERTIES AND GEOMETRICAL CHARACTERISTICS OF THE 9 9 PIXEL ANTENNA PROTOTYPE IN MULTIMODE OPERATION

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to M.-J. Lee and Prof. G. P. Li for their collaboration during the early stages of this work, to Profs. J. Romeu, L. Jofre, N. G. Alexopoulos, and H. Jafarkhani for their comments and constructive advice, and J. Rodriguez De Luis for his help in setting up the anechoic chamber. Finally, they would like to express their appreciation to the Ansoft, and Rogers Corporations. REFERENCES

summarizes the geometrical characteristics of the antennas ( ). Notice that for the mode, the is about 18.1% smaller than the theomapped radius predicted for a solid circular patch, in order to retical value compensate for the effects of pixelization, and biasing vias, and discretization errors (in agreement with the values estimated in Section II.C). The effect of these factors is more severe on the feeding location, in which the adjustment is about 25%. The simulated and measured data presented in this section demonstrate the multiple functionalities of the proposed PIXEL antenna (multimode, multifrequency and polarization agile operation), and its feasibility using MEMS switches. V. CONCLUSION In this work we have demonstrated the design of a multifunctional MEMS-reconfigurable pixeled antenna intended to operate within the 4–7 GHz frequency band, that we define as the PIXEL antenna. The PIXEL antenna is a highly reconfigmaurable planar surface radiator composed of an trix of metallic pixels interconnected through MEMS switches. We have demonstrated that the proposed antenna can function as a multimode antenna for pattern/polarization diversity systems, as a frequency tunable antenna to be used in multiband wireless communication systems, and as a polarization agile antenna to be used in polarization-sensitive applications such as in phased array architectures. A functional prototype using integrated MEMS switches was fabricated on a printed circuit board substrate to demonstrate the principles of operation, and also to show its fabrication feasibility and the impact of real MEMS switches during operation. Simulated and measured data on the electrical and radiation characteristics of the antenna are presented. Good agreement was found between the simulated and measured data. Nevertheless, the impact of the NiCr lines on the antenna radiation efficiency and the necessity of a reliable package are important aspects of the design that will need to be addressed in a final SDA solution. ACKNOWLEDGMENT The authors wish to express their appreciation to P. Balsells, and Prof. R. Rangel for their support. They extend their thanks

[1] E. Brown, “RF-MEMS switches for reconfigurable integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 11, pp. 1868–1880, Nov. 1998. [2] G. H. Huff and J. T. Bernhard, “Integration of packaged RF MEMS switches with radiation pattern reconfigurable square spiral microstrip antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 464–469, Feb. 2006. [3] D. Anagnostou, G. Zheng, M. Chryssomallis, J. Lyke, G. Ponchak, J. Papapolymerou, and C. Christodoulou, “Design, fabrication, and measurements of an RF-MEMS-based self-similar reconfigurable antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 422–432, Feb. 2006. [4] S. Nikolaou, R. Bairavasubramanian, C. Lugo, I. Carrasquillo, D. Thompson, G. Ponchak, J. Papapolymerou, and M. Tentzeris, “Pattern and frequency reconfigurable annular slot antenna using PIN diodes,” IEEE Trans. Antennas Propag., vol. 54, pp. 439–448, Feb. 2006. [5] D. Ressiguier, J. Costantine, Y. Tawk, and C. Christodoulou, “A reconfigurable multi-band microstrip antenna based on open ended microstrip lines,” in Proc. 3rd Eur. Conf. on Antennas and Propagation, EuCAP, Mar. 2009, pp. 792–795. [6] F. Yang and Y. Rahmat-Samii, “Switchable dual-band circularly polarised patch antenna with single feed,” Electron. Lett., vol. 37, no. 16, pp. 1002–1003, Aug. 2001. [7] H. Aissat, L. Cirio, M. Grzeskowiak, J.-M. Laheurte, and O. Picon, “Reconfigurable circularly polarized antenna for short-range communication systems,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 2856–2863, June 2006. [8] A. Grau, J. Romeu, M.-J. Lee, S. Blanch, L. Jofre, and F. De Flaviis, “A dual-linearly-polarized MEMS-reconfigurable antenna for narrowband MIMO communication systems,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 4–17, Jan. 2010. [9] J. Sor, C.-C. Chang, Y. Qian, and T. Itoh, “A reconfigurable leakywave/ patch microstrip aperture for phased-array applications,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 8, pp. 1877–1884, Aug. 2002. [10] Y. Qian, B. Chang, M. Chang, and T. Itoh, “Reconfigurable leakymode/multifunction patch antenna structure,” Electron. Lett., vol. 35, no. 2, pp. 104–105, Jan. 1999. [11] B. Cetiner, H. Jafarkhani, J.-Y. Qian, H. J. Yoo, A. Grau, and F. D. Flaviis, “Multifunctional reconfigurable MEMS integrated antennas for adaptive MIMO systems,” IEEE Commun. Mag., vol. 42, no. 12, pp. 62–70, Dec. 2004. [12] L. Pringle, P. Harms, S. Blalock, G. Kiesel, E. Kuster, P. Friederich, R. Prado, J. Morris, and G. Smith, “A reconfigurable aperture antenna based on switched links between electrically small metallic patches,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1434–1445, Jun. 2004. [13] G. Knowles and E. Hughes, “Electromagnetic energy coupling mechanism with matrix architecture control” U.S. Patent 7151506, Dec. 2006 [Online]. Available: http://www.freepatentsonline.com/7151506.html [14] W. H. Weedon, W. J. Payne, G. M. Rebeiz, J. S. Herd, and M. Champion, “MEMS-switched reconfigurable multi-band antenna: Design and modeling,” in Proc. Allerton Antenna Applications Symp., 1999, vol. 1, pp. 203–231. [15] G. M. Rebeiz, RF MEMS: Theory, Design, and Technology. New York: Wiley, 2003. [16] H. P. Chang, J. Qian, B. A. Cetiner, F. D. Flaviis, M. Bachman, and G. P. Li, “Design and process considerations for fabricating RF MEMS switches on printed circuit boards,” J. Microelectromechan. Syst., vol. 14, no. 6, pp. 1311–1322, Dec. 2005. [17] C. Waldschmidt and W. Wiesbeck, “Compact wide-band multimode antennas for MIMO and diversity,” IEEE Trans. Antennas Propag., vol. 52, no. 8, pp. 1963–1969, Aug. 2004. [18] A. Forenza and J. R. Heath, “Benefit of pattern diversity via two-element array of circular patch antennas in indoor clustered MIMO channels,” IEEE Trans. Commun., vol. 54, no. 5, pp. 943–954, May 2006.

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[19] A. Grau, H. Jafarkhani, and F. De Flaviis, “A reconfigurable multipleinput multiple-output communication system,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pt. I, pp. 1719–1733, May 2008. [20] R. G. Vaughan, “Two-port higher mode circular microstrip antennas,” IEEE Trans. Antennas Propag., vol. 36, no. 3, pp. 309–321, Mar. 1988. [21] C. A. Balanis, Antenna Theory, Analysis and Design, 2nd ed. New York: Wiley, 1997. [22] H. Bolcskei and A. Paulraj, “Space-frequency coded broadband OFDM systems,” presented at the IEEE Wireless Communications and Networking Conf., Chicago, IL, Sep. 23–28, 2000. [23] A. Derneryd, “Analysis of the microstrip disk antenna element,” IEEE Trans. Antennas Propag., vol. 27, no. 5, pp. 660–664, Sep. 1979. [24] G. Rebeiz and J. Muldavin, “RF MEMS switches and switch circuits,” IEEE Microw. Mag., vol. 2, no. 4, pp. 59–71, Dec. 2001. [25] C. W. Jung and F. De Flaviis, “RF-MEMS capacitive series switches of CPW and MSL configurations for reconfigurable antenna application,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 3-8, 2005, vol. 2A, pp. 425–428. [26] H. P. Chang, J. Qian, B. Cetiner, F. De Flaviis, M. Bachman, and G. Li, “RF MEMS switches fabricated on microwave-laminate printed circuit boards,” IEEE Electron Device Lett., vol. 24, no. 4, pp. 227–229, 2003. [27] D. Hong and M. El-Gamal, “Low operating voltage and short settling time CMOS charge pump for MEMS applications,” in Proc. Int. Symp. on Circuits and Systems, ISCAS’03, May 2003, vol. 5, pp. V-281–V284, vol. 5. [28] M. Innocent, P. Wambacq, S. Donnay, W. Sansen, and H. De Man, “A linear high voltage charge pump for MEMs applications in 0.18 m CMOS technology,” in Proc. 29th Eur. Solid-State Circuits Conf., ESSCIRC’03, Sep. 2003, pp. 457–460. [29] M. Zhang and N. Llaser, “On-chip high voltage generation with standard process for MEMS,” in Proc. 14th IEEE Int. Conf. on Electronics, Circuits and Systems, ICECS’07, Dec. 2007, pp. 18–21. [30] C. Goldsmith, J. Ehmke, A. Malczewski, B. Pillans, S. Eshelman, Z. Yao, J. Brank, and M. Eberly, “Lifetime characterization of capacitive RF MEMS switches,” in Proc. IEEE MTT-S Int. Microwave Symp. Digest, May 20–24, 2001, vol. 1, pp. 227–230. [31] ANSYS [Online]. Available: http://www.ansoft.com [32] D. Piazza, N. Kirsch, A. Forenza, R. Heath, and K. Dandekar, “Design and evaluation of a reconfigurable antenna array for MIMO systems,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 869–881, Mar. 2008.

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

Franco De Flaviis (SM’07) was born in Teramo, Italy, in 1963. He received the Laurea degree in electronics engineering from the University of Ancona, Italy, in 1990, and the M.S. degree and Ph.D. degree in electrical engineering from the University of California at Los Angeles, in 1994 and 1997, respectively. In 1991, he was an Engineer at Alcatel performing research in the area of microwave mixer design. In 1992, he was a Visiting Researcher at the University of California at Los Angeles, working on low intermodulation mixers. Currently, he is a Professor in the Department of Electrical and Computer Engineering, University of California Irvine. His research interests are in the field of computer-aided electromagnetics for high-speed digital circuits and antennas, and microelectromechanical systems (MEMS) for RF applications fabricated on unconventional substrates such as printed circuit board and microwave laminates.

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Design, Simulation, Fabrication and Testing of Flexible Bow-Tie Antennas Ahmet Cemal Durgun, Student Member, IEEE, Constantine A. Balanis, Life Fellow, IEEE, Craig R. Birtcher, and David R. Allee, Member, IEEE

Abstract—Design, simulation, fabrication and measurement of two different novel flexible bow-tie antennas, a conventional and a modified bow-tie antenna with reduced metallization, are reported in this paper. The antennas are mounted on a flexible substrate fabricated at the Flexible Display Center (FDC) of Arizona State University (ASU). The substrate is heat stabilized polyethylene naphthalate (PEN) which allows the antennas to be flexible. The antennas are fed by a microstrip-to-coplanar feed network balun. The reduction of the metallization is based on the observation that the majority of the current density is confined towards the edges of the regular bow-tie antenna. Hence, the centers of the triangular parts of the conventional bow-tie antenna are removed without compromising significantly its performance. The return losses and radiation patterns of the antennas are simulated with HFSS and the results are compared with measurements, for bow-tie elements mounted on flat and curved surfaces. The comparisons show that there is an excellent agreement between the simulations and measurements for both cases. Furthermore, the radiation performance of the modified bow-tie antenna is verified, by simulations and measurements, to be very close to the conventional bow-tie. Index Terms—Bow-tie antenna, broadband antenna, flexible antenna, heat stabilized PEN, microstrip-to-coplanar feed network balun.

I. INTRODUCTION HE Flexible Display Center (FDC) at Arizona State University (ASU) was founded in 2004 as a partnership between academia, industry, and government to collaborate on the development of a new generation of innovative displays and electronic circuits that are flexible, lightweight, low power, and rugged [1]. Due to the increasing need for flexible and lightweight electronic systems, FDC aims to develop materials and structural platforms that allow flexible backplane electronics to be integrated with display components that are economical for mass-production [2]. Processing of thin film transistors (TFT) at low temperatures that are compatible with flexible plastic substrates opened a new path for flexible circuitry. For instance, digital logic is readily

T

Manuscript received February 15, 2011; revised April 26, 2011; accepted June 06, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. This work was supported by the ASU Advanced Helicopter Electromagnetics (AHE) Program and ASU Flexible Display Center (FDC) under US Army agreement W911NF-04-2-0005. The authors are with the School of Electrical, Computer and Energy Engineering, Arizona State University (ASU), Tempe, AZ 85287 USA (e-mail: [email protected], [email protected], [email protected], [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165511

achieved as demonstrated by the programmable logic array implementing the control logic for a microcontroller [3]. Moreover, volatile and non-volatile digital memories and integrated display source drivers [4] have been successfully fabricated as well as a variety of digital building blocks such as counters, flip flops, and a standard cell library. In addition, in order to achieve complex and first time correct circuits, standard integrated circuit industry design tools and practices are being applied to flexible circuit design [5]. Flexible electronics also promises new areas of applications including light weight, rugged, large area sensing arrays to detect x-rays, radioactive particles (neutrons), and bio-chem agents. Large area arrays might coat the wings of aircraft detecting air pressure/flow for enhanced performance or detecting minute crack formation for prognostics. Substantial cost savings is possible if large equipment is only removed from service when service is needed as opposed to regular, time consuming inspections. Other potential applications include “smart” medical bandages that monitor the healing of a wound or a medical triage patch for remote monitoring of a patient’s vital signs. A natural extension of this technology to the design, fabrication and testing of conformal antennas is of great interest. Currently, the FDC is focusing on the incorporation of antenna structures which can function cooperatively with the other flexible integrated circuit elements. Flexible antennas, as a part of flexible electronic circuits, may have a wide spectrum of applications in wireless communication which can allow antennas to be integrated with the human torso. Design of flexible linear wire dipole antennas with liquid metal parts [6] and inkjet-printed antennas [7] have been reported in the literature. The antennas in [6] are reversibly deformable, and they can be mechanically tuned by stretching and releasing them. The inkjet-printed antennas in [7] are part of a conformal RFID module. Design of a flexible bow-tie antenna, which has a wider bandwidth than linear wire dipoles [8], has also been reported by the authors of this paper [9], [10]. The radiation characteristics of this flexible bow-tie has also been discussed in [11], when the antenna is bent in the form of a cylinder. Moreover, it has been verified that most of the metallization of the bow-tie antennas can be removed without affecting the radiation performance significantly [12]. In this paper, the work performed on the conventional and modified bow-tie antennas has been improved, and it is compared to what was reported in [10]–[12]. In addition to the radiation characteristics of the antennas, the effects of the feeding structure and the conductor losses on the radiation performance

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Fig. 1. Cross section schematic of the Flexible Display Center low temperature a-Si:H TFT process. The gate dielectric is silicon nitride. There are three metallic layers.

of the antennas are discussed. The impact of the curvature on the radiation performance is also examined. The organization of the paper is as follows. In the second section, an extensive discussion on the basic properties of the FDC substrate is outlined together with the fabrication process. This is followed, in the third section, by the design procedure of the antennas and their feed network. In the fourth section, return loss, amplitude radiation pattern, and gain simulations and measurements are reported. The antennas are modeled and simulated with Ansys HFSS [13], and the simulations are compared with measurements performed in the ASU Electro Magnetic Anechoic Chamber (EMAC). The differences between the radiation performance of the antennas when they are mounted on flat and curved surfaces are also examined in this section. Finally, concluding remarks are summarized. II. THE FLEXIBLE SUBSTRATE Due to the compatibility issues, the fabrication process of the antennas is almost the same as the processing of TFTs. Therefore, a discussion on the fabrication of TFTs would also disclose the main points of antenna fabrication. On the other hand, the details of the TFT fabrication is out of the scope of this paper. Hence, only the guidelines will be summarized here. The display technology of FDC is mainly based on amorphous silicon (a-Si:H) TFT that can be fabricated on plastic substrates. Traditionally, displays have been fabricated on glass with high temperature deposition processes, which is a mature technology. However, glass is inherently fragile and heavy making it unsuitable for portable, field applications. In the last few years, there have been substantial development efforts in reducing the a-Si:H TFT processing temperatures to be compatible with a plastic substrate (heat stabilized polyethylene naphthalate or PEN) and for handling flexible substrates in standard processing equipment. The TFTs are processed at 180 C on flexible substrates such as heat stabilized PEN. The gate metal and dielectric are molybdenum and silicon nitride, respectively. Source/drain metal is sputtered on as an N amorphous silicon-aluminum bilayer. Metallization of indium tin oxide and molybdenum (ITO) is then applied. Fig. 1 illustrates a cross section schematic of the low temperature a-Si:H TFT process. Since it is expensive and time consuming to design a new fabrication process, particularly for the antennas, they have to be processed simultaneously and fabricated on the same wafer with the TFTs and other circuitry. Therefore, the materials used in the fabrication of antennas are the same with the ones used for TFT fabrication. However, some of the steps related to TFTs are omitted for the antennas. As mentioned previously, the substrate

Fig. 2. Simplified model for the flexible substrate which best approximates the electrical properties of the actual substrate.

is a thin plastic (heat stabilized PEN), allowing the antenna to be flexible. The substrate is covered with a very thin silicon nitride layer which is the gate dielectric [14]. The conducting material used for the feed network, balun and the antenna element is aluminum. Fig. 2 shows a simplified model for the flexible substrate which best approximates the electrical properties of the actual substrate by using the materials in the HFSS library. III. ANTENNA DESIGN Antenna design using flexible substrates is a new research topic for the ASU FDC. Hence a broadband element was selected to initiate this research. A bow-tie antenna [8] was selected as a first design because of its basic geometry, broadband characteristics, and variety of applications when compared to a linear wire and a printed dipole. Furthermore, bow-tie antennas are expected to be more directive than conventional dipole antennas because of the larger radiating area [15]. They are also used in size reduction applications of patch antennas to achieve lower operating frequencies without increasing the overall patch area [16]. Printed bow-tie antenna designs, because of their attractive characteristics, have been examined by others [15]–[24]. Those in the literature are basically of three types: • Microstrip patch ([16], [17]); • Coplanar ([15], [18], [20]); • Double-sided ([21]–[24]). From these types of antennas, the coplanar bow-ties require a balanced feed network, so that they can be fed by a microstrip line or a coplanar waveguide with the use of a balun. Although they require an additional balun, the authors selected the coplanar bow-tie antenna because it is more appropriate for the interest and fabrication process of FDC. Coplanar waveguide (CPW)-to-coupled strip line baluns include air-bridges at the discontinuities of the balun to suppress the non-CPW modes and balance the lines [20], [25], [26]. These air-bridges may hinder the flexibility of the antennas. Besides, the connection of the air-bridges and aluminum traces of the antennas, which cannot be soldered, could be problematic. Therefore, the authors selected to feed the bow-tie with a microstrip line, although other feeding designs may be more appropriate for the fabrication process. Two different antennas were designed and fabricated: a conventional and an outlined bow-tie with reduced metallization. The latter one consists of a strip that outlines the conventional bow-tie. Hence, starting from this point, the former and new bow-tie antennas will be denoted as the solid and outline bow-tie antennas, respectively.

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Fig. 3. Detailed design geometry of the entire structure including the solid bow-tie and the balun.

A. Solid Bow-Tie The design procedure of a printed bow-tie antenna is similar to the design of rectangular microstrip patches. There is a set of design equations, which are obtained by modifying the semiempirical design equations for rectangular patches given in [8]. The resonant frequency of a bow-tie patch, for the dominant mode, can be obtained using the equations that follow [16], [17] (1) (2) (3) (4)

Fig. 4. Current density on the bow-tie antenna surface at different frequencies. The most intense surface current is concentrated along the edges of the bow-tie. (a) f : GHz, (b) f : GHz, (c) f : GHz.

=70

=74

=78

(5) In this set of design equations, the thickness, relative and ef, and fective permittivity of the substrate are denoted by , respectively. The other geometrical parameters are defined in Fig. 3. Although these equations were derived for microstrip patch type bow-ties, they were used to obtain an initial design of a coplanar bow-tie antenna. Afterwards, the antenna design was fine-tuned and finalized by numerical simulations. There was a slight difference between the initial and the final values because of the inaccuracy of the design equations and the existence of the microstrip to coplanar feed network balun. B. Outline Bow-Tie The current density on the surface of the solid bow-tie antenna was simulated with HFSS and it was observed, as expected, that most of the surface current was concentrated along the edges of the bow-tie. On the other hand, the magnitude of the current density toward the interior of the triangular parts of the element is very low as illustrated in Fig. 4. This is a result of the repulsion of electrons on the metallic surface which tend to migrate toward the edges. Since the majority of surface current was concentrated toward the edges, the performance of the antenna should not be altered significantly in terms of gain, bandwidth and

Fig. 5. Model of the outline bow-tie antenna. Most of the metallization is removed from the centers of the triangles to speed up the prototyping process.

center frequency if most of the metallization is removed from the centers of the triangles. Indeed, this is a phenomenon which has also been observed in log-periodic antennas of saw-tooth planar designs where the interior metallization of the plates is removed to reduce the weight and wind resistance [8]. Hence, with the guidance of these observations, a new bow-tie antenna was designed which is shown in Fig. 5. In this new design, the antenna consists of a strip of width 0.2 mm that outlines the conventional bow-tie antenna. In some applications, the total surface area of the metallic parts of the antenna can become a major design constraint. Therefore, it may be important to obtain a specified radiation performance by using an antenna with less metallization, as in

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the case of log-periodic antennas, sometimes at the expense of decreased gain. On the other hand, our basic intent to reduce the metallization of the antenna was to speed up the prototyping and fabrication process. The reduction of metal content in the antennas allows more rapid fabrication with serial printing techniques. Although the fabrication of TFTs and circuits with ink jet printers offers low cost and rapid prototyping, being inherently serial, the filling of large structures by this approach is problematic. Hence, reducing the metal content of otherwise solid antennas speeds prototyping while utilizing less metallization. C. Microstrip to Coplanar Feed Network Balun As it is mentioned previously, in order to properly feed the bow-tie antenna, a microstrip-to-coplanar feed network (CPFN) transition is necessary. For this purpose, a microstrip-to-CPFN balun was designed which provides an odd mode in the coupled microstrip line while it suppresses the even modes [27], [28]. This balun introduces a 180 phase difference between the coupled microstrip lines near the center frequency. The length of the phase shifter is a very important parameter for the balun design. The lengths of the two branches of the microstrip line should be adjusted such that their difference is equal to quarter of the guided wavelength at the center frequency [28]. Another critical parameter is the gap between the coplanar strip lines which can be adjusted to optimize the balun performance [27]. The design parameters of the balun were optimized via numerical simulations. The balun consists of four parts. The first part is the 50 microstrip line section which has a line width of 0.32 mm. The second part is the 25 microstrip line with a width of 0.84 mm which acts as an impedance transformer. The third part is a symmetric Tee junction which splits equally the power between the two arms of the phase shifter with minimum loss. Finally, the last part is the phase shifter which introduces a 180 phase difference between the coupled lines. The balun was designed to have an almost constant return and insertion loss characteristics within the entire operational band. The return and insertion losses of the balun are shown in Fig. 6. It is evident that the insertion loss is slightly larger than 3 dB at both of the output ports because of the power division, and the power is almost equally divided between the two branches of the balun. The return loss of the balun remains between 9 and 10 dB over the entire band. Another important parameter related to the balun performance is the phase difference, which is illustrated in Fig. 7, introduced between the two branches of the phase shifter. The center frequency of the balun is 7.48 GHz at which the phase difference is equal to 180 . However, this is subject to change after the loading of the balun with the coupled strip line and the antenna. Although the balun is broadband in terms of power division, this is not the case for the phase difference. The phase difference between the two arms of the balun changes linearly from 203 to 155 within the frequency interval 6.5–8.5 GHz. This rapid change in the phase shift has a significant effect on the overall performance of the radiating element. In fact, these types of baluns are known to be narrow band microwave devices.

Fig. 6. Return and insertion losses of the balun. The balun was designed to have an almost constant return and insertion loss characteristics within the operational band.

Fig. 7. Phase difference between the two output ports of the balun.

Fig. 8. The flexible bow-tie antennas: (a) Solid bow-tie antenna. (b) Outline bow-tie antenna.

The photos of the overall fabricated structures, including the antennas and the balun, are shown in Fig. 8, which are fingerbent to illustrate their flexibility.

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Fig. 9. Feeding structure and the metallic support of the antennas. A copper tape was used as the ground plane of the microstrip line. The antenna was fed by a coaxial line. The connection between the coax and microstrip line was maintained with a triangular brass support structure.

IV. SIMULATIONS AND MEASUREMENTS After the fabrication of the antenna at the FDC facilities, the return losses, radiation patterns, and gains of the antennas were measured in the ASU EMAC. In order to verify the validity of the design, the simulation results were compared with measurements. In addition, measured and simulated return loss, center frequency, bandwidth and gain of the outline bow-tie were compared with those of the solid bow-tie. Initially, measurements and simulations were performed when the antennas were flat. Later, the antennas were flexed in the form of a cylinder, and the impact of flexibility on the radiation performance of the antennas was examined. A. Flat Antennas The antennas were fed with a coaxial line which was connected to the microstrip side of the balun. A copper tape was utilized as the ground plane of the microstrip line. To stabilize the coaxial line-to-microstrip transition, a metallic support structure was soldered to the ground plane as shown in Fig. 9. However, it was experimentally observed, and later numerically verified, that the support structure has a significant effect on the center frequency of the antenna. This issue will be discussed in detail in subsequent sections of the paper. In this case, due to the imperfections of the soldering, there was a small gap between the ground plane and the support structure which introduced some capacitance to the overall system. Consequently, this extra capacitance resulted in a frequency shift between the simulated and measured return loss. Initially, the comparison of the measured and simulated return loss was performed without taking into account the metallic support structure and conduction losses of the antenna. This resulted in differences of the return loss characteristics of the measured and simulated data. However, when the support structure and surface impedance of the antenna were taken into account, a very good agreement was obtained between simulations and measurements, as illustrated in Fig. 10. The center frequency of the solid and outline bow-tie are 7.66 and 7.40 GHz, respectively, where the return loss is maximum. It is worth to note that the bandwidth of the solid bow-tie, return loss level, was with respect to 15 dB decreased from 15% to 8.75% after the inclusion of the balun. This is due to the rapid change of the phase shift, introduced by the balun, with respect to frequency. This observation verifies once more that the balun is the critical device of the design in

Fig. 10. Comparison of simulated and measured return loss of the: (a) Solid bow-tie. (b) Outline bow-tie.

terms of the bandwidth, and it is the balun which determines the bandwidth of the overall design. It is also important to mention the effects of the ground plane to the return loss of the antennas. It was experimentally and numerically verified that the length of the copper tape has a significant effect on the return loss level at the resonant frequency. On the other hand, resonant frequency is independent of the length of the ground plane. If the ground plane is truncated just at the connection point of the balun and the coupled strip lines, then the return loss at the resonant frequency turns out to be around 17 dB. Thus, for a reasonable return loss level, the ground plane should be truncated slightly after the connection point. It is obvious from Fig. 10 that the center frequencies of the solid and outline bow-ties are different. The slight decrease in the center frequency of the outline bow-tie can be attributed to the less intense surface current density distributed along the center part of the triangles. However, since the metallization at the center of the triangles is removed from the outline bow-tie, the surface currents are restricted to travel along the strip that outlines the triangles. The obstruction of surface currents to flow through the center of the triangles increases the average distance

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Fig. 11. Proposed new antenna configuration which resonates between the center frequencies of the outline and solid bow-tie antennas.

Fig. 12. Comparison of simulated return loss characteristics of the new bow-tie configuration shown in Fig. 11 with solid and outline bow-ties.

traveled in the outline design. Hence, the outline bow-tie is electrically longer than the solid bow-tie, which results in a decrease in the center frequency. Indeed, if a strip were to connect the vertical side of the triangle to its vertex, as shown in Fig. 11, this new configuration would resonate between the center frequencies of the outline and solid designs because it provides an additional path for the current to flow. As it can be observed from Fig. 12, the simulated resonant frequency of this new antenna is 7.52 GHz which is between the resonant frequencies of the outline (7.40 GHz) and solid (7.66 GHz) bow-ties. In addition to return loss, simulated and measured amplitude radiation patterns of the antennas were performed, in three planes: principal H-plane - plane), principal E-plane ( plane) and secondary E-plane ( - plane). The patterns are displayed and compared in Figs. 13 and 14, at the measured center frequencies (7.40 GHz for the outline and 7.66 GHz for the solid bow-tie). The secondary E-plane ( - plane) is defined as the one along which the E-field is parallel to it but does not pass through the overall field maximum. The coordinate system is the same as the one illustrated in Fig. 3. It can be seen that the measured radiation patterns are in excellent agreement with the simulated ones in all of the three planes. Although the pattern in the secondary E-plane is very close to the pattern of an ideal dipole, the patterns in the principal E- and H-planes are noticeably distorted. The back lobes of the patterns are approximately 10 dB lower compared to the forward lobes. This difference is due to the presence of the ground plane which “pushes” the radiation pattern against itself and towards the bow-ties. This structure can also be considered as a 2-element Yagi antenna which

Fig. 13. Comparison of simulated and measured normalized radiation patterns of the solid bow-tie in the: (a) Principal H-plane x-z plane). (b) Principal E-plane x-y plane). (c) Secondary E-plane y -z plane).

is composed of the bow-tie dipole and the ground plane. The ground plane acts as the reflector of the Yagi antenna which has a decreased backward radiation. Therefore, the direction of the peak gain is away from the ground plane. Since the ground plane lies only beneath the feed network, the antennas are allowed to radiate into the both hemispheres; above and below the plane on which the antennas lie. The absolute gains and other radiation parameters are summarized in Table I. The gains that are listed in Table I are in

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TABLE I COMPARISON OF THE RADIATION PARAMETERS OF THE SOLID AND OUTLINE BOW-TIE ANTENNAS

larger than the bandwidth of the solid bow-tie, which is practically negligible. Hence, we can conclude that the bandwidths of the antennas are basically the same. The conductor losses have a considerable effect on the radiation efficiency. As it is mentioned before, the metal layer used in the fabrication process is aluminum which is extremely thin. Since the thickness of the aluminum is much smaller than its skin depth within the operating frequency band (the skin depth m), conof aluminum at 7.5 GHz is approximately ductor losses play an important role in the radiation efficiencies of the antennas. The surface impedance of the thin aluminum was measured to be 0.18 /square. On the other hand, the ground plane and the support structure, which are thicker than the FDC metal, are made of copper and brass, respectively. Therefore, the conductor losses due to the ground plane and the support structure is insignificant compared to the one due to the thin aluminum. In order to see the effects of the surface impedance on the absolute gain of the antennas, the thickness of m) the solid bow-tie was increased to twice the skin depth ( and an additional simulation was performed. The gain of the solid bow-tie turned out to be 4.1 dBi which is considerably larger than the measured gain of the antenna. This result verified the lossy behavior of the FDC metal and relatively low absolute gains of the antennas. B. Flexed Antennas

Fig. 14. Comparison of simulated and measured normalized radiation patterns of the outline bow-tie in the: (a) Principal H-plane (x-z plane). (b) Principal E-plane (x-y plane). (c) Secondary E-plane (y -z plane).

the direction . However, the overall peak gain, which is 0.1 dB greater than the listed gain, occurred at and . Similar to the decrease in the center frequency of the outline design, there is also a slight decrease in the antenna gain. This decrease, as expected, is due to the smaller effective aperture of the outline bow-tie because of the reduced metallization. The fractional bandwidths tabulated in Table I are based on . It can be a return loss of 15 dB or larger observed that the bandwidth of the outline bow-tie is slightly

So far, we have only discussed the radiation characteristics of the bow-tie antennas when they are flat. However, a significant advantage and novelty of our antennas is their flexibility which makes the FDC substrate and bow-tie elements good candidates for conformal antenna applications. Hence, the radiation performances of our antennas when they are flexed are of particular interest. To investigate this, similar to the flat case, the return losses, radiation patterns and absolute gains of the flexed antennas were examined when they were flexed in the form of a cylindrical surface with different radii, and the results were compared with those of the flat ones. To be able to flex the antenna without damaging the conducting traces on the substrate, a new support structure was required, which allows the antenna to bend freely. For this reason, a cylindrical brass was proposed as a support structure. This brass tube had a small diameter and was soldered to the outer ( mm) semi-rigid coaxial cable conductor of the with which the antenna was fed. The part of the brass tube, that was soldered to the coax, was cut back so that the upper portion was close to the center conductor of the coax. Silver paint was used to connect the copper tape ground plane of the balun

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Fig. 15. A side-view schematic drawing of the feed/support geometry of the flexible bow-tie antenna.

Fig. 17. Measurement setup of the flexed solid bow-tie.

Fig. 16. Comparison of the simulated return losses of the bow-tie antennas for different degrees of flection. The return losses of the antennas are almost independent of the radius of curvature when it is larger than the antenna dimensions.

to the brass tube and the center conductor of the coax to the antenna feed point. A drawing of this new geometry is shown in Fig. 15 where the brass tube supports the flexible substrate as well as forming a ground path. This structure allows the flexible substrate to freely bend about the axis of the coaxial line. It also minimizes the probability of any distortion in the exposed metallization of the antennas which is inherently very fragile. However, in contrast to the triangular support, the cylindrical structure introduces an inductance to the overall system. This inductance can be attributed to the currents circulating within the cylindrical brass tube which behaves like a solenoid. As a result, the center frequency of solid bow-tie shifted to 7.24 GHz which is lower than its design frequency of 7.40 GHz. Fig. 16 illustrates the comparison of the simulated return loss of the flat and flexed antennas. It is obvious that the center frequencies of the antennas do not change dramatically, up to a certain extent, with curvature because bending does not change the electrical lengths of the elements. However, the characteristic impedances of the transmission lines are subject to change with bending, affecting the radiation properties of the overall system [29]. After the simulations, the solid bow-tie was bent over a poly(50.8 mm), as styrene which had a radius of curvature of illustrated in Fig. 17 and its return loss was measured. Since the dielectric constant of the polystyrene is very close to unity, dielectric loading of the material was kept at a minimum. Mea-

Fig. 18. Comparison of the return loss of the flat and flexed solid bow-tie with the cylindrical support structure.

sured and simulated return losses of the flexed antenna was compared with those of the flat one in Fig. 18. It is apparent that there is a very good agreement between the simulations and measurements for both types of support structures. Besides, as expected, there is not a significant difference between the return losses of the flat and flexed antennas. The center frequency of the antenna remained basically constant. However, there was approximately 4 dB increase in the return loss at the resonant frequency. Fig. 19 shows simulated and measured normalized radiation patterns of the flexed bow-tie in the principal planes. It is evident that there is a very good agreement between simulations and measurements. Except for the null filling in the secondary E-plane pattern of the curved antenna, the shapes of the patterns are basically identical. The null filling occurs because, in the secondary E-plane, the entire surface of the bow-tie is not parallel to its principal axis where an ideal null would otherwise have been formed. In addition to the normalized patterns, the absolute gains of the flat and flexed antennas were found to be very close to each other. The measured realized gains of the antennas are listed in Table II for each principal plane. The maxfor the principal E- and imums occurred at for the secondary E-plane. H-planes and at

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secondary E-plane is very smooth. Since the secondary E-plane pattern is the plane cut that has its rotation axis parallel to the feed cable, the RF cable does not protrude into the plane of the measurement, and consequently there is no ripple in the pattern due to scattering from the cable. The axes of rotation for the principal E- and H-planes are normal to the plane of the antenna substrate. Part of the RF cable is in the plane of the measurement, resulting in scattering, constructive and destructive interference, and ripple superimposed on the measurement. The same measurements and simulations were also repeated for the flexed outline bow-tie antenna and a similar behavior was obtained. Therefore, those results of the flexed outline bow-tie will not be repeated here, due to space limitations. V. CONCLUSION

Fig. 19. Comparison of the normalized radiation patterns of the flat and flexed bow-tie. (a) Principal H-plane (x-z plane). (b) Principal E-plane (x-y plane). (c) Secondary E-plane (y -z plane). TABLE II MEASURED REALIZED GAINS OF FLAT AND FLEXED BOW-TIE ANTENNAS

One can observe in Fig. 19 that although there are some ripples in the principal E- and H-plane patterns, the pattern in the

Two bow-tie antennas, a conventional and a novel antenna with reduced metallization, were designed and fabricated using the unique flexible material of the ASU FDC. The HFSS simulations were compared with measurements performed in the ASU EMAC. The radiation pattern, return loss and absolute gain comparisons indicated that the measurements and the HFSS simulations are in very good agreement. The comparison of the return loss and radiation parameters of the solid and outline bow-tie antennas verified that there is not a dramatic difference between the radiation performance of the two. The resonant frequency of the outline bow-tie was lower than that of the solid bow-tie because of its increased electrical length. In addition, as expected, the gain of the outline bow-tie turned out to be smaller than the gain of the solid bow-tie. However, the modified antenna can be prototyped more rapidly with serial printing techniques. Hence, there is a trade-off between the gain and the reduced metallization of the antenna. The slight decrease in the gain is acceptable for some applications where less metallization and rapid prototyping are desired. It was also verified that the conduction losses of the FDC metal plays an important role on the gain of the antennas. The performance of the antennas can be improved by using a thicker conductor in the fabrication process. The type of the support structure, which was used to stabilize the coax-microstrip transmission line connection, was verified to have an impact on the resonant frequency of the antennas. Although, due to the gap between the ground plane and brass, the triangular support introduced a capacitance resulting in an increase in the resonant frequency, the cylindrical brass tube had an inductive effect. This inductance was due to the currents circulating within the brass tube. Consequently, the resonant frequency of the antennas decreased after the assembly of the new support structure. Finally, it was observed that the return loss and radiation patterns of the flat and flexed antennas were very close to each other, if the radius of curvature was larger than the antenna dimensions. These affirmative results of the bow-tie antenna designs on flexible substrates indicates that the FDC technology can be a promising candidate for the design of reconfigurable flexible antenna arrays.

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REFERENCES [1] G. B. Raupp, S. M. O’Rourke, D. R. Allee, S. Venugopal, E. J. Bawolek, D. E. Loy, S. K. Ageno, B. P. O’Brien, S. Rednour, and G. E. Jabbour, “Flexible reflective and emissive display integration and manufacturing (invited paper),” Cockpit Future Displ. Def. Secur. vol. 5801, no. 1, pp. 194–203 [Online]. Available: http://link.aip.org/link/ ?PSI/5801/194/1 [2] F. Zenhausern and G. B. Raupp, “Pre-production display R&D facility: A framework for developing flexible display systems reducing the warfighters,” in SPIE Symp. Proc., 2004, vol. 5443, pp. 13–20. [3] R. Shringarpure, L. T. Clark, S. M. Venugopal, D. R. Allee, and S. G. Uppili, “Amorphous silicon logic circuits on flexible substrates,” presented at the IEEE Custom Integrated Circuits Conf., San Jose, CA, 2008. [4] S. M. Venugopal and D. R. Allee, “Integrated A-SI:H source drivers for 4 QVGA electrophoretic display on flexible stainless steel substrate,” IEEE J. Display Technol., vol. 3, pp. 57–63, 2007. [5] S. G. Uppili, D. R. Allee, S. M. Venugopal, L. T. Clark, and R. Shringarpure, “Standard cell library and automated design flow for circuits on flexible substrates,” presented at the Flexible Electronics and Displays Conf., Phoenix, AZ, 2009. [6] J. So, J. Thelen, A. Qusba, G. J. Hayes, G. Lazzi, and M. D. Dickey, “Reversibly deformable and mechanically tunable fluidic antennas,” Adv. Funct. Mater., vol. 19, no. 22, pp. 3632–3637, Oct. 2009. [7] L. Yang, R. Zhang, D. Staiculescu, C. Wong, and M. Tentzeris, “A novel conformal RFID-enabled module utilizing inkjet-printed antennas and carbon nanotubes for gas-detection applications,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 653–656, 2009. [8] C. A. Balanis, Antenna Theory Analysis and Design. Hoboken, NJ: Wiley, 2005. [9] C. A. Balanis, C. R. Birtcher, V. Kononov, A. Lee, and M. S. Reese, Advanced Electromagnetic Methods for Helicopter Applications. Tempe, AZ: Telecommunications Research Center, Jun. 2004. [10] A. C. Durgun, M. S. Reese, C. A. Balanis, C. R. Birtcher, D. R. Alle, and S. Venugopal, “Flexible bow-tie antennas,” in Proc. IEEE Antennas and Propagation Society Int. Symp. (APSURSI), Jul. 2010, pp. 1–4. [11] A. C. Durgun, C. A. Balanis, C. R. Birtcher, and D. R. Alle, “Radiation characteristics of a flexible bow-tie antenna,” in Proc. IEEE Antennas and Propagation Society Int. Symp. (APSURSI), Spokane, WA, Jul. 3-8, 2011, pp. 1239–1242. [12] A. Durgun, M. Reese, C. Balanis, C. Birtcher, D. Allee, and S. Venugopal, “Flexible bow-tie antennas with reduced metallization,” in Proc. IEEE Radio and Wireless Symp. (RWS), Jan. 2011, pp. 50–53. [13] [Online]. Available: http://www.ansoft.com/products/hf/hfss/ [14] K. R. Wissmiller, J. E. Knudsen, T. J. Alward, Z. P. Li, D. R. Allee, and L. T. Clark, “Reducing power in flexible A-SI digital circuits while preserving state,” in Proc. IEEE Custom Integrated Circuits Conf., Sep. 2005, pp. 219–222. [15] A. A. Eldek, A. Z. Elsherbeni, and C. E. Smith, “Wideband microstrip-fed printed bow-tie antenna for phased-array systems,” Microw. Opt. Technol. Lett., vol. 43, no. 2, pp. 123–126, Oct. 2004. [16] J. George, M. Deepukumar, C. Aanandan, P. Mohanan, and K. Nair, “New compact microstrip antenna,” Electron. Lett., vol. 32, no. 6, pp. 508–509, Mar. 1996. [17] B. Garibello and S. Barbin, “A single element compact printed bowtie antenna enlarged bandwidth,” in Proc. SBMO/IEEE MTT-S Int. Conf. on Microwave and Optoelectronics, Jul. 2005, pp. 354–358. [18] M. Rahim, M. A. Aziz, and C. Goh, “Bow-tie microstrip antenna design,” in Proc. 13th IEEE Int. Conf. on Networks, Jointly Held With the IEEE 7th Malaysia Int. Conf. on Communication, 2005, vol. 1, pp. 17–20. [19] R. Compton, R. McPhedran, Z. Popovic, G. Rebeiz, P. Tong, and D. Rutledge, “Bow-tie antennas on a dielectric half-space: Theory and experiment,” IEEE Trans. Antennas Propag., vol. 35, no. 6, pp. 622–631, Jun. 1987. [20] Y.-D. Lin and S.-N. Tsai, “Coplanar waveguide-fed uniplanar bow-tie antenna,” IEEE Trans. Antennas Propag., vol. 45, no. 2, pp. 305–306, Feb. 1997. [21] A. Eldek, A. Elsherbeni, and C. Smith, “A microstrip-fed modified printed bow-tie antenna for simultaneous operation in the C and X-bands,” in Proc. IEEE Int. Radar Conf., May 2005, pp. 939–943.

[22] K. Kiminami, A. Hirata, and T. Shiozawa, “Double-sided printed bow-tie antenna for UWB communications,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 152–153, 2004. [23] Y.-D. Lin and S.-N. Tsai, “Analysis and design of broadside-coupled striplines-fed bow-tie antennas,” IEEE Trans. Antennas Propag., vol. 46, no. 3, pp. 459–460, Mar. 1998. [24] G. Zheng, A. A. Kishk, A. W. Glisson, and A. B. Yakovlev, “A broadband printed bow-tie antenna with a simplified balanced feed,” Microw. Opt. Technol. Lett., vol. 47, no. 6, pp. 534–536, Dec. 2005. [25] D. Anagnostou, M. Morton, J. Papapolymerou, and C. Christodoulou, “A 0–55 GHz coplanar waveguide to coplanar strip transition,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 1, pp. 1–6, Jan. 2008. [26] K. Tilley, X.-D. Wu, and K. Chang, “Coplanar waveguide fed coplanar strip dipole antenna,” Electron. Lett., vol. 30, no. 3, pp. 176–177, Feb. 1994. [27] N. Kaneda, Y. Qian, and T. Itoh, “A broad-band microstrip-to-waveguide transition using quasi-Yagi antenna,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2562–2567, Dec. 1999. [28] Y. Qian and T. Itoh, “A broadband uniplanar microstrip-to-CPS transition,” in Proc. Asia-Pacific Microwave Conf., Dec. 1997, vol. 2, pp. 609–612. [29] K. L. Wong, Design of Nonplanar Microstrip Antennas and Transmission Lines. New York: Wiley, 1999.

Ahmet Cemal Durgun (S’09) received the B.S.E.E. and M.S.E.E. degrees from Middle East Technical University, Turkey, Ankara, in 2005 and 2008, respectively, where he completed the double major program in mathematics and received the B.S. degree in 2006. He is currently working toward the Ph.D. degree at Arizona State University, Tempe. His research interests include flexible antennas, high impedance surfaces and computational electromagnetics.

Constantine A. Balanis (S’62–M’68–SM’74–F’86 –LF’04) received the B.S.E.E. degree from Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, in 1964, the M.E.E. degree from the University of Virginia, Charlottesville, in 1966, and the Ph.D. degree in electrical engineering from Ohio State University, Columbus, in 1969. From 1964–1970 he was with the NASA Langley Research Center, Hampton, VA, and from 1970–1983 he was with the Department of Electrical Engineering, West Virginia University, Morgantown. Since 1983 he has been with the Department of Electrical Engineering, Arizona State University, Tempe, where he is now Regents’ Professor. His research interests are in computational electromagnetics, flexible antennas and high impedance surfaces, smart antennas, and multipath propagation. He is the author of Antenna Theory: Analysis and Design (Wiley, 2005, 1997, 1982), Advanced Engineering Electromagnetics (Wiley, 2011, 1989) and Introduction to Smart Antennas (Morgan and Claypool, 2007), and editor of Modern Antenna Handbook (Wiley, 2008) and for the Morgan & Claypool Publishers, series on Antennas and Propagation, and series on Computational Electromagnetics. Dr. Balanis is a Life Fellow of the IEEE. He received in 2004 a Honorary Doctorate from the Aristotle University of Thessaloniki, the 2005 IEEE Antennas and Propagation Society Chen-To Tai Distinguished Educator Award, the 2000 IEEE Millennium Award, the 1996 Graduate Mentor Award, Arizona State University, the 1992 Special Professionalism Award from the IEEE Phoenix Section, the 1989 IEEE Region 6 Individual Achievement Award, and the 1987–1988 Graduate Teaching Excellence Award, School of Engineering, Arizona State University. He has served as Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION (1974–1977) and the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING (1981–1984), as Editor of the Newsletter for the IEEE Geoscience and Remote Sensing Society (1982–1983), as Second Vice-President (1984) and member of the Administrative Committee (1984–85) of the IEEE Geoscience and Remote Sensing Society, and as Distinguished Lecturer (2003–2005), Chairman of the Distinguished Lecturer Program (1988–1991) and member of the AdCom (1992–95, 1997–1999) of the IEEE Antennas and Propagation Society.

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Craig R. Birtcher received the B.S.E.E. and M.S.E.E. degrees from Arizona State University, Tempe, in 1983 and 1992, respectively. He has been at Arizona State University since 1987, where he is now an Associate Research Professional in charge of the ElectroMagnetic Anechoic Chamber (EMAC) facility. His research interests include antenna and RCS measurement techniques and near-field to far-field (NF/FF) methods.

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David R. Allee (M’90) received the B.S. degree in electrical engineering, from the University of Cincinnati, OH, in 1984, and the M.S. and Ph.D. degrees in electrical engineering from Stanford University, Stanford, CA, in 1984 and 1990, respectively. While at Stanford University, and as a Research Associate at Cambridge University, he fabricated field effect transistors with ultra-short gate lengths using custom e-beam lithography and invented several ultra-high resolution lithography techniques. Since joining Arizona State University, his focus has been on mixed signal integrated circuit design. Currently, he is a Professor of electrical engineering at Arizona State University. He is also Director of Research for Backplane Electronics for the Flexible Display Center (flexdisplay.asu.edu) funded by the Army, and he is investigating a variety of flexible electronics applications. He has been a regular consultant with several semiconductor industries on low voltage, low power mixed signal CMOS circuit design. He has coauthored over 100 archival scientific publications and U.S. patents.

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Spiral Leaky-Wave Antennas Based on Modulated Surface Impedance Gabriele Minatti, Francesco Caminita, Massimiliano Casaletti, and Stefano Maci, Fellow, IEEE

Abstract—Different kinds of spiral planar circularly polarized (CP) antennas are presented. These antennas are based on an interaction between a cylindrical surface-wave excited by an omnidirectional probe and a inhomogeneous surface impedance with a spiral pattern. The surface impedance interaction transforms a bounded 0 surface wave into a circularly polarized leaky wave with almost broadside radiation. The problem is studied by adiabatically matching the local 2D solution of a modulated surface-impedance problem to the actual surface. Analytical expressions are derived for the far-field radiation pattern; on this basis, universal design curves for antenna gain are given and a design procedure is outlined. Two types of practical solutions are presented, which are relevant to different implementations of the impedance modulation: i) a grounded dielectric slab with a spiral-sinusoidal thickness and ii) a texture of dense printed patches with sizes variable with a spiral-sinusoidal function. Full wave results are compared successfully with the analytical approximations. Both the layouts represent good solutions for millimeter wave CP antennas.

TM

Index Terms—Leaky-wave antennas, millimeter waves, spiral antennas, surface impedance.

I. INTRODUCTION

A

new typology of planar circularly polarized leaky wave (LW) antennas excited by a single-point feed is presented. The basic structure is constituted by variable, spiral-shape modulated surface impedance. A vertical probe excites a cylindrical surface wave (SW) on the impedance surface, and the latter converts it into a circularly polarized LW. The phenomenology of the local interaction between an angular portion of cylindrical SW wavefront and an angular sector of the spiral is described by means of a 2D problem of a sinusoidal reactance excited by a planar SW. The latter is treated by means of the Oliner-Hessel method described in [1]. In [2], a similar approach is used for realizing both linearly and circularly polarized antennas. There, the local SW-impedance interaction is described by means of a holographic principle [2]; this description is linked in [3] to a leaky-wave radiation mechanism.

Manuscript received January 06, 2011; revised April 03, 2011; accepted May 06, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. This work was supported in part by the European Space Agency (ESA-ESTEC, Noordwijk, The Netherlands) under grant “Holoant.” The part of the paper relevant to the corrugated dielectric lens antenna was developed inside a European Science Foundation (ESF) project called “NEWFOCUS.” The authors are with the Department of Information Engineering, University of Siena, Siena 53100, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165691

The antennas presented here are completely different from conventional spiral antennas based on the active region concept [4], [5]. Our solutions exhibit high-gain (20–25 dB) and moderate bandwidth (10%) in contrast with conventional spiral antennas, characterized by broad bandwidth and low gain. This paper is structured as follows. Section II presents the case of a cylindrical surface wave propagating on a homogeneous inductive reactance, and excited by a vertical dipole placed on the surface. This section is preparatory for the successive Section III, which presents a formulation for a SW interacting with a spiral surface reactance. The surface reactance possesses a radial period equal to the wavelength of the surface wave running on the average uniform impedance. Section III also provides an analytical approximation of the aperture field and of the far-field radiation pattern. These approximations are functions of the following three parameters: 1) average impedance; 2) modulation index of the surface impedance; and 3) complex propagation constant of the cylindrical leaky wave. The latter parameter is derived as a function of the first two parameters on the basis of the solution of the local 2D sinusoidal surface-impedance presented in Section IV. The analytical approximation is used in Section V to calculate universal curves for gain and efficiency, which can be used in the design procedure. Sections VI and VII presents practical examples of implementation, based on a grounded slab with modulated thickness and a texture of dense printed patches with gradually modulated sizes. The conclusive Section VIII discusses potential advantages and future research topics. II. TM SURFACE WAVE EXCITED ON HOMOGENEOUS SURFACE IMPEDANCE Let us consider a vertical elemental dipole with moment placed on an infinite plane at . Assume that the latter is subjected to the impedance boundary condition (1) is a positive reactance, and are the transwhere verse-to-z components of the electric and magnetic fields, respectively. From here on, bold characters will denote vectors which denote the unit vectors associated to except for the cylindrical coordinates . The dipole excites on the -type SW which, at certain distance from reactive surface a the dipole, assumes the form

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(2) (3)

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and are related each other by (4). The constant is that denoted hereinafter as impedance modulation index. This index varies from zero to unity, thus implying that the reactance assumes positive values everywhere. A. Aperture Field Approximation The aperture field generalization of (7); i.e.,

Fig. 1. Geometry of the surface impedance illuminated by a vertical dipole and sketch of the excited surface wave.

where and are the free-space impedance and wavenumber, and are the Hankel functions of second respectively, kind and order zero and one, respectively. The radial propagarespects the dispersion equation tion constant (4) where is the attenuation constant in the z direction. Note that this TM-SW can actually be excited only if the reactance is positive. It can be demonstrated [6] that the complex amplitude is related to the dipole moment through (5) The dipole also radiates a space wave, which is approximated in the far zone as [6] (6) In the following, we will consider the tangent electric field at the surface which is rewritten here for convenience as a function of the equivalent current , namely

(7) Note that the dipole radiation is not directive because . When the surface is finite and circular (Fig. 1), the same surface wave diffract at the edge and produces additional contribution that can be estimated by properly modifying the method presented in [7]. III. SURFACE WAVE INTERACTION WITH A INHOMOGENEOUS SPIRAL IMPEDANCE We now assume that a vertical dipole source is placed on a inhomogeneous, slowly varying positive reactance (8) This reactance is shaped as a sinusoidal Archimedean spiral and represents the average values of the impedance. Its radial period is equal to the wavelength of the SW excited on an equivalent homogeneous surface impedance of value . This means

is approximated by an appropriate

(9) In (9), is a complex small correction to the propand . agation constant, which actually depends on The wave in (9) can be attributed to the interaction between the modulated surface and the incident SW in (7) (i.e., the SW ). This interlaunched by a dipole on the average surface action produces a Bragg radiation effect, which leads to an atand a small correction to the SW tenuation constant , wavenumber. We emphasize that complex deviation and average reactance should be conmodulation index sistent each other at a given frequency. This consistency will be determined in the next section by the solution of the 2D problem that matches the local radial impedance variation. in the relevant exDecomposing the sin function in ponential terms and approximating the Hankel function with its asymptotic expression for large argument, leads to -indexed cylindrical LWs with wavenumbers (10) The ( 1) indexed wavenumber belongs to the visible region of the aperture radiation and leads to an almost broadside beam. Actually, the beam with a single peak at broadside is on the deverge of splitting into two distinct peaks, because of the viation; however, maximum power density is radiated at broadside. A similar situation occurs for a general class of LW antennas comprised of a grounded slab covered with a partially reflecting surface [8]. The main characteristic of the ( 1) indexed LW is that it is improper, i.e., its field increases exponentially in the z-direction. However, this wave is physical only outside a conical region with symmetry around z and vertex at the source (see Fig. 5 in [9]). Although the LW itself never reaches the far field, its contribution dominated the aperture field distribution and can be used to estimate the far field by Fourier transformation (see Section III-D). B. Intuitive Explanation of the Circular Polarization An intuitive explanation of the fact that the main beam is circularly polarized is the following (Fig. 2). The periodicity of the modulation along each ray is equal to . Therefore, two rays separated by 90 intercept a at distances from the origin spiral line . This implies that that differ each other of any pair of elemental sectors separated by 90 gives rise to orthogonal and quadrature-phased components, thus leading to the broadside circular polarization.

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(14) where

(15)

Fig. 2. Surface-wave excitation of spiral-type circular surface reactance and visual explanation of the circular polarization radiated at broadside. The inset shows the dimension of the periodicity wrt the wavelength of the exciting surface.

and . We have used the primed coordinates for the aperture field integration. The above function, due to the , is an almultiplication for the exponential factor most non-oscillating function. This is apparent from its asymp. totic approximation in (15), considering that The field in (11)–(15) should be completed by the space wave contribution radiated by the exciting dipole. It is reasonable to approximate the latter as the field radiated by the vertical dipole (namely in over a homogeneous surface of average value (6)). In many cases this contribution has a small impact on the total field. in (12) is the right-hand circularly polarized For small , in (13) is the left-hand circularly (co-polar) contribution and polarized (cross-polar) contribution. To extend the definition to any value of , in the following, we will use

(16) (17) Fig. 3. Behavior of the amplitude of p (k ) as a function of k a. The normalization is chosen to obtain a unit maximum of p (k ). The visible region k < k is dominated by p and p , while the other spectra only give a small perturbation. In the inset: interpretation in terms of a demodulation process.

C. Analytical Expression of the Far Field Radiated by the LW on a Circular Aperture radiated in the far region is calculated by the The field radiation integral of the aperture distribution in (9). The latter can be expressed in terms of Fourier-Bessel (FB) spectra as

(11)

(12)

(13)

as definition of co-polar and cross-polar pattern, respectively, and the asterisk denotes conjugate. where in (14) are shown Typical spectral distributions of in Fig. 3. The dominant contributions in the visible range are (co-polar) and (cross-polar). The contribution is the FB spectrum of order of the non-oscillating ; as such, it is concentrated around the origin of function vanishes at broadside the -plane. The cross-polar term and it is small on the half-power beam angle of the co-polar pattern. The remaining co-polar and cross-polar contributions do not affect significantly the visible region since the exponential factors and in the FB integrand imply translations of the main spectral contributions outside the . visible region The mechanism just described is analogous to the demodulation process in a super-heterodyne system (see inset of Fig. 3). The currents , proportional to , spectrum is condo not intrinsically radiate, since its larger than . The surface reactance centrated around a acts as a demodulator and translates the spectrum into the visible region, also generating of additional spurious demodulation terms located outside the visible region. The radiation operator (inset of Fig. 3) acts as a low-pass filter and cut-off these spurious contributions.

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increases. However, we will the lateral lobes decrease when implies a decrease of the show in Section V that increasing antenna gain. IV. APPROXIMATION OF THE LW WAVENUMBER

Fig. 4. Normalized radiation pattern in the far zone for a unity normalized reactance and different values of modulation index. The aperture radius is equal to 4 free space wavelengths.

As seen in Section III-A, the modulation of the reactance acinto the complex wavenumbers tually transforms ; thus, implying that the aperture field is dominated by the ( 1) indexed LW with wavenumber . This wavenumber can be estimated by considering the local 2D problem of a surface with sinusoidal modulated reactance which matches the actual geometry at each azimuthal angle (see Fig. 5). The surface is assumed as infinite, with sinusoidal modulation along , and invariant along the orthogonal direction; TM plane-SW is propagating along and the surface impedance is given by (18)

Fig. 5. Geometry of the spiral impedance surface and local 2D problem.

where we have understood and suppressed the dependence on the azimuthal angle. The solution of this problem was presented by Oliner and Hessel in [1] and summarized in Appendix A. The final conis the first zero of the continued fraction clusion is that the determinantal function

(19)

=( 0

)

Fig. 6. Behavior of =k =k and  =k as a function of M for X = . different values of average normalized reactance 

=

D. Impact of the Modulation in Index on the Normalized Radiation Pattern The radiation pattern lateral lobes are influenced from the modulation index . Increasing implies an increase of the and then a reduction of the aperture taattenuation constant pering; the latter is associated to a reduction of lateral lobes. The and physical information about the relationship between will be specified in the next section. We anticipate here results obtained by this relationship. Fig. 4 shows normalized co-polar for an aperture of radius radiation patterns in the plane , for modulation indexes ranging from 20% to 50% and normalized average reactance . As expected,

are defined in Appendix A. The representation in where and six-seven terms (19) is rapidly convergent for in the fractions are usually sufficient to obtain the first zero ) with sufficient accu(i.e., the propagation wavenumber racy. In our case, (see (8)), the periodicity is taken as , namely coincident with the value of the wavelength of the SW propagating on the average sur; thus, reducing the solution to searching the null of face . To this the function . end, one can use as initial guess the value is assumed independent on frequency, the solution beIf comes scalable in wavelength. The final solution is thus parameand . terized in terms of the normalized-to- values for values Fig. 6 shows these parameters as a function of ranging from of the average normalized reactance 0.7 to 2. The obtained curves are very regular and can be easily and . In the range approximated as a polynomial form in and , the curves in Fig. 6 fit with an error less of 5% the least-mean square polynomial approximations

(20)

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Fig. 8. Dielectric spiral antenna with modulated thickness, its local 2D problem, and the equivalent transmission line model for the TM dominant = k =k; Z =  k =k ). surface wave (Z

following section, where the desired reactance is implemented by two different methods. VI. CORRUGATED DIELECTRIC LENS ANTENNA

Fig. 7. Design curves of gain (G) and efficiency (" ) as a function of the aper = 2 (0:5 0 1:5). (a) ture radius in free-space wavelengths for  = X M = 0:3 and (b) M = 0:5.

V. ANTENNA GAIN AND DESIGN CURVES The antenna gain in absence of losses (i.e., the directivity) can be approximated as

(21) where with defined in (14). The and through and . In gain implicitly depends on deriving (21), only the dominant contributions and in (14) have been considered (see Fig. 4). The pattern therefore becomes circularly symmetric and the integration in leads to , thus simplifying the expression of the gain in (21). The antenna . tapering efficiency is given by Fig. 7(a) and (b) show the design curves of gain and efficiency as a function of the aperture radius in free-space wavelengths. ranging from The curves are parameterized with (Fig. 7(a)) and (Fig. 7(b)). 0.5 to 1.25 at These design curves are always the same for any frequency, if independent on frequency. one assumes The efficiency is quite low, especially for gains over 20 dB. This is expected by the fact that the antenna is based on a LW phenomenon. However, the solution is useful for millimeter wave applications, where reducing the physical size of the antenna is not the principal issue. Examples are shown in the

In this first example, the surface impedance is obtained by varying the thickness of a grounded dielectric slab as a func. This leads to the corrugated dielectric antenna tion of shown in Fig. 8. The same figure also shows the local 2D problem and the equivalent transmission line for the dominant TM-SW. The latter is used to find the local dispersion equation. It is worth noting that in this case the solution of the 2D problem might be found as in [10]; we indeed adopt successfully the simpler surface impedance model. that matches the Our objective is to find the value of in (8). This is done by the three surface impedance steps described below. 1) First, we impose the maximum substrate height that ensures a mono-modal propagation on the dielec, where is the tric slab; i.e., is the relative permittivity free-space wavelength and , a maximum of the dielectric slab. Starting from value for the surface impedance is determined , where , by and is the TM-SW propagation con. stant of a grounded slab with uniform thickness is obtained by solving the dispersion The value of equation

(22) derived from the equivalent transmission line in Fig. 8. The above process allows one to have the maximum possible value of average reactance. 2) Next, one chooses an initial value of the modulation index (suggested value is from 0.3 to 0.5). In doing that, one imply higher latshould consider that lower values of eral lobes (see Fig. 4). Once and the maximum values of reactance have been fixed, the average of the surface re. actance is given by , the plots in Fig. 7 allows 3) On the basis of and one to fix the aperture radius to obtain a desired gain

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Fig. 10. Directivity diagrams for co-polar (a) and cross polar (b) components for the 24 dB gain antenna at 18 GHz. The results from the full-wave analysis (no absorbing boundary condition at the termination, continuous line) are compared with those from the closed form solution shown in (11)–(15) (dashed lines).

Fig. 9. Directivity diagrams for both co-polar (continuous line) and cross-polar (dotted lines) components for the 22 dB gain antenna at 18 GHz; (a) and (b): full wave results without absorbing boundary conditions in two orthogonal plane cuts (see the inset); (c) and (d): full-wave results with absorbing boundary conditions. The insets of Fig. (b) shows the bandwidth of gain around the center frequency.

G. If the found value of is too large one should restart from point 2 reducing the values of M. 4) Finally, one should find the value of the local thickness that provides the obtained reactance by means of the transmission line resonance equation

(23)

where , and are found in the previous steps. The is found by inversion of (23). If local value of the modulation index of the impedance is lower than 0.3 the numerical lens profile can be approximated as , where , and are obtained by solving (23) for maximum and minimum value of impedance, respectively. With the previous steps, two lenses at 18 GHz have been deand signed on a substrate relative permittivity , and with gain 22 and 23.7 dB, respectively (losses are neglected). In both cases a small transition flat zone is set around the dipole source for better controlling the initial launching of surface wave (see insets of Fig. 9). For the 22 dB antenna we , which leads to an have chosen a modulation index . To obtain the average normalized surface impedance desired gain of 22 dB, from the plots of Fig. 7 we have found . Since , this corresponds to around 6 periods. As predicted by the universal diagram in Fig. 7, the aperture illumination efficiency is around 18%. Fig. 9(a) and (b) shows the antenna directivity patterns obtained by full-wave results (HFSS solver) on 2 orthogonal cuts, for both co-polar CP (continuous line) and cross-polar components (dashed lines). The frequency bandwidth of the gain is presented in the inset of Fig. 9(b). A reduction of gain less than 3 dB is obtained in a 10% bandwidth; furthermore,

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Fig. 11. Printed patch holographic antenna, its local 2D problem, and the equiv= alent transmission line model for the TM dominant surface wave (Z k =k; Z =  k =k ).

the gain-frequency diagram is almost flat (reduction of gain less than 0.5 dB) in a bandwidth of 5,5%. Very similar values of bandwidth have been found in the other configuration tested numerically. The irregularity of the pattern is due to the strong reflections at termination, which are significant due to the high-values of the dielectric constant and to the relatively small antenna radius. These edge effects are mitigated by using absorbing boundary conditions, as can be observed from the results in Fig. 9(c) and (d). VII. PRINTED PATCH HOLOGRAPHIC ANTENNA The surface impedance can be synthesized by printing a dense texture of square metal patches on a grounded slab with constant period and variable sizes (Fig. 11). In [2] this antenna is called holographic antenna. The patches have slowly varying sizes. To analyze the coupling of a single patch with the rest of the structure, we assume that each patch is embedded in a locally uniform periodic FSS-like lattice. That is, we identify the local texture and relevant equivalent reactance at a certain point, with the periodic printed structure that matches adiabatically the local geometry. With difference to FSS, here the patches are small in terms of a wavelength (around one tenth of wavelengths). This forms a dense, smoothly varying spiral texture, where the lower impedance levels are represented by smaller patches and the higher impedance levels by larger patches. The final visual impression is a spiral image described by pixels. The design steps are the same as that for the dielectric spiral lens shown in Section VI. The level of average reactance is chosen as large as possible compatibly with the absence of higher order modes. To associate the impedance to the structure, a parametric dispersion analysis is carried out by using the TM transmission-line of the dominant Floquet mode (right-left corner in Fig. 11). The FSS-pixel texture is represented by an in parallel to the transmisadditional impedance sion line. This impedance, which depends on the ratio between

Fig. 12. Directivity diagrams for co-polar (a) and cross polar (b) components for the 20 dB gain antenna at 13 GHz. The results from the full-wave analysis (continuous line) are compared with those from the closed form solution shown in (14) (dashed lines).

the size of the local patch and the local periodicity , is obtained by using the pole-zero matching method presented in [11] with the method of moments (MoM) analysis in [12]. An example of 20 dB gain antenna at 13 GHz has been designed. The antenna involves 3323 square metal patches of variable size, printed on a grounded dielectric slab with permittivity and thickness . The modulation index and , respecand average reactance are tively. To get the desired 20 dB gain, it is found from the dia. grams in Fig. 7 that the antenna radius is Fig. 12(a) and (b) shows the co-polar and the cross-polar components of the directivity pattern of such antenna provided by using a non-commercial code based on the article in [13] (see acknowledgements). The numerical full wave results are compared with those from the analytical solution in (11)–(15). The agreement is satisfactory and confirms the adequacy of the design methodology. VIII. CONCLUSIONS Leaky-wave circularly polarized antennas based on variable surface impedance are presented. The surface impedance is inductive and modulated by a sinusoidal spiral function. Under this condition, we have provided analytical closed form approximations of the LW aperture field and of the relevant radiated field. This approximation is a function of the complex obtained from a 2D local problem LW wavenumber through the Oliner-Hessel method. This complex parameter, normalized wrt the free-space wavenumber, only

MINATTI et al.: SPIRAL LEAKY-WAVE ANTENNAS BASED ON MODULATED SURFACE IMPEDANCE

depends on the average value of reactance and on the modulais fixed, “universal” gain and tion index. Whenever efficiency diagrams versus the aperture normalized radius to wavelength are obtained. The design procedure is illustrated with two different types of implementation of the spiral-reactance: a corrugated grounded dielectric slab and a dense texture of printed patches with variable sizes. The latter surface creates a “pixel-type” picture of the spiral, which corresponds to a particular case of the holographic antennas introduced in [2]. Many other technological solutions can be invented; however, the design procedure is always the same. The final antennas exhibit a gain that can be well controlled in a range from 20 to 26 dB with lateral lobes less than 20 dB. The cross-pol pattern has a null at the center and peaks around 10 dB gain down wrt the maximum of the co-polar contribution; thus, suggesting a use of the antenna for monopulse radar [14]. The operational frequency bandwidth is not treated here at theoretical level. However, numerical results have shown that a maximum deviation of gain around 3 dB is obtained in a 10% bandwidth; furthermore, the gain-frequency diagram is almost flat (deviation less than 0.5 dB around the center frequency) in a bandwidth of 5%. The main disadvantage is the low aperture efficiency (from 15% to 20%), which is however accompanied by a low value of side lobe co-polar levels. The advantages of antennas presented here are that they are extremely flat and have a single point feed. Therefore it sounds interesting for millimeter wave applications. To these end, the impact of losses in the various technological configurations is a future research topic. The typology of feed (not treated here) is a further matter of investigation, especially concerning the efficiency in terms of surface-wave versus space-wave power. APPENDIX A We briefly synthesize here the Oliner-Hessel procedure. We make reference to the 2D periodic structure in Fig. 5, described by the impedance variation in (18). Due to the periodicity of the structure, the field representation can be given in terms of summation of Floquet plane-waves with wavenumbers

(A1) , the transverse electric and In order to find and are expanded in terms of Floquet magnetic fields modes with wavenumbers as in (A1). Next a mode matching method is applied by imposing average (through Floquet wave . This allows projections) boundary conditions one to set up an infinite set of equations among the coefficient and of the FW expansion of the magnetic and electric . field, respectively; i.e., This equation states that each transverse field Floquet mode is coupled only to the nearest lower and higher mode. The determinantal equation that allows to obtain the unknown complex is found by the transverse resonance TM conwavenumber . The latter, inserted in the predition vious equation, leads to

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(A2) The latter is an infinite set of linear homogenous equations with an infinite number of unknowns . This set possesses a nontrivial solution when the infinite determinant of the set vanishes. In [1] it is shown that the first zero of the determinant wrt is the same of the zero of the continued fraction the variable determinantal function defined in (19). ACKNOWLEDGMENT The authors wish to thank P. De Vita of Ingegneria dei Sistemi (IDS), S. Piero a Grado, Pisa, for providing numerical full-wave simulations in Fig. 12(a) and (b). They also thank M. Sabbadini of ESA-ESTEC for comments and suggestions about the design process. REFERENCES [1] A. Oliner and A. Hessel, “Guided waves on sinusoidally-modulated reactance surfaces,” IRE Trans. Antennas Propag., vol. 7, no. 5, pp. 201–208, Dec. 1959. [2] B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L. Visher, and D. F. Sievenpiper, “Scalar and tensor holographic artificial impedance surfaces,” IEEE Trans. Antennas Propag., vol. 58, no. 10, pp. 3212–3221, Oct. 2010. [3] M. Nannetti, F. Caminita, and S. Maci, “Leaky-wave based interpretation of the radiation from holographic surfaces,” in Proc. IEEE AP-S Int. Symp., Honolulu, HI, Jun. 9–15, 2007, pp. 5813–5816. [4] J. A. Kaiser, “The Archimedean two-wire spiral antenna,” IRE Trans. Antennas Propag., vol. 8, no. 3, pp. 312–323, May 1960. [5] H. Nakano, K. Kikkawa, N. Kondo, Y. Iitsuka, and J. Yamauchi, “Lowprofile equiangular spiral antenna backed by an EBG reflector,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1309–1318, May 2009. [6] K. Sarabandi, M. D. Casciato, and I.-S. Koh, “Efficient calculation of the fields of a dipole radiating above an impedance surface,” IEEE Trans. Antennas Propag., vol. 50, no. 9, pp. 1222–1235, Sep. 2002. [7] S. Maci, L. Borselli, and L. Rossi, “Diffraction at the edge of a truncated grounded dielectric slab,” IEEE Trans. Antennas Propag., vol. 44, no. 6, pp. 863–873, 1996. [8] G. Lovat, P. Burghignoli, and D. R. Jackson, “Fundamental properties and optimization of broadside radiation from uniform leaky wave antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1442–1452, May 2006. [9] S. Maci and A. Neto, “Green’s function of an infinite slot printed between two homogeneous dielectrics-Part II: Uniform asymptotic solution,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 666–676, Mar. 2004. [10] S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech., vol. 23, no. 1, pp. 123–133, Jan. 1975. [11] S. Maci, M. Caiazzo, A. Cucini, and M. Casaletti, “A pole-zero matching method for EBG surfaces composed of a dipole FSS printed on a grounded dielectric slab,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 70–81, Jan. 2005. [12] S. Maci and A. Cucini, “FSS-based EBG surfaces,” in Electromagnetic Metamaterials: Physics and Engineering Aspects, R. W. Ziolkowski and N. Engheta, Eds. New York: Wiley Interscience, 2006, ch. 13. [13] P. De Vita, A. Freni, F. Vipiana, P. Pirinoli, and G. Vecchi, “Fast analysis of large finite arrays with a combined multiresolution—SM/AIM approach,” IEEE Trans. Antennas Propag., vol. 54, no. 12, pp. 3827–3832, Dec. 2006. [14] M. Sierra-Castañer, M. Sierra-Perez, M. Vera-Isasa, and J. L. Fernández-Jambrina, “Low-cost monopulse radial line slot antenna,” IEEE Trans. Antennas Propag., vol. AP-51, no. 2, pp. 256–263, Feb. 2003.

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Gabriele Minatti was born in Florence, Italy, on 1982. He received M.S. degree (cum laude) in electronic engineering from the University of Florence, in 2008. He is currently working toward the Ph.D. degree at the University of Siena, Italy. His Ph.D. project is mainly focused on analysis and design of holographic antennas. He also dealt with scattering phenomena by rough surfaces analyzed with a statistic UTD diffraction approach and substrate integrated waveguides numerical modeling. Mr. Minatti was a co-recipient of the Best Paper Award on Antenna Theory at the 5th European Conference on Antennas and Propagation (EuCAP-2011), Rome, Italy, 2011.

Francesco Caminita was born in Montevarchi, Arezzo, Italy, in 1976. He received the Laurea degree (cum laude) in telecommunications engineering and the Ph.D. degree in information engineering from the University of Siena, Italy, in 2005 and 2009, respectively. He is presently a Research Associate at the University of Siena. Since 2006, he has been involved in projects funded by the European Space Agency (ESA) and the European Union (EU) concerning software antenna modeling. His research interests are in the area of periodic structures, frequency selective surfaces, electromagnetic band-gap structures, artificial surfaces and holographic antennas. Dr. Caminita was a co-recipient of the Best Paper Award on Antenna Theory at the 5th European Conference on Antennas and Propagation (EuCAP-2011), Rome, Italy, 2011.

Massimiliano Casaletti was born in Siena, Italy, in 1975. He received the Laurea degree in telecommunications engineering and the Ph.D. degree in information engineering from the University of Siena, Italy, in 2003 and 2007, respectively. From September 2003 to October 2005, he was with the research center MOTHESIM, Les Plessis Robinson (Paris, FR), under EU grant RTN-AMPER (RTN: Research Training Network, AMPER: Application of Multiparameter Polarimetry). Since 2006, he has been Research Associate at the University of

Siena, Italy. His research interests include electromagnetic band-gap structures, polarimetric radar, rough surfaces, numerical methods for electromagnetic scattering and beam methods. Dr. Casaletti was a co-recipient of the Best Poster Paper Award at the 3rd European Conference on Antennas and Propagation (EuCAP-2009), Berlin, Germany, and the recipient of an Honorable Mention for Antenna Theory at EuCAP-2010, Barcelona, Spain, and the Best Paper Award on Antenna Theory at EuCAP-2011, Rome, Italy.

Stefano Maci (S’98–F’04) received the Laurea degree (cum laude) in electronic engineering from the University of Florence, Italy. Since 1998, he has been with the University of Siena, Italy, where he is presently a Full Professor, Director of the Ph.D. School of Engineering and Head of the Laboratory of Electromagnetic Applications (LEA). 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 analysis of large truncated periodic structures. He was responsible and the international coordinator of several research projects funded by the European Union (EU), by the European Space Agency (ESA-ESTEC), by the European Defence Agency, and by various European industries. He was the founder and is currently the Director of the European School of Antennas (ESoA), a post-graduate school that comprises 30 courses on antennas, propagation, and EM modelling though by 150 teachers coming from 30 European research centers. He is the principal author or coauthor of 110 papers published in international journals, (among which 70 are in IEEE journals), 10 book chapters, and about 350 papers in proceedings of international conferences. Prof. Maci is member of the Finmeccanica. He was an Associate Editor of the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, twice Guest Editor and an Associate Editor of IEEE TRANSACTION ON ANTENNAS AND PROPAGATION (IEEE-TAP). He is presently a member of the IEEE AP-Society AdCom, a member of the Board of Directors of the European Association on Antennas and Propagation (EuRAAP), a member of the Executive Team of the IET Antennas and Propagation Network, a member of the Technical Advisory Board of the URSI Commission B, and a member of the Italian Society of Electromagnetism. He was the recipient of several national and international prizes and best paper awards.

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Radiation Bandwidth Enhancement of Aperture Stacked Microstrip Antennas Homayoon Oraizi, Senior Member, IEEE, and Reza Pazoki

Abstract—The radiation pattern of aperture stacked patch (ASP) antennas is improved. In the proposed geometry, the patch shape is designed such that the effective propagation constant of the antenna is decreased and therefore pattern degradation due to higher order modes is eliminated. The results show that the operational bandwidth of the proposed structure is increased from 68% to 76% compared to the traditional ASP. Index Terms—Microstrip antennas, radiation bandwidth.

I. INTRODUCTION

D

UE to the appealing characteristics of microstrip antennas such as low cost, light weight and ease of fabrication, they have many applications in communication systems. However, their low VSWR-bandwidth is a primary barrier to their implementation in many applications. Accordingly, increasing the VSWR-bandwidth has been the focus of research in microstrip antennas [1]–[6]. Among microstrip antennas, aperture stacked patch (ASP) antennas [7]–[10] have the widest VSWR band) which is 79% [8]. Although width (defined by ASP antennas have such a wide VSWR bandwidth, they cannot maintain an acceptable radiation behavior there. It will be shown that the radiation pattern of the antenna is degraded at higher frequencies, although the antenna has an acceptable return loss at these frequencies. Hence the radiation-bandwidth of the ASP antennas is less than their VSWR-bandwidth which limits their overall operational bandwidth. In comparison with increasing the VSWR-bandwidth of microstrip antennas, much less work is devoted to improving their radiation bandwidth. There are some efforts in the literature to enhance the radiation characteristics of microstrip antennas [11]–[13], but they are just a scratch on the surface and when the radiation improvement is attempted, it is mainly focused on increasing the gain of the antenna. In this paper, different sources of pattern deterioration in the ASP antennas are explained and compared. It is concluded that the dominant source of pattern degradation in ASP antennas at the upper frequency band is the generation of higher order modes. Then, a patch configuration is proposed such that these higher order modes are suppressed and the overall operational bandwidth is increased. Here, the radiation bandwidth is defined as the frequency band, where the on-axis Manuscript received July 12, 2010; revised March 24, 2011; accepted June 02, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. The authors are with the Department of Electrical Engineering, Iran University of Science and Technology, Narmak, Tehran 1684613114, Iran (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2165478

Fig. 1. Geometry of an ASP antenna.

pattern depression is less than 1 dB and the gain bandwidth is defined as the frequency interval in which the antenna gain has a 3 dB drop. In [8], two ASP antennas in the frequency range of 1–2 GHz and 2–4 GHz were fabricated. To compare the performance of the proposed structure to that of [8], a parametric study is performed in the frequency range of 1–2 GHz. CST-Microwave Studio simulation software is used for all simulations. An ASP antenna with a modified geometry is designed and fabricated in the frequency range of 2–4 GHz and the results are compared to that of [8]. The results show that the proposed structure has a wider operational bandwidth compared to the antenna results presented in [8]. Here, the antennas in [8] are called the reference antennas. II. THE APERTURE STACKED PATCH ANTENNA The aperture stacked patch (ASP) antenna is a microstrip antenna which utilizes a resonant aperture with stacked patches. This antenna has a wide VSWR-bandwidth [7]–[10]. Its structure is shown in Fig. 1 as an exploded view of a general multilayered configuration having N dielectric layers, where the lower patch is placed directly above layer N1 and the upper patch is placed directly above layer N2. The difference between this type of antenna and the aperture-coupled antennas is that the ASP antenna has a larger aperture and thicker substrate than that of the aperture coupled antennas. In [7] a detailed parameter study of the ASP antenna geometry is performed in order to improve its VSWR-bandwidth. In this paper the effect of the ASP antenna parameters on its radiation characteristics is investigated. III. ANTENNA RADIATION PARAMETER STUDY (1–2 GHZ) In the microstrip antennas there are four major sources of radiation pattern degradation: surface waves, polarization cur-

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TABLE I THE REFERENCE ANTENNA DIMENSIONS [8]

Fig. 2. Geometry of the ASP antenna for parameter study.

Fig. 3. Region i of Fig. 2 to calculate its effective permittivity.

rents, coupling of adjacent radiator elements and higher order modes. Surface waves are associated with the dielectric thickness as well as dielectric constant of the substrate. The dielectric constant also affects the polarization currents [11], [12]. The coupling of adjacent elements in patch arrays and parasitic patches is another factor which impacts on the antenna radiation pattern. In [12], three major sources of couplings are discussed: near field coupling, far field coupling and surface-wave coupling. In ASP antennas, the coupling is by the near-field where the patch spacing affects it. The patch-shape as well as patch dimensions are parameters which bear on the generation of higher order modes. With due regard to orientation of the rectangular slot and its dimensions in Fig. 2, the resonance frequency of the dominant TM10 mode is approximated by (1) where is the width of the patch i, is the effective permittivity of the patch i, and c is the speed of light in free space. In order to calculate the effective permittivity according to [14], we divide Fig. 2 into two regions namely region 1 between the ground plane and lower patch and region 2 between the lower and higher patches. Therefore, each region consists of two dielectric layers according to Fig. 3. The effective permit) may be obtained as [14] tivity of layer (for (2) where and are the dielectric constants of layers 1 and and are the thicknesses of layers 1 and 2 in region i and 2 in region , respectively. To perform a parameter study, the geometry of Fig. 2 is considered. The operational frequency of the structure is 1–2 GHz

Fig. 4. The VSWR of the antenna in different cases.

and the measured data of the VSWR-optimized structure are available in [8]. Table I shows the antenna dimensions. In what follows, the impact of different antenna parameters on the radiation pattern is investigated. A. The Dielectric Thickness We consider three cases to investigate the effect of substrate thicknesses. For the first case, we simulate the reference antenna [8]. For the second case we increase the substrate thickness (d2) of patch 1 of reference antenna from 3.17 mm to 5.47 mm, which amounts to a 4 percent increase of the effective dielectric constant of the layers under patch 1. According to (2), where . According to (1), the ratio of resonance frequency of patch 1 in case 2 to that of case 1 is equal to s. For the third case, the substrate thickness of patch 2 (namely d4) is increased from 1.58 mm to 2.68 mm, which leads to a 4 percent increase of the effective dielectric constant where of substrate under patch 2 namely . The VSWR of antennas are shown in Fig. 4. The radiation patterns of antennas in three cases are shown in Fig. 5(a) to (c) at different frequencies. Observe that the variation of substrate thicknesses leads to change of VSWR. However the VSWR bandwidths of these three cases have not changed. Evidently, if the substrate thicknesses are changed much more, the VSWR may be altered at some points, but the frequency response will not be shifted. On the other hand, observe that by increasing the substrate thicknesses, the responses of antenna radiation patterns will

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TABLE II DIRECTIVITY OF ANTENNAS IN 3 CASES

lower band. However in the upper band too the directivity has changed but it has not shifted and the antenna radiation bandwidth has decreased. The comparison of radiation responses of cases 1 and 3 in Table II, shows that the antenna radiation pattern of case 3 has shifted by the factor s to the lower frequencies in the upper band. However, the radiation pattern has not changed in the lower band. Consequently we may infer that the effective permittivity of the upper patch affects the radiation response in the upper band and that the effective permittivity of the lower patch affects the antenna performance both in the lower and upper bands. Observe that the radiation bandwidth of ASP antennas (similar to most microstrip antennas) starts at a lower frequency than the VSWR bandwidth. Consequently, the antenna radiation response shifts upwards in frequency leading to an increase of the antenna operational bandwidth. The increase of substrate thickness of the lower patch decreases the coupling between the lower patch and the aperture. By increasing the aperture size, the coupling between the aperture and the lower patch is increased. Therefore, to maintain the same coupling and hence the same input impedance, by increasing the feed substrate thickness, the aperture size should be increased. The substrate thickness of the upper patch is inversely proportional to the coupling of theupper and lower patches. This means that to have a constant coupling, by increasing the upper substrate thickness, the size of patches should be increased [7]. Fig. 5. Comparison of the effect of upper and lower patch dielectric thick: mm, d : mm (b) nesses on the antenna directivity. (a) d d : mm, d : mm, (c) d : mm, d : mm.

= 5 47

= 1 58

= 3 17 = 3 17

= 1 58 = 2 68

be shifted to the lower frequency bands. Consequently the change of substrate thicknesses does not lead to similar trends in the variations of VSWR and radiation patterns. The value of VSWR depends not only on the substrate characteristics and patch shapes, but also to a large degree on the coupling among the substrates. However, the propagation constant of each layer is the determining factor of the antenna radiation behavior. In order to compare the radiation responses of the antennas in the three cases, their directivities are calculated at three frequencies in the lower band and three frequencies in the upper band with the . They are shown in Table II. Observe that the radiation response of the antenna of case 2 has shifted to the lower frequencies by the ratio s in the

B. The Patch Dimensions In order to study the effect of patch dimensions on the radiation pattern, we first simulate the reference antenna as case 1. We then decrease the width of the lower patch of reference antenna from w1 to 0.9 w1 as case 2. Finally, we decrease the length of upper patch of reference antenna from w2 to 0.9 w2 as case 3. The radiation patterns of the three cases are drawn in Fig. 6(a) to (c) at frequencies in the upper band. Their VSWR values are drawn in Fig. 7. Observe that their values change as the patch width decreases, but they are not shifted in frequency. However, the radiation patterns are shifted to the upper frequencies, but their displacements are not exactly inversely proportional to w, as in (1). The resonance frequency of the second transverse mode TM02 of patch is obtained from (3)

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Fig. 7. VSWRs of the antennas in the three cases mentioned in Fig. 6.

Fig. 8. Radiation pattern of the reference antenna.

Fig. 6. Radiation patterns of the ASP antenna for three different patch sizes. (a) Case 1: The reference antenna (b) case 2: The lower patch width of the reference antenna (w1) is changed to 0.9 w1 and (c) case 3: The upper patch width of the reference antenna (w2) is changed to 0.9 w2.

Where is the length of the patch i, in Fig. 2. Observe that these frequencies are close to the dominant mode. This means that the antenna radiates through cross polarization in the TM02 mode, besides the radiation in the dominant mode TM10. The TM02 radiation shows itself in the increase of side lobe levels. Note that it is not the absolute dimensions of patches which affect the antenna input impedance behavior, rather their relative size is the determining factor [7]. C. The Aperture Dimensions To study the effect of aperture dimensions on the antenna radiation pattern, a comparison scenario is made as follows. First,

Fig. 9. Radiation pattern of the antenna with scaled aperture dimensions with different scaling factors.

the antenna radiation pattern with the primary dimensions of Table I is observed at the two frequencies 1.8 GHz and 2 GHz. The results are illustrated in Fig. 8. Note that the pattern is degraded at frequency 2 GHz. If this degradation is due to the aperture, by scaling the slot dimensions (width and length) by a (i.e., ), factor of the antenna radiation pattern at the frequency of 1.8 GHz should be degraded. Fig. 9 shows the antenna radiation pattern with different slot scalings. It is seen that the antenna radiation pattern . Therefore, it can is almost the same even for the scaling be deduced that the quick drop of the gain is not due to the aperture size. Varying the aperture length has a major effect on the antenna VSWR in the lower frequency band. Increasing the aperture

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Fig. 11. The proposed structure.

TABLE III THE PROPOSED ANTENNA DIMENSIONS

Fig. 10. The surface current distribution on top of patches of the ASP structure in different frequencies (a) f = 1:2 GHz, (b) 1.5 GHz and (c) f = 1:9 GHz.

IV. THE PROPOSED STRUCTURE (2–4 GHZ) size increases the coupling between the lower patch and the aperture [7]. D. Current Distribution on Patches The radiation mechanism of the ASP antenna is basically similar to that of the other microstrip antennas. The antenna radiation may be described according to the cavity model [15], transmission line model [15], and electric current distribution model on the patches [15]. In the cavity model the radiation is due to the equivalent magnetic currents on the substrate height over the patch periphery. In the transmission line model, the ground plane and the patches (with aperture feed) may be considered as a set of coupled lines. In the current distribution model, the induced electric current on the patch surfaces produce the radiation. With reference to the coordinate axes shown in Fig. 2, the surface currents on the patches perpendicular to its radiation edges generate the radiation. These current distributions are depicted in Fig. 10(a), (b), (c) for frequencies of 1.2 GHz, 1.5 GHz and 1.9 GHz. Observe that the concentration of surface currents is on the lower and upper patches for frequencies 1.2 GHz and 1.5 GHz, respectively. However, the radiation characteristics at 1.9 GHz are degraded due to the inappropriate surface current distribution, which lacks component at its radiative edges.

Considering the effect of the antenna geometrical dimensions on the antenna radiation, a modification of the patch shape is proposed within the frequency band of 2–4 GHz and the results are compared with those in [8]. The proposed structure is shown in Fig. 11 and the antenna dimensions are given in Table III. Note that an extra patch is added to the antenna, and the radiating edges of the two upper patches are jagged. However, in the proposed geometry, the addition of an extra jagged patch can compensate for the impedance mismatch. By jagging the parasitic patches, the effective patch width is reduced and hence according Section III.B, the radiation bandwidth of the antenna is improved. The surface current distribution on the jagged patch as shown in Fig. 12(a)–(c) show an increased density at its tip and sharp edges, which lead to enhanced radiation characteristics of the proposed antenna. The distribution of on each patch is co-directional at frequencies 2.5 and 3.5 GHz as shown in Fig. 10(a) and (b) which generate strong radiation. However at 4.5 GHz, the appearance of the transverse second mode (contribution of current) as shown in Fig. 12(c), leads to poor radiation. We now investigate the effect of the third extra patch on the ASP antenna. We consider four configurations: (1) ASP antenna with jagged patch (without an extra patch); (2) addition of an extra jagged patch to the antenna in case (1); (3) addition of an extra rectangular patch to the reference antenna configuration

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Fig. 12. The surface current distribution on top of patches of the proposed structure in different frequencies (a) f = 2:5 GHz, (b) 3.5 GHz and (c) f = 4:5 GHz.

in [8]; and (4) addition of an extra jagged patch to the reference antenna. The gains of the four configurations are shown in Fig. 13(a). Observe that the gain of cases (3) and (4) have the lowest 3 dB gain bandwidths. Observe that jagging one of the main patches (namely the second one) is the major cause of increasing the radiation bandwidth, and not the extra patch. Furthermore, observe that the return loss (and equivalently the input impedance matching) of the proposed antenna configuration is improved by the addition of the extra patch, as shown in Fig. 13(b). Note that in cases 3 and 4, the extra patches are added to the reference antenna which its parameters are optimized. Since addition of an extra patch does not considerably change the antenna performance, it can be deduced that the four cases are antennas with optimized parameters. The co- and cross-polar radiation patterns of the reference antenna and the proposed antenna are drawn in Figs. 14 and 15 for comparison. Observe that the cross polar radiation level of the proposed antenna is lower than that of the reference antenna. The cross polar radiation is due to the non-radiating edges of the antenna and the y-component of the surface magnetic curon the jagged patch periphery. The antenna co-polar rents on the radiradiation is due to the magnetic surface currents ating periphery. However, the magnetic surface current components on the surfaces of the jagged periphery add up to produce component and cancel the component. Furthermore, an

Fig. 13. Comparison of (a) gains of the ASPs with four different cases: case 1: double modified patch, case 2: addition of an extra patch to the case 1, case 3: addition of an extra rectangular patch to the reference antenna in [8] and case 4: addition of an extra jagged patch to the reference antenna in [8] (b) S comparison of cases 1 and 2.

Fig. 14. Co-polar radiation patterns of the proposed and reference ASP antenna at 3 GHz (in H-plane ( = 0 )).

the

component on the radiating periphery partly cancels the component on the non-radiating peripheries. and widths For the comparison of effective lengths of the rectangular and jagged patches, we consider two cases. They are a probe fed rectangular and a jagged patch with and as defined in Table III, with physical dimensions , and , respectively. The resonance frequencies of the two cases are shown in Fig. 16, where the input is drawn versus frequency. Observe that the impedance first resonance frequency is decreased by jagging the patch,

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Fig. 18. VSWR of the proposed ASP antenna and the ASP antenna in [8]. Fig. 15. Cross-polar radiation patterns of the proposed and reference ASP antenna at 3 GHz (in H-plane ( = 0 )).

Fig. 16. Resonant frequencies of probe fed rectangular and jagged patches with physical dimensions L and W as defined in Table III.

Fig. 17. Photograph of the fabricated antenna.

which is inversely proportional to the effective length . However the second resonance frequency of the jagged patch is increased and is inversely proportional to its effective width, . In fact jagging the rectangular patch increases its effective length and decreases its effective width. V. THE FABRICATED ANTENNA Fig. 17 shows a photograph of the fabricated antenna. The measured VSWR of the proposed antenna and the VSWR of antenna in [8] are sketched in Fig. 18. It is seen that the impedance bandwidth of the proposed antenna is 2.03–5.15 GHz (86%) and has a much wider impedance bandwidth compared to the results of [8]. The measured E and H plane radiation patterns of

Fig. 19. Measured E-plane ( = 90 ) and H-plane ( = 0 ) radiation patterns @3 GHz.

Fig. 20. Measured E-plane ( = 90 ) and H-plane ( = 0 ) radiation patterns @4 GHz.

the proposed ASP structure are shown in Figs. 19–23. To investigate the radiation behavior of the antenna at the upper band, the measurements are focused on the antenna upper band i.e., 4 GHz. It is seen that the gain bandwidth of proposed antenna is 2–4.4 GHz, while the simulations show that the antenna of [8] has a gain bandwidth of 2–3.9 GHz. Fig. 24 illustrates the measured and simulated gain of the proposed antenna structure and the measured results of the antenna of [8]. It should be mentioned that the gain measurements are performed using the 3-antenna system [15]. It is obvious that the proposed antenna has a higher gain than that of [8]. It is seen that the upper frequency band of the proposed structure is

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4.5 GHz, while that of [8] is 4.05 GHz, hence the operational bandwidth of the proposed structure is 76% (2.03–4.5 GHz) which compared to the antenna of [8], (2–4.05) some 8% increase is achieved. VI. CONCLUSION

Fig. 21. Measured E-plane ( = 90 ) and H-plane( = 0 ) radiation patterns @4.1 GHz.

To enhance radiation bandwidth, a modified geometry of the ASP antenna has been proposed. To investigate the radiation behavior of the antenna, a parametric study of different antenna parameters was made. It was concluded that the radiation section of the ASP antenna which includes parasitic patches, and patch substrates have impact on the radiation pattern of the antenna and among them, the patch-widths play the most important role to the antenna radiation pattern. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their constructive comments to improve the paper. REFERENCES

Fig. 22. Measured E-plane ( = 90 ) and H-plane ( = 0 ) radiation patterns @4.2 GHz.

Fig. 23. Measured E-plane ( = 90 ) and H-plane ( = 0 ) radiation patterns @4.3 GHz.

Fig. 24. Comparison of the measured and simulated gain of the proposed structure with the gain of [8] antenna.

[1] C. H. Tsao, Y. M. Hwang, F. Killburg, and F. Dietrich, “Aperture coupled patch antenna with wide bandwidth and dual polarization capabilities,” in IEEE Antennas Propagat. Soc. Symp. Dig., New York, Jun. 1988, pp. 936–939. [2] F. E. Gardiol and J. F. Zurcher, “Broadband patch antennas—A SSFIP update,” in IEEE Antennas Propag. Soc. Symp. Dig., Baltimore, MD, Jul. 1996, pp. 2–5. [3] F. Croq and D. M. Pozar, “Millimeter wave design of wide-band aperture-coupled stacked microstrip antennas,” IEEE Trans. Antennas Propag, vol. 39, pp. 1770–1776, Dec. 1991. [4] J. F. Zurcher, “The SSFIP: A global concept for high performance broadband planar antennas,” Electron. Lett., vol. 24, pp. 1433–1435, Nov. 1988. [5] F. Croq and A. Papiernik, “Wide-band aperture coupled microstrip antenna,” Electron. Lett., vol. 26, pp. 1293–1294, Aug. 1990. [6] S. D. Targonski and D. M. Pozar, “Design of wide-band circularly polarized aperture coupled microstrip antennas,” IEEE Trans. Antennas Propag, vol. 41, pp. 214–220, Feb. 1993. [7] S. D. Targonski, R. B. Waterhouse, and D. M. Pozar, “Design of wide-band aperture-stacked patch microstrip antennas,” IEEE Trans. Antennas Propag., vol. 46, pp. 1245–1251, Sep. 1998. [8] J. K. Ghorbani and R. B. Waterhouse, “Ultrabroadband printed (UBP) antenna,” IEEE Trans. Antennas Propag., vol. 50, pp. 1697–1705, Dec. 2002. [9] W. S. T. Rowe and R. B. Waterhouse, “Reduction of backward radiation for CPW fed aperture stacked patch antennas on small ground planes,” IEEE Trans. Antennas Propag., vol. 51, pp. 1411–1413, Jun. 2003. [10] K. Ghorbani and R. B. Waterhouse, “Dual polarized wide-band aperture stacked patch antennas,” IEEE Trans. Antennas Propag., vol. 52, pp. 2171–2175, Aug. 2003. [11] M. M. Nikolic, A. R. Djordjevic, and A. Nehorai, “Microstrip antennas with suppressed radiation in horizontal directions and reduced coupling,” IEEE Trans. Antennas Propag., vol. 53, pp. 3469–3476, Aug. 2005. [12] M. M. Nikolic and A. R. Djordjevic, “Improving radiation pattern of microstrip antennas,” presented at the 1st Eur. Conf. on Antennas and Propagation, 2006. [13] R. Q. Lee and K. F. Lee, “Gain enhancement of microstrip antennas with overlaying parasitic directors,” Electron. Lett., vol. 24, pp. 656–658, May 1988. [14] C. S. Lee, V. Nalbandian, and F. Schwering, “Planar dual-band microstrip antenna,” IEEE Trans. Antennas Propag., vol. 43, no. 8, pp. 892–894, Aug. 1995. [15] C. A. Balanis, Antenna Theory, Analysis and Design, 2nd ed. New York: Wiley, 1977.

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Homayoon Oraizi (SM’98) received the B.E.E. degree from the American University of Beirut, Beirut, Lebanon, in 1967, and the M.Sc. and Ph.D. degrees in electrical engineering from Syracuse University, Syracuse, NY, in 1969 and 1973, respectively. From 1973 to 1974, he was a teacher with the K. N. Tousi University of Technology, Tehran, Iran. From 1974 to 1985, he was with the Communications Division, Iran Electronics Industries, Shiraz, Iran, where he was engaged in various aspects of technology transfer mainly in the field of HF/VHF/UHF communication systems. In 1985, he joined the Department of Electrical Engineering, Iran University of Science and Technology, Tehran, where he is currently a Full Professor of electrical engineering. He teaches various courses in electromagnetic engineering and supervises theses and dissertations. He has conducted and completed numerous projects in both industry and with universities. From July 2003 to August 2003, he spent a two-month term with Tsukuba University, Ibaraki, Japan. From August 2004 to February 2005, he spent a six-month sabbatical leave at Tsukuba University. He has authored or coauthored over 200 papers in international journals and conferences. He has authored and translated several textbooks in Farsi. His research interests are in the area of numerical methods for antennas, microwave devices, and radio wave propagation. Dr. Oraizi is a Fellow of the Electromagnetic Academy and of the Japan Society for the Promotion of Science. His translation into Farsi of Antenna Analysis and Design, 4th ed. (Iran Univ. Sci. Technol., 2006) was recognized as the 1996 Book of the Year in Iran. In 2006, he was elected an exemplary nationwide university professor in Iran. He is an Invited Professor of the Electrical Engineering Group, Academy of Sciences of Iran, Tehran, Iran. He is listed as an Elite Engineer by the Iranology Foundation. He was listed in the 1999 Who’s Who in the World.

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Reza Pazoki received the B.S. degree from the Khajeh Nassir Toosi University of Technology, Tehran, Iran, in 2002 and M.S. degree from Iran University of Science and Technology (IUST), in 2004, where he is currently working towards the Ph.D. degree. For several years, he served as an antenna design engineer in different research centers. He has designed, modified and fabricated various types of antennas such as waveguide fed slot, spiral, omnidirectional slant polarized, horn, biconical, bow-tie, loop, reflector, and LPDA, lens, Yagi, and microstrip antennas. In addition to the antenna design, he is also interested in numerical techniques in electromagnetics, especially FDTD, parabolic equation and spectral domain methods. His current research topics are coupling compensation in antenna arrays, modeling of HF propagation in irregular terrain and phased arrays with circular polarization.

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Compact Metallic RFID Tag Antennas With a Loop-Fed Method Peng H. Yang, Yan Li, Lijun Jiang, Member, IEEE, W. C. Chew, Fellow, IEEE, and Terry Tao Ye

Abstract—Several compact, low profile and metal-attachable RFID tag antennas with a loop-fed method are proposed for UHF RFID systems. The structure of the proposed antennas comprise of two parts: (1) The radiator part consists of two shorted patches, which can be treated as two quarter-wave patch antennas or a cavity. (2) A small loop printed on the paper serves as the feeding structure. The small loop provides the needed inductance for the tag and is connected to the RFID chip. The input impedance of the antenna can be easily adjusted by changing loop dimensions. The antenna has the compact size of 80 mm 25 mm 3.5 mm, and the realized gain about 3 6 dB. The measured results show that these antennas have good performance when attached onto metallic surfaces. Index Terms—Compact antenna, feeding network, low profile, metallic surface, RFID tag antenna.

I. INTRODUCTION ADIO FREQUENCY identification (RFID) tag has been widely used recently in supply chain and logistics applications to identify and track goods. Tag antenna is one of its key technologies. In order to reduce the cost, most existing RFID systems use modified dipole antennas as tags; these dipole-type antennas can be printed on paper or plastic materials and then pasted on products. They have the merits of small size and are easy to fabricate. However, dipole-type antennas are sensitive to the environment due to their omni-directional radiation characteristics [1], [2]. For example, dipole antennas often show high performance when pasted on paper or plastic boxes, but they do not work when pasted on metal surfaces or bottles with liquid in it. Microstrip antenna is a good choice for metal-attachable RFID tags because of the ground plane in its structure. In [3], the author investigated the performance of microstrip-type tag antennas using the cheapest dielectric FR4 as substrate. These

R

Manuscript received September 13, 2010; revised May 13, 2011; accepted June 21, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. This work was supported in part by Hong Kong R&D Centre for Logistics and Supply Chain Management Enabling Technologies (LSCM) and in part by Innovation and Technology Fund (ITF, No. ITS/159/09), the government of the Hong Kong SAR. P. H. Yang was with the Department of Electrical and Electronic Engineering, University of Hong Kong, Hong Kong, China. He is now with University of Electronic Science and Technology of China (UESTC), Chengdu, China (e-mail: [email protected]). Y. Li, L. Jiang, and W. C. Chew are with the Department of Electrical and Electronic Engineering, University of Hong Kong, Hong Kong, China (e-mail: [email protected]). T. T. Ye is with the Hong Kong R&D Centre for Logistics and Supply Chain Management Enabling Technologies, Hong Kong, 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.2011.2165484

antennas are easy to fabricate but are large in size. Hence, it is unsuitable for RFID tag applications. In [4], a compact microstrip antenna with some slots loaded is proposed. Also, a simple feeding structure is used and the input impedance can be adjusted easily by changing the length of the feeding line alone. However, this structure requires a via-hole to connect the patch and the ground, which increases the fabrication cost. In [5], [6], patches are fed by a small loop. The merit of this inductively coupled feeding technique is that the imaginary part of the input impedance can be easily changed by tuning the loop size and the distance between the loop and the patches. In order to further reduce the size, the planar inverted-F antenna (PIFA) is also proposed for RFID tag antenna designs [7]. However, because PIFA antennas often need coaxial probe feedings, this type of structure requires embedding the RFID chip vertically between the ground plane and the radiation patch. Hence, it is difficult to fabricate. In [8], [9], the authors proposed to use a slot or aperture antenna as the radiator. A RFID chip is put on the center of the slot as the feeding source. In [10], the RFID chip is connected to a small dipole as a coupling source of the slot. The input impedance of this tag can be adjusted by changing the location of the small dipole. The shortcomings of these slot-type antennas are their size seem still large for RFID tags. Recently, many people consider using artificial magnetic conductor (AMC) or electronic band gap (EBG) structures for tag antenna designs. Because these new artificial structures have the character of zero-reflection phase, the dipole-type tag antenna can be put very close to them. In [11], a dipole is put on a 5 3 AMC plate as the tag antenna. This antenna has the advantage of the high gain (about 4.5 dB) but the drawbacks of large size and high cost. It is suitable for reader antennas instead of tag antennas. There are some other compact and low-profile AMC structures [12], but the complicated structures and narrow bandwidth (just a few megahertz) limit its application. Most RFID tags are disposable, which is acceptable for dipole-type antennas because of their low cost. However, for microstrip-type antennas, disposable designs will lead to a big waste. The microstrip-type antennas introduced above have relatively higher cost compared to dipole-type tags. Meanwhile their structures are unchangeable, implying that a fixed structure can only be used for one RFID chip. If we want to replace the chip, the entire antenna must be re-designed and re-fabricated. Therefore, a simple, reusable microstrip antenna is attractive for low-cost metal-attachable RFID systems. In [13], we have proposed an idea for the reusable microstrip tag antenna design. This antenna shows good performance when

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attached on metallic objects. But the physics of its feeding structure was not very clear and the size was relatively large. In this paper, several compact structures are proposed, analyzed and tested. In Section II,the motivation and principle of the design are introduced. Then, in Section III, several key parameters of the structure are analyzed and discussed. In Section IV, prototypes are fabricated and tested. This kind of designs is summarized in Section V. II. ANTENNA DESIGN Two issues motivated us to design a flexible feeding structure for low cost metallic RFID tags: First, unlike conventional 50 Ohms antennas, RFID tag antenna has complex input impedance because it should be a conjugate match to the RFID chip. There is no unified standard for RFID chips. Different chips have different impedance. It is impossible to design an antenna that can match all kinds of chips. Second, most tag antennas are disposable design because the information in the chip is unique for a certain product. If the chip is integrated into the antenna, the antenna can be used only once. For the reasons mentioned above, we propose to design the feeding network and the radiator separately. The radiator consists of two symmetrical patches, which are mounted on a dielectric substrate. One edge of the patch is shorted to the ground, and there is a gap between the two patches. This gap is the radiating slot. The radiator part can be seen as two quarter-wave patch antennas sharing a common radiating slot, or a cavity with a slot loaded. For traditional patch antennas, whose radiating slots are on the side edges of the patch, the ground is usually larger than the patch. The advantage of our design is to move the radiating slots from the edge to the center, which can reduce the total size of the antenna effectively. The coaxial probe can be put at a proper location of one of the two patches to feed this structure [14], [15]. Then this patch is regarded as the primary patch. Another one is the parasitic patch. This is the simplest feeding method. Unfortunately, it is not suitable for the RFID tag antenna. Another problem is that the parasitic patch has a phase delay compared to the primary patch due to the asymmetric feeding. To feed two patches simultaneously, we can connect the two patches directly by a RFID chip [8], [9], [16], [17]. This is straightforward but has some drawbacks, such as the inflexibility and difficulties of impedance matching. A feeding method of putting a small dipole on the top of the gap is proposed to feed the structure [10], [13]. The length of the dipole is far less than the operating wavelength of the antenna. To couple more energy into the cavity, the dipole should be put . This structure can be regarded very close to the slot as a cavity-backed slot antenna. If the dimension of the cavity is mode will be excited and resonate in designed properly, the cavity. The dipole and the cavity have a strong coupling at the resonant frequency, and then the energy can be transformed from the dipole to the cavity through the slot. Usually, the input impedance of a RFID chip has a small real part and a large negative imaginary part (capacitive). Hence, for conjugate matching, a loop (inductive) antenna is preferred than the small dipole as the feeding network. The geometry of the proposed loop-fed antenna is shown in Fig. 1(a).

Fig. 1. A small loop put on a pair of shorted patches. (a) The geometry. (b) : ohms, L nH, C : pF, C : Equivalent circuit with R pF, C : pF, R ohms, C : pF and L nH. (c) Antenna parameters and current distribution on the patches with L ;W ; ;W ; w ; d : and h . The strip width of g ;L the loop is 1 (all dimensions are in mm).

=02 = 5 = 16

=03 = 24 = 850 =77 = 12 = 4 = 0 5

= 0 02 =4 = 39 =3

= 0 15 = 25

Fig. 1(b) shows the equivalent circuit of the antenna. To model the small loop, a capacitance is put in parallel with resistance and inductance . The capacitance accounts for the distributed capacitance between the sides of the loop. Note that a loop with an uniform current distribution would have no capacitance, since there would be no charge along the conductor of the loop. For the loop-fed structure, if the loop is extremely small, the component current on the loop have the same magnitude but opposite directions so they cancel each other. Hence, the cavity cannot be excited in this case. However, when the loop becomes larger and larger, the component current have different magnitude and opposite directions. Hence, there is a net current along the axis. This net current looks like an electric dipole and can be used as the excitation source. The and inductance of the small rectangular loop resistance can be estimated approximately by [1], [21] (1) (2) where is the perimeter of the loop and is the width of the loop strip. Because the thickness of the cavity is very thin, according to the cavity model [18], [19], the two quarter-wave patch an-

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tennas (or cavity) can be represented as two parallel circuits. of the patch can be estimated roughly by The capacitance using the parallel plate wave guide model [2] (3) where is the intrinsic impedance of the medium between the parallel plates. Because the resonant frequency is , we can obtain (4) and is the guided wavehere we suppose that length in the cavity. Through (3) and (4), the equivalent capacand inductance of each patch are 7.7 pF and 4.0 itance of single patch can nH, respectively. The radiation resistance also be estimated using cavity model. Since the energy is coupled from the gap into the cavity, the location of feeding point in cavity model should be chosen close to the gap (the radiation is about 850 ohms. edge). The estimated resistance can be estimated approximately by [20] The capacitance

Fig. 2. Simulation results of full wave method and equivalent circuit model for the loop-fed antenna.

(5)

If the width of the gap mm, then pF. Note that these parameters are just approximate values. When the small loop is put close to the patches, these values would change due to the coupling between the loop and the patches. It is very hard to determine these values analytically if coupling effects are taken into consider. But they can be extracted by comparing the results of circuit model and full wave method (Volume Surface Integral Equation codes developed by our group). After a little as well as the coubit tuning, the precise values of and can be determined and the results of circuit pling capacitance model can match well to the full wave results. The ultimate parameters of these lumped elements and the antenna are given in Fig. 1(b) and (c), respectively. The input impedance of full wave method and equivalent circuit models are shown in Fig. 2. Good agreement is achieved. To evaluate the performance of the proposed antenna when mounted on metallic objects, we pasted this loop-fed tag antenna on a 200 mm 200 mm metallic plate. The dielectric has a relative permittivity of 4.2 and loss tangent of 0.02. The medium between the small loop and the cavity was set as air to simplify the simulation. Fig. 3(a) is the gain pattern of the antenna. The dB near 915 simulated maximum realized gain is about MHz, which is enough for most low-cost RFID tags. The current distribution on the loop and the patches are shown in Fig. 3(b). Note that the current at point “ ” and point “ ” are not symmetrical. Just as the current direction shown in Fig. 1(c), at point “ ”, the component of the current have the opposite direction. Hence they cancel each other. However, at point “ ”, the current have the same direction then add together.

Fig. 3. Performance study of the proposed loop-fed antenna. (a) Gain pattern of the proposed antenna at 915 MHz. (c) Current distributions at 915 MHz.

III. PARAMETERS STUDY In this section, we focus on several key parameters of the structure to see their effects to the performances of the antenna. , the distance These parameters include: The loop size between the loop and the edge of the patches and the gap between the two patches. In all simulations, the tags were supposed to be mounted on a 200 mm 200 mm metallic plate. A. Loop Size Our goal is to design a separable feeding network which can match different RFID chips by changing the size or structure

YANG et al.: COMPACT METALLIC RFID TAG ANTENNAS WITH A LOOP-FED METHOD

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Fig. 5. Input impedance for different value w . The loop size was fixed at 14 mm 14 mm. The parameter w was changed from 0 mm to 10 mm.

2

Fig. 4. Input impedance and S for different loop size. (a) Input impedance. (b) S . The blue line to match the RFID chip with impedance of 6 j127 at 915 MHz, the red line to 13 j140, the black line to 7 j170 and the pink line to 30 j200. Other parameters are: L = 39; W = 25; g = 5; d = 0:5 and h = 3. The strip width of the loop is 1 (all dimensions are in mm).

0

0

0

0

of the feeding network while keeping the radiator part unchanged. To simplify the problem, we fixed others parameters only. Four types of RFID and adjusted the loop size chips were chosen to do the simulation. Around 915 MHz, their [22], (Alien-H2), impedance are: [23] and (Alien-H3). To make this tag work in the north American frequency range (from 900 MHz to 930 MHz) , the with these chips, we can change the loop size results are shown in Fig. 4. Fig. 4(a) are the input impedance of the tag with different loop size: The blue line is to match the RFID chip with impedance at 915 MHz, the red one is to , the black of and the pink one is to . It was found one is to that with the perimeter of the loop becomes larger and larger, the resonance becomes stronger. Also, the resonant frequency tends to become lower because big loop will induce a large net current along the axis, which is equal to the increase of the coupling capacitance . From circuit theory, it will decrease . It can be seen the resonant frequency. Fig. 4(b) shows the that all of the four RFID chips can be matched well within the operating frequency range.

Fig. 6. Input impedance for different g . The loop size was fixed at 14 mm mm. The parameter g was changed from 2 mm to 10 mm.

2 14

It is worth to say that in this example, to change the input impedance, we only adjusted the loop size. Hence, the freedom is very limited. To add more freedoms into the feeding network, more complex feeding network structures, such as the T-mach or Gamma-match [1] can be designed, and then more RFID chips might be matched. B. The Distance Patches

Between the Loop and the Edge of the

In all of the aforementioned examples, the loops were put at the center of the slot. In fact, the location of the loop is not sensitive to the input impedance. To approve this, we fixed the at 14 mm 14 mm, other parameters were loop size the same to last example except . Then was changed from 0 mm to 10 mm. It is clear to see from Fig. 5 that the parameter will not affect the input impedance. C. The Gap

Between the Two Patches

In this example, we analyzed the effects of parameter to the input impedance. Similar, we fixed the loop size at 14 mm 14 mm, other parameters are the same to last two examples except

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Fig. 7. The fabricated prototype of the loop-fed double patches antenna, the total dimension is 80 mm 25 mm 3.5 mm.

2

2

Fig. 9. The loop-fed single patch antenna. The total dimension is 45.5 mm mm 3.5 mm. (a) The geometry. (b) The prototype.

2

2 20

more lower. Though small value of can reduce the total size of the antenna, it will also decrease the gain because of the strong mutual coupling between the two patches. Hence, cannot be set too small. IV. FABRICATION AND MEASUREMENTS

Fig. 8. The input impedance and S of the loop-fed double patch antenna (Type A). The antenna was put on a 200 mm 200 mm metallic plate.

2

. It is found in Fig. 6 that with the value of decreases, the resonant frequency becomes lower. The reason can be explained as follow: The center of the antenna can be regarded as a vircan be seen as the sum of tual ground and the capacitance two open ended capacitances (the capacitance between the radiof ating edge of the patch and the virtual ground). The length the patch will be a little bit shorter than quarter-wave dielectric length because the open ended capacitance effects. When decreases, the two open ended capacitances will increase. It means looks more for the same resonant frequency, the patch length shorter, or, for the same length , the resonant frequency looks

The advantage of the proposed idea is that the feeding network and the radiator can be designed separately. From the discussion above, the working mode of the antenna is determined by the resonant mode of the cavity. If we choose the size of the cavity properly, the antenna can work well at the mode. On the other hand, the feeding structures can be used as the excitation and impedance matching network. Different feeding structures will affect the input impedance significantly. This gives us an inspiration to design a disposable feeding network for tag antennas. This feeding structure should be simple, easy for fabrication and low cost. In the aforementioned examples, in order to simplify the simulation, we use the air as the substrate between the feeding network and the patches. In practice, we should find a proper material as the substrate. Paper is a good choice because it is very cheap and easily available. The most important point is that it is easy to print feeding circuits on the paper by using the conductive ink or copper foil. In our designs, we use some common paper, such as those used for name cards, as the substrate. The estimated relative permittivity constant of the paper is around

YANG et al.: COMPACT METALLIC RFID TAG ANTENNAS WITH A LOOP-FED METHOD

Fig. 10. The input impedance and S of the loop-fed single patch antenna (Type B). The antenna is put on a 200 mm 200 mm metallic plate.

2

3.2–3.5 [24], and the loss tangent is about 0.08. The drawback of paper is its high loss. But it is acceptable for tag antennas because the read range requirement of most passive RFID tag applications are just few meters. A. Loop-Fed Double Patch Tag Antenna (Type A) The loop-fed double patch antenna was fabricated and tested. The fabrication prototype is shown in Fig. 7. The antenna used the Alien’s RFID chip, whose impedance is about 30–200j at 915 MHz. The dielectric in the cavity is FR4. Its measured permittivity is about 4.2 and loss tangent is about 0.02 [10], [17]. The parameters of the antenna are (all in mm): and . The strip width of the loop is 1 mm. With the method proposed in [25], [26], the antenna was measured through the Agilent’s four ports vector network analyzer. Fig. 8 shows the of measured results. This tag antenna input impedance and was mounted on a 200 mm 200 mm metallic plate. It is clear to see that the antenna can match well around 920 MHz. B. Loop-Fed Single Patch Tag Antenna (Type B) The antenna proposed above has a symmetrical structure: The cross section ( plane) at the center of the antenna can be seen as a perfect electric conductor (PEC), see Fig. 1. Hence, it is possible to split the antenna along plane to reduce the total

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Fig. 11. The loop-fed single patch antenna with slots loaded. The total dimension is 30 mm 20 mm 3.5 mm. (a) The geometry. (b) The prototype.

2

2

dimension. Fig. 9(a) shows the geometry, where and (all in mm). The width of the loop strip is 1 mm. Fig. 9(b) is the of prototype. Fig. 10 is the measured input impedance and this antenna when mounted on a 200 mm 200 mm metallic plate. Here the FR4 has a relative permittivity about 4.2 and loss tangent about 0.04. The paper is the same as Type A. It can match well around 925 MHz by adjusting the parameters carefully. C. Loop-Fed Single Patch Tag Antenna With Slots Loaded (Type C) In order to further reduce the size of the tag, two slots were added on the patch. These slots can bend the patch surface current paths to achieve a lower fundamental resonant frequency. The geometry and prototype of the compact tag antenna are shown in Fig. 11(a) and (b), respectively. The parameters in Fig. 11(a) are: and (all in mm). The width of the loop strip is 1 mm. The parameters of FR4 and paper are the same as Type B. The measured results of the compact antenna mount on a 200 mm 200 mm metallic plate are shown in Fig. 12. From the results we can see that it can match well around 915 MHz.

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TABLE I READ RANGE OF THE PROPOSED ANTENNAS

objects but poor performance in free space because the small ground will lead to a large back radiation. Also, the resonant frequency in free space would be lower than the case when they are mounted on a large metallic plate. Finally, we compared our tags with other designs. In [17], the structure is very similar to ours but with a direct feeding method. The gain of [17] is about dB and the maximum read range is 3.1 meter when mounted on a metallic sheet. In [27], the maximum read range of the commercialize tag can reach 7 meter when mounted on a 300 mm 300 mm metallic plate. The performance of [27] is very good but also with the drawback of large assembled size: 9.45 cm 7.2 cm 1 cm. V. CONCLUSION

Fig. 12. The input impedance and S of the loop-fed single patch antenna with slots loaded (Type C). The antenna is put on a 200 mm 200 mm metallic plate.

2

Fig. 13. The simulated realized gain of the three type tag antennas (The antennas are put on a 200 mm 200 mm metallic plate).

2

D. Read Performance and Comparisons With Other Tags The realized gains of these tags were also investigated and the results are shown in Fig. 13. The maximum gain within the dB, dB and dB operating frequency rang are for Type A, Type B and Type C, respectively. The main reason for the low gain is due to the high loss of the paper we used. The read ranges of these prototypes were tested using Impinj reader IPJ-R1000. The operating frequency hops from 900 MHz to 930 MHz. We fixed the output power to 30 dBm and measured the read distances of these tags (summarized in Table I). These antennas have a good performance when mounted on metallic

A type of loop-fed compact UHF band RFID tag antennas for metallic objects is presented in this paper. These antennas use the quarter-wave patch structure (or a cavity) as the radiator and a small loop as the feeding network. The cavity determines the resonant mode while the feeding part is adjustable to match the required input impedance. The feeding network and the radiator are designed separately. The feeding part is simple with the low cost. It can be printed on paper using some copper foil or conductive ink. Hence, it is a disposable design. Thanks to the separable idea, the radiator part (the quarter-wave structure) can be used many times for different RFID chips or feeding networks, which will lower the total cost of metallic tag antennas. Three designs are proposed. The smallest one has the size of 30 mm 20 mm 3.5 mm, which is just 1/10 wavelength in free space. The merits of their compact, low cost, and metal-attachable properties make the proposed antennas well suitable for packaging RFID applications with metallic objects. ACKNOWLEDGMENT The authors would like to thank Dr. H. L. Zhu, Dr. J. T. Xi, Mr. F. Lu and Mr. X. S. Chen for the antenna fabrication and testing. REFERENCES [1] C. A. Balanis, Antenna Theory: Analysis and Design. New York: Wiley, 2005. [2] D. M. Dobkin, The RF in RFID: Passive UHF RFID in Practice. The Netherlands: Elsevier, 2008. [3] M. Eunni, M. Sivakumar, and D. D. Deavours, “A novel planar microstrip antenna design for UHF RFID,” J. Syst., Cybern., Informat., vol. 5, no. 1, pp. 6–10, 2007. [4] B. Lee and B. Yu, “Compact structure of UHF band RFID tag antenna mounted on metallic objects,” Microw. Opt. Technol. Lett., vol. 50, no. 1, pp. 232–234, 2008.

YANG et al.: COMPACT METALLIC RFID TAG ANTENNAS WITH A LOOP-FED METHOD

[5] B. Yu, S. J. Kim, and B. Jung, “RFID tag antenna using two-shorted microstrip patches mountable on metallic objects,” Microw. Opt. Technol. Lett., vol. 49, no. 2, pp. 414–416, 2007. [6] J. Dacuna and R. Pous, “Low-profile patch antenna for RF identification applications,” IEEE Trans. Microwave Theory Tech., vol. 57, no. 5, 2009. [7] M. Hirvonen, P. Pursula, K. Jaakkola, and K. Laukkanen, “Planar inverted-F antenna for radio frequency identification,” Electron. Lett., vol. 40, no. 14, 2004. [8] C. Occhiuzzi, S. Cippitelli, and G. Marrocco, “Modeling, design and experimentation of wearable RFID sensor tag,” IEEE Trans. Antennas Propag., vol. 58, no. 8, pp. 2490–2498, 2010. [9] G. Marrocco, “RFID antennas for the UHF remote monitoring of human subjects,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1862–1870, 2007. [10] K. H. Lin, S. L. Chen, and R. Mittra, “A capacitively coupling multifeed slot antenna for metallic RFID tag design,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 447–450, 2010. [11] B. Gao, C. H. Chen, and M. M. F. Yuen, “Low cost passive UHF RFID packaging with electromagnetic band gap (EBG) substrate for metal objects,” in Proc. Electronic Components and Technology Conf. ECTC’07, 2007, pp. 974–978. [12] D. Kim and Y. Yeo, “Low-profile RFID tag antenna using compact AMC substrate for metallic objects,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 718–720, 2008. [13] P. H. Yang, J. Z. Huang, W. C. Chew, and T. T. Ye, “Design of compact and reusable platform-tolerant RFID Tag antenna using a novel feed method,” in Proc. Asia-Pacific Microw. Conf., 2009, pp. 641–644. [14] C. H. See, R. A. Adb Alhameed, D. W. Zhou, and P. S. Excell, “A planar inverted-F-L antenna (PIFLA) with a rectangular feeding plate for lower-band UWB applications,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 149–151, 2010. [15] C. H. See, R. A. Adb Alhameed, D. W. Zhou, and P. S. Excell, “Dualfrequency planar inverted F-L-antenna (PIFLA) for WLAN and short range communication system,” IEEE Trans. Antennas Propag., vol. 56, no. 10, pp. 3318–3320, 2008. [16] S. L. Chen, K. H. Lin, and R. Mittra, “A low profile RFID tag designed for metallic objects,” in Proc. Asia-Pacific Microw. Conf., 2009, pp. 226–228. [17] S. L. Chen and K. H. Ling, “A slim RFID tag antenna design for metallic object applications,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 729–732, 2008. [18] W. F. Richards, Y. T. Lo, and D. D. Harrison, “An improved theory for microstrip antennas and applications,” IEEE Trans. Antennas Propag., vol. 29, no. 1, pp. 38–46, 1981. [19] Q. Zhu, F. Kan, and T. Z. Liang, “Analysis of planar inverted-F antenna using equivalent models,” in Proc. IEEE Antennas and Propagation Soc. Symp., 2005, pp. 142–145. [20] Mike and Golio, The RF and Microwave Handbook. Boca Raton: CRC Press LLC, 2001. [21] F. W. Grover, Inductance Calculations: Working Formulas and Tables. New York: D. Van Nostrand, 1946. [22] H. W. Son, J. Yeo, F. Y. Choi, and C. S. Pyo, “A low-cost, wideband antenna for passive RFID tags mountable on metallic surfaces,” in Proc. IEEE Antennas and Propagation Soc. Symp., 2006, pp. 1019–1022. [23] M. Hirvonen, P. Pursula, K. Jaakkola, and K. Laukkanen, “Planar inverted-F antenna for radio frequency identification,” Electron. Let., vol. 40, no. 14, pp. 848–850, 2004. [24] L. Yang, A. Rida, and M. M. Tentzeris, Design and Development of Radio Frequency Identification (RFID) and RFID-Enabled Sensors on Flexible Low Cost Substrates. London: Morgan & Claypool, 2009. [25] K. D. Palmer and M. W. van Rooyen, “Simple broadband measurements of balanced loads using a network analyzer,” IEEE Trans. Instrum. Meas., vol. 55, no. 1, 2006. [26] X. M. Qing, C. K. Goh, and Z. N. Chen, “Impedance characterization of RFID tag antennas and application in tag co-design,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1268–1274, 2009. [27] K. V. S. Rao, S. F. Lam, and P. V. Nikitin, “UHF RFID tag for metal containers,” in Proc. Asia-Pacific Microw. Conf., 2010, pp. 179–182.

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Peng H. Yang was born in Kunming, Yunnan, China. He received the B.S. degree and M.S. degree in electronic engineering from the University of Electronic Science and Technology of China (UESTC), in 2001 and 2008, respectively, where he is currently working toward the Ph.D. degree. From January 2009 to November 2010, he was a Research Assistant with the Department of Electrical and Electronic Engineering, University of Hong Kong. His research interests include microstrip antenna theory and design, metameterials, and smart antenna systems.

Yan Li was born in Shaanxi, China. He received the B.S. degree in electronic engineering and the M.S. degree in microwave engineering from the University of Electronic Science and Technology of China (UESTC), in 2007 and 2010, respectively, where he is currently working toward the Ph.D. degree. From February 2010 to August 2011, he was a Research Assistant with the Department of Electrical and Electronic Engineering, University of Hong Kong. His research interests include antenna theory and design, antenna array optimization, and passive microwave circuits design.

Lijun Jiang (S’01-M’04) received the B.S. degree in electrical engineering from the Beijing University of Aeronautics and Astronautics, China, in 1993, the M.S. degree from Tsinghua University, China, in 1996, and the Ph.D. degree from the University of Illinois at Urbana-Champaign, in 2004. From 1996 to 1999, he was an application Engineer with Hewlett-Packard. From 2004 to 2009, he was a Postdoctoral Researcher, research staff member, and Senior Engineer at the IBM T. J. Watson Research Center, New York. Since the end of 2009, he has been an Associate Professor with the Department of Electrical and Electronic Engineering, University of Hong Kong. His research interests focus on electromagnetics, IC signal/power integrity, antennas, multidisciplinary EDA solutions, RF and microwave technologies, and high performance computing (HPC), etc. Prof. Jiang received the IEEE MTT Graduate Fellowship Award in 2003 and the Y.T. Lo Outstanding Research Award in 2004. He is an IEEE Antennas and Propagation Society (AP-S) Member, and a Sigma Xi Associate Member. He was the Semiconductor Research Cooperation (SRC) Industrial Liaison for several academic projects. Since 2009, he has been the SRC Packaging High Frequency Topic TT Chair. He also serves as a Reviewer of IEEE TRANSACTIONS and other primary electromagnetics and microwave related journals.

Weng Cho Chew (S’79–M’80–SM’86–F’93) received the B.S. degree in 1976, both the M.S. and Engineer’s degrees in 1978, and the Ph.D. degree in 1980, from the Massachusetts Institute of Technology, Cambridge, all in electrical engineering. He is serving as the Dean of Engineering at The University of Hong Kong. Previously, he was a Professor and the Director of the Center for Computational Electromagnetics and the Electromagnetics Laboratory at the University of Illinois. He was a Founder Professor of the College of Engineering, and previously, the First Y.T. Lo Endowed Chair Professor in the Department of Electrical and Computer Engineering, University of Illinois. Before joining the University of Illinois, he was a Department Manager and a Program Leader at Schlumberger-Doll Research. His research interests are in the areas of waves in inhomogeneous media for various sensing applications, integrated circuits, microstrip antenna applications, and fast algorithms for solving wave scattering and radiation problems. He is the originator several fast algorithms for solving electromagnetics scattering and inverse problems. He has led a research group that has developed parallel codes that solve dense matrix systems with tens of

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millions of unknowns for the first time for integral equations of scattering. He has authored the book, Waves and Fields in Inhomogeneous Media, coauthored two books, Fast and Efficient Methods in Computational Electromagnetics and Integral Equation Methods for Electromagnetic and Elastic Waves, and authored and coauthored over 300 journal publications, over 400 conference publications and over ten book chapters. Dr. Chew is a Fellow of the IEEE, OSA, IOP, Electromagnetics Academy, Hong Kong Institute of Engineers (HKIE), and was an NSF Presidential Young Investigator (USA). He received the Schelkunoff Best Paper Award from the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, the IEEE Graduate Teaching Award, UIUC Campus Wide Teaching Award, and IBM Faculty Awards. In 2008, he was elected by the IEEE AP Society to receive the Chen-To Tai Distinguished Educator Award. He served on the IEEE Adcom for the Antennas and Propagation Society as well as the Geoscience and Remote Sensing Society. From 2005 to 2007, he served as an IEEE Distinguished Lecturer. He served as the Cheng Tsang Man Visiting Professor at Nanyang Technological University in Singapore in 2006. In 2002, ISI Citation elected him to the category of Most-Highly Cited Authors (top 0.5%). He is currently the Editor-in-Chief of JEMWA/PIER journals, and is on the board of directors of the Applied Science Technology Research Institute, Hong Kong.

Terry Tao Ye received the Bachelor of Science degree in electronic engineering from Tsinghua University, Beijing, China and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA. He has more than 15 years of industry and research experience in RFID, wireless communication and VLSI. He had been actively involved in the maturation and implementation of EPCGlobal RFID standards and protocols as well as the development of the world’s first RFID Gen2 tag chip and reader system. Besides his expertise in RFID, he also has extensive experience in VLSI ASIC designs, electronic design automation (EDA) and systems-on-chip (SoC). Prior to LSCM, he had held various engineering and consulting roles in Impinj Inc, Synopsys Inc., Magma Design Automation Inc., Silicon Architects Inc and many other Silicon Valley companies. He is supervising the center’s research and development activities on RFID hardware and systems since joining LSCM. He recently initiated many LSCM center’s new research projects in different areas that include RFID IC designs, communication security, RFID anti-counterfeit application and application-specific RFID tag and reader development. He has also established collaborations with many research institutes and companies in both Hong Kong and mainland China.

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Mode Excitation in the Coaxial Probe Coupled Three-Layer Hemispherical Dielectric Resonator Antenna Anandrao B. Kakade and Bratin Ghosh, Member, IEEE

Abstract—In this paper, a coax-probe excited multilayer dielectric resonator antenna (DRA) structure is rigorously analyzed from the modal perspective. The full-wave Green’s function approach is presented for the analysis of such multilayer structures with an arbitrary number of layers with greatly reduced computational overhead. Additional reduction in computation time is demonstrated for a centered probe. Also, the modes of the multilayer DRA can be identified from the analysis, which can be used to explain the broadband nature of the coupling. The layer permittivities are optimized for broadband operation of the coax-fed DRA. The bandwidth enhancement for a centered and offset probe is seen to be due to a combination of several DRA modes and the probe resonance. Frequency tuning of the antenna structure is also demonstrated by exciting the antenna at a higher order mode, maintaining the broadband characteristics. The radiation characteristics of the antenna are also investigated. Index Terms—Dielectric resonator antenna (DRA), DRA mode, Green’s function, mode excitation, multilayer DRA, probe coupled.

I. INTRODUCTION

T

HE dielectric resonator antenna (DRA), originally proposed by Long et al. as a radiating element in [1], is characterized by low loss due to the absence of conductor loss. One of the techniques to extend the bandwidth of a single DRA element has been the multilayer DRA configuration. A broadband hemispherical DRA topology consisting of an inner resonator surrounded by a dielectric shell slot coupled to the microstrip line has been reported in [2] and [3]. A two-layer broadband DRA excited by a coaxial probe was presented in [4] and [5]. In the following work, the Green’s function approach is presented for the analysis of a hemispherical DRA with arbitrary number of layers, taking into account all higher order modes. The analysis of the -layer DRA structure is accomplished by separation of the source terms from the field-matching equations at the dielectric interfaces and incorporating them in the potential Green’s function. The resultant matrix equation can be easily Manuscript received August 02, 2010; revised June 06, 2011; accepted June 16, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. A. B. Kakade was with the Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology, Kharagpur, West Bengal 721 302, India. He is now with the Department of Electronics and Telecommunication Engineering, Rajarambapu Institute of Technology, Sakharale, Islampur 415414, India (e-mail: [email protected]). B. Ghosh is with the Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology, Kharagpur, West Bengal 721 302, India (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165480

solved. The analysis, which to the best of our knowledge has not been reported elsewhere, can also be used to identify the modes of the multilayer DRA [6]. The identification of the modes also explains the broadband nature of the coupling. The Green’s function technique used in this paper is distinctly different from the Green’s function approach used in [7] and [8] to analyze multilayer spheres. It should be noted that the matrix formulation developed in this work leads directly to the identification of modes in the multilayer DRA, unlike that in [7] and [8]. In addition, the Green’s function approach in this work is developed from the fundamentals taking into account the configuration of the antenna structure and feed, and as such, all components of the dyadic Green’s function need not be computed [9], [10]. This contributes to the efficiency of our technique compared to the approach in [7] and [8]. It is seen that the approach also needs significantly less computational time and memory compared to a finite-element-based electromagnetic software like the high-frequency structure simulator (HFSS) [11], particularly when the number of dielectric layers increase. The simulation time and memory requirements are almost independent of the number of layers, which is one of the main advantages of this approach over other simulation softwares. In addition, the double-summation representation of the homogeneous Green’s function has also been extended for the multilayer case, with a large reduction in computation time. Also, a further reduction in computation time is achieved for the centered probe by a suitable dissociation of the homogeneous Green’s function and expressing the homogenous impedance matrix elements as a product of two single integrations. The technique is used to design a broadband coax-fed multilayer DRA topology. It is shown that the layer permittivities should be appropriately chosen to minimize the quality factor of the antenna mode and to achieve optimum broadband behavior of the multilayer structure. A bandwidth of 65.60% is obtained with the centrally located probe, contributed by the DRA modes and the probe resonance. Frequency tuning and the effect of probe offset are also investigated for the antenna structure. II. FORMULATION OF THE PROBLEM A. Offset Probe Case The antenna configuration is shown in Fig. 1, where the -layer DRA is excited by a -directed probe of length and radius located at a displacement along the -axis from the center of the DRA. The DRA rests on an infinite ground plane.

0018-926X/$26.00 © 2011 IEEE

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Bessel function of the first kind, the spherical Bessel function of the second kind, and the spherical Hankel function of the second kind, all of the Schelkunoff type and order . All other symbols have the usual meanings. The three homogeneous potential Green’s functions , and have unknowns. These unknowns can be obtained by matching the tangential electric and magnetic fields at the dielectric layer interfaces at . Thereafter, orthogonal properties of the exponential and associated Legendre functions can be used to evaluate the unknowns. This generates three sets of equations for the three potential Green’s functions, each set having equations. The set of equations corresponding to is as follows: Fig. 1. Coaxial probe-coupled

-layer hemispherical DRA.

The innermost hemispherical DRA of the -layer configuration with radius and permittivity and permeability and , respectively, is surrounded by hemispherical shells with the th layer being free space. The outer radius of each shell is , where . The permittivity and the permeability of the th shell are and , respectively. A time dependence of is assumed and suppressed throughout. Also, in the formulation, and refer to the field and source points, respectively. In the derivation of the Green’s function of the probe-coupled single-layer hemispherical DRA [12] and of the two-layer DRA [4], [5] for a -directed electric current source, the Green’s function has been divided into particular and homogeneous solutions. In the present case of the -layer DRA, the particular part of the solution remains unchanged and is not repeated here. In order to evaluate the homogeneous part, the -directed electric current is decomposed into and components [12]. Thereafter, three potential functions , and are used to compute the fields in the multilayer DRA. The expression for is given as (1), shown at the bottom of the page, where

(4)

(5)

(6)

(7)

(8)

(2) (3) The expressions for the other two potential functions can be written down in a similar fashion replacing the unknown coefficients ( to ) in (1) by coefficients and for and , respectively. is the associated Legendre function of the first kind with order and degree . , and are, respectively, the spherical

.. .

(9) on the It might be noted that the terms involving right-hand side (RHS) of (4) and (7) are the contributions from the particular solution. In order to facilitate the extraction of Green’s function for the -layer DRA, the term

(1)

KAKADE AND GHOSH: MODE EXCITATION IN THE COAXIAL PROBE COUPLED THREE-LAYER HEMISPHERICAL DRA

involving the source which is common on the RHS of (4) and (7) is incorporated into the potential Green’s function with associated unknowns ( to ) from the particular and homogeneous solutions, respectively. Thus, no source terms need to be considered in the evaluation of the unknowns in (4)–(9). As a result, the number of unknowns in the homogeneous Green’s function reduces from to . In addition, this results in the matrix formulation as a result of which the Green’s function need not be rederived when the probe is placed in another layer, in contrast to [2]–[5]. This simplifies the solution to the -layer DRA. The source-free resonant frequencies of the TM [13] modes of the -layer DRA can be obtained by computing the zeroes of the impedance matrix. The corresponding matrix for the TE modes of the -layer DRA can be obtained by replacing in the previous matrix by , from which the resonant frequencies of the TE modes can be obtained. The double summation representation of the homogeneous Green’s function [14], [15] has also been extended to the multilayer case, resulting in a reduction in computation time for the antenna input impedance by a factor of about 100. B. Centered Probe Case For the probe located at the center of the multilayer DRA configuration, the integration with respect to and variables in the homogeneous impedance matrix in the method of moments can be performed independently and analytically, dissociating the homogeneous Green’s function as . Also as a result, the summation over the modal index can be eliminated, resulting in significant reduction in computational overhead. The final expression of is obtained as

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Fig. 2. Calculated, simulated, and measured return loss of the probe-coupled 12.5 three-layer DRA with centered probe: 15.5 mm, 20.7 mm, 5.9 mm, 0.63 mm, 2.0 mm, mm, 0 mm. and

(10) and refer to the piecewise sinusoidal basis In (10), and testing functions, respectively, with the unknown constants. It can be noted that (10) has been reduced to a form using a product of two single integrations over and . As a result, the computational time to evaluate (10) is further reduced. An additional reduction in computation time is achieved using the thin-wire approximation for the probe and using analytical integration [16]. III. RESULTS A. Centered Probe Case In this section, the coax-fed three-layer hemispherical DRA (HDRA) structure is investigated from a rigorous modal perspective, for excitation with a centered probe. As will be seen, this gives us an extremely thorough insight into the coupling and radiation phenomenon for this structure and also the mechanism of bandwidth enhancement. Fig. 2 shows the computed, simulated, and measured return loss of the three-layer probe coupled HDRA for wideband operation. The permittivity of the first layer is chosen as 9. This enables us to obtain maximum bandwidth using reasonable values of permittivities and for the second and third layers, respectively. The dimensions 12.5 mm, 15.5 mm, and 20.7 mm are chosen to excite the antenna in the TM mode and optimized for broadband operation. A very good agreement is seen between the theoretical, simulated, and measured results. It is observed that the first dip in the return loss characteristics in Fig. 2 is at 4.68 GHz, which is close to the source-free resonant frequency of TM mode at 4.46 GHz. The second dip in the return loss plot at 6.08 GHz is due to the loaded probe resonance which occurs at 6.20 GHz. The loaded probe resonance is obtained by computing the resonant frequency of the probe antenna in an infinite dielectric medium of permittivity (i.e., contribution of the homogeneous Green’s function is set to zero). The third dip in the return loss at 7.88 GHz

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Fig. 3. Variation in layer permittivities: mm.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 12, DECEMBER 2011

-factor of the TM mode of the three-layer DRA with 12.5 mm, 15.5 mm, and 20.7

Fig. 5. Calculated maximum bandwidth versus number of layers for different 0.63 mm, 2.0 mm, 0 mm. D1: . D2: layer widths. 3.0 mm, . D3: 12.5 mm, . 3.0 mm, 5.2 mm, . D5: D4: 12.5 mm, 5.2 mm, . D6: 12.5 3.0 mm, . mm,

Fig. 4. Calculated maximum bandwidth versus number of layers for different 0.63 mm, 2.0 mm, 0 mm. Case C2: dielectric constants. 12.5 mm, 15.5 mm, . Case C3: 12.5 mm, 15.5 mm, . Case C4: 12.5 mm, 15.5 mm, 20.7 mm, . Case C5: 12.5 mm, 15.5 mm, 20.7 mm, . Case C6: 12.5 mm, 15.5 mm, 20.7 mm, .

is caused by the TM mode with a source-free resonant frequency of 6.89 GHz. The wideband behavior is thus seen to be due to a combination of the above DRA modes together with the probe resonance. The 10-dB impedance bandwidth is measured at 65.60%. The variation in the quality factor of the TM mode with the layer permittivities is investigated next. Fig. 3 shows the variation in -factor of the TM mode with change in the permittivities and of the second and third layers, respectively. The variation in the impedance bandwidth characteristics with the number of layers and the layer permittivities is shown in Fig. 4. In Fig. 4, case C1 refers to a single-layer DRA, cases C2 and C3 to the two-layer DRA, and cases C4, C5, and C6 to the three-layer DRA. It is observed from Fig. 3 that the minimum TM mode -factor is attained for the . It is also seen that for , the minimum TM mode -factor

occurs at . The permittivity possesses a similar -factor at . However, it has been verified from the impedance bandwidth characteristics that the bandwidth for the layer permittivities is higher than that for the case . Thus, the optimized permittivity combinations , and are chosen which are also easy to obtain. It might be noted that if the permittivity of the first layer is less than 9, a permittivity value of 2 has to be used in the second or third layer to obtain high bandwidth, which is difficult to obtain. It can also be observed from case C5 in Fig. 4 that, similar to Fig. 3, the bandwidth is maximized for . It might also be noted that the maximum bandwidth with for case C5 almost coincides with that for and is thus not shown for clarity. The probe length was also optimized for each value of , and to obtain the maximum bandwidth in the figure. The maximum computed impedance bandwidth for the permittivity combination , and is at 62.67%, which can be compared to the measured bandwidth of 65.60% for Fig. 2. For the two-layer case, the computed maximum bandwidth is observed at 46.28% for while for the singlelayer case, the maximum bandwidth is computed at 35.28% with a DRA permittivity of 8. For the single-layer case C1, the DRA radius has to be adjusted for each permittivity to maintain a fixed resonant frequency. The variation in the impedance bandwidth with the number of layers and the layer widths are shown in Fig. 5, for the permittivities corresponding to the maximum bandwidth in Fig. 4. For Fig. 5, case D1 refers to a single-layer DRA, cases D2 and D3 to the two-layer DRA, and cases D4, D5, and D6 to the three-layer DRA. The maximum bandwidth is seen to be achieved with 12.5 mm, 3.0 mm, and 5.2 mm for the three-layer case. For the two-layer case, the bandwidth is seen to be maximized with 12.5 mm, 3.0 mm while a maximum bandwidth is obtained with 16 mm for a single-layer DRA.

KAKADE AND GHOSH: MODE EXCITATION IN THE COAXIAL PROBE COUPLED THREE-LAYER HEMISPHERICAL DRA

Fig. 6. Calculated and simulated return loss of the probe-coupled three-layer DRA with centered probe excited at a higher order mode : 12.5 mm, 15.5 mm, 17.5 mm, 3.8 mm, 0.63 mm, 2.0 mm, and 0 mm.

Fig. 7. Calculated, simulated, and measured radiation pattern in the plane for the centered probe-fed three-layer DRA at 6.32 GHz corresponding to Fig. 2.

It is shown next that the resonant frequency of the antenna can be tuned by exciting the antenna at a higher order mode. The antenna dimensions in this case are 12.5 mm, 15.5 mm, 17.5 mm with the same layer permittivities as in Fig. 2 and 3.8 mm with the probe at the center. Fig. 6 shows the return loss performance of the antenna when the antenna is excited at the TM mode. The first dip in the return loss at 8.16 GHz in Fig. 6(a) is near the TM source free resonance of the three-layer DRA at 8.12 GHz. The second return loss dip at 9.98 GHz corresponds to the loaded probe resonance at 10.06 GHz. The computed 10-dB impedance bandwidth for this case is 34.09%, with the wideband behavior caused by the TM mode and the probe resonance. The computed, simulated, and measured radiation patterns for the antenna structure in Fig. 2 are shown in Fig. 7 at the center frequency of 6.32 GHz. It can be observed from Fig. 7 that the radiation pattern for the centered probe is monopole-like with low cross-pol levels. The computed cross-pol levels are below 50 dB and are not visible. The gain was measured at 3.28 dBi. It was also seen that the radiation characteristics for Fig. 6 are preserved with frequency tuning and are similar to that of Fig. 2. It has also been observed that for the centered probe-fed DRA of Figs. 2 and 6, the radiation pattern is stable with low cross-pol levels across the impedance bandwidth. The radiation pattern across the impedance bandwidth for the case of Fig. 2 is shown in Fig. 8.

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Fig. 8. Simulated radiation pattern in the plane for the centered probe-fed three-layer DRA across impedance bandwidth corresponding to Fig. 2.

Fig. 9. Calculated, simulated, and measured return loss of the probe-coupled 12.5 three-layer DRA with offset probe: 14.0 mm, 20.0 mm, 6.0 mm, 0.63 mm, 2.0 mm, mm, 4 mm. and

B. Offset Probe Case The broadband behavior for a multilayer DRA caused by the probe offset is investigated next. Fig. 9 shows the return loss for the antenna configuration with 12.5 mm, 14.0 mm, 20.0 mm, 6.0 mm, and 4 mm. The first resonant dip in the return loss characteristics at 4.65 GHz in this case is due to the loaded TM mode with a resonant frequency of 4.56 GHz. The second return loss dip at 5.90 GHz is caused by the resonant frequency of the TM mode at 6.12 GHz together with the loaded probe resonance at 6.10 GHz. The third dip at 7.17 GHz is due to the TM mode with a resonant frequency of 7.04 GHz. It can be seen from Fig. 9 that the loaded TM mode and the probe resonances occur simultaneously and cannot be distinguished from each other. The merging of the above three DRA modes together with the probe resonance result in the bandwidth extension for the antenna structure. The measured 10-dB bandwidth for the antenna in Fig. 9 is at 54.51%. The radiation characteristics for the offset-probe fed case are shown in Fig. 10 at the center frequency of 6.06 GHz. A broadside pattern is observed in this case due to the probe displacement. In addition, due to the probe displacement along the -axis, the co-pol pattern in the plane is asymmetric, with the pattern symmetry preserved in the plane. A high cross-pol level is also observed in the plane as a result of

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Fig. 12. Convergence of the return loss with number of terms in the homogeneous Green’s function at 5.90 GHz corresponding to Fig. 9.

in the plane co-pol improves slightly at the lower and upper band edges with a decrease in broadside gain. The peak radiation occurs at about 60 , with low cross-pol levels. For the plane, an increase in the cross-pol level is observed at the lower and upper band edges relative to the co-pol radiation. Thus, the pattern stability is better for the centered probe-fed case compared to the offset probe-fed case. Fig. 10. Calculated, simulated, and measured radiation patterns for the offset plane. probe-fed three-layer DRA at 6.06 GHz corresponding to Fig. 9. (a) plane. (b)

Fig. 11. Simulated radiation patterns for the offset probe-fed three-layer DRA plane. (b) across impedance bandwidth corresponding to Fig. 9. (a) plane.

the -offset of the probe with low cross-pol level in the plane. The measured gain was at 5.65 dBi. The radiation pattern across the impedance bandwidth for this case is shown in Fig. 11. It is observed that the pattern symmetry

IV. COMPUTATION The convergence of the return loss with the number of terms in the homogeneous Green’s function is shown in Fig. 12 for the case of Fig. 9, where it is observed that a good convergence is achieved with six modal terms. The code was written in MATLAB 7.1 and run on a Intel(R) Core(TM)2 Duo CPU with 2 GB of RAM and 2.53-GHz clock speed. The average time taken by the code using the double-summation representation of the homogeneous Green’s function to compute the return loss and input impedance for a single frequency point using three basis functions was 5.12 s. An additional reduction in computation time is obtained for the centered probe, with 1.64 s required for the computation of input impedance using (10), corresponding to a reduction of about 67.97% compared to the previous case. It was found that using the single-summation representation of the homogeneous Green’s function, it takes 514.56 s for the same number of basis functions and sample points in the integration. Correspondingly, results using HFSS for the same structure could be obtained after 3 h, 49 min, and 55 s in this machine using a lambda refinement of 0.2 and . The peak RAM utilization by HFSS was 1.05 GB. It can be noted that the computation time and memory resources required by HFSS increase exponentially with increase in the number of layers in the DRA. As an example, the simulation of a five-layer DRA using HFSS is beyond the capability of the current machine. In contrast to this, the execution time and memory requirements of the MATLAB code with the single- or double-summation representation of the homogeneous Green’s function is almost independent of the number of layers in the DRA. Table I shows the computational time at a single frequency point for the calculation of input impedance for a single-layer, a three-layer, and a five-layer DRA excited by a probe. It can be seen that the computation time is almost invariant with the increase in the number of DRA layers.

KAKADE AND GHOSH: MODE EXCITATION IN THE COAXIAL PROBE COUPLED THREE-LAYER HEMISPHERICAL DRA

TABLE I COMPUTATIONAL TIME AT A SINGLE FREQUENCY POINT FOR THE EVALUATION OF INPUT IMPEDANCE OF COAXIAL PROBE COUPLED MULTILAYER DRA USING THE PREVIOUS [12, (19b)] AND CURRENT HOMOGENEOUS GREEN’S FUNCTIONS

V. CONCLUSION The analysis of multilayer DRA with an arbitrary number of shells excited by a coaxial probe using the Green’s function approach has been presented. The formulation enables us to identify the modes of the multilayer DRA and explain the broadband nature of coupling. Also, one of the main advantages of this method is that the computational time and memory requirements are almost independent of the number of DRA layers. The above technique was employed to investigate the enhancement of bandwidth of a three-layer DRA rigorously in terms of the modes of the multilayer structure. A maximum bandwidth of 65.60% was obtained for the antenna using a centered probe. The broadband behavior is caused due to a combination of the TM and the TM modes together with the probe resonance. Frequency tuning, preserving the broadband characteristics, was also demonstrated for the antenna structure, together with mode excitation for the offset probe. REFERENCES [1] S. A. Long, M. W. Mcallister, and L. C. Shen, “The resonant cylindrical dielectric cavity antenna,” IEEE Trans. Antennas Propag., vol. AP-31, no. 5, pp. 406–412, May 1983. [2] N. C. Chen, H. C. Su, K. L. Wong, and K. W. Leung, “Analysis of a broadband slot-coupled dielectric-coated hemispherical dielectric resonator antenna,” Microw. Opt. Technol. Lett., vol. 8, pp. 13–16, Jan. 1995. [3] N. C. Chen and K. L. Wong, “Input impedance of a slot-coupled multilayered hemispherical dielectric resonator antenna,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Newport Beach, CA, Jun. 1995, vol. 4, pp. 1796–1799. [4] K. L. Wong and N. C. Chen, “Analysis of a broadband hemispherical dielectric resonator antenna with a dielectric coating,” Microw. Opt. Technol. Lett., vol. 7, pp. 73–76, Feb. 1994. [5] K. L. Wong, N. C. Chen, and H. T. Chen, “Analysis of a hemispherical dielectric resonator antenna with an airgap,” Microw. Guided Wave Lett., vol. 3, pp. 355–357, Oct. 1993.

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[6] K. W. Leung and K. K. So, “Theory and experiment of the wideband two-layer hemispherical dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1280–1284, Apr. 2009. [7] C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory. New York: IEEE Press, 1994, ch. 10. [8] L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE Trans. Microw. Theory Tech., vol. 42, pp. 2302–2310, Dec. 1994. [9] K. W. Leung, K. M. Luk, K. Y. A. Lai, and D. Lin, “Theory and experiment of an aperture-coupled hemispherical dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 43, no. 11, pp. 1192–1198, Nov. 1995. [10] K. W. Leung, “Conformal strip excitation of dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 961–967, Jun. 2000. [11] Ansoft Corporation, Pittsburgh, PA, HFSS ver. 10.2. [12] K. W. Leung, K. M. Luk, K. Y. A. Lai, and D. Lin, “Theory and experiment of a coaxial probe fed hemispherical dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 41, no. 10, pp. 1390–1398, Oct. 1993. [13] M. Gastine, L. Courtois, and J. L. Dormann, “Electromagnetic resonances of free dielectric spheres,” IEEE Trans. Microw. Theory Tech., vol. 15, pp. 694–700, Dec. 1967. [14] A. B. Kakade and B. Ghosh, “Efficient technique for the analysis of microstrip slot coupled hemispherical dielectric resonator antenna,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 332–336, 2008. [15] A. B. Kakade and B. Ghosh, “Efficient technique for the analysis of coaxial probe-coupled hemispherical dielectric resonator antenna,” Microw. Opt. Technol. Lett., vol. 52, pp. 1588–1591, Jul. 2010. [16] K. W. Leung, “General solution of a monopole loaded by a dielectric hemisphere for efficient computation,” IEEE Trans. Antennas Propag., vol. 48, no. 8, pp. 1267–1268, Aug. 2000.

Anandrao B. Kakade received the B.E. degree in electronics engineering from Shivaji University, Kolhapur, India, in 2001 and the M.Tech. and Ph.D. degrees in RF and microwave engineering from the Indian Institute of Technology, Kharagpur, India, in 2005 and 2010, respectively. He is currently an Assistant Professor in the Department of Electronics and Telecommunication Engineering, Rajarambapu Institute of Technology, Sakharale, Islampur, India. His research interests are novel antennas, metamaterials, and guided-wave components.

Bratin Ghosh (M’05) received the B.E. degree in electronics and telecommunication engineering from Jadavpur University, Kolkata, India, in 1990, the M.Tech. degree in microwave engineering from the Indian Institute of Technology, Kharagpur, India, in 1994, and the Ph.D. degree in applied electromagnetics from the University of Manitoba, Winnipeg, MB, Canada, in 2002. He did his postdoctoral research from the Royal Military College, Kingston, ON, Canada from September 2002 to December 2003. He is currently an Associate Professor in the Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology. His research interests are efficient antennas, antenna miniaturization, metamaterials, guided-wave components, and numerical techniques.

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Compact Disc Monopole Antennas for Current and Future Ultrawideband (UWB) Applications Mohamed Nabil Srifi, Symon K. Podilchak, Member, IEEE, Mohamed Essaaidi, Senior Member, IEEE, and Yahia M. M. Antar, Fellow, IEEE

Abstract—Circular disc monopole antennas are investigated for current and future ultrawideband (UWB) applications. The studied antennas are compact and of small size (25 mm 35 mm 0.83 mm) with a 50- feed line and offer a very simple geometry suitable for low cost fabrication and straightforward printed circuit board integration. More specifically, the impedance matching of the classic printed circular disc UWB monopole is improved by introducing transitions between the microstrip feed line and the printed disc. Two particular designs are examined using a dual and single microstrip transition. By using this simple antenna matching technique, respective impedance bandwidths dB) from 2.5 to 11.7 GHz and 3.5 to 31.9 GHz are ( 11 obtained. Results are also compared to a classic UWB monopole with no such matching network transitions. Measured and simulated reflection coefficient curves are provided along with beam patterns, gain and group delay values as a function of frequency. The transient behavior of the studied antennas is also examined in the time domain.



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Index Terms—Impedance bandwidth, monopole disc antenna, ultrawideband (UWB) applications.

I. INTRODUCTION LTRAWIDEBAND (UWB) technology has become a very promising solution for indoor wireless radio, imaging and radars [1], [2]. Such applications can feature very high-speed data rates, low power consumption and good immunity to multipath effects. One component in these UWB systems is the front-end antenna unit, engineered to send and receive short pulse trains with minimal distortion. Thus there has been a considerable interest by the electromagnetics community to design efficient and compact UWB antennas to operate over significant bandwidths (BWs), particularly the 3.1 to 10.6 GHz spectrum allocated by the Federal Communication Commission (FCC) in the United States for wireless transmission [3].

U

Manuscript received January 10, 2011; revised April 11, 2011; accepted June 02, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. M. Nabil Srifi is with the Telecommunication Systems Laboratory, National School of Applied Sciences, Ibn Tofail University, Kenitra, Morocco (e-mail: [email protected]). M. Essaaidi is with the Electronics and Microwaves Group, Faculty of Sciences, Abdelmalek Essaadi University, Tetouan, Morocco (e-mail: [email protected]). S. K. Podilchak and Y. M. M. Antar are with Royal Military College of Canada (RMC) Kingston, ON K7K 7B4, Canada and also with Queen’s University at Kingston, Kingston, ON K7L 3N6, 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.2011.2165503

One of the strongest contenders in terms of achieving good impedance BWs, radiation efficiencies, and omnidirectional far field beam patterns are the circular and elliptical disc monopoles [4]–[16]. These designs can be made printed and can allow for low-cost fabrication and simple integration with associated UWB electronics. Antenna operation is generally limited within the FCC defined UWB frequency range for these antenna designs. Typical feeding techniques include simple microstrip lines [4], coplanar waveguide feeds [17], and slotted structures [18]. But with increasing demands for improved performances, higher bit rate transmission speeds, and the desire for synonymous operation with several different technologies, there may be the need for new and future UWB wireless schemes. Antenna operation could thus be required to function beyond the 10.6 GHz upper frequency band limit currently allocated by the FCC. One main design goal for these new UWB antennas is a good 50- impedance match over the desired operating BW. Fortunately numerous matching and miniaturization techniques have been reported in the literature and presented concepts may prove to be helpful. Techniques include feedgap optimization [5], bevels [6], ground plane slits and shaping [12], [13], multiple feeding configurations and orientations [14], [15], variations in monopole shape [9], [16] and size reduction [19]–[21]. In addition, other important antenna design goals include minimal dispersion effects and minor group delay variations, constant gain values as a function of frequency, good impulse responses in the time domain, and in some cases, general omnidirectional radiation behavior. In this work we study printed UWB antennas with increased impedance matching beyond the 10 GHz upper band limit typically observed for planar microstrip fed monopoles [8]. By introducing simple microstrip transitions between the 50feed line and the printed circular discs, the impedance BW of the planar monopole can be extended beyond 30 GHz. Specifically, two structures are investigated using a dual and single microstrip line transition: Designs A and B. By this added impedance matching, measured BWs ( dB) of 2.5 to 11.7 GHz and 3.5 to 31.9 GHz are respectively obtained. To evaluate antenna performances results are also compared to those of a classic UWB monopole antenna with no such matching network transitions: Design C ( dB for 3.3–10.3 GHz). Photographs of the three fabricated and measured UWB antenna structures, Designs A, B, and C, are shown in Figs. 1 and 2 while dimensions are outlined in Table I and Fig. 3. Section II discusses the design methodology and compares the operation of the proposed antennas in the

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TABLE I DIMENSIONS OF THE CIRCULAR DISC MONOPOLE ANTENNAS

Fig. 1. Layout and photograph of the planar monopole using a dual-microstrip transition for increased 50- impedance matching beyond 30 GHz.

Fig. 2. Investigated planar disc UWB antennas using a single microstrip transition and just a 50- feed line for comparison, (a) and (b) respectively.

Fig. 4. Simulated reflection coefficient of the UWB planar monopoles [10].

Fig. 3. Dimensions of the investigated planar disc UWB monopole antennas.

frequency domain, while Section III describes the respective time domain responses. Results are also compared against simulations using commercial solvers in the frequency and time domains. Section IV provides a brief conclusion of the presented material. II. ANTENNA DESIGN PRINCIPLES AND OPERATION IN THE FREQUENCY DOMAIN The radiation mechanism of planar circular disc monopoles is an involved topic and has been investigated by many UWB antenna researchers [7], [8]. One method for analyzing such structures can be in the frequency domain where wide band monopole operation is explained by the overlapping of closely distributed minimums in the reflection coefficient, sometimes referred to as resonances [8]. This response in the broadband dB impedance BW. matching is responsible for the Furthermore, at lower frequencies monopole antennas can be thought to function in an oscillating or standing wave mode, and with an increase in frequency, operation develops into a hybrid of both standing and traveling waves.

Operation at higher frequencies for these classic microstrip fed printed circular disc monopoles is generally limited to 10 GHz [4], [8]. Good antenna matching can be troublesome and very challenging to achieve in practice. Typically diminished gain and reduced radiation performances can result for these simple, single-input designs with increased ringing in the time domain due to multiple reflections along the feed line. Numerous factors contribute to this impedance mismatch such as the configuration of the ground the plane, the substrate selection, the feed line orientation, and the dimensions of the printed monopole disc. But by proper configuration of these parameters, good antenna matching may be achieved beyond 30 GHz [10]. By introducing the aforementioned microstrip transitions between the 50- feed line and the printed circular discs, impedance bandwidths can be improved as shown in Figs. 4–9. It should be mentioned that other feed line structures were investigated by the authors, but the presented microstrip transitions and ground plane configurations offered a very low cost solution for the simple antenna designs, while also offering good performance values in terms of 50- impedance matching and radiation behaviors. Essentially, the dimensions of the microstrip transitions were optimized by completing a , and while maintaining parametric analysis of

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Fig. 5. Simulated input impedance, Z , of the UWB planar monopoles. Fig. 7. VSWR of the UWB antenna, Design A, using the dual-microstrip feed 0 dB) configuration. A horizontal line defining a VSWR of 1.92 (jS j is shown. Measured [Simulated] VSWR values are below 1.92 from 3.5–31.9 [3.5–28.6] GHz.

= 10

Fig. 6. Simulated current distributions (in A/m) overlaid with the electric field within the substrate at 5.2, 7.0, 9.4, 12.7, 17.7, and 23.6 GHz for Design A. The electric field (in V/m) is described by arrows with color defining field strength (red defines a maximum while blue defines a minimum, same color scale for current) and orientation defining phase along the antenna structure.

Fig. 8. VSWR of the planar monopole using only a single-microstrip feed configuration, Design B. Measured [Simulated] values are below 1.92 from 2.5–11.7 [3.2–10.5] GHz offering a 50- impedance BW of 9.2 [7.3] GHz.



the width of input feed line, the circular shape of the disc, and the characteristics of the utilized substrate. A. Simulated Reflection Losses & Antenna Operation for the investigated anBy observing the minimums in tenna designs (A, B, and C), more insight into their UWB operation can be obtained [4], [8]. Simulated return loss curves are plotted in Fig. 4 and values are highlighted in Table II. At around 5 GHz the first minimums can be observed for the three monopoles. The first minimum of Design A occurs at a higher frequency (5.2 GHz) when compared to Design C (4.1 GHz). The first minimum of Design B occurs between Designs A and C (4.6 GHz). In addition, a new second minimum can be observed at 7.0 GHz for Design A and this can be thought to give rise to the very good impedance match over the 5–10 GHz range; dB. With these added transitions the minimums ie.



Fig. 9. VSWR of the planar monopole using only a 50- feedline, Design C. Measured and simulated values are below 1.92 from 3.3–10.3 GHz offering an impedance BW of 7.0 GHz.

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TABLE II REFLECTION COEFFICIENT MINIMUMS

in are now more equally spaced in frequency and thus help to contribute to the good return loss values. It should be mentioned that the 10 GHz limit of the classic monopole is extended by the dual-microstrip transition in Design A. For example, the original third minimum of Design C increased to 16.8 GHz. A new minimum is also introduced at 12.7 GHz. There is also a very good overlapping and reason, giving rise to able separation for the first 5 minimums in the extended impedance BW as shown in Fig. 4. It is also shown maintains a good impedance match until that Design A [B] GHz. 28.6 [10.5] Simulated input impedances “ ” are also plotted in Fig. 5. For Design A it can be observed that from 3.5–10.3 GHz. This can be thought to give rise to the low reflection losses. At 10.8 and 13.8 GHz, two distinct maxima are shown in the input resistance (84.0 and 89.4- , respectively), but the associated reactances are small and change from positive and to negative values near these frequencies ( - [ and - ] at 10.7 and 13.7 GHz [10.8 and 13.8 at 12.7 GHz. GHz]) contributing to the fourth minimum in Conversely, for both Designs B and C, the input resistances are below 32.1- at this same frequency and the simulated VSWR approaches 2.5 as shown in Figs. 8 and 9. In general, the planar discs can be thought to act as a frequency dependent load in series with the added transmission line matching sections and thus the developed design strategy can be described as follows. When the real part of the input impedance is observed to be - ), small reactances ( - ) high ( that change sign with frequency are helpful in achieving a good antenna match. This controlled resonance was achieved by the added microstrip transitions in Design A. Current and electric field distributions are illustrated in Fig. 6 . Additional plots for Design A at the first six minimums in are also shown in Fig. 9 of [10] for Design A. It can be observed that the currents are mainly concentrated near the edge of the ground plane (closest to the disc), while on top of the structure currents are primarily distributed along the periphery of the disc edge and feed line [4], [8], [10]. Radiating slots can be thought to form between the lower edge of the disc and ground plane [22].

Fig. 10. Measured beam patterns for the dual transition UWB planar monopole. Normalized values are shown for Design A (Fig. 1) in dB.

Thus the radiated far fields originate from these main current distributions. These current maxima in Fig. 6 also increase in number with frequency. For instance, at 5.2 GHz one distinct maxima can be observed at the junction of the feed line and the disc, while at 17.7 GHz five maxima are visible in total. Similar results are shown in Fig. 9 of [10]. Furthermore, these currents and electric field distributions can also signify particular modes of antenna operation. For example at 5.2 GHz the electric field is mainly directed away from the disc edges and the ground plane, while at the next minimum at 7.0 GHz, the electric field has a different orientation: from the outer edge of the disc towards the ground plane. More complex electric field distributions can also be observed in Figs. 6(c–f).

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Fig. 11. Measurement and simulations of the co- and cross-polarized beam patterns in the

H -plane for the monopole with the dual microstrip feed configuration.

E x0y

Fig. 12. Maximum observed realized gain in the ( ) plane for the planar monopoles. (a): Design A, and (b): Design B. Similar values are observed for Design C (monopole with no transitions and just the 50- feed line) as the results for the single transition structure (Design B), subplot (b) of this figure.

At higher frequencies when antenna operation has developed into a hybrid of both standing and traveling waves, phase propagation along the radial disc aperture can be inferred by observation of the current distribution (please refer to Fig. 9 of [10] at 32.0 GHz). This can signify traveling wave operation of the antenna. Moreover, at 32.0 GHz the major dimension of the antenna structure, , is large in comparison to the free space wavemm, mm). length ( B. Fabrication and Reflection Loss Measurements The UWB circular disc monopoles (with mm) were fabricated on 30 mm 35 mm dielectric slabs ( mm) and partial ground planes (30 mm 15.6 mm) were maintained on the underside of the antenna substrates. The feed lines were then soldered with 50- K-Connectors for reflection loss measurements in a calibrated anechoic chamber using a Anritsu 37377C Vector Network Analyzer (VNA). Results are compared to simulated values in Figs. 7–9. Good dB agreement in terms of the impedance match ( ) is observed. Deviations may be attributed to or substrate variations over frequency, fabrication tolerances, feed connector misalignment, and difficulty in modeling the metal thicknesses near the ground planes, the circular discs, and the feed line edges due to the fabrication process. In addition, the mechanical details of the 50- K Connectors were not included in the simulations in order to simplify the modeling.

Regardless, the measurements and the simulation results are in agreement and this suggests that the simple microstrip feed transitions can increase the 50- impedance BW of classic monopole antennas. C. Radiation Patterns Beam pattern measurements were completed in the frequency domain for all three monopole designs in an anechoic chamber. Measures were sampled in magnitude and phase. All trials were completed in receive mode and the appropriate calibration calculations were completed to negate cable and free space losses, chamber effects, and the contributions of the reference antennas , or nor[23]. Thus the received frequency response, malized antenna transfer functions [24], [25] for the antennas under test were observed. Measurements were also completed - , and - planes and for both co- and cross-polarin the izations. In addition, 150 samples were recorded and averaged by the VNA for each frequency measure in an attempt to minimize any high frequency noise and multiple reflections due to cable bending and twisting. It should be noted that some preliminary antenna gain patterns were reported earlier in [10] and agreement was observed between the measurements and simulations. Specifically, measurements were provided in the - and - planes for Designs A and B in dBi. In this work additional results are provided in Figs. 10–12. Fig. 10 shows co-polarized measurements of the

SRIFI et al.: COMPACT DISC MONOPOLE ANTENNAS FOR CURRENT AND FUTURE UWB APPLICATIONS

Fig. 13. Measurements of the relative group delay in the

H (x 0 y) plane for  = 0

TABLE III MAXIMUM ENVELOPE VALUES

TABLE IV TIME DOMAIN CHARACTERISTICS AND COMPARISONS

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and 90 incidence. (a): dual, and (b): single microstrip feed configuration.

show a similar response. This gain decrease may correspond to the somewhat high impedance reflections (Fig. 7) also observed and centered at 17.5 GHz. However, the VSWR is still less than 1.92 in this range. For a practical UWB system this result may be acceptable. Performances could be improved by further tuning, additional microstrip transitions and the selection of a higher performance connector. The authors wish to stress that antenna measurements were difficult to complete in the - and - planes due to the available azimuth range on the rotating antenna tower. In and addition, measures in the range may have reduced accuracy due the possible interference and positioning of the metallic tower and measurement cables. Absorber was also placed on the metallic antenna tower in an effort to minimize any unwanted interference. Despite these practical difficulties, results are in agreement with the simulations and a good proof of concept for the three UWB antenna structures is presented. D. Group Delay Small variations of the antenna phase response, or group delay 1, are important frequency domain characteristics for , where UWB antennas. Relative group delay values,

(1) defined as the deviation of from the mean group delay, [26], were plotted in Fig. 13 for Designs A and B. Minor group delay variations are observed for Design A in the operating frequency range of the antenna up to 30 GHz, while noticeably high values are observed for Design B between 2.0–2.8 GHz ns). This could be caused by and 11.0–12.5 GHz ( the high reflections observed in the VSWR of Design B below 2.5 GHz and at 12.0 GHz as respectively shown in Fig. 8. The high group delay variations from 3.9–4.7 GHz may also be related to unwanted energy storage or other dispersive effects. In brief, a similar phase response was also observed for Design C normalized beam patterns in the - , and - planes. Beam plane are shown in Fig. 11 and good patterns in the H agreement can be observed with the simulations. Measured realized gain values are also plotted in Fig. 12. A reduction in gain of 4 dB can be observed in Fig. 12(a) for Design A at 17.5 GHz. Realized antenna gain simulations do not

1The group delay is defined as the negative derivate of the antenna phase angle with respect to frequency, ( ) = = 2 . Small variations in group delay, defining a flat response or linear phase within a particular frequency range, suggest that waveform distortions in the time domain of transmitted or received pulses will be small; ie. a constant implies good UWB antenna operation [26]. Conversely, a nonlinear suggests unwanted resonant behavior and in the time domain this can result in ringing and unwanted oscil). lations in the antenna impulse response, (

 !

0@'=@!

 h t; ;

0@'= @f



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Fig. 15. Envelope of the measured impulse response (jh (t;  = 0 ;  = 90 )j) for the dual microstrip feed configuration and the design with only a 50- microstrip feedline. (a): Co-Pol response, (b): X-Pol response.

Fig. 14. Measured impulse response for the UWB monopole (Design A) with the dual microstrip feed line: Co-Pol. [X-Pol.] incidence—[ – – –].

as in Fig. 13(b). Thus these relative group delays suggest Design A has an increased performance when compared to both Designs B and C.

Fig. 16. Comparison of the three planar monopoles in the H (y 0 z ) plane: normalized peak envelope jh (;  = 90 )j, FWHM, and ringing time.

III. TIME DOMAIN ANALYSIS The previous section provided a frequency domain characterization of the examined antenna designs, but UWB systems are generally implemented using an impulse-based technology, and as such time domain effects are equally as important [7], [8]. For example, in an UWB system antenna behavior can be compared to that of a bandpass filter with constant group delay values and flat gain or amplitude responses over the entire operating BW. Signal distortion is dependent on how the spectra of transmitted and received pulses is reshaped in the time domain. This section examines important time domain characteristics of the studied UWB antennas. Results are presented in Figs. 14–20 and Tables III and IV.

A. Antenna Impulse Response The impulse or transient responses, , of the investigated antennas were determined by taking the inverse Fourier . Received frequency measures were transform of zero padded and conjugate values were also included in the calculation to obtain full analytic responses in the time domain. Transient signals are plotted for various incident angles in Fig. 14 as a function of time for Design A. It can be observed is dependent on the angle of arrival and polarization. that For example Fig. 14(c), which corresponds to the received signal at end-fire, has one main upward pulse and a second downward pulse with reduced amplitude. Oscillatory behavior2

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Fig. 18. Ringing time and pulse width in the

H

E

Fig. 17. Normalized peak envelope in the - and -planes for Design A, plots (a) and (b), respectively. Results are shown in polar form in linear units.

(or ringing) is observed after these two pulses until about 1.5 ns, suggesting increased dispersion. Reduced ringing and an increased settling time is observed for the other impulse responses.

B. Envelope of the Impulse Response, Pulse Width, & Ringing

Another method to assess the dispersive quality of UWB antennas is to calculate the envelope of the analytic impulse re, where sponse,

(2) and refers to the Hilbert Transform of the impulse response can be a more useful tool in as[26]. The envelope sessing and quantifying antenna dispersion. Important time doincluding main characteristics can be calculated from , the width of the observed the peak value of the envelope, pulse at full width half maximum (FWHM) or pulse width,

H -plane for Design A.

, and the duration of the ringing time,2 . Maximum values of are desired as this quantity can signify the amount of radiated or received power in a linear wireless system. Reduced pulse widths are also advantageous for increased data transmission rates. Ideally the FWHM should not exceed a few hundred picoseconds while the ringing time should not be more than a few pulse widths [26]. Measured values of the envelope are shown as a function of time in Fig. 15 for Designs A and C. It can be observed that the UWB monopole with the dual transitions has an increased pulse peak for the co-polarized response (0.21 m/ns) when compared to the structure with just the single microstrip line (0.15 m/ns). An increase in the pulse width can also be observed for Design C. Maximum envelope peak values are compared in Table III for the three measured designs. It is interesting to note that Design B achieved a higher pulse peak maximum in the plane when compared to Design A, but the opposite is true in the plane. Reduced pulse peak values are observed for Design C, suggesting a more dispersive antenna. Simulations are also plotted in Fig. 16 for the co-polarized pulse peak along with the pulse width and ringing time (when the peak pulse is reduced to ). Agreement is observed with 22% of it maximum value, the measurements in Table III in that Design B also achieved a maximum pulse peak when compared to the two other designs . The simulated response for is also comat pared to measurements in polar form as a function of beam angle for Design A in Figs. 17 and 18. Good agreement is observed. Measured and simulated maximums of the analytic pulse peak with a general decrease at in the are both at -plane. The -plane responses in Fig. 17(b) have a null at and 180 suggesting more dispersive antenna behavior. This confirms the observations in Fig. 14(c) and the discussions in Section III.A regarding the increased ringing time at end-fire. Discrepancies increase near the backside of the antenna for



2Oscillations or the ringing after, the main pulse is defined as the quantity of time when j ( )j is reduced below a certain percentage from the peak maximum [26]. This ringing is unwanted and can be a result of energy storage or multiple reflections along the feed line and antenna structure. Energy contained in the ringing reduces the amount of radiated power, decreases peak envelope levels, and increases the pulse width of j ( )j.

h t

h t

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and are likely due to the aforementioned practical effects of the antenna tower and connecting cable. The ringing time and FWHM are also compared in the -plane in Fig. 18 as a function of angle in polar form. Agreement is shown for the observed pulse widths as mean values are both approximately 80 ps. Results are further compared in Table IV for all monopole antennas and good agreement is shown when the minimum values are observed. Differences in the ringing time are likely due to the practical challenges with such monopole antenna measurements: unwanted reflections from the anechoic chamber side walls, possible interference from the antenna test tower, and the metallic K-connector attached to the measurement cables. Despite these concerns a good proof of concept for the three UWB antenna structures is presented and agreement is observed in the measurements and simulations. Designs A and B offer reduced dispersion effects when compared to Design C.

Fig. 19. Normalized spectrum for two test pulses ((3)) incident on the receiving : ns and : ns. antenna terminals with

= 0 175

=01

C. Received Fidelity Due to an Incident Gaussian Waveform To further study the measured UWB antennas fidelity estimations, , were completed for Designs A and C using the cal, and the fourth derivative of a culated impulse responses, template Gaussian pulse in the time domain: [24], [20]

(3) where and (in ns) can characterize the pulse width of . Thus determined fidelity values can define the quality of the received waveform incident onto the antenna structure for . an ideal far field source transmitting The power spectrum density (PSD) of the Gaussian in (3) can comply with typical FCC indoor emission mask requirements ns [24]. Furthermore, for reduced values of with sharper time domain pulses are possible along with increased spectral content in the frequency domain. For example, the nor, is plotted in Fig. 19 for malized spectrum of this signal, and 0.1 ns. Both signal waveforms are investigated in this work as the proposed monopole antennas could be suitable for current and future UWB systems. Moreover, the FCC could designate new spectral guidelines for future UWB systems that allow for such increased BW utilization. The Gaussian wavens) could comply with forthcoming form of (3) (with emission requirements. Results of the fidelity estimations are shown in Fig. 20 using -plane measurements for antenna Designs A and C. If fidelity and the received waveform, , are values achieve unity, exactly the same in shape. This means that the antenna causes no distortion of the received pulse. Essentially, the incident signal was convolved in the time domain with the impulse response as a function of beam angle to determine the output waveform, , at the receiving antenna terminal, mainly (4)

= 0 175 ( 0)

=01

Fig. 20. Calculated fidelity for the two test pulses ( : ns and : ns) incident on the dual microstrip feed configuration and the design with only a 50- microstrip feedline. Results shown in the H y z plane using measured ; . data for  2



[0 180 ]

By linear system theory this procedure is analogous to multiand the antenna transfer functions, plying in the frequency domain and taking the Inverse Fourier Transform. For completeness both methods were verified by the authors, and as expected, identical values were observed. Fidelities were then evaluated by determining the correlation coeffi, and the received signal cient of the template Gaussian, . Further discussions on these procedures can be found in [20] and [24]. By analysis of Fig. 20, Design A achieved increased fidelity for both values when compared to Design C. Average values are as follows. Design A: and for and ns. Design C: and for and ns. It is also interesting to note that achieves minimum values . This observation is consistent for Design C for with the results of Fig. 16 and the discussions of Section III.B, in that the classic monopole (Design C) can exhibit increased . Calculated values are not dispersion effects near close to unity (likely due to the aforementioned practicalities and measurement challenges for such monopole antennas) but

SRIFI et al.: COMPACT DISC MONOPOLE ANTENNAS FOR CURRENT AND FUTURE UWB APPLICATIONS

the expected trend is observed. Design A exhibits less distortion effects for a Gaussian waveform incident at the receiving antenna terminal when compared to Design C. IV. CONCLUSION New compact circular disc monopole antennas for UWB applications were presented and a simple technique has been introduced to improve the performance of classic UWB planar monopole antennas. Microstrip transitions, with a characteristic impedance different than 50- , were arranged between the feed line and the printed discs. Calibrated measurements in an anechoic chamber show that the operating bandwidth of the proposed antennas, after introducing the single-microstrip [dual-microstrip] transition, increases from 3.3–10.3 GHz, for the classic UWB monopole, to 2.5–11.7 GHz [3.5–31.9 GHz]. Thus at most a BW of 28.4 GHz can be achieved. Improvements in these designs may be possible by additional microstrip transitions or by additional tuning techniques. Return loss measurements are provided along with beam patterns, gain and group delay values as a function of frequency. Transient behavior of the studied UWB antennas was also presented and results suggest that the designs with the added microstrip transitions can offer reduced dispersion effects when compared to the classic planar monopole. This proposed matching technique is also very simple to introduce in practice and could be attractive for current and future UWB applications.

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[13] X. L. Bao and M. J. Ammann, “Investigation on UWB printed monopole antenna with rectangular slitted groundplane,” Microw. Opt. Tech. Lett., vol. 49, no. 7, pp. 1585–1587, Jul. 2007. [14] E. Antonino-Daviu, M. Cabedo-Fabres, M. Ferrando-Bataller, and A. Valero-Nogueira, “Wideband double-fed planar monopole antennas,” Electron. Lett., vol. 39, no. 23, pp. 1635–1636, Nov. 2003. [15] M. J. Ammann and Z. N. Chen, “An asymmetrical feed arrangement for improved impedance bandwidth of planar monopole antennas,” Microw. Opt. Tech. Lett., vol. 40, no. 2, pp. 156–158, Dec. 2003. [16] J. Liu, K. P. Eselle, and S. S. Zhong, “A printed extremely wideband antenna for multi-band wiresless systems,” presented at the Antennas Propagation Society Symp. (APS-URSI), Toronto, Canada, Jul. 2010. [17] J. Liang, L. Guo, C. C. Chiau, X. Chen, and C. G. Parini, “Study of CPW-fed circular disc monopole antenna for ultra wideband applications,” IEE Proc. Microw. Antennas Propag., pp. 520–526, Dec. 2005. [18] P. Li, J. Liang, and X. Chen, “Study of printed elliptical/circular slot antennas for ultrawideband applications,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1670–1675, Jun. 2006. [19] Q. Wu, R. Jin, J. Geng, and M. Ding, “Printed omni-directional UWB monopole antenna with very compact size,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 896–899, Mar. 2008. [20] M. Sun, Y. P. Zhang, and Y. Lu, “Miniaturization of planar monopole antenna for ultrawideband radios,” IEEE Trans. Antennas Propag., vol. 58, no. 7, pp. 2420–2425, Jul. 2010. [21] R. Zaker and A. Abdipour, “A very compact ultrawideband printed omnidirectional monopole antenna,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 471–473, 2010. [22] R. Garg, P. Bhartia, I. Bahl, and A. Ittipiboon, Microstrip Antenna Design Handbook. Norwood, MA: Artech House, Inc., 2001. [23] C. A. Balanis, Antenna Theory, 3rd ed. Hoboken, NJ: Wiley, 2005. [24] T. Ma and S. Jeng, “Planar miniature tapered-slot-fed annular slot antennas for ultrawide-band radios,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1194–1202, Mar. 2005. [25] Y. Duroc, A. Ghiotto, T. P. Vuong, and S. Tedjini, “UWB antennas: Systems with transfer function and impulse response,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1449–1451, May 2007. [26] W. Wiesbeck, G. Adamiuk, and C. Sturm, “Basic properties and design principles of UWB antennas,” Proc. IEEE, vol. 97, no. 2, pp. 372–385, Feb. 2009.

REFERENCES [1] K. Siwiak, “Ultra-wide band radio: Introducing a new technology,” in Proc. IEEE Veh. Technol. Conf., May 2001, vol. 2, pp. 1088–1093. [2] D. G. Leeper, “Ultrawideband—The next step in short-range wireless,” in IEEE Radio Freq. Integrated Circuits Symp. Dig., Jun. 2003, pp. 493–496. [3] “First Report and Order, Revision of Part 15 of the Commissions Rule Regarding Ultra Wideband Transmission Systems,” Fed. Commun. Comm., FCC 02-48, Apr. 22, 2002. [4] J. Liang, C. C. Chiau, X. Chen, and C. G. Parini, “Study of a printed circular disc monopole antenna for UWB systems,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3500–3504, Nov. 2005. [5] M. John and M. J. Ammann, “Optimization of impedance bandwidth for the printed rectangular monopole antenna,” Microw. Opt. Tech. Lett., vol. 47, no. 2, pp. 153–154, Oct. 2005. [6] M. J. Ammann, “Control of the impedance bandwidth of wideband planar monopole antennas using a beveling technique,” Microw. Opt. Tech. Lett., vol. 30, no. 4, pp. 229–232, Jul. 2001. [7] H. Schartz, The Art and Science of Ultrawideband Antennas. Boston, MA: Artech House, 2005. [8] B. Allen, M. Dohler, E. Okon, W. Malik, A. Brown, and D. Edwards, Ultra Wideband Antennas and Propagation for Communications, Radar and Imaging. Hoboken, NJ: Wiley, 2007. [9] A. M. Abbosh and M. E. Bialkowski, “Design of ultrawideband planar monopole antennas of circular and elliptical shape,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 17–23, Jan. 2008. [10] M. N. Srifi, S. K. Podilchak, M. Essaaidi, and Y. M. M. Antar, “Planar circular disc monopole antennas using compact impedance matching networks for ultra-wideband (UWB) applications,” in Proc. IEEE Asia Pacific Microwave Conf., Dec. 2009, pp. 782–785. [11] J. Liang, C. C. Chiau, X. Chen, and C. G. Parini, “Printed circular disc monopole antenna for ultra-wideband applications,” Electron. Lett., vol. 40, no. 20, pp. 1246–1248, Sep. 2004. [12] C. Zhang and A. E. Fathy, “Development of an ultra-wideband elliptical disc planar monopole antenna with improved omnidirectional performance using a modified ground,” in Proc. IEEE Int. Antennas Propag. Symp., Alburqueque, NM, 2006, pp. 1689–1692.

Mohamed Nabil Srifi was born in Sidi Redouane Ouezzane, Morocco, in January 1978. He received the Licence de Physique degree in electronics from Ibn Tofail University, Kenitra, Morocco, in 1999, and the Deep Higher Studies Diploma DESA degree in telecommunications systems and the Ph.D. degree in electrical engineering from Abdelmalek Essaadi University, Tetuan, Morocco, in 2004 and 2009, respectively. He is currently an Assistant Professor of electrical engineering at Ibn Tofail University, Morocco. His research interests include biological effects of radiofrequency and microwave, bio-electromagnetics, biomedical engineering and antenna design. He holds two patents on antennas for ultra-wide band applications. Dr. Nabil Srifi is a recipient of national and international awards, and is the Vice-secretary of the Moroccan Association of Electricity, Electronics and Computers Engineering (AEECE).

Symon K. Podilchak (S’03–M’05) received the B.A.Sc. degree from the University of Toronto, ON, Canada, in 2005. He is currently working toward the Ph.D. degree at Queen’s University at Kingston (QU), ON, Canada. He is also a Research Associate at The Royal Military College of Canada (RMC), ON, Canada. During this same period, He was a Teaching Assistant and Fellow at QU and RMC where he contributed to the development and teaching of electromagnetics and circuits based courses at the graduate and undergraduate level. He has also had experience as a computer programmer, technology investment analyst, and assisted in the design of radomes for 77 GHz automotive radar. Recent industry experience also includes modeling the radar cross-section of military vessels for high frequency surface-wave radar, professional software design and implementation for measurements in anechoic chambers, and the

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research and development of highly compact circularly polarized antennas for microsatellites. His current research interests include the analysis and design of planar leaky-wave antennas, metamaterials, millimetre-wave CMOS integrated circuits, UWB antennas, and periodic structures. Mr. Podilchak has also been the recipient of more than 25 best paper awards, international travel grants, and scholarships; most notably an IEEE Antennas and Propagation Society Doctoral Research Scholarship and three Young Scientist Awards.

Mohamed Essaaidi (SM’00) is a Professor of electrical and computer engineering at Abdelmalek Essaadi University, Morocco. His research interests focus mainly on RF and microwave passive and active circuits and antennas for wireless communications and medical systems and Wireless Sensor Networks (WSN). He is the author and coauthor of more than 95 papers which appeared in refereed specialized international journals and conferences. He holds four patents on antennas for very high data rate UWB and multiband wireless communication networks and high resolution medical imaging systems. Moreover, he has supervised several Ph.D. and Masters theses and has been the principal investigator and the project manager for several international research projects dealing with different research topics concerned with his research interests mentioned above. Prof. Essaaidi is a member of the IEEE Microwave Theory and Techniques Society, IEEE Antennas and Propagation Society, IEEE Communications Society, IEEE Computer Society and European Microwave Association. He is the Founder and the General Chair of Mediterranean Microwave Symposium (MMS) since 2000. He is also the co-Founder and the current Coordinator of the Arab Science and Technology Foundation (ASTF) RD&I network on Electrotechology. He has been also a member of the Organizing and the Scientific Committees of several international symposia and conferences dealing with topics related with RF, microwaves and Information and Communication technologies and their applications. He was the Editor of the proceedings of several international symposia and conferences and a Special Issue of the European Microwave Association Proceedings. He has been the Editor-in-Chief of the International Journal of Information and Communication Technologies since 2007. Furthermore, he has been a referee of several international journals such as the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and IEEE

TRANSACTIONS ON ANTENNAS AND PROPAGATION. His biography was listed in Who’s Who in The World in 1999.

Yahia M. M. Antar (S’73–M’76–SM’85–F’00) received the B.Sc. (Hons.) degree from Alexandria University, Alexandria, Egypt, in 1966, and the M.Sc. and Ph.D. degrees from the University of Manitoba, Winnipeg, MB, Canada, in 1971 and 1975, respectively, all in electrical engineering. In May 1979, he joined the Division of Electrical Engineering, National Research Council of Canada, Ottawa, where he worked on polarization radar applications in remote sensing of precipitation, radio wave propagation, electromagnetic scattering and radar cross section investigations. In November 1987, he joined the staff of the Department of Electrical and Computer Engineering, Royal Military College of Canada in Kingston, where he has held the position of Professor since 1990. He has authored or coauthored over 160 journal papers and 300 Conference papers, and holds several patents. Dr. Antar is a Fellow of the IEEE and the Engineering Institute of Canada (FEIC). He is an Associate Editor (Features) of the IEEE Antennas and Propagation Magazine and served as Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IEEE Antennas and Wireless Propagation, and a member of the Editorial Board of the RFMiCAE Journal. In May 2002, he was awarded a Tier 1 Canada Research Chair position in Electromagnetic Engineering which has been renewed in 2009. In 2003, he received the 2003 Royal Military College Excellence in Research Prize. He was elected to the Board of the International Union of Radio Science (URSI) as Vice President in August 2008. He chaired conferences and has given plenary talks in many conferences, and supervised or co-supervised over 70 Ph.D. and M.Sc. theses at the Royal Military College and at Queens University, of which several have received the Governor General of Canada Gold Medal as well as best paper awards in major symposia. He served as the Chairman of the Canadian National Commission for Radio Science (CNC, URSI, 1999–2008), Commission B National Chair (1993–1999), holds adjunct appointment at the University of Manitoba, and has a cross appointment at Queens University in Kingston. He also serves, since November 2008, as Associate Director of the Defence and Security Research Institute (DSRI) and has been a member of the Defence Science Advisory Board (DSAB) since January 2011.

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A New Multimode Antenna for MIMO Systems Using a Mode Frequency Convergence Concept Julien Sarrazin, Yann Mahé, Member, IEEE, Stephane Avrillon, Member, IEEE, and Serge Toutain

Abstract—A new multimode antenna is proposed in this paper. The structure is based on a mode frequency convergence concept. A microstrip square patch antenna is modified in order to force different modes to resonate at the same frequency. Then, with a common/differential excitation, these modes can be fed independently. Thus, three radiation patterns are available simultaneously, thereby producing diversity. With dimensions less than a guided wavelength ( 0:88g ), the proposed structure is well suited to reduce the size of multiple antennas used in diversity applications such as MIMO systems. The antenna has been designed in the 2.45 GHz band for Wi-Fi applications. Index Terms—Envelope correlation, microstrip antennas, multimode antennas, multiple input multiple output (MIMO) systems, radiation pattern diversity.

I. INTRODUCTION ULTIPLE input multiple output (MIMO) systems can drastically improve wireless communication capacity and robustness by exploiting multipath effects [1]. Performances of these multiple antenna systems largely depend on the correlation between received signals. To maximize the capacity, this correlation must be as low as possible. Spatial diversity is commonly used to reduce it. However, this requires up to several a space between antennas of less than (depending on which kind of environment is considered). This is not always compatible with the limited volume available on a wireless terminal. That is why other kinds of diversity such as radiation pattern diversity are also investigated. By using antennas with different radiation patterns, it is possible to reduce the size of the multiple antenna system while keeping a low correlation between received signals. To achieve this pattern diversity, the concept of co-located antennas has been introduced [2]. The idea is to co-localize several antennas radiating different polarizations [3]–[5]. This concept, mainly based on polarization diversity, has been extended to amplitude pattern diversity in order to provide a compact system integrating a larger number of antennas [6]. More recently, promising solutions based on multimode antennas have been proposed [7]–[13]. These structures have different modes of

M

Manuscript received May 19, 2010; revised April 20, 2011; accepted May 09, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. J. Sarrazin is with the LTCI research institute, Telecom ParisTech, Paris, France (e-mail: [email protected]). Y. Mahé and S. Toutain are with the IREENA research institute, University of Nantes, France (e-mail: [email protected]). S. Avrillon is with the IETR research institute, University of Rennes I, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165472

resonance depending on how the excitation is performed. Since the generated modes produce uncorrelated radiation patterns, diversity is achieved. Thus, this kind of structure appears to be well suited for MIMO systems. However, most of these multimode antennas are based on spiral shapes and do not contain any ground plane. This can be problematic when the antenna has to be integrated with a front-end. In this paper, this problem is overcome by introducing a multimode antenna based on a new mode frequency convergence concept. The original structure is a microstrip dual-polarized square patch antenna. Hence, it has the advantages of having a low profile and a ground plane, thereby making it suitable for any integration in a multilayer front-end. However, the number of uncorrelated radiation patterns available on such kind of antenna is only two (one per polarization). So the idea is to modify the square patch structure in order to obtain different modes resonating at the same frequency. Then, by an appropriate excitation, the different modes are fed independently. Thus, radiation pattern diversity is achieved between the fields radiated by the different modes. The mode frequency convergence concept is explained in Section II. Using this approach, a dual-mode antenna is designed and is presented in Section III. Then, in Section IV, the concept is extended to both polarizations of the square patch in order to obtain a dual-mode dual-polarization antenna. S-parameter and radiation pattern measurements have been conducted and the results are shown in Section V. Diversity performances of the structure are discussed in Section VI by calculating envelope correlations. II. ANTENNA THEORY AND ANALYSIS A. Mode Frequency Convergence Concept The main idea of the antenna system presented in this paper is to take advantage of multiple resonant modes of a microstrip patch antenna in order to produce radiation pattern diversity. Indeed, a structure such as a patch antenna contains some resonant modes and so it has the potential to radiate with different electromagnetic field distributions. These multiple field distributions lead to different radiation patterns which can be used to produce radiation pattern diversity. However, two major problems must be taken into account. The first one is to find a way to independently feed every mode. The second one is to force these modes to resonate at the same frequency. Thus, the patch antenna will act as different antennas performing pattern diversity. B. The Dual-Mode Aspect In this study, we focus on two modes of the patch antenna: the TM and the TM . The reasons of this choice are the fol-

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Fig. 1. Current distributions of (a) TM mode with a common feeding and (b) TM mode with a differential feeding.

lowing. Firstly, resonance frequencies of these modes are quite close together and with some design modifications, can converge. Secondly, as we will see, their electromagnetic field distributions are suitable to feed them independently. mm and Let consider a square patch antenna of length thickness mm. The substrate permittivity is . According to the cavity model [14], resonance frequencies are GHz for the TM mode and GHz for the TM mode. Fig. 1(a) and (b) show the current distribution of TM and TM modes respectively. One possible location for the feeding accesses on the square patch antenna is also indicated. The short-circuit axis are also shown (s.c.). In Fig. 1(a), the + sign indicates the location of the two input accesses, here feeding the patch in phase. In Fig. 1(b), and signs indicate the inputs out of phase and consequently, feeding a differential mode. According to TM and TM field distributions, the electric field phase at probe locations are identical in the TM mode [Fig. 1(a)] and are opposite in the TM mode [Fig. 1(b)]. Thus, the common excitation feeds only the TM mode and the differential one, only the TM mode. Therefore, this technique requires a dual-feeding for each MIMO branch which can be a disadvantage compare to usual MIMO systems involving a single-feeding per branch. However, this limitation can be overcome with the use of a Rat-Race coupler such as the one used in Section III. By observing current distributions in Fig. 1(a) and (b) (the warmer the colour, the higher the current density), one can notice that the TM one is spatially more concentrated than the TM one. So by changing the patch design in the high density current areas indicated in Fig. 1(b), both resonance modes will be disturbed but the TM will be more affected than the TM . As it will be shown in the next section, inductive slots have been etched in these areas to slow down the current in order to decrease resonance frequencies of both modes. However, since the effect is stronger on the TM mode than on the TM one, both resonance frequencies converge down to a single one. So TM and TM modes are able to produce two different radiation patterns at the same frequency.

Fig. 2. Feeding of the dual-mode dual-polarization square patch antenna.

high density current areas of both polarizations, the four different modes resonate at the same frequency. Consequently, four radiation patterns are available to produce diversity. The feeding system principle is presented in Fig. 2. The shared short-circuit axis of TM and TM modes are drawn as well as those of TM and TM modes. It appears that four short-circuit points are identical for the four modes. Labels 1 and 2 represent the location of TM and TM inputs and labels 3 and 4, the location of TM and TM inputs. Inputs 1 and 2 are located on TM and TM short-circuit axis and inputs 3 and 4 on TM and TM short-circuit axis. This has been done in order to avoid mutual coupling between the two orthogonal polarizations. III. DUAL MODE ANTENNA A. Antenna Overview The design of the proposed structure is given in Fig. 3 and and , are its dimensions in Table I. Two coaxial probes, used to feed the antenna with a common and a differential mode. Consequently, the structure is able to resonate respectively in the TM mode or in the TM mode. Four metallic via holes are located on the shared short-circuits in order to make easier finding out 50 locations for the probes (it has also the effect to bring resonance frequencies down). We firstly consider only one polarization of the square patch antenna. Height clusters of eleven inductive slots have been etched on the patch. Their locations are in the TM mode’s high density current areas [as indicated in Fig. 1(b)] but also of the TM mode, since both polarizations will be used later. B. Simulated S-Parameters

C. Dual-Polarization Aspect On a square patch, TM and TM modes resonate naturally at the same frequency, as well as TM and TM ones. So, using four input accesses and etching inductive slots in the

The structure has been simulated with CST Microwave Studio. Common and differential feedings have been achieved directly with the software. Firstly, a parametric study on the has been conducted. The resonance frequency slot length

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Fig. 5. Simulated S-parameters.

Fig. 3. Design of the proposed dual-mode antenna.

TABLE I PARAMETERS VALUES OF THE ANTENNA STRUCTURE

Fig. 6. Dual-mode antenna with its Rat-Race coupler.

Fig. 4. Variation of the resonance frequency according to the slot length.

of both the TM and TM modes has been recorded and plotted in Fig. 4. As expected from the earlier discussion, the resonance frequency of the TM mode is less affected by the exists for slots than the TM one. Consequently, a value of which both the resonance frequencies converge. This value is mm. One can also notice that the slots found to be lead to a miniaturization of the structure. Simulated S-parameters are given in Fig. 5. Reflection coefficients show that common and differential modes resonate GHz, with a bandwidth at the same frequency, of 0.25% and 0.13%, respectively. The frequency bandwidth is narrow as expected. In fact, inductive slots as well as metallic via holes disturb the resonating modes thereby acting as miniaturization techniques. For such techniques, narrow bandwidths are common. The coupling is very low (about dB at the operating frequency), which is typical from idealistic simulations.

Another simulation has been performed to validate the dualmode antenna concept. Instead of implementing the common and differential feeding mode directly with the software, an hybrid coupler (Rat-Race) has been designed to perform this function. Furthermore, the simulation has been conducted with Ansoft HFSS. Since the meshing process in HFSS and in Microwave Studio are different and since the bandwidths involved are very narrow, a comparison of the results obtained with both software seems to be appropriate. The dual-mode antenna and its hybrid coupler are presented in Fig. 6. It is a two dielectric layers structure. The upper dielectric layer has the same characteristics as the previous simulation. The patch antenna is etched on the upper side and the ground plane takes place on the lower side (so between the two substrate layers). On the lower dielectric layer, a Rat-Race coupler has been designed following the procedure described in and of thickness [15] with a substrate of permittivity mm. The two inputs of the coupler are located on the edge of the substrates whereas the outputs are connected to the patch with metallic via-holes through the ground plane. Thus, depending on which coupler input is fed, the coupler outputs produce an in-phase or an out-of-phase excitation. The S-parameters obtained by the simulation are given in Fig. 7. S corresponds to the common excitation (TM mode), and S to the differential excitation (TM mode). Obtained results are in good agreement with the previous ones even though slight differences are there. This is naturally expected when the bandwidths involved are narrow. S is the coupling between the two modes. Results are much more realistic than those of previous simulation since the maximum coupling is now about dB. The increase of the coupling level in this simulation

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Fig. 7. Simulated S-parameters obtained with the Rat-Race coupler.

Fig. 9. Gain of common mode and differential mode.

spectively the TM and the TM , even though they are altered from those of an original square patch antenna. D. Simulated Radiation Pattern

Fig. 8. Electric field distribution of (a) TM

mode and (b) TM

mode.

may be due to the fact that the in-phase/out-of-phase excitation comes from a simulated Rat-Race coupler and not directly from the software. The common/differential excitation generated by the software is perfectly accurate over the whole bandwidth of interest but this is not the case with the simulated Rat-Race.

Gains of the common and differential modes are given in Fig. 9. Patterns are drawn in the planes in respect to the -axis presented in Fig. 8. Antenna’s broadside direction is at . The maximum gain of the common mode is 3 dB whereas the maximum gain of the differential mode is about 5.7 dB. The patterns obtained are not exactly similar to those expected from a rectangular patch resonating in the TM and TM mode. The difference is mainly due to the inductive slots which modify the field distribution and so the radiation. However, the radiation shape can be identified since the common mode’s pattern presents a null in the patch’s broadside direction (as the TM mode) and the differential’s one, a maximum in this same direction (as the TM mode). Both radiation patterns being different, radiation diversity is so achieved between common and differential modes. IV. DUAL-MODE DUAL-POLARIZATION ANTENNA

C. Simulated Field Distribution The electric field distributions obtained from the simulation with HFSS are given in Fig. 8(a) and (b) respectively for the common mode and the differential mode. The boxes which can be observed under the slots are only there to help for the meshing. The substrate is completely homogeneous and these boxes are a part of it. In Fig. 8(a), the electric field distribution of the TM mode can be observed. The variation of the field is in agreement with the one of a classical patch [14]. Of course the distribution itself is not homogeneous anymore because of the slot and metallic via hole effect but the mode can still be recognized. The Fig. 8(b) presents the electric field obtained by the differential feeding. This mode can be identified as the TM one, even if the distribution has been greatly altered by the slots. By observing carefully, it can be observed that the electric field phase shifts twice along -axis while it shifts only once along -axis. So by feeding the antenna either with common or differential mode, it is possible to excite two different resonating modes, re-

A. Antenna Overview We now propose to extend the dual-mode concept to the two polarizations available on the square patch antenna. The structure remains the same as the previous one except that two additional feeding inputs are added to the ones already present. Thus, a dual-mode dual-polarization antenna is obtained as it is shown in Fig. 10. Inputs F and F allow to feed common and differential modes of the first polarization, say TM and TM and F and F the common and differential modes of the second polarization, say TM and TM . B. Simulated S-parameters The simulation has been performed with CST Microwave Studio by using the direct common and differential excitation proposed by the software. The S-parameters results are presented in Fig. 11. The reflection coefficients [Fig. 11(a)] are given for both polarizations: S and S . For each polarization, common and differential mode results are distinguished. Hence,

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Fig. 10. Design of the dual-mode dual-polarization antenna.

corresponds to the reflection coefficient of the first polarization when inputs F and F are in-phase (TM mode) when F and F are out-of-phase (TM and S mode). Also, S is the reflection coefficient of the second polarization when inputs F and F are in-phase (TM mode) and S when F and F are out-of-phase (TM mode). The obtained results are in good agreement with the previous single-polarization antenna. The slight differences are probably due to the addition of the two inputs F and F . Theoretically, these inputs should be located on the short-circuit axis of the modes excited by inputs F and F (and F and F , on the short-circuit axis of the modes excited by F and F ). However, due to the slots, inputs are not located anymore exactly on these short-circuit axis. So their impact may explain the differences in reflection coefficient results. The Fig. 11(b) shows the transmission parameters between the four modes available on the antenna. The coupling between all the modes which reaches can be neglected except dB at the maximum. This parameter represents the coupling between common modes along both the polarizations (TM and TM ). This happens when all the inputs are fed in phase. In order to understand this strong coupling, the electric field distribution of these common modes is now discussed.

Fig. 11. Simulated S-parameters: (a) reflection coefficients and (b) transmission coefficients.

C. Simulated Field Distribution The electric field distribution of common modes along both polarizations is shown in Fig. 12. The field pattern in Fig. 12(a) occurs when F and F are fed in-phase and corresponds to the TM mode. The pattern of Fig. 12(b) is obtained by feeding F and F in phase and corresponds to the TM mode. Distributions of the modes are, except the phase, similar. However, by observing carefully in the low intensity field region, one can notice that these modes are not exactly identical, thereby making them identifiable from each other. The fact that TM and TM modes are quite similar is obviously due to the presence of the slots which largely modify their field distribution. To illustrate the mechanism, we refer to the electric potential shown in Fig. 12. Fig. 12(a) presents the electric potential of the TM mode and the Fig. 12(b), the TM one. According to

Fig. 12. Electric field distribution (left) and potential (right) of (a) TM and (b) TM mode.

mode

the field distribution presented in Fig. 12, we notice that the field is mainly concentrated in four regions. These four regions are indicated by four circles in Fig. 12. Now, if we consider the field pattern only in these four regions, it is similar for both modes. To summarize, the TM and TM modes are so modified by the slots that their respective field distribution became similar. Thus, rather than two different modes, TM and TM modes can also be seen as a single hybrid mode.

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Fig. 15. Dual-mode dual-polarization antenna prototype: (a) upper view; (b) lower view.

Fig. 13. Radiation pattern (a) TM

Fig. 14. Radiation pattern of (a) TM

mode and (b) TM

mode.

mode and (b) TM .

D. Simulated Radiation Pattern Radiation patterns have been obtained by simulation and are drawn along the two polarizations and . The pattern along is defined using the electric field projection on the vector u and the pattern along using the projection on the vector u of a spherical coordinate system. Fig. 13 presents the radiation patterns of the common modes in the same planes than in Fig. 9. Patterns of Fig. 13(a) is obtained when inputs F and F are fed in phase and patterns of Fig. 13(b) is obtained when inputs F and F are fed in phase. As expected regarding the field distribution of these two modes, their radiation patterns are similar. The radiation patterns of the differential modes are shown in Fig. 14. Patterns of Fig. 14(a) is obtained when inputs F and F are fed out of phase and patterns of Fig. 14(b), when F and F are out of phase. Patterns of Fig. 14(a) and (b) are similar except that they are completely orthogonal. So it means that the differential modes excited by the couple F and F and the couple F and F have orthogonal polarizations, which is highly suitable to achieve pattern diversity. V. FABRICATION AND MEASUREMENTS A. Prototypes Two prototypes have been fabricated, one with two inputs (dual-mode single-polarization antenna) and one with four inputs (dual-mode dual-polarization antenna). Since both are identical apart from the number of coaxial inputs, only the picture of the dual-mode dual-polarization antenna is presented in Fig. 15. The patch design has been etched on a FR4 epoxy

Fig. 16. Reflection coefficients of the (a) single-polarization antenna and the (b) dual-polarization antenna.

substrate of permittivity and thickness mm. The four metallic via holes have been manually welded and can be observed in Fig. 15(a). With the four coaxial accesses (Fig. 15(b)), it is possible to feed common and differential modes of both polarizations. B. S-Parameters Measurements The S-parameters have been measured for both the prototypes: the single-polarization antenna and the dual-polarization antenna. From a classical S-parameter measurement, the reflection coefficients of the common and differential modes of the antennas have been determined by using the method described in [16], [17]. Reflection coefficients of the single-polarization antenna are shown in Fig. 16(a) and those of the dual-polarization one in Fig. 16(b). With both antennas and for all the common and differential modes, a resonance can be observed around 2.45 GHz. However, impedance matching levels are not

SARRAZIN et al.: A NEW MULTIMODE ANTENNA FOR MIMO SYSTEMS USING A MODE FREQUENCY CONVERGENCE CONCEPT

Fig. 17. Radiation pattern of (a) TM

mode and (b) TM

mode.

as good as in simulation. Minima obtained with the single-podB for the common mode and larization antenna are about dB for the differential mode. For the dual-polarization prodB and dB. Measured totype, minima are between and simulated results are in good agreement regarding the frequency resonance but not regarding the reflection coefficient levels. Differences may be due to the fabrication tolerances. In fact, by considering the narrow bandwidth of the antenna and the fact that it is a miniaturized structure, it is not surprising that the design is very sensitive to the tolerance of fabrication. It has been observed in simulation that a 10 m variation of the slot length leads to notable changes in the S-parameter results, and the tolerance of the used chemical etching is precisely about 10 m. C. Radiation Pattern Measurements Radiation pattern measurements have been conducted in a near-field anechoic chamber. Then a near-field/far-field transformation has provided the radiation pattern along and in steradian range. Measured results are given for the duala polarization antenna only and are plotted in the same planes than previously ( steradian range results are used in Section VI for envelope correlation calculations). Common and differential excitation modes have been obtained by using a RatRace coupler to feed the coaxial inputs of the antenna. Fig. 17 shows the radiation patterns of the common modes. Ripples on the curves are probably due to the metallic stand located below the antenna during the measurement process. As expected from the simulation, both modes (TM and TM ) produce the same radiation pattern and consequently no diversity is achieved. Only one of these two modes should be used at the same time in a multiple antenna communication scheme. Differential mode results are presented in Fig. 18. Patterns of both modes (TM and TM ) being different, polarization diversity is achieved between them. If one common mode is considered in addition, three different modes on this single compact antenna structure can be used to perform radiation pattern diversity

Fig. 18. Radiation pattern of (a) TM

mode and (b) TM

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mode.

Fig. 19. Envelope correlation between modes with a uniform AoA.

(in amplitude as well as in polarization), thereby acting as three different antennas. To quantify the achieved diversity, envelope correlation calculations are conducted in the next section. VI. RADIATION PATTERN DIVERSITY ANALYSIS provides a measure of the radiThe envelope correlation ation pattern diversity performances. The lower the correlation, the better the antenna diversity performances. This coefficient is defined as the equation shown at the bottom of the page [18], are complex electric fields along where and radiated by two different antennas. The parameter X is the cross-polarization discrimination (XPD) of the incident field (where and represent and is defined as the average power along the spherical coordinates and ). The angle is defined by [0: ] in elevation and [0:2 ] in azimuth. and are the Angle-of-Arrival (AoA) distributions of incoming waves. Fig. 19 presents the envelope correlation between the measured patterns of the dual-mode dual-polarization antenna with a uniform AoA distribution. The mode 1 is the common mode of the polarization 1 (TM ), the mode 2 the common mode of the polarization 2 (TM ), the mode 3 the differential mode of the polarization 1 (TM ) and the mode 4 the differential mode

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certain values of . So adequate processing should be considered. However, as it has already been mentioned, only three modes have to be used simultaneously in a MIMO context. So only one branch will undergo power imbalance. This suggests that the processing may be less complex than in the case where power imbalance would occur with all the three branches. VII. CONCLUSION

Fig. 20. Envelope correlation between modes with an AoA distribution.

A new multimode antenna has been introduced. Our approach had led to modifications of the design of a microstrip square patch in order to obtain different modes resonating at the same frequency. Then, with an appropriate excitation, these modes can be fed independently. Through a common/differential excitation, the proposed structure is thus able to produce three different radiation patterns, thereby acting as three independent antennas suitable for MIMO systems or any systems based on antenna diversity. This concept has been validated in simulation as well as in measurement in the 2.45 GHz frequency band. In conclusion, the observation of the electromagnetic field of square patch modes has made possible the development of a new concept which may be applicable not only to the square patch antenna but also to other structures. REFERENCES

Fig. 21. Self-correlations (power imbalance).

of the polarization 2 (TM ). Thus, as an example, represents the correlation between the patterns of TM and TM modes. It can be observed that correlation levels between patterns of the different modes available on the antenna are below . This indicates that radiation pattern diver0.18 except for sity is achieved between the differential modes as well as between the differential and the common modes. Consequently, this structure can act as three different antennas in a diversity communication scheme. The correlation between the common is quite high as expected. In fact, the pattern of the modes common modes being very similar, it is logical that the envelope correlation is high. Hence, these two modes should not be used simultaneously in a diversity communication scenario. Fig. 20 shows the correlation with a Laplacian AoA distribution in the elevation plane as suggested in [19]. The angular deabout (with being the antenna’s viation is broadside direction). Even by taking into account an AoA distribution, correlation levels remain quite low. So pattern diversity is still efficient. It is well known that power imbalance in MIMO system branches leads to degradation of performances [20], [21]. For that purpose, Fig. 21 shows the self-correlation of each mode using the same previous Laplacian AoA distribution. Self-correlations are expressed in dB and show the differences between modes’ average power. Power imbalance between common and ) and between differential modes ( modes ( and ) never exceeds 2.5 dB. However, power imbalance up to 10 dB exists between common and differential modes for

[1] A. Paulraj, D. Gore, R. Nabar, and H. Bölcskei, “An overview of MIMO communications—A key to gigabit wireless,” Proc. IEEE, vol. 92, no. 2, pp. 198–218, Feb. 2004. [2] A. S. Konanur et al., “Increasing wireless channel capacity through MIMO systems employing co-located antennas,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 6, pp. 1837–1844, Jun. 2005. [3] H.-R. Chuang and L.-C. Kuo, “3-D FDTD design analysis of a 2.4 GHz polarization diversity printed dipole antenna with integrated balun and polarization-switching circuit for WLAN and wireless communication applications,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 374–381, Feb. 2003. [4] C.-Y. Chiu, J.-B. Yan, and R. Murch, “Compact three-port orthogonally polarized MIMO antennas,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 619–622, 2007. [5] J. Sarrazin, Y. Mahé, S. Avrillon, and S. Toutain, “Investigation on cavity/slot antennas for diversity and MIMO systems: The example of a three-port antenna,” IEEE Antennas Wireless Propag. Lett., vol. 7, no. 11, pp. 414–417, Nov. 2008. [6] J. Sarrazin, Y. Mahé, S. Avrillon, and S. Toutain, “Collocated microstrip antennas for MIMO systems with a low mutual coupling using mode confinement,” IEEE Trans. Antennas Propag., vol. 58, no. 2, pp. 589–592, Feb. 2010. [7] O. Klemp, M. Schultz, and H. Eul, “Novel logarithmically periodic planar antennas for broadband polarization diversity reception,” Int. J. Electron. Commun., vol. 59, pp. 268–277, Apr. 2005. [8] C. Waldschmidt and W. Wiesbeck, “Compact wide-band multimode antennas for MIMO and diversity,” IEEE Trans. Antennas Propag., vol. 52, no. 8, pp. 1963–1969, Aug. 2004. [9] D. Chew, I. Morfis, and S. Stavrou, “Quadrifilar helix antenna for MIMO system,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 197–199, 2004. [10] S. Ko and R. Murch, “Compact integrated diversity antenna for wireless communications,” IEEE Trans. Antennas Propag., vol. 49, no. 6, pp. 954–960, Jun. 2001. [11] A. Khaleghi, A. Azoulay, J. Bolomey, and N. Ribière-Tharaud, “Design and development of a compact WLAN diversity antenna for wireless communications,” presented at the Conf. JINA, Nov. 2004. [12] E. Rajo-Iglesias, O. Quevado-Teruel, and M. Pablo-Gonzales, “A compact dual mode microstrip patch antenna for MIMO applications,” presented at the IEEE Antennas and Propagation Int. Symp., Jul. 2006. [13] T. Svantesson, “An antenna solution for MIMO channels: The multimode antenna,” in Proc. 34th Asilomar Conf., 2000, vol. 2, pp. 1617–1621.

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[14] J. R. James, P. S. Hall, and C. Wood, “Microstrip antenna theory and design,” Inst. Elect. Eng., 1981. [15] M. Mandal and S. Sanyal, “Reduced-length rat-race couplers,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 12, pp. 2593–2598, Dec. 2007. [16] R. Meys and F. Janssens, “Measuring the impedance of balanced antennas by an S-parameter method,” IEEE Antennas Propag. Mag., vol. 40, no. 6, pp. 62–65, Dec. 1998. [17] T. Milligan, “Parameters of a multiple-arm spiral antenna from a single-arm measurement,” IEEE Antennas Propag. Mag., vol. 40, no. 6, pp. 65–69, Dec. 1998. [18] R. Vaughan and J. Andersen, “Antenna diversity in mobile communication,” IEEE Trans. Veh. Technol., vol. VT-36, no. 4, pp. 149–172, Nov. 1976. [19] K. Kalliola, K. Sulonen, H. Laitinen, O. Kivekäs, J. Krogerus, and P. Vainikainen, “Angular power distribution and mean effective gain of mobile antenna in different propagation environments,” IEEE Trans. Veh. Technol., vol. 51, no. 5, pp. 823–838, Sep. 2002. [20] K. Ogawa, S. Amari, H. Iwai, and A. Yamamoto, “Effect of received power imbalance on the channel capacity of a handset MIMO,” in Proc. IEEE Int. Symp. PIMRC, Sep. 2007, pp. 1–5. [21] O. Klemp and H. Eul, “Diversity efficiency of multimode antennas impacted by finite pattern correlation and branch power imbalances,” in Proc. Int. Symp. ISWCS, Oct. 2007, pp. 322–326.

Julien Sarrazin received the M.Tech. and M.S. degreesin electronics from the Ecole Polytechnique de l’Université de Nantes, Nantes, France, in 2005, and the Ph.D. degree from the Institut de Recherche en Electrotechnique et Electronique de Nantes Atlantique (IREENA), University of Nantes, in 2008. From 2008 to 2010, he was with the BK Birla Institute of Technology (BKBIET), Pilani, India. He is currently a Research Engineer at Telecom ParisTech, Paris, France. His research interests are focused on antennas dedicated to MIMO systems and highimpedance surfaces.

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Yann Mahé (M’01) was born on June 29, 1973. He received the Ph.D. degree in electrical engineering from the University of Nantes, Nantes, France, in 2001. He is currently an Assistant Professor at the Polytechnic School of the University of Nantes, Nantes, France. He also performs research with the Institut de Recherche en Electrotechnique et Electronique de Nantes Atlantique (IREENA), France. His research interests include antenna design and the development of miniaturized and tunable antenna for MIMO systems.

Stephane Avrillon (M’04) received the B.S. degree in electronics engineering from the Ecole Normale Superieure (ENS), Cachan, France, in 1998, the M.S. degree in electronics engineering from Polytech’Nantes, Nantes, France, in 2000, and the Ph.D. degree from the Institut de Recherche en Electrotechnique et Electronique de Nantes Atlantique (IREENA), Nantes, in 2004. Since 2005, he has been an Assistant Professor at Institut d’Electronique et Telecommunications de Rennes (IETR), Rennes, France. His research interests are focused on antenna design, characterization, and modelling applied to diversity, MIMO, and UWB systems.

Serge Toutain was born in 1948. He received the Engineer degree from the Ecole Nationale de Radio-Electricité Appliquée (ENREA), Paris, France, in 1970, and the Ph.D. degree from the University of Lille, Lille, France, in 1976. From 1984 to 1998, he was head of the Microwave Department, Telecom Bretagne. In 1998, he joined the University of Nantes, Nantes, France, where he was until 2009 a full professor, head of the IREENA research institute, University of Nantes, Nantes, France. His main research interests were telecommunications, design of passive and active microwave devices, antennas and propagation, but now, he spends his time playing bridge and doing photography during his peaceful retirement.

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Wearable Circularly Polarized Antenna for Personal Satellite Communication and Navigation Emmi K. Kaivanto, Student Member, IEEE, Markus Berg, Student Member, IEEE, Erkki Salonen, Member, IEEE, and Peter de Maagt, Fellow, IEEE

Abstract—Integrating antennas into fabrics is a potential way for facilitating many applications, such as health monitoring of patients, fire-fighting, rescue work, and space and military personal communications. This paper studies possibilities to construct a flexible, lightweight and mechanically robust textile antenna for dual-band satellite use: Iridium and GPS. Different textile materials were characterized and the most promising materials were used to design, construct, and test a rectangular patch antenna. The gain and axial ratio for both bands is compliant with specifications and relatively stable under most bending conditions. The developed antenna solution allows integration into clothing. Index Terms—Dual-band patch antenna, circularly polarized antenna, textile antenna, wireless body area network (WBAN).

I. INTRODUCTION

T

HE number of wireless applications is continuously increasing, while the size of the corresponding devices is decreasing. However, the trend towards miniaturization causes some challenges considering the correct functioning of the antenna and a tradeoff between antenna performance and size and battery life must be performed. Furthermore, one needs to consider the interfering effect of components in close proximity to the antenna. Integrating wireless electronics into clothing is a potential solution for many applications aiming to increase user safety, awareness, convenience, and operability. These applications include health monitoring of patients [1], fire-fighting [2], rescue work [3], and space and military personal communications [4]. The literature covers both on-body [5]–[7] and off-body [8], [9] solutions. In satellite communication systems, such as GPS, Galileo, and Iridium, circularly polarized antennas provide up to 3-dB better power levels compared to linearly polarized antennas. However, these antennas are often too large or complex to be implemented as an internal antenna in a mobile phone but can easily be hidden inside a sleeve of a jacket as an external antenna. Textile antennas can often be implemented without an extreme pressure towards miniaturization while even improving Manuscript received February 24, 2011; revised June 08, 2011; accepted June 16, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. This work was supported in part by the European Space Agency. E. Kaivanto, M. Berg, and E. Salonen are with the Electrical Engineering Department, Centre for Wireless Communications, University of Oulu, Oulu FIN-90014, Finland (e-mail: [email protected]). P. de Maagt is with the European Space Research & Technology Centre, NL 2200 AG Noordwijk, The Netherlands. Digital Object Identifier 10.1109/TAP.2011.2165513

comfort of use and allowing hands-free operation. Also, the use of multiple external antennas is possible, hence improving the overall coverage. As a textile antenna is meant to be attached to or integrated in user’s clothes, it is desirable to make an antenna conformable and as nonobtrusive as possible. When the user moves, the antenna should bend along with the clothes and still maintain its functionality. However, changes in antenna shape are a potential cause for degraded performance [10]. Designing the shape carefully and by positioning the antenna in places with reduced bending, such as the shoulder area, are ways to reduce these problems [11]. Also the selection of textile materials has an influence on both the technical performance and user comfort. Although noise temperature of the antenna has an effect on the quality of GPS signal and Iridium downlink signal, the effect is estimated to be of second order but would need to be taken into account for further design improvements. The antenna with conical radiation pattern towards “cold” sky has lower antenna noise temperature and is less sensitive to multipath effects than an omnidirectional antenna. This paper studies possibilities to construct a flexible and lightweight textile antenna for dual-band use. The antenna should be operational in an outdoor environment, and allow normal activities like walking, running, and sitting. The antenna must function for both Iridium and GPS bands, and preferably maintain right-hand circular polarization even under bending conditions. The feasibility of antenna diversity is examined by determining the coverage area and isolation between two antennas. II. ANTENNA DESIGN In order to achieve an adequate solution for integrating the antenna into clothing, a planar patch-type antenna was selected as the baseline for this research because of its inherent low profile. A ground plane of the patch should also reduce the user’s influence on antenna performance. The selected specific geometry was a square ring patch with a polygon-shaped slot due to its demonstrated circular polarization and wide band performance [14]. In this case, the wide band performance is needed in bending conditions when a frequency shift is probable. When selecting potential fabrics, the following aspects were considered to be of paramount importance. The design driver was the best possible performance, and low loss fabrics were thus preferred. The fabrics should also be mechanically durable and although they should be highly flexible they should show limited stretching and compression. In addition, the fabrics should return to their original form after dimensional deformations. The fabrics should also be nonhygroscopic, and tolerate

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TABLE I MEASURED ELECTRICAL PARAMETERS OF THE SUBSTRATE MATERIALS

temperatures required for soldering. Out of many materials studied, two substrate textiles, Cordura and another woven fabric, which fulfilled the mechanical qualification requirements, were selected. The electrical parameters of these dielectric fabrics were studied to confirm their suitability as antenna substrates. Since these fabrics are woven and thereby anisotropic, their electrical parameters were measured separately in all three orthogonal directions using a cavity resonance method. In Cartesian coordinates, the - and -axes are defined as warp and weft directions respectively, while the -axis is the out-of-plane direction. Measured effective permittivities and loss tangents are presented in Table 1. Both materials have reasonably low losses and qualify to be used as substrates. The uncertainty of the measurements is estimated to be approximately 5% for permittivity values and about 60% for loss tangent. The accuracy of the sample volume is the most significant error source in the measurements. Although low loss values are challenging to measure, the loss is so small that even 60% change in the value does not have significant effect on the antenna performance. The details of the cavity resonance measurement technique, analysis methods, and the measurement results for the other textiles will be presented elsewhere. In order to achieve an adequate thickness for the antenna to function properly, several fabric layers were stacked. Both measured dielectric fabrics were exploited to get the best possible structure for the antenna. The ground plane and the patch were woven, silver, and copper plated, low-loss nylon fabric, which surface resistance is less than 0.03 / . All the fabric layers were connected by sewing in order to keep the structure as bendable as possible and to avoid extra losses caused by adhesives. The antenna is fed via a SubMiniature version A (SMA) connector, which was punched through the antenna and soldered to the conductive fabrics. The antenna structure is illustrated in Fig. 1 in which the length and the width are both 65 mm. The feed point coordinates in millimeters when the origin is located at the center of the antenna are and . The total thickness of the antenna is approximately 3 mm. The antenna was designed, using CST Microwave Studio software, to simultaneously cover two different frequency bands of 1575 MHz (GPS) and 1621.35–1626.50 MHz (within Iridium). In addition, to assure adequate functioning even when bent, a guard band of some 30 MHz was included. Some target parameters for the antenna were preset (by the project) as follows. The lower limit for the gain was 2.5 dBi

Fig. 1. (a) The structure and (b) first prototype of the antenna.

while the upper limit was 7.5 dBi. To assure circular polarization, the highest limit for the axial ratio was 5 dB. The target for the coverage was 85 from the zenith. III. ANTENNA PERFORMANCE A circularly polarized wave is very susceptible to distortions in multipath environment wherein arbitrary reflections may degrade the polarization purity of the wave or even change the direction of rotation of the polarization. However, comparative studies in [12] and [13] show that GPS functions also with linearly polarized receiver antennas, thus right-hand circular polarization (RHCP) requirements for GPS, were considered less critical. In the case of a satellite mobile phone, the antenna is not only receiving but also transmitting. As the connection is sustained by an adaptive power control, the quality of the link has an effect on the battery consumption of the phone. In order to secure the connection and to reduce the power consumption, the RHCP requirements were considered to be stricter for Iridium as compared to GPS, and hence the center frequency of the antenna was designed to be closer to the Iridium band. A. Return Loss Return loss (S11) measurements were conducted on a prototype textile antenna using a HP8720ES vector network analyzer. The total 10-dB S11 bandwidth was some 80 MHz. Since the prototype antenna resonated at a lower frequency band than the simulated one, another prototype was made and measured to verify the results. After tuning the prototype antennas and using the real measured dimensions in the simulation there remains about 50-MHz difference between simulated and measured values as can be seen in Fig. 2. This is believed to be due

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Fig. 2. Simulated and measured S11 curves of the wearable antenna.

Fig. 3. Bending directions of the antenna:

45 , and

Fig. 5. Measured total efficiency curves of the wearable antenna unbent and bent in four directions.

45 .

Fig. 6. Measured RHCP gain curves of the wearable antenna unbent and bent 0 . in four directions at

Fig. 4. Measured S11 curves of the wearable antenna unbent and bent in four directions.

to the inaccuracy of the measured permittivity values and fabrication tolerances. To examine the effect of bending on the performance, the antenna was measured while bent in four directions: , , 45 , and 45 . The bending directions are depicted in Fig. 3. The bending radius used was 50 mm, corresponding to the typical size of a human arm. Should the antenna be attached to a sleeve of a jacket, the radius will be greater, and the chosen 50 mm represents the worst case situation. The bending measurements were performed using a Rohacell half cylinder as a support. Even though bending shifts the resonance frequency, the bandwidth is wide enough to secure a 10-dB band of 53 MHz (see Fig. 4). B. Radiation Pattern Measurements Radiation pattern measurements were performed using a Satimo Starlab measurement system. As the measured total

efficiency curves illustrated in Fig. 5 show, the total efficiency is relatively resistant against bending remaining around 70% over the band. It should be noted that bending in the -direction seems to be the most harmful, affecting the bandwidth more than the other bending directions. Fig. 6 illustrates the results from the boresight ( -direction) RHCP gain measurements as a function of frequency. The results indicate that the RHCP gain is better than 5 dBic for the GPS band and for the Iridium band approximately 6 dBic. These results are compliant with the preset requirements from 2.5 dBic to 7.5 dBic. In addition, the RHCP gain does not seem to be significantly influenced by bending. In order to examine the polarization purity of the antenna, its axial ratio and right- and left-hand circular polarization gains were measured. Fig. 7 illustrates the RHCP and LHCP gains for Iridium at 1625 MHz as a function of angle at -plane 0 . To keep the curves separable, only two bending directions and are shown. These directions were found to represent the least and most affecting bending directions. Regardless of bending, the RHCP gain remains significantly greater and consequently the antenna is clearly right-hand polarized. Also, bending in the -direction has the strongest effect on the gain and polarization properties of the antenna. Compared to the

KAIVANTO et al.: WEARABLE CIRCULARLY POLARIZED ANTENNA FOR PERSONAL SATELLITE COMMUNICATION AND NAVIGATION

Fig. 7. Measured right- and left-hand circular polarized gain curves of the wearable antenna unbent and bent in - and -directions at 1625 MHz at 0 .

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Fig. 9. Measured axial ratio curves of the wearable antenna unbent and bent in 0 . four directions at

Fig. 10. Measured axial ratio curves of the wearable antenna unbent and bent 0 . in - and -directions at 1575 and 1625 MHz at Fig. 8. Measured right- and left-hand circular polarized gain curves of the wearable antenna unbent and bent in - and -directions at 1575 MHz at 0 .

unbent case bending in the -direction causes only one decibel decrease in RHCP boresight gain but nearly 16-dB increase in LHCP gain. This means that the purity of circular polarization remarkably decreases when the antenna is bent in the -direction. Fig. 8 illustrates the RHCP and LHCP gains for GPS as a function of angle at 1575 MHz at -plane. For GPS, the polarization properties seem to be worse, but the RHCP gain remains greater than the LHCP gain even under bending conditions. As for the Iridium band, bending in the -direction reduces the RHCP gain more than the other bending directions. Measured boresight axial ratio curves are presented in Fig. 9 as a function of frequency. The axial ratio is defined as the relation between RHCP and LHCP electric fields or absolute value gains of the antenna as follows:

If the -direction is treated separately, there is a 28-MHz wide band where the axial ratio remains below 5 dB regardless of bending. Fig. 10 illustrates the axial ratios at 1575 MHz (for GPS) and at 1625 MHz (for Iridium) as a function of at -plane, both unbent and bent in - and -directions. At the Iridium frequency, the axial ratio remains better than 5 dB over a wide range, while bending in the -direction worsens the axial ratio to 6 dB. This is in agreement with the LHCP and RHCP gain analyses described above. Therefore, to secure circular polarization, bending in the -direction should be avoided. This can be arranged in practice by attaching the antenna, for example, to a user’s upper arm so that the -axis of the antenna runs along the arm. Fig. 11 presents the radiation pattern of the antenna at 1625 MHz both for unbent and bent in the -direction situations. When the antenna is bent, its maximum gain is reduced only by 0.3 dB and its maximum back lobe level at 120 increases from 13 to 7 dB but otherwise the shape of the pattern is preserved. C. Benefit of Multiantenna Utilization

(1)

Since the antenna radiates to a half hemisphere, the best coverage using single antenna is achieved when the boresight of the

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Fig. 11. Measured 3-D RHCP gain patterns of the wearable antenna (a) unbent and (b) bent in the -direction.

Fig. 13. RHCP and LHCP gain curves of two antennas measured one at a time 0 compared to one antenna, with phantom hand and head at 1625 MHz at which boresight is towards zenith.

Fig. 12. Radiation measurement setup with a phantom head and hand when the antenna is on the left.

antenna is towards zenith. However, if the antenna is located, for example, on the upper arm and thus the boresight is towards the horizontal plane, the coverage is reduced. To maintain an adequate link with satellites in all circumstances, the total coverage can be enhanced by employing multiple antennas. Fig. 12 illustrates the radiation measurement setup, in which a phantom head and a hand are employed in Satimo Starlab measurement system and antennas are measured in turn on the left side and on the right side of the head. The resulting curves in Fig. 13 show the benefit of using two antennas. Since RHCP gain of 2.5 dBic is the lowest acceptable gain value, the coverage angle of single antenna measured boresight towards zenith is approximately 140 while the total coverage angle of two antennas on upper arms is more than 180 . The improvement in RHCP gain at 81.8 from zenith, which is the limit angle for 100% probability of visibility of one Iridium satellite [15], is approximately 7 dB. It can also be seen that the highest gain value at each theta angle degree is right handed. D. Isolation Should multiple antennas be utilized, it is essential to ascertain adequate isolation between the antennas. To study the possible coupling, two antennas were attached on the upper part of both arms of a user and the S21 parameter was measured by using an HP8720ES analyzer. Also the worst case, where the antennas are side by side on the same arm, was measured. The measurement setups are shown in Fig. 14. Fig. 15 illustrates the result of these measurements. Isolation is around 50 dB for

Fig. 14. (a) Wearable antennas on both upper arms of the user and (b) wearable antennas side by side on the same upper arm.

both GPS and Iridium bands when the antennas were attached on the different arms and still better than 30 dB for the antennas side by side. Hence, the isolation is adequate to effectively enable using multiple antennas.Fig. 15 also shows S11 curves of these two antennas. It can be seen that the presence of a user has no significant effect on the total S11 bandwidth of the antennas. IV. CONCLUSION This paper studies possibilities to construct a flexible and lightweight textile antenna for dual-band satellite use, Iridium and GPS, covering both positioning and communication needs. The study includes analyzing the effects of bending and looks into the use of multiple antennas. It is also shown that the antenna is not sensitive to proximity of the user body. The antenna radiates well into a half-hemisphere with a wide coverage angle. The total efficiency remains better than 65% over approximately 75-MHz band and is relatively immune to bending. The RHCP gain is compliant with the preset requirements from 2.5 to 7.5 dBic. The developed antenna maintains RHCP over 53-MHz band even in bending conditions. Since Iridium is seen more critical in the polarization point of view, the center frequency of the antenna is tuned closer to Iridium band at the expense of GPS.

KAIVANTO et al.: WEARABLE CIRCULARLY POLARIZED ANTENNA FOR PERSONAL SATELLITE COMMUNICATION AND NAVIGATION

Fig. 15. Measured S11, S22, and S21 curves of two antennas placed on user’s upper arms and side by side.

The axial ratio for Iridium remains better than 5 dB (if bending in the -direction is excluded) whereas the polarization for GPS is elliptical. Noise temperature of the antenna and multipath effects between the links will be included in the future work. Improving the total coverage by utilizing two antennas is also investigated. Compared to a single antenna, even 7-dB increase in RHCP gain is seen at the limit angle of 81.8 from zenith, which is the case when at least one Iridium satellite is always visible. The isolation between antennas is good, approximately 50 dB when the two antennas are attached on the upper parts of both arms of the user, allowing the use of multiple antennas and hence potentially improving the total coverage. The developed antenna solution facilitates integration into clothing and consequently various applications. The presented textile antenna tolerates bending well and its dual-band characteristics enable a single antenna covering both positioning and communication needs.

ACKNOWLEDGMENT The authors would like to thank Patria for cooperation.

REFERENCES [1] A. Alomainy, Y. Hao, and F. Pasveer, “Numerical and experimental evaluation of a compact sensor antenna for healthcare devices,” IEEE Trans. Biomed. Circuits Syst., vol. 1, no. 4, pp. 242–249, Dec. 2007. [2] L. Vallozzi, P. Van Torre, C. Hertleer, H. Rogier, M. Moeneclaey, and J. Verhaevert, “Wireless communication for firefighters using dual-polarized textile antennas integrated in their garment,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1357–1368, Apr. 2010. [3] D. Curone, E. L. Secco, A. Tognetti, G. Loriga, G. Dudnik, M. Risatti, R. Whyte, A. Bonfiglio, and G. Magenes, “Smart garments for emergency operators: The ProeTEX project,” IEEE Trans. Inf. Technol. Biomed., vol. 14, no. 3, pp. 694–701, May 2010. [4] P. Salonen and Y. Rahmat-Samii, “Textile antennas: Effects of antenna bending on input matching and impedance bandwidth,” IEEE Aerosp. Electron. Syst. Mag., vol. 22, no. 12, pp. 18–22, Dec. 2007.

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[5] P. S. Hall, Y. Hao, Y. I. Nechayev, A. Alomalny, C. C. Constantinou, C. Parini, M. R. Kamarudin, T. Z. Salim, D. T. M. Hee, R. Dubrovka, A. S. Owadally, W. Song, A. Serra, P. Nepa, M. Gallo, and M. Bozzetti, “Antennas and propagation for on-body communication systems,” IEEE Antennas Propag. Mag., vol. 49, no. 3, pp. 41–58, Jun. 2007. [6] 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. [7] D. Psychoudakis and J. L. Volakis, “Conformal asymmetric meandered flare (AMF) antenna for body-worn applications,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 931–934, 2009. [8] 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. [9] S. L. Cotton and W. G. Scanlon, “Measurements, modeling and simulation of the off-body radio channel for the implementation of bodyworn antenna diversity at 868 MHz,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3951–3961, Dec. 2009. [10] T. Kellomaki, J. Heikkinen, and M. Kivikoski, “Effects of bending GPS antennas,” in Proc. Microw. Conf., Dec. 12–15, 2006, pp. 1597–1600. [11] E. Kaivanto, J. Lilja, M. Berg, E. Salonen, and P. Salonen, “Circularly polarized textile antenna for personal satellite communication,” in Proc. 4th Eur. Conf. Antennas Propag., Apr. 12–16, 2010, pp. 1–4. [12] V. Pathak, S. Thornwall, M. Krier, S. Rowson, G. Poilasne, and L. Desclos, “Mobile handset system performance comparison of a linearly polarized GPS internal antenna with a circularly polarized antenna,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jun. 22–27, 2003, vol. 3, pp. 666–669. [13] A. A. Serra, P. Nepa, G. Manara, and R. Massini, “A low-profile linearly polarized 3D PIFA for handheld GPS terminals,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1060–1066, Apr. 2010. [14] M. Ali, R. Dougal, G. Yang, and H.-S. Hwang, “Wideband (5–6 GHz WLAN band) circularly polarized patch antenna for wireless power sensors,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jun. 22–27, 2003, vol. 2, pp. 34–37. [15] H. Keller, H. Salzwedel, G. Schorcht, and V. Zerbe, “Comparison of the probability of visibility of the most important currently projected mobile satellite systems,” in Proc. IEEE 47th Veh. Technol. Conf., May 4–7, 1997, vol. 1, pp. 238–241.

Emmi K. Kaivanto (S’08) received the M.Sc. degree in electrical engineering from the University of Oulu, Oulu, Finland, in 2008, where she is currently working towards the Ph.D. degree in the field of antennas and radio channel properties in body area networks. During her studies, she was with Pulse Finland, Oulu, Finland, where she was involved in projects related to mobile phone antennas. In December 2008, she joined the Radio Engineering group in the Centre for Wireless Communications, University of Oulu. Her current research interests include ultrawideband WBAN antennas and radio channels for medical healthcare, printable antennas, antenna materials, and electromagnetic dosimetry.

Markus Berg (S’06) received the M.Sc. degree in electrical engineering from the University of Oulu, Oulu, Finland, in 2005, where he is currently working towards the Dr. Tech degree in the Centre for Wireless Communications. His research interests include frequency reconfigurable antennas and user effect compensation techniques.

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Erkki T. Salonen (A’11) received the M.Sc., Licentiate of Science, and Doctor of Science in Technology degrees from Helsinki University of Technology (TKK), Espoo, Finland, in 1979, 1986, and 1993, respectively. From 1984 to 1996, he was in charge of the radio wave propagation studies at the Radio Laboratory, Helsinki University of Technology. In 1997–2008, he was a Professor of Radio Engineering in the Telecommunication Laboratory, University of Oulu, Oulu, Finland. Currently, he is a Research Manager in the area of radio engineering at the Centre of Wireless Communication, University of Oulu. His main research interests include antennas and propagation in radio communications.

Peter de Maagt (S’88–M’88–SM’02–F’08) was born in Pauluspolder, The Netherlands, in 1964. He received the M.Sc. and Ph.D. degrees from Eindhoven University of Technology, Eindhoven, The Netherlands, in 1988 and 1992, respectively, both in electrical engineering. From 1992 to 1993, he was a Station Manager and Scientist with an INTELSAT propagation project in Surabaya, Indonesia. He is currently with the European Space Research and Technology Centre (ESTEC), European Space Agency (ESA), Noord-

wijk, The Netherlands. His research interests are in the area of millimeter and submillimeter-wave reflector and planar integrated antennas, quasi-optics, electromagnetic bandgap antennas, and millimeter- and submillimeter-wave components. He spent summer 2010 as a Visiting Research Scientist at the Stellenbosch University, Stellenbosch, South Africa. Dr. de Maagt was corecipient of the H. A. Wheeler Award of the IEEE Antennas and Propagation Society (IEEE AP-S) for the Best Applications Paper of 2001 and 2008. He was granted an ESA Award for Innovation in 2002 and an ESA award for Corporate Team Achievements for the Herschel and Planck Programme in 2010. He was corecipient of Best Paper Awards at the Loughborough Antennas Propagation Conference (LAPC) 2006 and the International Workshop on Antenna Technology (IWAT) 2007. He served as an Associate Editor for the IEEE TRANSACTION ON ANTENNAS AND PROPAGATION from 2004 to 2010 and was Co-Guest Editor of the November 2007 Special Issue on Optical and Terahertz Antenna Technology.

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24 GHz Balanced Doppler Radar Front-End With Tx Leakage Canceller for Antenna Impedance Variation and Mutual Coupling Han Lim Lee, Won-Gyu Lim, Kyoung-Sub Oh, and Jong-Won Yu, Member, IEEE

Abstract—A new balanced Doppler radar front-end architecture with Tx leakage canceller for antenna impedance variation and antenna mutual coupling is proposed. Consisting of two quadrature hybrids, two ring hybrids and 4-port feed antenna system, the proposed balanced structure achieves high transmit/receive (Tx/Rx) isolation stable to antenna impedance variation and antenna mutual coupling. The proposed 4-port feed antenna is configured by dual fed 2 1 patch array having 6.97 dBi and 5.65 dBi of measured Tx and Rx peak gain, respectively. In addition, the proposed architecture undergoes no power loss in both transmit and receive paths, and obtains the measured Tx/Rx isolation of more than 45 dB stable at 24 GHz regardless of the load variation. The proposed architecture that can determine the speed as low as 0.5 mm/s (0.078 Hz Doppler shift) is theoretically and experimentally analyzed for high isolation despite of antenna mismatch. Index Terms—Antenna isolation, antenna reflection, Doppler radar, load impedance variation, mutual coupling, Tx leakage.

I. INTRODUCTION HORT range radar using microwave technology has actively been applied in many applications such as speed/motion detectors, automotive sensors, healthcare sensors and collision avoidance [1]–[3]. These radar applications strongly emphasize high performance, compact size, simple structure and low cost [4]. Since continuous wave (CW) radars have a relatively simple circuit structure with compactness and cost-effective solutions, CW radar systems such as frequency modulated continuous wave (FMCW) or Doppler radars are widely applied in industrial, scientific and medical bands. Among various short range radar technologies, 24 GHz has been widely studied because of the favors in size reduction of passive and active components in high frequency band. Moreover, a single antenna radar structure is preferred over two-separate Tx/Rx antenna system since two antennas double the size of system comparing to a single antenna system. However, if the Tx/Rx signals are at the same frequency, a duplexer cannot be used and thus the leakage from Tx to Rx becomes the important

S

Manuscript received October 05, 2010; revised May 03, 2011; accepted June 02, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. H. L. Lee, K.-S. Oh, and J.-W. Yu are with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea (e-mail: [email protected]) W.-G. Lim is with the Korea Aerospace Research Institute (KARI), Daejun 305-600, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165486

Fig. 1. Conventional methods for Tx and Rx isolation by using: (a) quadrature hybrid, (b) circulator, (c) directional coupler, and (d) two antennas.

limit for receiver sensitivity. That is, it is very crucial to have high Tx/Rx isolation in a single antenna radar system. Conventional approaches to minimize the Tx leakage can be represented as shown in Fig. 1. In these structures, the receiver is still exposed to a substantial signal from transmitter due to the reflection from antenna and the inherent isolation performance of components such as circulator, directional coupler and hybrid coupler. A commercially available circulator, directional coupler or hybrid coupler generally provides the isolation of about 25 dB in a well-matched condition. Also, orthomode transducers (OMT) which utilize left and right polarization are used for providing the isolation of about 30 dB in a matched condition. If the antenna and transceiver are not well-matched, the Tx leakage then will be significantly increased and thus the conventional structures cannot guarantee sufficient receiver sensitivity for the antenna impedance variation. There have been several topologies based on quadrature hybrids and lange couplers to cancel the Tx leakage [5]–[8]. However, even these recently proposed radar front-ends with Tx leakage cancellers are only valid in a well-matched condition. That is, if the antenna impedance varies, the Tx leakage will no longer be cancelled and affect the receiver sensitivity. Moreover, the previously proposed radar topologies with Tx leakage cancellers using patch array antennas fed by multi-ports are influenced by the antenna mutual coupling among ports and

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Fig. 2. Proposed balanced front-end structure with high Tx/Rx isolation.

Fig. 3. Conventional use of quadrature hybrid with antenna (load) mismatched: (a) single quadrature hybrid and (b) cascaded quadrature hybrids.

have 3 dB loss in the receive path. That is, the Tx/Rx isolation of the previously proposed architectures is not efficient and stable. In this paper, a new balanced Doppler radar front end configured with ring hybrids, quadrature hybrids and Wilkinson power dividers for 24 GHz applications is presented as shown in Fig. 2. The proposed structure has no loss in both transmit and receive paths. In addition, Tx leakage is cancelled regardless of the antenna impedance variations and mutual couplings among antenna ports are also cancelled. II. BALANCED QUADRATURE-RING HYBRID DESIGN A. Tx Leakage Cancellation for Antenna Impedance Variation In Fig. 3(a), a single hybrid coupler is connected to a mismatched load (antenna). The scattering matrix of the single hybrid coupler with perfectly matched load can be represented as follows:

the connected antenna . Similarly, the Tx leakage of cascaded quadrature hybrids depends on the isolation of hybrid coupler and antenna reflection as in Fig. 3(b). However, recently proposed structures configured by multiple quadrature hybrids or lange couplers only take account of the Tx leakage due to hyby implementing brid or lange coupler’s isolation ( opposite phases in each path). That is, if the antenna load varies , the Tx leakage due to antenna reflection still remains and dominantly degrades the overall system sensitivity. Thus, new balanced quadrature-ring hybrid structure consisting of two ring hybrids, two quadrature hybrids and 4-port feed antenna as shown in Fig. 2 is proposed to consider both component isolation and antenna reflection for the case of antenna mismatching. and describe amplitude and phase errors of In Fig. 4, the circuits in path A and B, respectively. Also, and indicate the reflection coefficients corresponding to each antenna port. If the path errors are not considered, then the Tx leakage of the proposed architecture becomes

(1) (3) where and denote an incident wave and a reflected wave ) and I represent through at th port, respectively. Also, ( and isolation of the quadrature hybrid coupler. If the antenna is ) and port 3 is terminated with 50 , mismatched (i.e., then the Tx leakage of the single hybrid coupler can be simpli) fied as follows (under the assumption of (2) Equation (2) implies that the Tx leakage is determined by the isolation of hybrid coupler (I) and the reflection coefficient of

Equation (3) becomes 0 when the isolations in path A and B are identical in amplitudes and opposite in phases while the reflections from path A and B are also identical in amplitudes and opposite in phases. Considering the possible imbalance in the and are now taken proposed architecture’s symmetry, and (Fig. 4). Thus, the account such that scattering matrix for the balanced quadrature-ring hybrid strucand ) can ture with mismatched load (i.e., be represented by (4)

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Fig. 4. Proposed balanced structure with mismatched load and path error.

Fig. 6. (a) Simulated and (b) measured isolation of the proposed Tx leakage canceller according to load variation (load impedance of 50 Ohm is measured with the circuit having Tx/Rx antennas while other impedances are measured with termination load).

Fig. 5. Tx/Rx isolation measurement for load impedance variation.

From the above equation, the Tx leakage can be considered into 3 cases. The first case is that the electrical characteristics of path A and B are equal, and the reflection coefficients of the antennas and are unbalanced. ( ). The Tx leakage is represented as

where

(6)

Then, the Tx leakage of the balanced quadrature-ring hybrid coupler with mismatch load is as follows:.

(5)

The second case is that the electrical characteristics of path A and B are unequal and the reflection coefficients of the antennas and ). are balanced. ( Lastly, the third case is that the electrical characteristics of path A and B are equal and the reflection coefficients of the antennas and ). The are balanced ( Tx leakages for the second and third cases are expressed in (7) and (8), respectively (7)

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Fig. 7. Simulated isolation characteristics of the proposed structure for unbalanced path: (a) phase imbalance (  6=  ), (b) magnitude imbalance (j j 6=  j), (c) both phase and magnitude imbalance, and unbalanced reflection coefficients: (d) antenna reflection imbalance (0 6= 0 where RL = RL +RL ; + RL corresponding to 0 ; 0 ). RL = RL

j

(8) As seen from (6), (7), and (8), the Tx leakage depends on and . If these factors are in balance, maximally high isolation can be achieved. That is, since and are formed by symmetric implementation (i.e., balanced structure), the proposed architecture obtains robust and stable isolation for the antenna impedance variation. As shown in Fig. 5, the Tx/Rx isolation is measured by replacing the antenna with different load impedances. Since the circuit is designed with 50 Ohm, the case of 50 Ohm load impedance is measured by the Tx leakage canceller with patch antennas while other impedances are measured by different load terminations. Fig. 6 shows the simulated and measured isolation characteristics with respect to load (antenna impedance) variations. According to the measurement, an isolation of about 35 dB is maintained from 23.5 GHz to 24.5 GHz under the mismatched load condition while more than 45 dB is maintained at 24 GHz regardless of the antenna impedance variation. For the case of matched load, approximately 55 dB is achieved. Fig. 7 shows the simulation results for the imbalance occurred in the symmetrical path and reflection coefficients. Fig. 7(a), (b)

and (c) show the effect of unbalanced paths on the isolation characteristics of the proposed structure: presence of phase errors, magnitude errors, and both phase and magnitude errors, denotes the magnitude of the respectively. For Fig. 7(d), return loss, and the magnitude difference is investigated with dB as a reference. High isolation of about 40 dB is retained regardless of the asymmetry occurred in the balanced structure. B. Tx Leakage Cancellation for Antenna Mutual Coupling In the previous section, the isolation characteristic of the quadrature-ring hybrid structure without considering the mutual coupling among antenna ports has been discussed. Since Tx leakage also depends on the amount of field coupling between antenna ports, the mutual coupling among ports should also be considered when analyzing the Tx leakage. The proposed quadrature-ring hybrid balanced structure uses 2 1 patch array antenna with 4 feed ports. Fig. 8 describes the mutual coupling among ports, for which and denote the mutual coupling between port 1 (port 3) and port 2 (port 4), port1 (port 2) and port 3 (port 4), and port 1 (port 2) and port 4 (port 3), respectively. If indicates the mutual

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Fig. 8. Antenna mutual couplings among ports.

Fig. 9. 24 GHz Doppler front-end for radiation pattern measurement.

coupling from port n to port m and the patch array is assumed to be perfectly symmetric, the mutual coupling matrix can be equals to derived as follows since

(9)

Considering the Tx leakage due to the mutual coupling, the coupling occurred at each port can be expressed as follows where denotes the common phase delay. Equations (10), (11), (12) and (13) show the coupling occurred from port 1, 2, 3 and 4 to the Rx, respectively

Fig. 10. Simulated and measured 24 GHz patch array antenna radiation pattern for (a) Tx and (b) Rx, and (c) measured return loss for Tx and Rx channel.

(10)

(12)

(11)

(13)

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Fig. 11. Proposed 24 GHz balanced Doppler radar front-end with Tx leakage canceller.

According to (10), (11), (12) and (13), the phase delays outside the braces correspond to the delays due to the ring hybrid and quadrature hybrid from Tx viewpoint. The delays inside the braces are due to the mutual coupling paths through quadrature hybrid and ring hybrid from Rx viewpoint. The total amount of Tx leakage due to the mutual coupling to Rx is the summation of the couplings occurred at each port. Then, the total Tx leakage by mutual coupling is calculated as follows:

(14) According to (14), the effect of mutual coupling becomes 0. That is, the Tx-to-Rx leakage due to antenna mutual coupling is cancelled out by the proposed quadrature-ring hybrid front-end structure. III. EXPERIMENTAL ANALYSIS A. 24 GHz Patch Array Antenna Measurement Fig. 9 shows the fabricated 2 1 dual fed patch array antenna with the balanced quadrature-ring hybrid. The proposed 2 1 dual fed patch array antenna works as a Tx antenna when signal is inputted from Tx Port. Simultaneously, the patch array also operates as an Rx antenna when signal is received from Rx port. The proposed structure is configured by multi-layers conmm and ) and FR4 sisting of Rogers 3003 ( mm and ) substrates. The top and bottom ( layers on which the proposed structure is printed are Rogers

Fig. 12. Proposed 24 GHz balanced Doppler radar front-end fabrication.

3003 and the middle layer is FR4. The middle FR4 layer is used as a dummy layer since Rogers layers are very thin and flexible so that printed circuits are easily broken if only Rogers layers are used. Therefore, relatively strong FR4 board is inserted to implement a firm PCB. The quadrature hybrid and ring hybrid are designed to have a center frequency at 24 GHz. The size of single patch is 3.38 mm 3.38 mm and the total ground plane in Fig. 8 is 40 mm 40 mm. Fig. 10 presents the radiation pattern of the proposed antenna. As shown in previous section, the Tx leakage by mutual coupling between patch arrays is cancelled and as a result, cross-polarization is generated (x and y polarization). The radiation pattern is simulated and measured at 24 GHz, and shown

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TABLE I SUMMARY OF DOPPLER SHIFT AND SPEED MEASUREMENT

Fig. 13. Multi-layer structure of the proposed module implementation.

3003 and FR-4 substrates as previously mentioned and shown in Fig. 13. The front-end circuits and antenna are printed on the top layer, and DC bias lines and mixer output signal lines are printed on the bottom layer (through via). To verify the performance of the proposed system, the test-setup based on [6] is performed. A 23.8–24.8 GHz VCO with an output power of 15 dBm is used as a transmitter source and an LNA with a gain of 19 dB is used in the receiver path. The baseband quadrature outputs of the front-end module (from mixers) are filtered and amplified through Stanford Research System model SR560. The output signals are then converted to digital data with PCMCIA type NI DAQ card. The proposed module is tested with a moving metal plate (30 cm 30 cm) distant about 2 m away and the reflected signal received is read by LabView setup. Then, the speed of the target and Doppler shift can be obtained. Fig. 14 shows I/Q signal responses for the moving metal (toward with the speed of 5.0 mm/s and away with the speed of 7.5 mm/s from the proposed system), and Table I summarizes the measured target speed and Doppler shift. The proposed system can evaluate the speed as low as 0.5 mm/s which corresponds to about 0.078 Hz Doppler shift. IV. CONCLUSION

Fig. 14. Measured I/Q responses for the target moving (a) toward the proposed system with 5.0 mm/s and (b) away from the proposed system with 7.5 mm/s.

on yz-plane ( and ). The Tx antenna has the simulated and measured peak gains of about 7.74 dBi and 6.97 dBi. The Rx antenna has the simulated and measured peak gains of about 6.18 dBi and 5.65 dBi, respectively. The measured 3-dB beamwidths of Tx and Rx antennas are approximately 69 and 91 while the radiation efficiencies for Tx and Rx are 86% and 91%, respectively. Comparing to a single patch measurement (4.58 dBi and 92%), the proposed system showed a reasonable performance. B. 24 GHz Balanced Doppler Radar Front-End Measurement Figs. 11 and 12 show the system architecture for the proposed 24 GHz balanced Doppler radar front-end and its fabrication, respectively. The proposed module is implemented with Rogers

This paper presented a new architecture for 24 GHz balanced Doppler radar front-end with stable Tx leakage cancellation regardless of antenna impedance variation and antenna mutual coupling. The proposed architecture was configured with 2 1 dual fed patch array antenna having 6.97 dBi and 5.65 dBi of measured Tx and Rx peak gain. The isolation of more than 45 dB was maintained at 24 GHz despite of the antenna impedance variation and mutual coupling among antenna ports while 55 dB was obtained for a perfect matching condition. Having the robust and stable Tx leakage cancellation characteristic, the proposed module could determine the Doppler shift as low as about 0.078 Hz. REFERENCES [1] H. H. Meinel, “Commercial applications of millimeter waves history, present status, and future trends,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 7, pp. 1639–1653, July 1995. [2] M. Klotz and H. Rohling, “A 24 GHz short range radar network for automotive applications,” in Proc. IEEE Radar Conf., 2001, pp. 115–199. [3] M. E. Russell, C. A. Drubin, A. S. Marinilli, W. G. Woodington, and M. J. Del Checcolo, “Integrated automotive sensors,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 674–677, Mar. 2002. [4] L. Roselli, F. Alimenti, M. Comez, V. Palazzari, F. Placentino, N. Porzi, and A. Scarponi, “A cost driven 24 GHz Doppler radar sensor development for automotive applications,” in Proc. IEEE Radar Conf., 2005, pp. 335–338. [5] J. Kim, S. Ko, S. Jeon, J. Park, and S. Hong, “Balanced topology to cancel Tx leakage in CW radar,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 9, pp. 443–445, Sep. 2004.

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[6] C. Y. Kim, J. G. Kim, and S. Hong, “A quadrature radar topology with Tx leakage canceller for 24-GHz radar applications,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 7, pp. 1438–1444, Jul. 2007. [7] C. Y. Kim, J. G. Kim, D. H. Baek, and S. Hong, “A circularly polarized balanced radar front-end with a single antenna for 24-GHz radar applications,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 2, pp. 293–297, Feb. 2009. [8] T. H. Ho and S. J. Chung, “Design and measurement of a Doppler radar with new quadrature hybrid mixer for vehicle applications,” IEEE Trans. Microw. Theory Tech., vol. 58, no. 1, pp. 1–8, Jan. 2010.

Han Lim Lee received the B.A.Sc. degree in electronics engineering from Simon Fraser University, BC, Canada, in 2008 and the M.S. degree in electrical engineering from KAIST, Daejeon, South Korea, in 2010, where he is currently working toward the Ph.D. degree. His research interests are microwave/millimeter wave circuits and RF/wireless hybrid systems, RFIC and antenna.

Won-Gyu Lim received the B.S. degree in electrical engineering from Kyungpook National University, Daegu, South Korea, in 2002, and the M.S. and Ph.D. degrees in electrical engineering from KAIST, Daejeon, South Korea, in 2004 and 2008, respectively. During his studies for the M.S. degree, he focused on the packaging for microwave circuit on compound semiconductor substrate. For his Ph.D. studies, he emphasized RFID and UWB systems. He is currently with Korea Aerospace Research Institute (KARI), Daejun, Korea. His research interests are wireless transmitter/receiver front-end isolation, multilayer EMI/EMC analysis and small antenna.

Kyoung-Sub Oh received the B.S. degree from Chonbuk National University, Jeonju, Korea, in 1994, and the M.S. and Ph.D. degrees in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea, in 1997 and 2004, respectively. From 2004 to 2005, he worked at Hyundai Motor. He also served Maltani Lighting and Samung Electronics from 2005 to 2008 and from 2008 to 2010, respectively. Since 2010, he has been an Assistant Professor of electrical engineering and a Research Associate Professor at KAIST. His research interests emphasize microwave imaging, inverse scattering, wireless power transfer, wireless communication system and RFID/USN.

Jong-Won Yu (M’98) received the B.S., M.S., and Ph.D. degrees in electrical engineering from KAIST, Daejeon, South Korea, in 1992, 1994, and 1998, respectively. From 1995 to 2000, he worked at Samsung Electronics. From 2000 to 2001, he served as Head of the Hardware Team, IMT2000 Division, Wide Telecom, Korea, and from 2001 to 2004, as Researcher-in-Charge at Telson, USA. In 2004, he was an Assistant Professor in the Department of Electrical Engineering, KAIST where, since 2006, he has been an Associate Professor . His research interests emphasize microwave/millimeter wave circuit (MMIC, Hybrid), wireless communication system and RFID/USN.

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Sum, Difference and Shaped Beam Pattern Synthesis by Non-Uniform Spacing and Phase Control Homayoon Oraizi, Senior Member, IEEE, and Mojtaba Fallahpour, Student Member, IEEE

Abstract—A design procedure for the synthesis of non-uniformly spaced linear arrays is presented, which uses the Poisson sum expansion of the array factor introduced in the literature. By considering the nonzero phase term in the existent formula and using the appropriate line source pattern synthesis methods, a general design procedure is obtained to synthesize any type of pattern, such as sum, difference and shaped beams. This approach converts the nonlinear complex problem of pattern synthesis for non-uniformly spaced linear arrays to a simple problem, which makes it fast and easy to implement. Moreover, an extra optimization process is added to the synthesis procedure to improve final pattern and provide control on computed parameters. Index Terms—Linear arrays, non-uniformly spaced arrays.

I. INTRODUCTION

A

PPLICATION of non-uniformly spaced linear arrays instead of uniformly spaced linear arrays has been an attractive topic in antenna area for some years. Simplicity of the feeding network for non-uniformly spaced linear arrays is the most important feature of this type of array. Unz [1] in 1960 analyzed and developed a matrix formulation for non-uniformly spaced linear arrays. Harrington [2] developed an iterative method to reduce the sidelobe level of uniformly excited N-element linear array to about 2/N times the field intensity of the mainlobe by employing unequal spacings. Since the element positions occur in the argument of exponential function, the desired pattern synthesis by proper calculation of element positions is a nonlinear and complex problem. Regarding this fact, most of the previous works have attacked the problem by utilizing numerical and computer-based iterative [3] or stochastic optimization methods [4]–[6]. The development of non-uniform fast Fourier transform (NFFT) and its applications for sidelobe level reduction are among the current interesting research activities [7]. In most of these works, the main goal is only the sidelobe level reduction and they are not able to synthesize patterns with arbitrary shape. Although, there is a lack of analytical formulas for this type of array, yet in recent

Manuscript received January 12, 2010; revised January 17, 2011; accepted March 09, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. H. Oraizi is with the Department of Electrical Engineering, Iran University of Science and Technology (IUST), Narmak, Tehran, Iran (e-mail: [email protected]). M. Fallahpour was with the Iran University of Science and Technology (IUST), Narmak, Tehran, Iran. He is now with the Applied Microwave Nondestructive Testing Laboratory (amntl), Electrical and Computer Engineering Department, Missouri University of Science and Technology, Rolla, MO 65409 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165468

Fig. 1. Linear array with arbitrarily spaced elements.

years scarcely any work is reported in this area [8], [9]. But the first well-structured mathematical formulation was introduced by Ishimaru [10], who used the Poisson sum expansion of the array factor and some simplifying assumptions to derive a relationship between the pattern and element spacings. He used the derived relation to reduce the sidelobe level. This paper will consider a nonzero phase term in the Ishimaru’s formula to make it capable of synthesizing any type of pattern, such as sum, difference and shaped beams by non-uniform spacings and/or element phase control. The phase term allows the synthesis of asymmetrical sum pattern (i.e., sum pattern with sidelobes asymmetrically placed around the mainlobe) by non-uniform spacings and phase control which have not been addressed up to now. The proposed method utilizes the derived formulas to convert the nonlinear complex problem to a relatively simpler one (namely the line source pattern synthesis), and solve it by introducing an appropriate line source pattern synthesis method. This method is fast and easy to implement, despite the fact that the synthesis of patterns of any shape by non-uniform spacings and also element phase control (if necessary) is a complex problem and has not been considered in the literature. Furthermore, an additional genetic algorithm (GA)based optimization method is introduced to improve the synthesized pattern and provides a control on the calculated phase and/or spacing. This extra GA-based optimization process is necessary for the shaped beam pattern synthesis. II. THEORETICAL BACKGROUND A. Poisson Sum Expansion of Array Factor Assume a linear array of antennas composed of arbitrarily spaced N elements on the -axis as shown in Fig. 1. Its array factor can be defined as (1) is the ’th element current excitation, is the wave where number, is the position of ’th element relative to the origin, is the angle relative to -axis, and is the array factor.

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Here, the Ishimaru’s results from [10] are briefly summarized, and then a general case will be considered. Ishimaru shows that by good approximation for not-very-long arrays, the array factor using the Poisson’s sum formula can be written as [10]

where

(2)

will be obtained for positions of elements to synthesize asymmetrical sum patterns which is unacceptable. But, using the exin (8), one can generate real and symmetponential term rical values for , even for asymmetrical patterns. This point of view has not been addressed and no attempt has been made to synthesize asymmetrical sum patterns with arbitrarily specified sidelobe levels by non-uniform spacings and phase control.

(3)

B. Desired Pattern Synthesis With Pattern’s Null Manipulation

is the normalized source position function and is normalized source number . Also, is defined as

function

(4) where

is

. The actual position of th element is (5)

where the value of is provided in (20) and (21) of [10]. By defining as

In order to synthesize the desired pattern, and should be evaluated using line source pattern synthesis methods. Elliott’s method [11], as a well-known method, uses a null perturbation process to obtain some control on individual sidelobes, but his method has no control on the filled nulls. Here, Elliot’s basic ideas for the manipulation of nulls is applied, but by using GA. Also, the complex valued nulls are used instead of real valued nulls for the synthesis of shaped beam patterns. This method has control on both the deep and filled nulls as well as sidelobe levels and filled null areas (shaped beam areas). In the following, for each pattern type, appropriate pattern null manipulation method is introduced. 1) Sum Pattern: For the sum patterns a generic pattern is defined as follows [11]:

(6) equation (3) can be written as [10]

(9) To have full control on sidelobes, one should manipulate nulls of this generic pattern as follows [11]:

(7) Assuming a line source with amplitude of excitation equal to , then (7) provides its pattern. In the derivation of (7), it is assumed that the excitation of elements is equal to unity with zero phase. Here, we use a more general case by assuming that the element excitations have unit magnitude (to avoid feeding (to have more control complexity) but nonzero phase as on pattern). By defining as a continuous version of , (7) is modified to (8) Consequently, the nonlinear problem (pattern synthesis by non-uniform spacing and phase control) has been converted to finding and which are similar to the current distribution for line sources. Although (8) is reported in [10], but later, the phase term is assumed to have no variation in order to use available line source pattern synthesis methods. Here, a nonzero phase term will be used which leads to a new position/phase synthesis process instead of only position synthesis. To clarify the role of phase term, it should be noted that in the line source case, based on the Fourier analysis, the excitation current is always real and symmetrical around the center for symmetrical sum pattern [11]. But, for the asymmetrical sum pattern only the amplitude of current distribution is symmetrical around the center and the phase distribution is asymmetrical and non-zero [11]. Regarding this fact and referring to (3), a complex value

(10)

where R and show the right and left hand sides of the mainlobe for as variable, respectively [11]. The right side includes and the left side includes . Also, is the number of displaced nulls in the right hand side is the number of displaced nulls in the left hand side. and can be calculated, the Fourier expansion method as If nulls applied in [11] can be used to write and as (11) and then substituting (11) in (8) gives

(12) where, the integral in (12) for reduces to

is zero and for (13)

ORAIZI AND FALLAHPOUR: SUM, DIFFERENCE AND SHAPED BEAM PATTERN SYNTHESIS BY NON-UNIFORM SPACING

Since is zero for or , then applying (13) to (11) leads to a truncated Fourier series as (14)

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3) Shaped Beam Pattern: To generate this type of pattern, the generic pattern introduced for the sum pattern may be used, ’s) should be introduced and but here complex nulls ( manipulated as [13]

and are amplitude and phase of the right side summation, respectively. is defined as Now, we need to calculate . First (21)

(15) where is the ’th null of generic pattern, which is used as is the displacement of ’th null to its initial value, and be computed here by GA. More details about the cost function definition and the parameters of applied GA are provided in Appendix. 2) Difference Pattern: For the difference pattern, a generic pattern is defined as follows [11]:

where shows the number of displaced deep nulls and MF and shows the number of filled nulls. For this case, are derived as [13] (22)

C. Calculation of Position and Phase for Each Element (16)

Applying the computed equation, we have

from (22) to (6) and solving the

Again for full control, the nulls may be displaced by the following relation [11]: (23)

(17)

where may be defined as (15) and GA is used for their calculation. After the appropriate null calculations (see Appendix), and can be derived as (18) where

is defined as [12]

The integrations may be done by the trapezoidal rule. For the symmetrical pattern, some analytical relations may be obtained. may be derived by analytical inversing Then, the function techniques or geometrical method. Finally, the appropriate posiwill be calculated from , using tions of elements (20) or (21) of [10]. Having the positions then can be sampled to yield . III. THE APPLICATION OF PROPOSED SYNTHESIS METHOD Here, with some examples the capability of the proposed method to synthesize different types of patterns is demonstrated. A. Sum Pattern Synthesis

(19)

(20)

Here, an asymmetrical sum pattern with the mainlobe at (i.e., broadside pattern) is synthesized. Three innermost sidelobes (i.e., the first three sidelobes which are in the vicinity are set to be lower than dB of mainlobe) for dB. and the other sidelobes in this region are set to be under , a shape For the second half of the pattern, similar to the pattern of uniformly excited array is specified. The . As the first step, a line source number of elements is pattern should be synthesized. The GA (with the cost function explained in Appendix) manipulated the line source pattern’s nulls and found their appropriate positions to give the desired and are calculated by (14). By applying pattern. Then, to (23) and performing the integration, is calculated. , the values of are calculated After the computation of at , the element phases are and then by sampling derived. The achieved pattern for the array degrades from the

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TABLE I SPACING AND PHASE OF LEFT SIDE ELEMENTS OF DESIGNED ARRAYS

Fig. 2. The function y (x) relating the position of elements with their number is shown for synthesized pattern in Fig. 3.

line source pattern. The same event happens when one tries to uniformly sample the continuous line source distribution to feed a discrete array, when it varies a lot with position [11]. For uniformly spaced arrays, the uniform sampling theory determines the required minimum number of elements and maximum element spacing. But for non-uniformly spaced linear arrays, the situation is complex and the well-known uniform sampling theory will not work. The definition of average element spacing , enables in [6], as us to apply partly uniform sampling criteria to non-uniformly spaced linear arrays. One can treat the average element spacing like that in uniformly spaced linear arrays and set a constraint on it to satisfy the uniform sampling conditions. But this is not a general solution and one can start the pattern synthesis with the proposed method and then based on the results, take an additional modification procedure. This additional modification may be either an easy manipulation (e.g., by trial and error) to change the initial line source pattern to achieve the final desired pattern or an optimization procedure that directly varies the element spacings to find their optimum values. This additional modification is comparable to the iterative procedure reported in [11] to improve the uniformly spaced array pattern using sampled line source data. For the current example, just simple manipulation of the sidelobe levels of initial line source pattern could help. Three , the trial and errors showed that in the interval line source pattern should be set about 5 dB below the desired dB), in order to have the desired pattern value (e.g., about for non-uniformly spaced array. Using this line source, is calculated and drawn in Fig. 2 (due to the symmetry only the right side is depicted). The element spacings and their phases are listed in Table I (on the left section). In this paper, always the first element on the left side is numbered one, and the last one on the right side is N. The pattern of the synthesized array is shown in Fig. 3. Also, for comparison, the used line source pattern (after modification) is shown in Fig. 3. The CPU time on a 2 GHz Celeron PC was about 50 seconds (about 90% of this time was taken up by the GA run).

Fig. 3. The synthesized asymmetrical pattern by non-uniform spacing and phasing in comparison with initially used line source pattern.

Fig. 4. The synthesized difference pattern by non-uniform spacing and phasing in comparison with initially used line source pattern.

B. Difference Pattern Synthesis Here, for a 40-element array, a pattern with sidelobe level of dB is synthesized. Applying GA to (17) and following the same procedure as for the sum pattern synthesis, the position

and phase of each element are calculated and listed in Table I (middle section). Phase of right side elements are 180 . As compared in Fig. 4, the pattern of line source and non-uniformly

ORAIZI AND FALLAHPOUR: SUM, DIFFERENCE AND SHAPED BEAM PATTERN SYNTHESIS BY NON-UNIFORM SPACING

Fig. 5. The synthesized flat top pattern by uniform excitation and non-uniform spacing in comparison with initially used line source pattern.

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Fig. 6. The modified synthesized pattern in Fig. 5 by extra optimization in comparison with the synthesized pattern from beginning using GA with phase data from proposed method but 30% of value of positions are changed.

and it is zero elsewhere. Also, for spaced array are close enough. The CPU time for its design on the same PC is less than 30 seconds, and again 90% of this time is consumed by GA. C. Shaped Beam Pattern Synthesis Although, there are some works for the synthesis of shaped beam patterns such as csc and flat top patterns by calculating the phases and keeping the amplitudes constant or almost constant [14], [15], the non-uniform spacing may be seen as another candidate to synthesize these patterns. Here, by an example, we show that the proposed synthesis procedure can be used for this purpose. In this example, it is desired to have a flat top pattern dB and sidelobe level of with ripples of 2 dB centered on dB by an element array. Applying GA to (21) and using the described procedure, the optimum position of each element is calculated in less than 45 seconds. The synthesized pattern is drawn in Fig. 5. Due to the complexity of this type of pattern, the synthesized pattern degrades from the initially used line source. To overcome this problem, an extra optimization stage is added. are a In this procedure, the derived element positions and using GA to little bit disturbed by defining, to provide the desired pattern. Since, we find the appropriate are close to the optimum positions in the initial approach and so a limited variation is needed, then the GA algorithm converges very fast. The cost function for the current example is as (24) is the cost function for sidelobe level and is where defined on the sidelobe region (SR) as (25) also,

is defined on the shaped beam region (SB) as, (26)

where

for

is (27)

(28) and it is zero elsewhere. Moreover, DL is the desired pattern dB and dB). Also, level in SB region (here, for transition from SB to SR a monotonic pattern condition is imposed. Adding this new step to the design procedure, the calculated element positions are listed in the right section of Table I and the pattern is drawn in Fig. 6. The sidelobe level is dB and the ripple around dB is less than 1 dB which complies with the design goals. The CPU time for this new step is about 1 minute for convergence to the desired pattern. This extra optimization can apply the constraint on spacings and phases of elements and so provides more control on the design values. It is interesting to mention that the extra optimization procedure was fed by the new different initial values (different initial population). First, the phase was kept the same as before but 30% of element positions were changed. The optimizer (GA) could converge to an acceptable pattern in 30 minutes (for the initial population size of 400 and number of generations of 40). This pattern is shown in Fig. 6. Later, GA was fed with randomly valued initial phases and positions for all elements. For this case, after 10 hours with trying many different cost functions, GA did not converge to any acceptable answer. This shows that using GA to design the array from beginning does not work for complex patterns and also shows the important role of the proposed first procedure for approximate position/phase control. IV. CONCLUSION In this paper, the Ishimaru’s formulas are used with nonzero value for element’s excitation phase in order to synthesize any type of pattern, such as sum, difference and shaped beams. The array pattern synthesis is performed by non-uniform spacing and if necessary by phase control. Furthermore, an extra optimization procedure was applied on calculated values to modify them in order to improve the final pattern. This procedure provides more control on the spacings and phases by possibly applying constraints on them. The proposed method is very fast and easy to implement.

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APPENDIX COST FUNCTION DEFINITION AND FEATURES OF USED GA To synthesize a desired line source pattern, appropriate positions for nulls should be determined. To perform this, new positions of nulls are assumed to be related to initial values (generic pattern nulls) as (15) and with good accuracy one can assume sidelobes of array pattern to occur between two neighboring nulls as

(A1) . For the sum pattern case, at the where pattern has a peak but for the difference pattern a null is located there. Therefore, the sidelobe level can be calculated as (A2) Now, an appropriate cost function can be defined based on the design specifications. In general, the problem is to synthesize a pattern with its sidelobes individually adjusted to a desired value . Then, a cost function can be as (A3) where is the number of sidelobes. For the shaped beam pattern case, appropriate complex nulls should be calculated to achieve the desired pattern. In order to do this, nulls are manipulated as (A4) and As it is explained for (21), MF filled nulls deep nulls should be calculated to minimize the following cost function: (A5) is the cost function for sidelobe level and is where defined in the sidelobe region (including deep nulls) and is the is defined in the shaped same as (A3). Also, beam region (including filled nulls) as, (A6) (A7) where

is the desired ripple factor and (A8)

GA as a robust optimizer was selected to minimize the cost functions defined in (A3) and (A5). We set the crossover function as scattered, and mutation as adaptive. Number of population was 65 and number of generations was 20. Also, since the inverse function calculation is very fast, the CPU run time for GA is approximately equal to the time reported earlier for each example.

REFERENCES [1] H. Unz, “Linear arrays with arbitrarily distributed elements,” IRE Trans. Antennas Propag., vol. 8, no. 2, pp. 222–223, Mar. 1960. [2] R. F. Harrington, “Sidelobe reduction by nonuniform element spacing,” IRE Trans. Antennas Propag., vol. 9, no. 2, pp. 187–192, Mar. 1961. [3] F. Hodjat and S. A. Hovanessian, “Nonuniformly spaced linear and planar array antennas for sidelobe reduction,” IEEE Trans. Antennas Propag., vol. AP-26, no. 2, pp. 198–204, Mar. 1978. [4] Y. B. Tian and J. Qian, “Improve the performance of a linear array by changing the spaces among array elements in terms of genetic algorithm,” IEEE Trans. Antennas Propag., vol. 53, no. 7, pp. 2226–2230, July 2005. [5] 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. [6] H. Oraizi and M. Fallahpour, “Nonuniformly spaced linear array design for the specified beamwidth/sidelobe level or specified directivity/sidelobe level with mutual coupling considerations,” Progr. Electromagn. Res. M, vol. 4, pp. 185–209, 2008. [7] W. P. M. N. Keizer, “Linear array thinning using iterative FFT techniques,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2757–2760, Aug. 2008. [8] B. P. Kumar and G. R. Branner, “Design of unequally spaced arrays for performance improvement,” IEEE Trans. Antennas Propag., vol. 47, no. 3, pp. 511–523, Mar 1999. [9] B. P. Kumar and G. R. Branner, “Generalized analytical techniques for the synthesis of unequally spaced arrays with linear, planar, cylindrical or spherical geometry,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 621–634, Feb. 2005. [10] A. Ishimaru, “Theory of unequally-spaced arrays,” IRE Trans. Antennas Propag., vol. AP-10, pp. 691–702, Nov. 1962. [11] R. S. Elliott, Antenna Theory and Design. New York: Wiley, IEEE Press, 2003. [12] R. J. Mailloux, Phased Array Antenna Handbook. Boston, MA: Artech House, 2005. [13] F. Ares, R. S. Elliott, and E. Moreno, “Synthesis of shaped line-source antenna beams using pure real distributions,” Electron. Lett., vol. 30, no. 4, Feb. 1994. [14] A. Chakraborty, B. Das, G. Sanyal, and G. , “Determination of phase functions for a desired one-dimensional pattern,” IEEE Trans. Antennas Propag., vol. 29, no. 3, pp. 502–506, May 1981. [15] G. Franceschetti, G. Mazzarella, and G. Panariello, “Array synthesis with excitation constraints,” IEE Proc. H, Microw. Antennas Propag., vol. l35, no. 6, pp. 400–407, Dec. 1988.

Homayoon Oraizi (SM’98) received the B.E.E. degree from the American University of Beirut, Beirut, Lebanon, in 1967 and the M.Sc. and Ph.D. degrees in electrical engineering from Syracuse University, Syracuse, NY, in 1969 and 1973, respectively. From 1973 to 1974, he was a Teacher with K. N. Tousi University of Technology, Tehran, Iran. From 1974 to 1985, he was with the Communications Division, Iran Electronics Industries, Shiraz, where he was engaged in various aspects of technology transfer mainly in the field of HF/VHF/UHF communication systems. In 1985, he joined the Department of Electrical Engineering, Iran University of Science and Technology, Tehran, where he is currently a Full Professor of electrical engineering. He teaches various courses in electromagnetic engineering and supervises theses and dissertations. He has authored and translated several textbooks in Farsi. His translation into Farsi of Antenna Analysis and Design, 4th ed. (Iran Univ. Sci. Technol., 2006) was recognized as the 1996 book of the year in Iran. He has conducted and completed numerous projects in both industry and with universities. He has authored or coauthored over 200 papers in international journals and conferences. His research interests are in the area of numerical methods for antennas, microwave devices, and radio wave propagation. He spent a two-month term (July–August 2003) and a six-month sabbatical leave (August 2004–February 2005) with Tsukuba University, Ibaraki, Japan. Dr. Oraizi is a Fellow of the Electromagnetic Academy, USA and the Japan Society for the Promotion of Science. He is an Invited Professor of the Electrical Engineering Group, Academy of Sciences of Iran, and is listed as an Elite Engineer by the Iranology Foundation. In 2006, he was elected an Exemplary Nationwide University Professor in Iran. He was listed in the 1999 Who’s Who in the World.

ORAIZI AND FALLAHPOUR: SUM, DIFFERENCE AND SHAPED BEAM PATTERN SYNTHESIS BY NON-UNIFORM SPACING

Mojtaba Fallahpour (S’09) received the B.Sc. and M.Sc. degrees in electrical engineering from the Iran University of Science and Technology (IUST), Tehran, in 2005 and 2008, respectively. He is currently working toward the Ph.D. degree at the Missouri University of Science and Technology, Rolla. He is a Graduate Research Assistant in the Applied Microwave Nondestructive Testing Laboratory (amntl), Missouri University of Science and Technology. His research interests include nondestructive techniques for microwave and millimeter wave inspection and testing of materials (NDT), microwave measurement instruments design, array antenna pattern synthesis, ultra-wideband and miniaturized antenna design.

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Optimal Polarization Synthesis of Arbitrary Arrays With Focused Power Pattern Benjamin Fuchs, Member, IEEE, and Jean Jacques Fuchs, Member, IEEE

Abstract—The joint synthesis of the spatial power pattern and polarization of arbitrary arrays is addressed. Specifically, the proposed approach gives the solution to a frequently encountered problem, namely the array design (i.e., the determination of the radiating element weightings) to achieve a pattern that is arbitrarily upper bounded, while its polarization is optimized in a given angular region. Any state of polarization (elliptical, circular and linear) can be synthesized and there is no restriction regarding the array geometry and element patterns. The synthesis problem is rewritten as a convex optimization problem, that is efficiently solved using readily available software. This ensures the optimality of the proposed solution. Various numerical results are presented to validate the proposed method and illustrate its potentialities. The synthesis of a sequentially rotated array is first addressed. Then a linear array of equispaced randomly oriented dipoles is considered. Finally, a conformal and a planar array of patches, where the mutual coupling effects are considered, are synthesized to radiate a linear and a circular polarization. Index Terms—Array pattern synthesis, conformal arrays, convex optimization, waveform polarization.

I. INTRODUCTION XPLOITING the polarization of a waveform has many advantages. It enables, for instance, to improve the performance of active sensing systems, such as radars [1]. Polarimetric radar systems have thus been developed and efficiently used in various applications [2], [3]. Moreover, the polarization diversity has been shown to improve the performances of communication systems. It indeed significantly increases the capacity of wireless communications [4]. The polarization diversity is also effective to combat fading in mobile wireless communications [5], [6] and to compensate for polarization mismatch due to random handset orientation [7]. Investigations have also been led to show the interest of using polarized arrays for interference rejection in wireless communication systems [8], [9]. If the benefits of using polarized waveforms are well known, the way to generate them, i.e., the synthesis of polarized arrays, has been relatively few reported in the literature. Though the synthesis of antenna arrays has received much attention over the years [10], [11], most of these works focus on spatial power

E

Manuscript received November 05, 2010; revised April 08, 2011; accepted June 02, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. B. Fuchs is with the Laboratory of Electromagnetics and Acoustics, Ecole Polytechnique Fédérale de Lausanne, Switzerland on leave from the IETR/University of Rennes I, Rennes Cedex 35042, France (e-mail: [email protected]). J. J. Fuchs is with the IRISA/University of Rennes I, 35042 Rennes Cedex, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165492

pattern synthesis and thus consider that the field emitted by antenna arrays is scalar. These techniques are therefore not able to handle the design of arrays having any elliptical polarization. It is only quite recently that a few works dealing with the joint synthesis of power pattern and polarization have been reported in the literature. An iterative least square method is presented in [12] to synthesize a main beam with optimized circular polarization for conformal arrays. An analytic approximation of circular polarized patches is used to model the elements radiation pattern. The ideal shape of the far field radiation pattern and the targeted polarization are defined as the desired radiation pattern in a least square optimization procedure. An adaptive array approach is applied in [13] to a conformal antenna array to synthesize a main beam with optimized polarization employing dual polarized patch antennas as radiating elements. The aforementioned synthesis methods can not ensure that the optimum is reached, since the optimization problems they solve are not convex or not transformed into convex programs. The exploitation of convexity in array synthesis problems has been introduced by [14] and applied, later on, to various classes of problems as reviewed in [15]. Such a convex formulation has been proposed in [16] to optimally synthesize pencil beams. While the coupling is taken into account, arbitrarily defined upper bounds can be set on the co- and cross-polarization components of the field. The optimal synthesis of beampattern having any state of polarization via convex optimization has been addressed very recently in [17]–[19] using an array of vector antennas, i.e., antennas composed of orthogonal electric or magnetic dipoles. Specifically, the goal is to find the antenna weightings that achieve a pattern whose main beam pointing at a given direction has a specified polarization, while the sidelobe level is minimized. This vector array synthesis problem is cleverly transformed into a scalar one using an orthogonal transformation, which then makes the problem efficiently solvable. In this paper, a method to jointly synthesize the spatial power pattern and the polarization of arbitrary arrays is presented. Specifically, the proposed approach designs the array radiating element weightings to achieve a pattern whose power is subject to arbitrary upper bounds, while its polarization is optimized over an angular region. The search for the array element weightings is formulated as a convex optimization problem. This ensures the optimality of the proposed solution that is obtained by transforming the problem into a second order cone program (SOCP) [20]. From now on, it is thus possible to solve this frequently encountered synthesis problem in an efficient way.

0018-926X/$26.00 © 2011 IEEE

FUCHS AND FUCHS: OPTIMAL POLARIZATION SYNTHESIS OF ARBITRARY ARRAYS WITH FOCUSED POWER PATTERN

is the term-by-term product of the vectors where The electric field radiated by the array is:

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and .

(6) where is the -dimensional vector of complex element is thus composed of weightings. The weighting vector scalars that control the vectorial field radiated by the array elements. These complex weightings are the unknowns to determine. Fig. 1. Coordinate system and polarization ellipse. In far field, the electric field ^ '^). propagates along r^ and lies in a plane orthogonal to r^, that is spanned by (; The waveform polarization can be described by the polarization ellipse, with  the orientation angle and the ellipticity angle. cos and sin are the major and minor axis lengths of the ellipse respectively.

The paper is organized as follows. In Section II, the synthesis problem is precisely defined, formulated as a convex optimization problem and transformed into a SOCP ready to be solved. Various numerical results are presented in Section III to both validate and illustrate the potentialities of the proposed synthesis method. Conclusions are drawn in Section IV. II. PROBLEM FORMULATION AND RESOLUTION A. Antenna Array Let us consider an antenna array composed of elements . placed at arbitrary but known locations with The problem is described for a one-dimensional pattern synthesis. This synthesis is performed over the polar angle in a fixed azimuthal plane (see Fig. 1), that is omitted in the notations. The extension to a two-dimensional (2-D) pattern synthesis, i.e., a synthesis over both angular directions and , is straightforward and two examples of 2-D pattern synthesis are shown in Sections III.A and III.D. The array factor in the direction is: (1) where denotes the Hermitian transposition, is the free space is the unit vector in the direction (and wave number and azimuthal plane ). The element of the array radiates a vectorial far field pattern , which has, in general, both a and component: . This leads to the following vectors: (2) (3) Note that arbitrary arrays can be considered, since there is no restriction regarding the array geometry and the element patterns. The latter can indeed be obtained from either analytical formula, simulation results or measured data. The vectorial antenna array response is: (4) (5)

B. Waveform Polarization A brief reminder on the waveform polarization is presented to introduce the notations that are used later. The polarization of an electric field can be defined, see [21], by the orientation angle and the ellipticity angle represented and yield linear in Fig. 1. For instance, and circular polarizations respectively. Moreover, the axial ratio denoted AR, often used to evaluate the quality of the circular wave polarization, is equal to: (7) It must be pointed out, that there is a one-to-one relationship and the coefficients of the ratio between the angles if , as demonstrated in [22]. Specifically, is linked to by:

(8) where and . In the sequel, the waveform polarization is defined using . C. Array Pattern Synthesis The array pattern synthesis, performed over the polar angle in a plane (omitted for clearness reasons), is now detailed. The problem amounts to find the weighting vector in (6) to achieve a pattern that has both spatial power and polarization requirements. More precisely, the spatial power is taken care of by constraints whereas the polarization is the optimization goal. As represented in Fig. 2, the goal is to synthesize a pattern having: with sidelobes below a • a main beam in the direction over an angular region and given upper bound , optimized • a wave polarization, characterized by over an angular region . The regions and can overlap. The magnitude of the vector electric field is equal to:

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With the notations of (6), one gets:

(12)

=

Fig. 2. Schematic view of the pattern synthesis problem in a plane ' '. The magnitude of the electric field must reach E at the direction  and remain below an arbitrary upper bound   over the angular region S while the polarization ;  is optimized over P .

(

)

()

To further formalize (12), the constraints on the sidelobe level diand beam polarization are discretized. One introduces and directions rections that covers and respectively. The synthesis problem (12) becomes as follows:

A unique, potentially -dependent, bound will be imposed on this quantity over to handle the spatial power constraint. Let us now focus on the polarization constraint. A given pocan be synthesized over a range of directions larization by imposing: (9) where allows to tune the degree of accuracy with which the polarization is achieved. Since relation (9) is difficult to enforce, it is replaced by:

(10) In order to guaranty a constant upper bound over in (9), one in (10). Since the value of has to enforce a -dependent is not a priori known, it will be replaced by an estimate . The synthesis problem will thus be solved itdenoted in the right member of the first constraint eratively and at the first iteration and by the is replaced by outcome of the previous iteration afterwards. Convergence generally occurs after three iterations. The synthesis problem described in Fig. 2 can be written as:

(13) and where in terms of or the previous where the expression of optimal vector is not detailed. The optimization problem (13) is convex, which ensures the optimality of the design solution. It can be transformed, as detailed in the next Section, into a Second Order Cone Program (SOCP) [20], that is solved iteratively, as described below relation (10). Convergence generally occurs after a few (typically three) iterations. D. Resolution via Second Order Cone Program (SOCP) A SOCP has the following standard form: (14) represents second order cones. A second order cone where of dimension is of the form:

(15) (11) To justify the last constraint, one observes that if the polarization at the direction , i.e., of the electric field is equal to , then the magnitude of the total elec. tric field is: , one expects that the Fixing then , optimum in will yield a that is such that i.e., such that the phase of is zero at . Imposing this condition on is not a limitation, since is defined up to an arbitrary phase.

A SOCP can be seen as a generalization of a linear program. It has been established [20] that one can extend the linear program (theory and algorithms) to the conic program. To handle the polarization constraint, one introduces the and associates a complex number , for to . It yields triplet and one has consequently , for to . in The sidelobe level constraint (13), is taken into account by creating a second order cone of

FUCHS AND FUCHS: OPTIMAL POLARIZATION SYNTHESIS OF ARBITRARY ARRAYS WITH FOCUSED POWER PATTERN

dimension 5: It implies that:

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.

(16) which leads to for . Standard transformations, namely the use of slack variables, are then used to rewrite the inequalities of (11) into equalities, as required in (14). Note that the use of SOCP in (16) allows to constrain precisely the magnitude of the vector field: Fig. 3. Schematic view of the sequentially rotated array composed of 16 dipoles spaced by  =2.

This is indeed a much weaker constraint than imposing, as done in [13], [16], an upper bound on the two components of the field, and , with e.g., . Once the possibly delicate transcription of the synthesis problem into a SOCP (as described above) has been performed, the existence of free software that solve efficiently SOCP, without any further tuning, justifies largely its use. To solve the SOCP (14), the optimization toolbox SeDuMi [25] is used. More examples of synthesis problems using SOCP are given in [23], [24]. III. NUMERICAL RESULTS In this Section, numerical results are presented to validate and illustrate the potentialities of the proposed approach. Let in (10) can be us remind that any state of polarization synthesized, but in the examples below, only linear and circular polarization are considered. They corresponds respectively to and . In the case of the the minimization of linear polarization synthesis, relation (10) becomes then and the synthesis procedure requires therefore no iteration. First, a sequentially rotated array composed of dipoles is designed to radiate a circularly polarized wave in order to validate the method. Then, a linear array of dipoles is synthesized to radiate a pattern with a spatial power constraint and an optimized polarization. The synthesis of a conformal array, where the coupling effects are taken into account, is also presented. Finally, a two-dimensional (2-D) pattern synthesis problem is addressed for a planar array composed of randomly oriented patches. A. Validation: Sequentially Rotated Array The approach, described in Section II, is here extended in a straightforward way to the synthesis of 2-D patterns. A dependency has to be considered in the (1) to (13). It is well known that sequentially rotated arrays of linearly polarized elements can generate circularly polarized radiation [26], [27]. Let us consider such an array composed of 16 dipoles, as represented in Fig. 3. This array is synthesized to radiate a pattern having 2-D spatial power constraints: , • a main beam in the broadside direction since and i.e., , with sidelobe levels below

Fig. 4. Results of the circular polarization synthesis with a sequentially rotated array composed of 16 dipoles. (a) The magnitude of the total electric field is lower than 12 dB outside the white circle. The axial ratio on both axis u = 0 and v = 0 is plotted, when optimized over (b) the broadside direction (i.e., P = 0 ) and (c) the angular region P = [ 25 ; 25 ], i.e., for u or v [ 0:4; 0:4].

0

20

dB for , i.e., outside the white circle in Fig. 4(a) and a circular polarization: • optimized either in the broadside direction or over the angular range . The synthesized far field pattern and axial ratio are plotted in Fig. 4. While the power pattern complies to the requirements, the axial ratio plots, Fig. 4(b),(c), clearly show that a circular polarization is synthesized over the region . The optimized weightings are given in Table I. The 16-element array can be seen as 4 sub-arrays composed of 4 sequentially rotated dipoles, namely (1, 6, 11, 16), (2, 7, 12, 13), (3, 8, 9, 14) and (4, 5, 10, 15). Within each sub-array, the optimized weightings have almost the same magnitude and a phase shift close to 90 , which is consistent with the results shown in [26], [27]. B. Linear Arrays 1) Linear Polarization Synthesis: A linear array of ten equispaced dipoles, that are randomly oriented, is considered to synthesize a pattern having a broadside main beam, sidelobe levels dB for outside and an optimized linear below . A schematic view of the array polarization over is represented in Fig. 5(a) and the dipoles positions and specific

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TABLE I OPTIMAL WEIGHTINGS OF THE SEQUENTIALLY ROTATED ARRAY SYNTHESIS

orientations together with the optimal weightings are given in Table II. The synthesized far field patterns are shown in Fig. 5(b). While the magnitude of the total field complies to the sidelobe is at least 16.8 dB lower than over , level constraint, which confirms the linear polarization of the wave. 2) Circular Polarization Synthesis: A linear array of ten crossed dipoles, that are randomly oriented and spaced by , is considered to synthesize a main beam in the direcwith an optimized circular polarization over tion . The sidelobe levels are constrained to remain dB over . A schematic below view of the array is represented in Fig. 6(a) and the dipoles positions and orientations are given in Table III. The synthesized far field patterns Fig. 6(b) show that the spatial power constraints are well respected. Moreover, the and components of the field have the same magnitude over . This is confirmed by the axial ratio, plotted in Fig. 6(c), that is lower than 0.1 dB over , which establishes the good quality of the synthesized circular polarization. C. Conformal Array An array of five patches that are conformed on a cylinder, as represented in Fig. 7(a), is considered. Each square patch is fed by two coaxial probes and , as shown in Fig. 7(b). Each element is therefore dually polarized. The goal is to find the complex weightings of each coaxial probe in order to radiate a main beam in the diwith sidelobe levels below dB for rection and an optimized circular . polarization over The active element pattern method [28] is applied to calculate the pattern of the fully excited array. Each patch is simulated in the array environment with a full wave numerical software (Ansoft HFSS) to provide the array response and of (6). Using this method enables one to take the mutual coupling effects into account. The optimal weightings of the conformal array synthesis problem are given in Table IV. The synthesized far field patterns

Fig. 5. (a) Schematic view of the linear array composed of 10 dipoles that are randomly oriented (see angle  ) in the (xOy) plane and spaced by 0:5 . (b) Far field patterns of the linear polarization synthesis.

TABLE II SETTINGS AND RESULTS OF THE LINEAR POLARIZATION SYNTHESIS WITH A LINEAR ARRAY OF DIPOLES

and the resulting axial ratio are plotted in Fig. 7(c),(d). A very

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Fig. 7. (a) Schematic view of the conformal array composed of five patches. (b) Top view of the dual polarized square patch i fed by two coaxial probes iX and iY . Synthesized circular polarization with (c) the electric field magnitude and (d) the axial ratio.

TABLE IV OPTIMAL WEIGHTINGS OF THE CONFORMAL ARRAY SYNTHESIS PROBLEM

Fig. 6. (a) Schematic view of the linear array composed of 10 crossed dipoles that are randomly oriented (see angle  ) in the (xOy) plane and spaced by 0:75 . Results of the circular polarization synthesis: (b) the normalized far field patterns and (c) the axial ratio.

TABLE III SETTINGS AND RESULTS OF THE CIRCULAR POLARIZATION SYNTHESIS WITH A LINEAR ARRAY OF CROSSED DIPOLES

good circular polarization is obtained over level respects the requirements.

and the sidelobe

D. Planar Array for 2-D Pattern Synthesis A planar array is used to synthesize a 2-D pattern having spatial power constraints and an optimized polarization. This array is composed of 3 3 dual polarized patches with a random orientation, as represented in Fig. 8(a). The active element pattern method, described in Section III.C, is applied to calculate the array pattern radiation in order to consider the mutual coupling effects. 1) Circular Polarization Synthesis: The synthesis problem is the following: , i.e., • a main beam in the direction , with an upper bound of dB outside the circle in Fig. 8(b) and • a circular polarization optimized inside the circle in Fig. 8(c). The contour plots Fig. 8(b) show that the spatial power constraint is respected. The axial ratio, represented in Fig. 8(c), is lower than 0.2 dB inside the circle where the polarization has been optimized. The computation time of such synthesis problem (the matrix in (14) is of dimension larger than 1300 1600 for a number

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2

Fig. 8. (a) Top view of the planar array composed of 3 3 dual polarized patches with random orientation. (b) Contour plot of the synthesized total electric field. The sidelobe levels are below 10 dB outside the white circle. (c) Contour plot of the axial ratio. A circular polarization has been optimized inside the white circle, where AR is lower than 0.2 dB.

0

of unknowns equal to 18) is less than 5 seconds on a standard laptop. 2) Linear Polarization Synthesis: For this synthesis problem, the spatial power constraints are unchanged and dB outside the circle represented in Fig. 9(a). A linear polarization according to is optimized inside the circle plotted in Fig. 9(b),(c). The contour plots of the synthesized patterns confirm that, in this is at most 19 dB lower than . region, IV. CONCLUSION A synthesis method to design arrays that radiate a pattern, having both spatial power constraints (upper bounded sidelobes levels) and an optimized specified polarization over a given angular range, has been proposed. Arbitrary arrays, i.e., arrays of

Fig. 9. Far field patterns corresponding to the linear polarization synthesis with the planar array shown in Fig. 6(a). (a) The magnitude of the total electric field is lower than 10 dB outside the white circle. A linear polarization according to  is optimized inside the white circle plotted in (b) and (c). In this region, E is at most 19 dB lower than E .

0

j j

j j

any geometry composed of elements that can have arbitrary and differing radiation patterns, can be handled and any state of polarization can be synthesized. Using the polarization formulation proposed in [22], this synthesis problem has been written as a convex optimization problem, that requires to be iterated a small number of times. As opposed to other contributions, the desired polarization is optimized, not only in a single direction, but over a whole angular range, which is a frequently encountered requirement. The proposed synthesis method can easily be modified to yield an optimization problem that minimizes the side lobe levels, while guaranteeing a specified accuracy of the polarization over an angular range. Various numerical examples of joint power pattern and polarization synthesis have been presented to validate the proposed

FUCHS AND FUCHS: OPTIMAL POLARIZATION SYNTHESIS OF ARBITRARY ARRAYS WITH FOCUSED POWER PATTERN

approach and to show its potentialities. The design of a sequentially rotated array has first been considered to ascertain that the proposed approach recovers the expected weightings. Arbitrary linear and planar arrays as well as a conformal array have been synthesized to radiate both linear and circular polarization to show the flexibility of our approach. The mutual coupling effects between elements have been also taken into account in the synthesis procedure by applying the active element pattern method. Finally, the extension to the synthesis of two-dimensional power pattern with optimized polarization has been addressed to show the efficiency of the approach in terms of computation load. ACKNOWLEDGMENT The authors would like to thank the reviewers for their comments that helped improve the paper as well as M. Kupka Dias da Silva for fruitful discussions. REFERENCES [1] D. G. Giuli, “Polarization diversity in radars,” Proc. IEEE, vol. 74, pp. 245–269, Feb. 1986. [2] M. A. Sletten and D. B. Trizna, “An ultrawideband, polarimetric radar for the studies of sea scatter,” IEEE Trans. Antennas Propag., vol. 42, no. 11, pp. 1461–1466, Nov. 1994. [3] J. Vivekanandan, V. N. Bringi, M. Hagen, and P. Meischner, “Polarimetric radar studies of atmospheric ice particles,” IEEE Trans. Geosci. Remote Sensing, vol. 32, no. 1, pp. 1–10, Jan. 1994. [4] M. R. Andrews, P. P. Mitra, and R. De Carvalho, “Tripling the capacity of wireless communications using electromagnetic polarization,” Nature, vol. 409, pp. 316–318, Jan. 2001. [5] W. C. Y. Lee and Y. S. Yeh, “Polarization diversity system for mobile radio,” IEEE Trans. Commun., vol. 26, pp. 912–923, Oct. 1972. [6] C. B. Dietrich, K. Dietze, J. R. Nealy, and W. L. Stutzman, “Spatial, polarization, and pattern diversity for wireless handheld terminals,” IEEE Trans. Antennas Propag., vol. 49, no. 9, pp. 1271–1281, Sept. 2001. [7] D. C. Cox, “Antenna diversity performance in mitigating the effects of portable radio telephone orientation and multipath propagation,” IEEE Trans. Commun., vol. 31, no. 5, pp. 620–628, May 1983. [8] R. T. Compton, Jr., “The tripole antenna: An adaptive array with full polarization flexibility,” IEEE Trans. Antennas Propag., vol. 29, no. 6, pp. 944–952, Nov. 1981. [9] A. J. Weiss and B. Friedlander, “Performance analysis of diversely polarized antenna arrays,” IEEE Trans. Signal Processing, vol. 39, no. 7, pp. 1589–1603, Jul. 1991. [10] A. Schell and A. Ishimaru, “Antenna pattern synthesis,” in Antenna Theory, R. E. Collin and F. J. Zucker, Eds. New York: McGraw-Hill, 1969, ch. 10, pt. 1. [11] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design. New York: Wiley, 1981, ch. 10. [12] L. I. Vaskelainen, “Iterative least-squares synthesis methods for conformal array antennas with optimized polarization and frequency properties,” IEEE Trans. Antennas Propag., vol. 45, no. 7, pp. 1179–1185, Jul. 1997. [13] C. Dohmen, J. W. Odendaal, and J. Joubert, “Synthesis of conformal arrays with optimized polarization,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2922–2925, Oct. 2007. [14] H. Lebret and S. Boyd, “Antenna pattern synthesis via convex optimization,” IEEE Trans. Signal Processing, vol. 45, no. 3, pp. 526–531, Mar. 1997. [15] O. M. Bucci, M. D’Urso, and T. Isernia, “Exploiting convexity in array antenna synthesis problems,” presented at the IEEE Radar Conf., Roma, Italy, May 2008.

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[16] L. Caccavale, T. Isernia, and F. Soldovieri, “Methods for optimal focusing of microstrip array antennas including mutual coupling,” IEE Microw. Antennas Propag., vol. 147, no. 3, pp. 199–202, Jun. 2000. [17] J.-J. Xiao and A. Nehorai, “Optimal beam pattern synthesis of a polarized array,” presented at the IEEE Statistical Signal Processing Workshop, Madison, WI, Aug. 2007. [18] J.-J. Xiao and A. Nehorai, “Performance of beampattern synthesis using high-dimensional vector antenna arrays,” presented at the Sensor, Signal and Information Processing Workshop, Sedona, AZ, May 2008. [19] J.-J. Xiao and A. Nehorai, “Optimal polarized beampattern synthesis using a vector-antenna array,” IEEE Trans. Signal Processing, vol. 57, no. 2, pp. 576–587, Feb. 2009. [20] A. Ben-Tal and A. S. Nemirovski, “Lectures on modern convex optimization: Analysis, algorithms, and engineering applications,” in Society for Industrial and Applied Mathematics. Philadelphia, PA: MPS-SIAM, 2001. [21] G. A. Deschamps, “Part II—Geometrical representation of the polarization of a plane electromagnetic wave,” Proc. IRE, vol. 39, pp. 540–544, May 1951. [22] A. Nehorai and E. Paldi, “Vector-sensor array processing for electromagnetic source localization,” IEEE Trans. Signal Processing, vol. 42, no. 2, pp. 376–398, Feb. 1994. [23] S. P. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [24] B. Fuchs and J. J. Fuchs, “Optimal narrow beam low sidelobe synthesis for arbitrary arrays,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 2130–2135, Jun. 2010. [25] SeDuMi [Online]. Available: http://sedumi.ie.lehigh.edu/ [26] T. Teshirogi, M. Tanaka, and W. Chujo, “Wideband circularly polarized array antenna with sequential rotations and phase shift of elements,” in Proc. of ISAP, 1985, pp. 117–120. [27] J. Huang, “A technique for an array to generate circular polarization with linearly polarized elements,” IEEE Trans. Antennas Propag., vol. 34, no. 9, pp. 1113–1124, Sep. 1986. [28] D. F. Kelley and W. L. Stutzman, “Array antenna pattern modeling methods that include mutual coupling effects,” IEEE Trans. Antennas Propag., vol. 41, no. 12, pp. 1625–1632, Dec. 1993. Benjamin Fuchs (S’06–M’08) received the Electronics Engineering degree and the M.S. degree in electronics from the National Institute of Applied Science (INSA) of Rennes, France, in 2004 and the Ph.D. degree from the University of Rennes 1, France, in 2007. In 2008, he was a Postdoctoral Research Fellow at the Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland. In 2009, he joined the Institute of Electronics and Telecommunications of Rennes (IETR), as a Researcher at the Centre National de la Recherche Scientifique (CNRS). Since 2011, he is on leave at EPFL. His research interests include mode matching techniques, millimeter-wave antennas, focusing devices (lens antennas) and array synthesis methods.

Jean Jacques Fuchs (M’81) was born in France in 1950. He graduated from the “École Supérieure d’Électricité” Paris, France, in 1973 and received the M.S. degree in electrical engineering from the Massachusetts Institute of Technology, Cambridge, in 1974. After a short period in industry with Thomson-C.S.F., he joined the “Institut de Recherche en Informatique et Systèmes Aléatoires” (IRISA), France, in 1976. Since 1983, he is a Professor at the “Université de Rennes 1” France. His research interests shifted from adaptive control and identification, in which he obtained the “Thèse d’Etat” in 1982, towards signal processing. He is now involved in array processing and sparse representations.

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Low Sidelobe Phased Array Pattern Synthesis With Compensation for Errors Due to Quantized Tapering Will P. M. N. Keizer, Member, IEEE

Abstract—Recently, the iterative Fourier transform (IFT) method has been introduced for the synthesis of low sidelobe patterns for array antennas with periodic element arrangement. The IFT method makes use of the property that for an array with periodic element spacing, an inverse Fourier transform relationship exists between the array factor and the element excitations. This property is used in an iterative way to derive the array element excitations from the prescribed array factor using a direct Fourier transform. With some simple additions, the same IFT technique is also applicable to mitigate the degradation of sidelobe performance of array antenna caused by quantization of its taper across the array when using discrete control devices. Numerical examples of low sidelobe pattern synthesis applying compensation for taper quantization errors are presented. The results refer to both phase as well as amplitude tapering of linear arrays of various sizes. Index Terms—Array antennas, low sidelobes, quantized tapering, pattern synthesis.

I. INTRODUCTION

M

ANY phased-array antennas make use of phase and amplitude control schemes with discrete levels instead of a continuum of phases and amplitudes. This simplifies the control circuitry, but results in periodic or systematic phase and amplitude errors across the array aperture. Since these quantization errors are highly correlated, they can produce large, well-defined sidelobe pattern errors called quantization lobes. The occurrence of quantization lobes will degrade the array low sidelobe performance, lowers directivity, and results in beam pointing errors. Periodic phase errors come into play when beam steering is arranged by discrete binary phase shifters. Discretization of the phases of the array element weights by binary phase shifters causes that the required linear phase taper to steer the main beam is attained by a staircase approximation with the result of periodic triangular phase errors that produce additional grating lobe like sidelobes. A suitable mean to suppress the occurrence of phase quantization lobes is provided by the phase-added method [1]. The essence of this method is to introduce a known random offset to each array element and then subsequently to compensate these offsets with new settings of the phase shifters resulting in randomized quantization errors. Quantized tapers created by discrete control devices, i.e., phase shifters or attenuators, suffer Manuscript received February 07, 2011; revised May 03, 2011; accepted June 03, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. The author, retired, was with TNO Physics and Electronics Laboratory, 2597 AK, The Hague, Netherlands. He is now at Clinckenburgh 32, 2343 JH Oegstgeest, The Netherlands (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165509

as well from quantization errors and experience therefore degraded sidelobe performance. It turns out that randomization of quantization errors is an effective mean to mitigate sidelobe deterioration when tapers are realized by discrete control devices. Haupt [2] reported quantized phase taper results pertaining to a 31-element linear array. He performed simulations using a hybrid optimization method based on the combination of a genetic algorithm and a local optimizer, the Nelder–Mead downhill simplex algorithm. In [3], simulation results were published for 100- and 200-element linear arrays applying quantized phase tapering. These results were obtained with a quantized particle swarm optimization method. In this paper, the effectiveness of randomization of quantization errors will be investigated in combination with the iterative Fourier transform (IFT) method that can synthesize low sidelobe tapers for linear and planar array antennas [4], [5]. This new method is called quantized IFT (QIFT). The compensation of errors due to quantized tapering in order to enhance sidelobe performance will be explored for both phase-only and amplitude–only tapers. The presented numerical results refer to linear arrays of various sizes and will be compared with available published data. II. COMPUTATIONAL APPROACH of a linear array with isotropic eleThe array factor ments equally spaced at distance can be written as (1) where is the complex excitation of the th element, the wavenumber the operating wavelength, the direction cosine, and the pattern angle measured from broadside of the array. Since (1) represents a finite Fourier series that relates the exof the linear array to its array factor citation coefficients that can be calculated through a inverse fast Fourier transform (FFT), a direct FFT applied on will yield the excitation coeffi. This property is used in an iterative way to calcucients late the excitation coefficients when the array factor determined with (1) is adapted to the prescribed sidelobe requirements. The MATLAB program of the IFT method presented in [5] is restricted to amplitude–only low sidelobe synthesis. To perform a phase-only low sidelobe synthesis combined with a compensation for errors due to quantized phase tapering, the original MATLAB program requires three modifications. The first modification concerns the initialization of the phase of all element excitation coefficients at the start of the program which must be

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selected to be random instead of uniform. The second modification involves the amplitudes of the element calculated from the adapted array factor using the direct FFT which all must be made equal before a new iteration can be executed. The third modification concerns the implementation of the phase-added method to randomize phase quantization errors. The phase of the th element after quantization carried out by its phase shifter and including the phase-added randomization follows from (2) the phase where denotes rounding to the nearest integer, of the th element computed from the direct FFT carried out the known random phase on the adapted array factor the least significant phase bit of the discrete offset and phase shifter. The quantization of the element phases has to take place each time before a new iteration with the QIFT method will be executed. To make the original IFT method suitable to minimize error effects induced by quantized amplitude tapering with the aim to find the taper with the lowest maximum SLL, the original MATLAB program in [5] needs only one major modification. This modification involves the implementation of the quantiof the th element by its diszation of the amplitude crete variable attenuator and the randomization of the associated quantization error before the next iteration starts. and This operation results in the quantized amplitude obeys the expression

Fig. 1. Array factor of the 31-element array resulting from a 4-bit binary phase taper corrected for phase quantization errors and designed with the QIFT method.

(3) the amwhere denotes rounding to the nearest integer, plitude of the th element computed from the direct FFT perthe random amplitude formed on the adapted array factor offset and the least significant bit of the discrete variable atand in (3) are all tenuators. The values of in decibels. III. SIMULATION RESULTS A. Quantized Phase-Only Low Sidelobe Synthesis To demonstrate the effectiveness of the QIFT method to find the lowest maximum SLL in case of quantized phase tapering using binary phase shifters, various simulations have been performed for linear array antennas consisting of 31, 61, 100, and 200 elements, respectively. These arrays, provided with spacing, are equipped isotropic elements positioned at with binary phase shifters having 3-, 4-, 5-, 6-, and 8-bit phase accuracy, respectively. All arrays operate with symmetric phase weights. Fig. 1 shows the normalized array factor of the 31-element linear array operating with a phase taper featuring a maximum 17 dB sidelobe level (SLL) obtained by randomizing quantization errors of its 4-bit phase shifters. Fig. 2 shows the convergence rate during the run of the QIFT method that produced the design of Fig. 1. The rapid fluctuations in the maximum SLL in this figure are the result of the randomization of the element phases using (2).

Fig. 2. Convergence rate of the run of the QIFT method that was responsible for the result of Fig. 1.

Table I lists the phase weights of the 31 elements of the array for the considered bit sizes of the phase shifters. The table includes furthermore the phase weights for 3- and 4-bit phase shifters obtained by rounding off the optimum continuous phase weights. Rounding off corresponds to rounding the required phase to the nearest bit. The maximum SLL results obtained for the four arrays operating with phase shifters of various bit sizes are summarized in Table II and refer to phase error quantization compensation as well as to rounding off the optimum continuous phase weights. One can see from this table that compensation for discrete phase errors is mainly effective for the 31-, 61-, and 100-element arrays when operating with phase shifters having a resolution less than 8 bits. Similar design simulations for quantized phase tapers pertaining to a 31-element linear array were performed by Haupt in [2]. He used a hybrid optimization method based on the combination of a genetic algorithm and a local optimizer, the Nelder–Mead downhill simplex algorithm, to get a maximum SLL of 16.8 dB for a 4-bit binary phase taper (QIFT method 17.0 dB) and 16.9 dB maximum SLL for the continuous

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TABLE I PHASE WEIGHTS OF THE 31-ELEMENT LINEAR ARRAY FOR VARIOUS PHASE SHIFTER BIT SIZES

TABLE II MAXIMUM SLL RESULTS FOR VARIOUS LINEAR ARRAYS FOR A RANGE OF PHASE SHIFTER BIT SIZES

phase taper, (QIFT method 17.6 dB). In [3] quantized phase taper simulation results were published for 100- and 200-element linear arrays. The results in [3] were obtained with a quantized particle swarm optimization method and pertained to the use of 4-bit phase shifters with a maximum phase control range of 114.5 . For the 100-element linear array a 18-dB maximum SLL was reported (QIFT method 19.8 dB) and for the 200-element linear array a value of 21 dB, (QIFT method 21.5 dB). The maximum SLL results in Table II when the phase weights are free from quantization errors (30-bit resolution) are for all four arrays in line with the maximum SLL results in [6], ([7], dB for the 61-elFig. 8). The maximum SLL result of ement array operating with phase weights realized with 30-bit resolution corresponds quite well with the maximum SLL of 19.43 dB reported in [8] for a 60-element linear array using a continuous phase-only taper. B. Quantized Amplitude-Only Low Sidelobe Synthesis To demonstrate the effectiveness of the QIFT method to find the lowest maximum SLL in case of quantized amplitude tapering using discrete variable attenuators, various simulations have been performed for two linear array antennas consisting of

Fig. 3. Array factor of the 120-element array resulting from a 5-bit binary amplitude taper corrected for amplitude quantization errors and designed with the QIFT method. (a) Array factor. (b) Corresponding amplitude taper.

40 and 120 elements, respectively. The considered arrays were equipped with discrete variable attenuators with an amplitude control range of 20 dB and having 4-, 5-, or 6-bit resolution, respectively. Both arrays consist of isotropic elements and were operated with symmetric equi-spaced apart at amplitude weights. Sum and difference patterns have been evaluated. Fig. 3(a) shows the normalized array factor of the sum pattern of the 120-element linear array produced by quantized amplitude taper compensated for quantization errors. The taper responsible for the result of Fig. 3(a) with a maximum SLL of 42.35 dB is shown in Fig. 3(b) and is created by discrete attenuators with 5-bit resolution. The displayed taper is synthesized with the QIFT method. Fig. 4(a) shows the array factor of the same array when no compensation is applied for quantization errors induced by the 5-bit attenuators while Fig. 4(b) shows the associated quantized taper. This taper is responsible for a maximum SLL of 40.60 dB. Similar simulations as for the array of Figs. 3 have been performed for the difference pattern with variable discrete attenuators having a 20 dB amplitude control range and featuring again

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TABLE III MAXIMUM SLL RESULTS FOR VARIOUS LINEAR ARRAYS FOR A RANGE OF ATTENUATOR BIT SIZES

Fig. 4. Array factor of the 120-element array resulting from a 5-bit binary amplitude taper obtained by rounding off of the optimum continuous weights. (a) Array factor. (b) Corresponding amplitude taper.

4-, 5-, and 6-bit resolution, respectively. The simulations using the QIFT method for the difference pattern involved the same attenuator resolutions as used for the sum pattern. Identical simulations have been performed for the 40-element array. Table III summarizes the maximum SLL results of all simulations carried out for both arrays including those of 30-bit attenuators since these last results are representative for the best SLL performance of continuous amplitude weights. In order to evaluate the effect of the randomization of attenuator errors on the maximum SLL performance, Table III contains furthermore maximum SLL results pertaining to simulations with quantized tapers based on rounding off continuous amplitude weights. The tapers used for this comparison are indicated by the values between parentheses in the columns of Table III under the heading “No”. These tapers revealed the lowest maximum SLL when the bit setting of the discrete attenuators was done by the rounding off method. One can notice that randomization of taper quantization errors lowers the maximum SLL with a value between 1.2 and 2.9 dB in comparison to the rounding off method. Numerical information about the taper efficiency of the synthesized patterns is included in Figs. 1(a), 3(a), and 4(a). Taper

efficiency hardly changed when compensation for discretization errors was applied as can be noted from comparing taper efficiency results of Figs. 3(a) and 4(a). For the considered difference patterns the taper efficiency was of the order of about 0.48. Responsible for this much lower value compared to the taper efficiency of the sum patterns is the presence of two main lobes for the difference pattern. The computations were carried out on a PC equipped with an Intel Core i7-2600K quad core processor @3.4 GHz, 12 GB RAM memory and running 64-bit Windows 7. The QIFT simulations for the 200-element array were performed with 2048 points forward and inverse FTTs and for the 31-element array with 512 points FFTs. A high number of points for the FTTs were needed to locate precisely the position of the first null along the main beam and therefore the start of the SLL region. The computation time to perform 20 000 iterations with the QIFT method using 2048 points FFTs was about 8 seconds. The required number of iterations to get the specified maximum SLL varied for the majority of the considered examples between a few hundred and about 15 000. The presented results apply to linear arrays featuring an identical embedded element pattern for all elements. For finite arrays the condition of identical embedded element patterns is in general not met. Elements near both edges of a finite linear array will have a different embedded element pattern compared to that of the elements located in the center of the array due to a different array environment. Variation in embedded element pattern will degrade the maximum SLL performance of an array. The extent of such SLL degradation will depend on the size of the array and the amount of element tapering near both edges. Strong reverse tapering near both array edges will certainly have a negative effect on the SLL performance especially when the array is small, while large arrays with very low edge tapering will hardly experience a SLL degradation. Fortunately, two solutions exist to cure the problem of the variation in embedded element pattern for finite arrays. One solution is to add at both edges of the array a few dummy elements. These dummy elements, passively terminated into a matched load, are intended to simulate an infinite array environment to the active edge elements [9]. The second solution is only applicable to arrays oper-

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ating with digital beamforming at the element level and involves a compensation of mutual coupling to be performed in the digital domain [10]. With such compensation it is possible to create for each element the same embedded element pattern identical to that of an isolated element. Experimental results presented in [10] have demonstrated the effectiveness of the mutual coupling compensation and prove that even for very small linear arrays low sidelobes are feasible. IV. CONCLUSION A new method was presented that mitigates the degradation of sidelobe performance of array antennas caused by quantization of its taper using discrete control devices. The improvement of the maximum sidelobe level was obtained by randomization of the quantized taper errors introduced by discrete control devices. Simulations performed on various linear arrays demonstrated the effectiveness of the proposed method for both quantized phase and amplitude tapers. Comparison with comparable data from literature revealed that the QIFT method outperforms the quantized phase taper results obtained with nature inspired stochastic methods. REFERENCES [1] M. S. Smith and Y. C. Guo, “A comparison of methods for randomizing phase quantization errors in phased arrays,” IEEE Trans. Antennas Propag., vol. 31, no. 6, pp. 821–828, Nov. 1983. [2] R. Haupt, “Antenna design with a mixed integer genetic algorithm,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 577–582, Mar. 2007. [3] T. H. Ismail and Z. M. Hamici, “Array pattern synthesis using digital phase control by quantized particle swarm optimization,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 2142–2145, Jun. 2010. [4] W. P. M. N. Keizer, “Fast low-sidelobe synthesis for large planar array antennas utilizing successive fast Fourier transforms of the array factor,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 715–722, Mar. 2007.

[5] W. P. M. N. Keizer, “Low sidelobe pattern synthesis using iterative Fourier techniques coded in MATLAB,” IEEE Antennas Propag. Mag., vol. 51, no. 2, pp. 137–150, Apr. 2009. [6] J. F. DeFord and O. P. Gandhi, “Phase-only synthesis of minimum peak sidelobe patterns for linear and planar arrays,” IEEE Trans. Antennas Propag., vol. 36, no. 2, pp. 191–201, Feb. 1988. [7] D. G. Kurup, M. Himdi, and A. Rydberg, “Synthesis of uniform amplitude unequally spaced antenna arrays using the differential evolution algorithm,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2210–2217, Sep. 2003. [8] S. K. Goudos, K. Siakavara, T. Samaras, E. E. Vafiadis, and J. N. Sahalos, “Self-adaptive differential evolution applied to real-valued antenna and microwave design problems,” IEEE Trans. Antennas Propag., vol. 59, no. 4, pp. 1286–1298, Apr. 2011. [9] L. Stark, “Microwave theory of phased-array antennas—A review,” Proc. IEEE, vol. 62, no. 12, pp. 1651–1704, Dec. 1974. [10] H. Steyskal and J. S. Herd, “Mutual coupling compensation in small array antennas,” IEEE Trans. Antennas Propag., vol. 38, no. 12, pp. 1971–1975, Dec. 1990.

Will P. M. N. Keizer (M’99) received the M.S. degree in electrical and electronics engineering (cum laude) from the Eindhoven University of Technology, Eindhoven, The Netherlands in 1970. From 1971 to 2002, he was with TNO Physics and Electronics Laboratory, The Hague, The Netherlands, as Research Scientist and Technical Manager in the field of microwave components, antennas, propagation, and radar. He has been involved in the NATO Anti Air Warfare System (NAAWS) study directed by the U.S. Navy and aiming at an advanced sensor suite for future warships. He was one of the key persons in the initiation of the APAR naval multi-function active phased array radar based on the NAAWS concept that is now in operational use with The Netherlands and German navies. He was responsible for the design of the APAR antenna aperture consisting of waveguide radiators featuring a 60 degree scan angle capability in all directions over a 30% bandwidth at X-band. His research interests include phased array antennas, near-field antenna testing, microwaves, low-angle radio-wave propagation, multi-function phased array radar and synthetic aperture radar. He authored or coauthored more than 40 papers in journals and conference proceedings. He has been retired since 2003.

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Interference Suppression in Uniform Linear Arrays Through a Dynamic Thinning Strategy Paolo Rocca, Member, IEEE, Randy L. Haupt, Fellow, IEEE, and Andrea Massa, Member, IEEE

Abstract—Thinned arrays are designed to have low average sidelobe levels. The randomness in selecting the elements that are turned on/off to achieve the low sidelobes implies that there are several arrangements of the thinned aperture that have the same or nearly the same average sidelobe level. If the same number of elements are always turned on, but the elements that are turned off change, then the array directivity does not change, but the nulls and sidelobes do. This paper presents a technique for dynamically altering the thinning configuration of a linear array in order place low sidelobe and nulls in desired directions. A set of representative results is reported and discussed to show the effectiveness of the proposed approach. Index Terms—Dynamic thinning, interference suppression, thinned array, uniform linear array.

I. INTRODUCTION

YNTHESIZING array antennas needs reduced sidelobe levels (SLLs) besides an adequate directivity and mainlobe beamwidth. To minimize the SLL when dealing with regular lattices and uniformly-spaced elements, tapering the amplitude weights from the center to the edges of the aperture has shown being an effective and reliable solution. Towards this purpose, several strategies have been proposed to radiate optimal patterns with either equi-ripple [1] or tapered [2] sidelobes as well as patterns with arbitrary upper bounds [3]. More recently and thanks to the growing efficiency of modern computers, evolutionary-based iterative algorithms have been also used [4]–[8] to exploit the easy inclusion of unconventional constraints in the design process. Notwithstanding their potentialities, one of the main drawback of such an architectural solution is the complexity of the feeding network that requires dedicated control points (i.e., amplifiers/attenuators), one at each element, with a non-negligible power consumption and increased costs. Array thinning is an alternative for reducing the SLL while keeping limited costs. Thinned arrays are uniform linear or planar arrangements where a subset of elements is connected

S

Manuscript received January 25, 2011; revised May 15, 2011; accepted June 21, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. P. Rocca and A. Massa are with the ELEDIA Research Group, Department of Information Engineering and Computer Science, University of Trento, Povo 38123 Trento, Italy (e-mail: [email protected]; [email protected]). R. L. Haupt is with the Haupt Associates, Boulder, CO 80303 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165506

to the beam forming network (BFN), while those belonging to the complementary subset are connected to matched loads or removed. The advantages of array thinning over amplitude tapering are a simpler BFN, no amplitude weights/attenuators, and a reduced number of active elements for synthesizing the same pattern in terms of mainlobe beamwidth and SLL of the uniform array with equal size. For these reasons, thinned arrays have had a widespread diffusion in both satellite [9] and terrestrial [10], [11] systems. Approaches for thinning arrays are based on statistical or analytical or optimization-based strategies. Originally, statistical methods [12] have been used where the density of active elements is proportional to the amplitude of a continuous reference distribution (e.g., Taylor [13]). More recently, other approaches based on evolutionary optimization algorithms, namely genetic algorithms (GAs) [14], particle swarm optimizer (PSO) [15], and ant colony optimizer (ACO) [16] have gained a great attention. Because of their high computational burden, mainly when dealing with large antennas, hybrid techniques [17], [18] have been proposed to better address the convergence issues. Sub-optimal, but much more faster, deterministic strategies [19] have been investigated, as well. By exploiting the relationships between the samples of the power pattern and the Fourier transform of the autocorrelation of the element distribution, massively-thinned very large arrays have been synthesized by reformulating the original pattern designs as the definition of binary sequences with suitable autocorrelation functions [20]. However, thinning regular lattices does not guarantee satisfactory/reliable control of the pattern sidelobes with reduced performances of the communication/radar system when interfering signals or jammers impinge away from the main lobe region. As a matter of fact, adaptive nulling strategies have generally considered amplitude tapering [21], [22] and/or phase tapering [23], [24] of all element excitations or only a subset of them [25], [26]. Although guaranteeing effective suppression capability of the interfering signals, unavoidable complex antenna structures and BFNs are required [27]. In order to properly tackle such a situation by using thinned arrays, adaptive thinning techniques have been proposed in the literature [28], [29]. In such a framework, this paper presents a dynamic strategy for the suppression of undesired signals impinging within the sidelobe region of the receiving pattern of the array. The on-off status of the array elements is modified by acting on a set of radio frequency (RF) switches until a null or a low sidelobe region is generated in correspondence with the direction of arrival of the jammer. More specifically, the thinned sequence is changed to minimize the total output power at the receiver [23], while keeping unaltered the reception of the desired signal supposed

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probability that a pattern sample of the th sequence is smaller turns out to be [31] than a sidelobe threshold

(3)

Fig. 1. Sketch of the dynamic thinned array.

entering the main beam. Towards this end, each sequence belongs to a subset of pre-computed configurations with the same number of elements turned on in order to have the same main beam gain and avoiding the modulation of the received signal [30]. The paper is organized as follows. The synthesis problem is mathematically formulated in Section II where the dynamic thinning strategy is described, as well. A set of representative numerical results are reported and discussed in Section III. Finally, some conclusions are drawn (Section IV). II. MATHEMATICAL FORMULATION Let us consider the antenna schematically depicted in Fig. 1. The array is characterized by a set of elements with a uniform spacing along the -axis. In the BFN, a radio-frequency (RF) connects the th array element eiswitch, ther to the feeding line or to a matched load. When the element is connected to the feeding line [“on” state— ], it can either transmit or receive signals. Otherwise [“off” ], the signal is dissipated through the state— matched load. Whether all switches are on, the antenna turns out being a fully-populated uniform array. Mathematically, the array factor of the th arrangement of the RF switches can be expressed as (1) is the free-space wavenumber ( being where the wavelength), and is the angle from the array axis. There possible array factors, even though not all are unique, synthesized by the different on-off sequences, . As for the pattern features and since the excitations are binary, the average power level of the sidelobes of the th thinned , is statistically proportional to the number arrangement, of turned on (i.e., ) elements of the array [31] (2) being the number of active elements of the sequence. Moreover, by assuming that all sidelobes deviate from at most for three-times the standard deviation, the

is the mainlobe region, where being the first null beamwidth. Equation (3) also provides statistical information on the percentage of the whole angular range . for which the th power pattern is below Since it is unlikely that a single thinned sequence is able to such suppress a jammer whatever its angular direction [i.e., ], it is profitable to that dynamically change the sequence (i.e., to exploit more thinned sequences—see the Appendix) until a suitable array pattern is generated with sufficient deep sidelobe or a null along the direction of arrival of the undesired signal at hand. If some a-priori information on the maximum power of the interfering signals is available or it can be inferred, the set of thinned sequences (“Control-Sequence” Set) can be computed off-line fitting the following constraints: (a) each control sequence synthesizes a power pattern with amplitude lower than a specified outside ; threshold, (b) the number of control configurations, , is relatively ) to minimize the time of the enumerative small ( , [i.e., the search of the optimal one, arrangement whose pattern guarantees a good signal-plus-interference noise ratio (SINR)] within the same set; (c) the directivity of the antenna has to remain almost constant when switching among the control sequences in order to avoid some undesired modulations of the desired signal at the receiver. Accordingly, the on-off sequences are off-line determined with the following procedure: • Step 0 (Initialization)—Select the number of antenna eland the range of the aperture efficiency (i.e., ements and being the maximum and the minimum value of the aperture efficiency, respectively) to satisfy the condition (c) since the peak directivity of the array is given by [10] (4) , to Choose the value of the rejection threshold, properly receive the desired signal, while reducing the noise of the interference signal, thus yielding a reliable communication system (a); • Step 1 (Generation of the Admissible-Sequence is small, exhaustively sample the Set)—Whether set of thinned configurations to select sequences (“Reference-Sequence” Set, such that ), being the aperture efficiency of the th thinning given by [32] configuration

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(5) Let us notice that, the number of configurations for each value of

is equal to

. In order to avoid unsuitable sequences (e.g., those with all the elements turned off in the tails of the antenna leading to a uniform array of elements or those with a high number of switched-off elements in the center of the array), keep only the thinned configurations (“Admissible-Sequence” Set, ) whose are smaller peak sidelobe levels, than those of the corresponding uniform fully-populated linear arrays with the same number of active elements

(6)

Such a choice allows one to avoid patterns with high secondary lobes, but also limit the sequence selection within the pool of patterns with main beams of similar shapes (c) since the array elements are distributed on the whole aperture of each th arrangement. is large, the procedure proposed in [12] is used. When Starting from a reference distribution (e.g., Taylor [13]), statistically thin the array elements only maintaining the admissible sequences for which and ; • Step 2 (Definition of the Nulling Regions)—For each th sequence, store in a vector the angular samples where the corresponding power pattern is below

(7) , being the th sample of the whole angular and ). The angular directions are called nulling directions, while is denoted as the nulling region of the th thinned arrangement; • Step 3 (Definition of the Control-Sequence Set)—Rank sequences according to the extension of their the “null-pattern” coverage given by the number of angular samples in . The pattern of the best-ranked sequence, , allows one the rejection of the interferences over the widest angular range, while the , synthesizes a pattern with the smallest region latter, with . and iteratively ( being Set the iteration index) perform the following loop to define for the the set of sequences, adaptive control: range (

N

Fig. 2. Example #1 ( = 8)—Directivity patterns (a) and peak directivity, = ( )c , (b) for the different thinned sequences.

D D

K

— Step 3.a (Control-Sequence Selection)—Set the th , to the th sequence with sequence, the maximum number of nulling directions not already . Mathematically, present in (8) where ; — Step 3.b (Updating)—Update the global nulling coverage vector and the iteration ; index — Step 3.c (Convergence Check)—If the nulling region covers the whole angular region outside the mainlobe ) goto (Step 4), otherwise repeat (i.e., (Step 3.a) and (Step 3.b); • Step 4 (Array Architecture Setup)—Locate a RF switch only in correspondence with the th array element whose status differs at least between two sequences in (i.e., ), otherwise remove . The total number of switches amounts to it . During the on-line working of the antenna array, the th on-off sequence that provides the minimum output power

(9)

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N

Fig. 3. Example #1 ( ( = 1) configuration.



= 8)—Power pattern of the fully-populated array

N



Fig. 4. Example #1 ( = 8)—Relative level of the secondary lobe at = 69 of the power patterns generated by the whole set of thinned sequences.

K

is enabled by properly setting the corresponding ,( ) and ( ) being the switch sequence strength and direction of the desired and (unknown) interference signal, respectively. III. NUMERICAL RESULTS The first example of the numerical validation is aimed at further motivating the use of the dynamic thinning approach. For the sake of simplicity and without any loss of generality, elements it deals with a small uniform linear array of where each element is connected equally-spaced by to the BFN through a RF switch. It follows that thinned sequences can be generated whose array patterns are shown in Fig. 2(a). Their peak directivities vary from a maxdB to a minimum value of imum value of dB [Fig. 2(b)] proportionally to . With reference to the fully-populated array, let us suppose that in correspondence an interfering signal impinges at dB with respect to with the highest sidelobe the main beam (Fig. 3). The received interference power can be reduced by thinning the array with the RF switches in order to . Fig. 4 shows the values synthesize a pattern with lower of the different thinned configurations ranging of dB. from 0 dB down to almost

N

;

:

Fig. 5. Example #1 ( = 8 = 0 875)—Plot of the power patterns of the sequences belonging to the “Reference-Sequence” Set and of the corresponding nulling regions.

By setting and quences with aperture efficiency of

R

thinned se-

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N = 32; J = 61;  = 0:75; SLL =

Fig. 6. Example #2—Experiment #1 ( dB)—Nulling coverage of the thinned sequences of the “AdmissibledB). Sequence” Set (

020

A SLL = 020

J

TABLE I

EXAMPLE #2 (

N = 32; J = 61;  = 0:75)—PERCENTAGE OF SIDELOBE NULLING COVERAGE

Fig. 7. Example #2—Experiment #1 (N = 32; J = 61;  = 0:75; SLL = 020 dB)—Thinned sequences (a),(b) and corresponding power patterns (c) of the control configurations q = 1 (a),(c) and q = Q = 2 (b),(c).

satisfy the condition and four different power patterns are synthesized (Fig. 5) since pairs of sequences generate the same beam. Whether the (minimum) sidedB, Fig. 5 lobe rejection threshold is fixed at shows the arising nulling regions pointing out that only the two configurations and [Fig. 5(b)] can be effectively used since the level of the sideis dB. lobes along The second example is aimed at illustrating the behavior of the proposed approach. Towards this end, let us considered a elements symlinear array antenna with metrically distributed with respect to the center. The desired value of the aperture efficiency has been fixed to such that is constant whatever the thinned configuration in . In the first experiment, the nulling threshold has been set to dB, while only half (i.e., ) elements have been considered in the synthesis process by virtue of the an, the number of active elements tenna symmetry. Since for each half of the array must be equal to . Within the

se-

only bequences of the “Reference-Sequence” set long to since their are below dB (6) (Step 1). The sequences are then ranked according to the number of nulling directions present in . Fig. 6 shows the percentage of the sidelobe angular region where the control patterns have an amplitude smaller than . As an example, the best thinning sequence

Fig. 8. Example #2—Experiment #1 (N = 32; J = 61;  = 0:75; SLL = 020 dB)—Plot of the rejection probability when SLL = 020 dB.

(i.e., ) is able to reduce of at least 20 dB an undesired signal within almost 83% of the sidelobe region. The iterative procedure for completing the control set terminates after since just different sequences only one iteration of the “Admissible-Sequence” set are enough to guarantee the required suppression threshold within the whole angular range outside the main beam (Table I). The control sequences and the corresponding radiation patterns are shown in Fig. 7 where the solid dots indicate turned-on elements, while the empty circles correspond to switched-off RFs. As it can be observed, at least in the sidelobe one pattern presents a value smaller than region and, in some directions, both patterns reject the undesired signal. In order to quantitatively estimate the “rejection efficiency” of the adaptive thinning strategy, Fig. 8 gives the is synthesized by probability that a sidelobe lower than the patterns in as a function of the angular coordinate. Let

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N = 32; J = 61;  = 0:75; SLL = 020 dB)—Architecture of the dynamic thinned array when SLL = 020 dB.

Fig. 9. Example #2—Experiment #1 (

TABLE II

EXAMPLE #2 (

J = 61;  = 0:75)—PERFORMANCE INDEXES

Fig. 11. Example #2—Experiment #3 (N = 32;J = 61; = 0:75;SLL = 030 dB)—Nulling coverage of the J thinned sequences of the “AdmissibleSequence” Set A (SLL = 030 dB).

N = 32;J = 61; = 0:75;SLL = q = 1 and (b) q =

Fig. 10. Example #2—Experiment #1 ( dB)—Power patterns of the control configurations (a) with dipoles and mutual coupling effects.

020 Q=2

us notice that means that the rejection along the direction is enabled by just half of the configurations, while indicates an interthat all patterns are able to reduce of at least ference impinging from . As for the synthesized antenna, the sketch of the array structure is shown in Fig. 9. Only 8 array have a RF switch, 4 elements are never elements over used (Fig. 9—shaded dots), while the remaining are always active (Fig. 9—dark dots). Although the elements always off can be theoretically removed from the antenna, they are maintained on the aperture to avoid edge effects and obtain a more constant mutual impedance [11]. On the other hand, the corresponding connections to the feeding line are avoided to reduce the system complexity. In order to assess the stability of the proposed approach, the mutual coupling (MC) effects between the array elements have been taken into account to evaluate their impact on the radiated

power patterns and their null coverage capability. Towards this aim, an array of -dipoles and the MC model in [33] have been considered as benchmark case. Fig. 10 shows the relative power patterns and the corresponding nulling regions when MC is present and the thinning sequences in Fig. 7 are applied. As compared to the ideal case [Fig. 7(c)], it turns out that setting the thinning sequences by considering ideal isotropic radiators is a conservative choice since the null coverage capability as well as the rejection probability improve in case of real systems thanks to the higher directivity of the real elements and the fact that the behavior of the secondary lobes is only slightly perturbed. In the second and third experiments, the rejection threshold dB and has been reduced down to dB, respectively, while the set has been kept unaltered with sequences. By analyzing the nulling covthe same (Table II), the best-ranked sequence in the erage of the set two cases has a nulling region covering almost 55% and 39% of the secondary lobe region (i.e., respectively 28% and 44% less than the best one of the previous experiment). The full covwith erage is obtained (Step 3) by selecting the control set and (Tables I and II) different on-off sequences when dB and dB is set, respectively. As compared to the first experiment (i.e., dB), the number of thinning sequences to guarantee a complete coverage of the sidelobe region does not proportionally

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N = 32; J = 61;  = 0:75; SLL = 030 dB)—Thinned sequences (a)–(g) and power pattern with nulling regions q = 1 (a),(h), q = 2 (b),(i), q = 3 (c),(l), q = 4 (d),(m), q = 5 (e),(n), q = 6 (f),(o), and q = Q = 7 (g),(p).

Fig. 12. Example #2—Experiment #3 ( (h)–(p) of the control configurations

N = 32; J = 61;  = 0:75; SLL = 030 dB)—Architecture of the dynamic thinned array when SLL = 030 dB.

Fig. 13. Example #2—Experiment #3 (

grow with the increment of the sidelobe rejection capability. As for the complexity of the antenna (Table II), the number and , of switches of the BFN increases to respectively. These outcomes confirm that the overall system complexity (i.e., number of thinning sequences and number of switches) grows quickly when low sidelobes are required. As a representative example, the results of the third experidB) are then reported. Fig. 11 shows ment (i.e., that the admissible sequences provide nulling regions ranging control sequences from 39% down to almost 13%. The as well as the corresponding radiation patterns are shown in RF switches Fig. 12, while the array structure with is displayed in Fig. 13. Finally, Fig. 14 gives the behavior of the rejection probability. As it can be noticed, the rejection probability is equal to 1 in a few directions, even though multiple sequences can be equivalently exploited to suppress interferences in a wider angular range of directions of arrival. IV. CONCLUSION An innovative strategy for the suppression of undesired signals that exploit dynamically thinned arrays has been presented. The approach is based on the dynamic thin of the array carried

Fig. 14. Example #2—Experiment #3 (N = 32;J = 61; = 0:75;SLL = 030 dB)—Plot of the rejection probability when SLL = 030 dB.

out with a set of adaptively-controlled RF switches connecting the array elements to the BFN. The proposed method has shown being able to: • generate a suitable rejection sidelobe level along the direction of arrival of the undesired signal impinging on the array from whatever direction outside the main beam;

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• modify the “rejection region” very quickly by just switching the RF state within a limited set of precomputed configurations; • deal with various electromagnetic scenarios characterized by different levels of interference suppression; • keep constant the directivity of the main beam in order to avoid undesired amplitude modulations of the desired signal at the receiver; • determine simple antenna structures where only a sub-set of the array elements is connected to the BFN through a RF switch. The results have confirmed the effectiveness of the proposed thinning strategy whose main advantages can be summarized as follows: • unlike classical nulling strategies, the dynamic thinning is more robust to the temperature variations or environmental conditions since the synthesis of sufficient low sidelobe levels is required instead of the synthesis of pattern nulls exactly along the directions of arrival of the undesired signals; • the computational burden is reduced as compared to optimization strategies based on (usually stochastic) iterative methods; • other electromagnetic effects (e.g., mutual coupling) can be easily taken into account for defining the “Control-Sequence” Set . APPENDIX Let be the unknown direction of arrival of a jammer. In order to guarantee the rejection of the jammer, it is necessary for which to synthesize an “equivalent pattern” (10) This is asymptotically enabled by the superposition of patterns generated by different thinned sequences

(11) as shown in the following. Let us consider the addition law of probability

(12) as applied to thinned sequences randomly generated in an independent fashion (13)

when

where since fore,

being

. There-

(14)

The same holds true when ) since

(i.e.,

(15)

REFERENCES [1] C. L. Dolph, “A current distribution for broadside arrays which optimizes the relationship between beam width and sidelobe level,” in Proc. IRE, Jun. 1946, vol. 34, no. 6, pp. 335–348. [2] A. T. Villeneuve, “Taylor patterns for discrete arrays,” IEEE Trans. Antennas Propag., vol. 32, no. 10, pp. 1089–1093, Oct. 1984. [3] T. Isernia, P. Di Iorio, and F. Soldovieri, “An effective approach for the optimal focusing of array fields subject to arbitrary upper bounds,” IEEE Trans. Antennas Propag., vol. 48, no. 12, pp. 1837–1847, Dec. 2000. [4] V. Murino, A. Trucco, and C. S. Regazzoni, “Synthesis of equally spaced arrays by simulated annealing,” IEEE Trans. Signal Process., vol. 44, no. 1, pp. 119–122, Jan. 1996. [5] F. J. Ares-Pena, J. A. Rodriguez-Gonzalez, E. Villanueva-Lopez, and S. R. Rengarajan, “Genetic algorithms in the design and optimization of antenna array patterns,” IEEE Trans. Antennas Propag., vol. 47, no. 3, pp. 506–510, Mar. 1999. [6] D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 771–779, Mar. 2004. [7] E. Rajo-Lglesias and O. Quevedo-Teruel, “Linear array synthesis using an ant-colony-optimization-based algorithm,” IEEE Antennas Propag. Mag., vol. 49, no. 2, pp. 70–79, 2007. [8] Y. Chen, S. Yang, and Z. Nie, “The application of a modified differential evolution strategy to some array pattern synthesis problems,” IEEE Trans. Antennas Propag., vol. 56, no. 7, pp. 1919–1927, Jul. 2008. [9] G. Toso, C. Mangenot, and A. G. Roederer, “Sparse and thinned arrays for multiple beam satellite applications,” in Proc. Eur. Conf. Antennas Propag. (EuCAP 2007), Edinburgh, England, Nov. 11–16, 2007, pp. 1–4. [10] R. J. Mailloux, Phased Array Antenna Handbook, 2nd ed. Norwood, MA: Artech House, 2005. [11] R. L. Haupt, Antenna Arrays: A Computational Approach. Hoboken, NJ: Wiley, 2010. [12] M. I. Skolnik, J. W. Sherman, and F. C. Ogg, “Statistically designed density-tapered arrays,” IEEE Trans. Antennas Propag., vol. 12, no. 4, pp. 408–417, Jul. 1964. [13] T. T. Taylor, “Design of line-source antennas for narrow beam-width and low side lobes,” Trans. IRE Antennas Propag., vol. 3, pp. 16–28, 1955. [14] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 42, no. 7, pp. 993–999, Jul. 1994. [15] J. Nanbo 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. [16] 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, no. 1, pp. 349–352, Dec. 2006.

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[17] S. Caorsi, A. Lommi, A. Massa, and M. Pastorino, “Peak sidelobe level reduction with a hybrid approach based on GAs and difference sets,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 1116–1121, Apr. 2004. [18] M. Donelli, A. Martini, and A. Massa, “A hybrid approach based on PSO and Hadamard difference sets for the synthesis of square thinned arrays,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2491–2495, Aug. 2009. [19] G. Oliveri, M. Donelli, and A. Massa, “Linear array thinning exploiting almost difference sets,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3800–3812, Dec. 2009. [20] G. Oliveri, L. Manica, and A. Massa, “ADS-based guidelines for thinned planar arrays,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 1935–1948, Jun. 2010. [21] H. M. Ibrahim, “Null steering by real-weight control—A method of decoupling the weights,” IEEE Trans. Antennas Propag., vol. 39, no. 11, pp. 1648–1650, Nov. 1991. [22] K. Guney and M. Onay, “Amplitude-only pattern nulling of linear antenna arrays with the use of bees algorithm,” Progr. Electromagn. Res., vol. 70, pp. 21–36, 2007. [23] R. L. Haupt, “Phase-only adaptive nulling with a genetic algorithm,” IEEE Trans. Antennas Propag., vol. 45, no. 6, pp. 1009–1015, Jun. 1997. [24] R. Vescovo, “Reconfigurability and beam scanning with phase-only control for antenna arrays,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1555–1565, Jun. 2008. [25] D. Morgan, “Partially adaptive array techniques,” IEEE Trans. Antennas Propag., vol. 26, no. 6, pp. 823–833, Nov. 1978. [26] R. L. Haupt, “Element selection for partial adaptive nulling,” in Proc. IEEE Int. Symp. Antennas Propag. (APSURSI 2010), Toronto, Canada, Jul. 11–17, 2010, pp. 1–4. [27] H. Steyskal, R. A. Shore, and R. L. Haupt, “Methods for null control and their effects on the radiation pattern,” IEEE Trans. Antennas Propag., vol. 34, no. 3, pp. 404–409, Mar. 1986. [28] J. T. Mayhan, “Thinned array configurations for use with satellitebased adaptive antennas,” IEEE Trans. Antennas Propag., vol. 28, no. 6, pp. 846–856, Nov. 1980. [29] P. Lombardo, R. Cardinali, D. Pastina, M. Bucciarelli, and A. Farina, “Array optimization and adaptive processing for sub-array based thinned arrays,” in Proc. Intern. Conf. on Radar (RADAR 2008), Rome, Italy, May 26–20, 2008, pp. 197–202. [30] P. Rocca and R. L. Haupt, “Dynamic array thinning for adaptive interference cancellation,” in Proc. European Conf. Antennas Propag. (EuCAP 2010), Barcelona, Spain, Apr. 12–16, 2010, pp. 1–3. [31] E. Brookner, “Antenna array fundamentals—Part 1,” in Practical Phased Array Antenna Systems. Norwood, MA: Artech House, 1991. [32] R. L. Haupt, “Interleaved thinned linear arrays,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2858–2864, Sep. 2005. [33] I. J. Gupta and A. K. Ksienski, “Effects of mutual coupling on the performance of adaptive arrays,” IEEE Trans. Antennas Propag., vol. 31, no. 5, pp. 785–791, May 1983. Paolo Rocca (M’09) received the M.S. degree in telecommunications engineering and the Ph.D. degree in information and communication technologies from the University of Trento, Italy, in 2005 and 2008, respectively. He is currently an Assistant Professor in the Department of Information Engineering and Computer Science, University of Trento, and a member of the ELEDIA research group. He has been a visiting student at the Pennsylvania State University and at the University “Mediterranea” of Reggio Calabria.

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His main interests are in the framework of antenna array synthesis and design, electromagnetic inverse scattering, and optimization techniques for electromagnetics. Dr. Rocca has been awarded from the IEEE Geoscience and Remote Sensing Society and the Italy Section with the best Ph.D. thesis award IEEE-GRS Central Italy Chapter.

Randy L. Haupt (F’00) received the B.S.E.E. degree from the United States Air Force Academy, CO (1978), the M.S. degree in engineering management from Western New England College, Springfield, MA, (1982), the M.S.E.E. degree from Northeastern University, Boston, MA (1983), and the Ph.D. degree in EE from The University of Michigan, Ann Arbor (1987). He is an RF Staff Consultant at Ball Aerospace & Technologies Corp., and was Senior Scientist and Department Head at the Applied Research Laboratory of Penn State, Professor and Department Head of ECE at Utah State, Professor and Chair of EE at the University of Nevada Reno, and Professor of EE at the USAF Academy. He was a Project Engineer for the OTH-B radar and a Research Antenna Engineer for Rome Air Development Center early in his career. He is coauthor of the books Practical Genetic Algorithms (2 ed., Wiley, 2004), Genetic Algorithms in Electromagnetics (Wiley, 2007), and Introduction to Adaptive Antennas (SciTech, 2010), as well as author of Antenna Arrays a Computation Approach (Wiley, 2010). Dr. Haupt was the Federal Engineer of the Year in 1993 and is a Fellow of the Applied Computational Electromagnetics Society (ACES). He is a member of the IEEE Antenna Standards Committee, IEEE AP-S Fellows Committee, and ACES Fellows Committee. He serves as an Associate Editor for the “Ethically Speaking” column in the IEEE Antennas and Propagation Magazine.

Andrea Massa (M’03) received the “Laurea” degree in electronic engineering and the Ph.D. degree in electronics and computer science from the University of Genoa, Genoa, Italy, in 1992 and 1996, respectively. From 1997 to 1999, he was an Assistant Professor of electromagnetic fields at the Department of Biophysical and Electronic Engineering, University of Genoa, teaching the university course of Electromagnetic Fields 1. From 2001 to 2004, he was an Associate Professor , University of Trento. Since 2005, he has been a Full Professor of electromagnetic fields at the University of Trento, where he currently teaches electromagnetic fields, inverse scattering techniques, antennas and wireless communications, and optimization techniques. At present, he is the Director of the ELEDIALab at the University of Trento and Deputy Dean of the Faculty of Engineering. His research work since 1992 has been focused on electromagnetic direct and inverse scattering, microwave imaging, optimization techniques, wave propagation in presence of nonlinear media, wireless communications and applications of electromagnetic fields to telecommunications, medicine and biology. Prof. Massa is a member of the PIERS Technical Committee, the Inter-University Research Center for Interactions Between Electromagnetic Fields and Biological Systems (ICEmB), and is the Italian representative in the general assembly of the European Microwave Association (EuMA).

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Connecting Spirals for Wideband Dual Polarization Phased Array Régis Guinvarc’h, Member, IEEE, and Randy L. Haupt, Fellow, IEEE

Abstract—A technique is presented to design dual polarization spiral antenna phased arrays using mono polarization spirals in an alternating configuration. The proposed technique consists of connecting cross polarized neighboring spirals. Thus, the currents in the arms of one spiral flow into the arms of the adjacent spirals. In addition, the currents transmitted to the neighboring spirals radiate the same polarization, as a center fed spiral and an externally fed cross spiral radiate the same polarization. The optimization of the shape of the connection is also presented. The lowest operating frequency of the new array design is 1.8 times lower than an array of isolated spirals while being dual polarized and steerable 30 . Index Terms—Phased arrays, spiral antennas, polarization, wideband array.

I. INTRODUCTION

R

ADAR applications, such as FOliage PENetration (FOPEN), airborne reconnaissance, and synthetic aperture radar mapping require wideband dual polarized phased arrays. An array of spiral antennas is broad band and circularly polarized: right-hand circularly polarized (RHCP) if the spiral winds in the counter clockwise direction and left-hand circularly polarized (LHCP) if the spiral winds in the clockwise direction. As the diameter of a spiral increases, its lowest frequency of operation decreases. As the diameter of the spirals in an array increases, however, the frequency at which grating lobes appear decreases. Interleaving a RHCP array with a LHCP array where every other element is LHCP and the others are RHCP results in very large element spacing and grating lobes in the antenna pattern before the spirals start to radiate. West and Steyskal have described a mono polarization spiral array [1] using square spirals. Guinvarc’h and Haupt [2] recently introduced a technique to build wideband dual polarization spiral arrays, steerable to , using simple mono polarization spirals. They used a Genetic Algorithm (GA) to optimize the positions of the spirals in two interleaved sparse subarrays. The resulting arrays have lower peak sidelobes and delay the introduction of grating lobes due to the nonuniformly spaced elements. Their technique is not applicable to small arrays, because thinning requires many elements in order to break up the periodicity that leads to higher sidelobes. Manuscript received October 23, 2009; revised September 01, 2010, April 28, 2011; accepted May 11, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. R. Guinvarc’h is with SONDRA, Supelec, 91192 Gif-Sur-Yvette France (e-mail: [email protected]). R. L. Haupt is with Haupt Associates, Boulder, CO 80303 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165464

Many authors have worked on miniaturizing the size of the spirals by using inductive, resistive or capacitive loadings [3], [4]. If a spiral can be made smaller for the same bandwidth, then they can be placed closer together in an array which delays the introduction of grating lobes. We are unaware of any research reporting the use of miniaturized spirals in a dual polarized array. This paper presents a technique to greatly improve the bandwidth of a dual polarized array of alternating RHCP-LHCP spirals. The arms of the neighboring (cross polarized) spirals are connected to avoid reflections from the ends of the spiral arms. A similar concept was presented in [5] applied to a mono polarized array to enhance the gain. We will demonstrate that for dual polarization (and only for dual polarization) it helps to increase both the axial ratio (AR) and the voltage standing wave ratio (VSWR). Section II describes the isolated center-fed two-arm spiral used as the array element. Its basic operation is explained in order to understand the design choices made for the new array. Section III presents the basic design of a dual polarized linear spiral array and the idea of connecting adjacent spirals. Finally, Section IV presents results from optimizing the connections between adjacent spirals. II. A SELF COMPLEMENTARY ARCHIMEDEAN SPIRAL ARRAY ELEMENT The array presented in this paper has 28 mm in diameter spiral elements with five turns. The spiral is made from a perfect electric conductor 0.55 mm wide with a gap between two strips that is also 0.55 mm wide (self-complementary). It is center fed with a 1 V source. The spiral is broken into triangles with a max. The spiral is in free space and radiimum edge length of ates out both sides. It is simulated using a commercial method of moment software package FEKO [6]. Its VSWR is below dB starting at 2 starting at 3.4 GHz, while the AR is over 4 GHz (corresponding to a rejection of the cross polarization of more than 15 dB). The lowest frequency in the bandwidth is estimated from [7] (1) with D the diameter of the spiral, which yields 3.4 GHz for this case. The low frequency limit of a spiral is determined by its size: the larger the spiral the larger the bandwidth. At high frequencies, the spiral’s active region is a small circular region about its center as shown by the current intensity on the spiral (see [8] for instance). Its AR and its VSWR are good. As the frequency decreases, the diameter of the active region increases. At lower

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frequencies, the current reaches the end of the arms and is reflected. This reflected current also radiates but in the opposite polarization. This leads to a bad AR. At the lowest frequency, some of the current returns to the generator, leading also to a bad VSWR. In this paper, the axial ratio and voltage standing wave ratio are used to determine the bandwidth of a spiral. However, it is clear from the above explanation that the AR is a stronger criterion. III. DUAL POLARIZATION WIDEBAND LINEAR SPIRAL ARRAYS Designing a wideband spiral array consists of a trade-off between the low frequency limit and the high frequency at which grating lobes appear. The size of the spiral is proportional to its bandwidth, as shown in (1). Thus, increasing the bandwidth forces the element spacing to get larger which in turn causes grating lobes to appear at smaller steering angles. For instance, , the grating lobes appear at: for a beam steered (2) In other words, improving the low limit of the bandwidth means degrading the upper limit. The dual polarized case is worst as spirals of both polarization have to be included in the array. This can be done in an alternating configuration: RHCP, LHCP, RHCP, LHCP, RHCP, LHCP, RHCP, LHCP and so on, which forces the spacing to , the maxdouble. The consequence is, for a beam steered imum frequency becomes (3) This has assumed that the diameter D could be equal to the spacing, which is actually wrong. The real is therefore . The solution is then either to increase larger than or to decrease . Increasing was investigated in [2]. It works well but requires a large number of elements for the thinning to break is usually addressed by deup the periodicity. Decreasing signing a specific spiral. As demonstrated in [9], [2], one way to overcome this problem is to thermically dissipate this current that would otherwise be reflected at the ends. This can be done through the addition of lossy loads for instance. Other solutions exist with other loading (reactive, resistive or inductive [3], [4]). The use of some meander like solutions has also been investigated [10], [11]. All these solutions basically aim at attenuating the currents on the elementary spiral in order to reduce reflections from the ends. The approach presented in this paper is totally different but complementary and can therefore be used with or without the other solutions. A. Solution: Working at the Array Level Instead of working on the design of an isolated spiral element, our approach is to design the spiral in the presence of other spiral elements. As previously explained, the bandwidth limitation is due to the reflection of the current at the ends of the arms of the spirals. Instead of trying to attenuate the current at the ends, our

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Fig. 1. Straight connection between adjacent cross polarized elements.

solution is to connect the arms of adjacent spirals in order to allow the current to flow to neighboring spirals. As the excess current of the fed spiral is no longer reflected at its end, its VSWR and its AR are enhanced. In addition, the cross polarized spiral, with its own feed off, radiates the co-polarization. The reason is that a LHCP/RHCP center fed spiral radiates the same polarization as a RHCP/LHCP end-fed spiral, thanks to their windings. So at low frequencies, without the connections, some of the current is reflected and leads to a bad AR and a bad VSWR. On the contrary, when spirals of opposite polarizations are connected, these currents contributes to radiate the correct polarization. Furthermore, the connections are easy to design when adjacent elements are of opposite polarizations. It should also be noted that, for the reasons explained before, this technique should not be applied to a mono polarized array. B. Example of a Dual Polarized 16-Spiral Array A dual polarized spiral array with 16 standard Archimedean spirals (8 RHCP and 8 LHCP) was simulated with FEKO. The spirals are 28 mm in diameter while the inter-element spacing is 34 mm. The two subarrays (LHCP and RHCP) are anti symmetrical, but as the radiation patterns are symmetric, both radiation patterns are actually symmetric. As explained above, with no connections between spirals, this array has grating lobes starting at 2.94 GHz before the radiation of the spirals should theoretically occur (VSWR lower than 2 starting at 3.4 GHz). The connections are simple straight segments of the same width than the spirals strips, as shown on Fig. 1. 1) RSLL Bandwidth: The grating lobes define the upper limit of the bandwidth. In this case, GHz for a . Consequently, the highest frequency of the beam steered computer models is 3 GHz. 2) VSWR Bandwidth: Fig. 2 shows the VSWR (matched to 200 ohms) for 4 different spiral elements of the array when adjacent elements are not connected. The VSWR at an element is calculated when all 8 elements having the same polarization are excited and the 8 cross-polarized elements are not excited. Elements 1 and 16 are edge elements. Elements with an odd number are fed. Antenna 1 is located on one edge, while antenna 15 is almost at the other edge of the array. Antenna 7 and antenna 11 are not located on the edge. For all antennas, the VSWR is below 2 only above 4.4 GHz, far from the theoretical 3.4 GHz. It is interesting to note that, for a mono polarized array, the VSWR bandwidth starts at 3.06 GHz. So, for a mono polarized array, we observe the classical improvement of the VSWR bandwidth thanks to coupling in the array. But, for a dual polarized array, there is a strong degradation of this VSWR bandwidth (when there is no connection). Fig. 3 is a plot of the VSWR for the dual polarized array when adjacent element arms are connected as shown in Fig. 1. The

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Fig. 2. VSWR (matched to 200 ohms) of different elements of an alternate dual polarized array, without connections.

Fig. 3. VSWR (matched to 200 ohms) of different elements of an alternate dual polarized array, with connections.

VSWR significantly decreased and is now below 2 for almost all of the antenna elements starting at 2.5 GHz. 3) AR Bandwidth: Fig. 4(a) shows the Axial Ratio of this array without connections and Fig. 4(b) with connections for four main beam steering angles. The array does not function without the connections. With the connections, the array has a very narrow bandwidth limited by the AR. 4) Summary of the Bandwidths: The criteria are a VSWR less than 2, a RSLL greater than 10 dB, and an AR less than 3 dB. The results for a mono polarized array appear in Fig. 5. The bandwidth is 3.6–5.8 GHz (from (3) because the spacing is here half of the spacing for the dual polarization case). A dual polarized array, without connections, has its RSLL upper limit cut in half and the AR bandwidth is increased (Fig. 6). As a consequence, there is no longer a frequency band where the three criteria are simultaneously met. However, the addition of the connections between neighboring spirals decreases the lower limits of the AR and of the VSWR bandwidth. We then have a dual polarized array functioning between 2.72 GHz and 2.9 GHz, as shown on Fig. 7. Although the bandwidth is small, it is obtained with a simple straight segment as the connection. In

Fig. 4. Comparison of the Axial Ratio of a dual polarized 16-spiral array whether it is connected or not. (a) Axial Ratio when not connected (b) Axial Ratio when connected.

Fig. 5. Sum up of the bandwidth for a mono polarized array, without connections.

Fig. 6. Sum up of the bandwidth for a dual polarized array, without connections.

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Fig. 7. Sum up of the bandwidth for a dual polarized array, with straight connections.

Fig. 9. Triangular and optimized shapes of the connection. (a) Triangular shape, (b) optimized shape. TABLE I PARAMETERS OF THE CONNECTION TO BE OPTIMIZED

Fig. 8. Generic shape of the connection to be optimized.

the next section, the optimization of the shape of the connection is presented in order to extend the bandwidth. IV. OPTIMIZATION OF THE CONNECTION SHAPE A. Optimization Setup The previous section demonstrated that connecting the spirals helps to improve the AR and the VSWR bandwidths. In addition, the upper limit is clearly set by the grating lobes at when the array is in this alternating dual polarization configuration. The goal of optimizing the shape of the connections must therefore focus on frequencies below . The FEKO optimization tool is used to perform the optimization of the connection shape, with a genetic algorithm. We experimented with several different connecting shapes and optimization variables. A first trial without optimization used the triangular shape in Fig. 9(a) and demonstrated better performance than the straight connection. But the addition of a second triangle led to worse results. We therefore chose to use the generic shape defined in Fig. 8. The corresponding parameters to be optimized are listed in Table I, along with their limits. The idea is to keep a roughly circular shape to have smooth transitions. In addition the connections should not be too large, for two reasons. First, we must minimize the radiation of the connections, as its polarization will be roughly linear and, as a consequence, maximize the radiation by the neighboring spirals, with the circular polarization of interest. Second, it is always of interest to keep the array as small as possible. dB, corresponding to The design criterion is an AR above a rejection of the cross polarization of more than 15 dB. From our experience, if the AR objective is met, then the VSWR objective is also met.

B. Results The optimization bandwidth is over a 2–3 GHz frequency range with a 100 MHz frequency step. The array has only 5-spirals in order to keep the simulation time as reasonable as possible. Since there are an odd number of spirals, one polarization has 3 spirals and the other has 2. Consequently, the 2 arrays have different radiation patterns. The optimum shape is shown in Fig. 9(b). We first have a transition region to connect to a larger strip, so that the current easily flows from the spiral arm to the connection of the spiral. The middle of this strip narrows in order to impede the current flow. In this way, the current flowing into the transition region of the cross polarized spiral is limited. Otherwise, the current reflects into this transition region. Fig. 10 shows the AR clearly improved with the addition of the straight connections, by lowering the cutoff frequency at 2.65 GHz instead of 2.72 GHz, so an AR bandwidth of 250 MHz instead of 180 MHz. There is almost no effect on the VSWR. 1) Current and Symmetry: As shown on Fig. 11, the current flows from a RHCP spiral arm into a LHCP spiral arm into the neighboring spirals. It is also interesting to note the evolution of the symmetry: the current on a spiral is symmetric with respect to the center of the spiral for any frequency. The current on the spiral on the left of Fig. 11 is not symmetric. Instead, as we are now working at the array level, this is the current on the entire array which is now symmetric with respect to the center spiral. C. A Further Improvement The addition of lossy loads further improved the connection by dissipating the current. This is similar to what is presented in [9], [12] for instance. In the present case, the resistance is

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Fig. 12. Sum up of the bandwidth for a dual polarized array, with optimized connections.

connections. Adding more loads to the connection could improve the results, but it would also lower the gain. D. Larger Antisymmetrical Array

Fig. 10. Optimized shape with or without one load per connection compared to the straight connection for a 5-spiral array. (a) Comparison of the Axial Ratio at broadside, (b) comparison of the VSWR.

Fig. 11. Current on a 5-element array when elements 1, 3, and 5 are excited for RHCP.

placed in the middle of the connection, and not on the spirals. Optimizing the 5-spiral array with a new parameter, the value (real) of the impedance obtained is 88 . Fig. 10 shows the new AR bandwidth starts at 2.2 GHz (exdB, so a rejection of cept from 2.3 to 2.35 GHz where it is the cross polarization of 13.5 dB). The lower frequency bound is 2.72 GHz before optimization and 3.4 GHz for the isolated spiral (1.5 times less). The associated bandwidth is given in Fig. 12. It has to be noted that, despite the addition of the loads, the broadside copolarized gain is still over 5 dB over the whole bandwidth. Compared to separated spirals, there is no loss of copolarized gain as the loads are not on the spirals but on the

Since the 5-spiral array does not have its RHCP subarray identical to its LHCP subarray, their radiation patterns are slightly different. The optimized loaded connection is thus applied to the same 16-element array used previously. An additional load is put between each spiral and the corresponding connection to suppress the small decrease of the AR at 2.3 GHz. We thus have three loads per connections. The radiation pattern . The corresponding AR is computed when steered up to and VSWR are shown in Fig. 13. dB The VSWR is as good as usual. The AR is above starting at 2.2 GHz, for the three steering angle, cf. Fig. 13(a). It has to be noted that the AR increases as the steering angle increases. The effect of the loads on the copolarized gain is small. The gain is over 8 dB from 2.2 GHz and above 10 dB from 2.5 GHz (with 8 spirals fed). It is actually interesting to note that the co polarized gain is higher with this connection with three loads than with the connection with one load. The next paragraph will explain this. 1) A Last Word on the Currents: The initial assumption of this paper is that it is better to let the currents flow onto the neighboring (cross polarized) arms than having them reflected from the end of the arms. Fig. 14 shows the currents at 2.3 GHz for different connections. The cross polarized spiral with the most currents is the one with the straight connection, cf. Fig. 14(a). However, for that connection at that frequency as can be seen on Fig. 3 and 4(b), both the AR and the VSWR are not good. In this case the currents are strong enough to reach the center of the cross polarized spirals and continue on the other arm of the cross polarized, radiating then the cross polarization. They even continue on the next co polarized spiral and reach its generator, leading to a bad VSWR. The case in which less currents flow into the cross polarized spiral is with the optimized connections with three loads, see Fig. 14(c). For the latter case, both the AR and the VSWR are actually good. The second case, cf. Fig. 14(b), is an intermediate case where the VSWR is good but not the AR. This means that the currents do not reach the generator of the next co polarized spiral but reach the second arm of the cross polarized spiral. In order to properly design a connected dual polarization alternate array, one has to ensure that an optimum amount of current flows to the adjacent cross polarized spiral.

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Fig. 14. Zoom on the currents at 2.3 GHz of a cross polarized spiral (a) with a straight connections (b) with an optimized connections and one load (blue dot) per connection (c) with an optimized connection and three loads (blue dots) per connection.

Fig. 13. Optimized loaded connections on a 16-spiral array. (a) Axial Ratio with the optimized connection, (b) VSWR with the optimized connection, (c) gain and rejection of the cross polarization with the optimized connection.

E. Effect on the Feeding We now look at the effect of these connections on the feeding, if any. Feeding an Archimedean spiral requires both a balanceto-unbalance device and an impedance matching circuit. We have chosen to use a balun transformer, as introduced by Duncan

[13]. It is a transition from a coaxial line to a balanced, two-conductor line using a continuous taper described by Klopfenstein [14]. It covers a broad bandwidth (nearly 100:1). Its disadvanat the lowest tage is the length of the transition: it can be frequency of operation. For our case study, this large size could lead to additional interactions (with the connections). We have designed the balun to go from 50 ohms to 220 ohms. It starts working at 1.2 GHz with a reflection coefficient less than dB. We have fed a 6-element array (a 16-element array would required too many memory), without connection and with the optimized connections. Three spirals are fed (the second, the fourth and the sixth) with the feeds being waveguide port at the input of the baluns. The VSWR is plotted in Fig. 16, for 50 ohms (thanks to the balun). The three dashed lines plot the VSWR of the three fed

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Fig. 17. Gain of a 6-element array including the feeding structures, with and without connections. The plain lines are for the connected cases, the dashed lines for the disconnected cases. No marker means LHCP and diamond markers are for RHCP.

Fig. 15. View of two connected spirals with their baluns. The top left spiral is fed at the input of the balun (indicated by a red ring on the figure). The bottom right spiral is not fed. The load in the middle of the connection is in blue.

V. CONCLUSION This paper has demonstrated a new technique to enhance the VSWR and the AR bandwidths of a dual polarization spiral array by simply connecting together the neighboring spirals of an alternating array. The resulting array starts to radiate at a frequency 1.8 times lower (considering either the VSWR bandwidth or the AR bandwidth) than an isolated spiral, while being . dual polarized and steerable Thanks to the connections, the currents can flow into the neighboring (cross polarized) spirals instead of being reflected and will radiate the same polarization. We have also demonstrated that the currents must be attenuated when going to the neighboring spirals. It must be highlighted that this paper has focused on simple standard Archimedean spirals but the connections can be used in combination with any other techniques usually applied on spirals for extending the bandwidth (loading, meandering ). REFERENCES

Fig. 16. VSWR for 50 ohms of a 6-element array including the feeding structures, with and without connections. The plain lines are for the connected cases, the dashed lines for the disconnected cases. No marker means the first fed element (second spiral), diamond markers are for the center element (spiral number 4) and the star markers indicates the edge element (spiral number 6).

spirals when they are not connected (similar to Fig. 2). The three plain lines indicates the three fed spirals for the connected case. The improvement is obvious. One can also note that the baluns function efficiently. Without the connections, the array actually radiates the wrong polarization (as the VSWR is very bad), cf. Fig. 17. When we add the connections, the spirals have a good co polarization gain and a good rejection of the cross polarization. The latter is not as good as for the case without the balun, especially between 2.25 and 2.65 GHz. This is obviously due to the radiation of the baluns. However, despite the effect of the baluns, the connections improve both the VSWR and the gain.

[1] J. West and H. Steyskal, “Analysis and feeding of a spiral element used in a planar array,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 1931–1935, July 2009. [2] R. Guinvarc’h and R. L. Haupt, “Dual polarization interleaved spiral antenna phased array with an octave bandwidth,” IEEE Trans. Antennas Propag., To appear in. [3] M. Nurnberger and J. Volakis, “New termination for ultrawide-band slot spirals,” IEEE Trans. Antennas Propag., vol. 50, no. 1, pp. 82–85, Jan. 2002. [4] Z. Zhang and J. T. Bernhard, “Two-arm archimedean spiral with filter based reactive loading,” in Proc. IEEE Antennas Propagation Society Int. Symp., Jul. 2006, pp. 3677–3680. [5] R. J. Barton, P. J. Collins, P. E. Crittenden, M. J. Havrilla, and A. J. Terzuoli, “A compact passive broadband hexagonal spiral antenna array,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2007, pp. 4401–4404. [6] Feko [Online]. Available: www.feko.info [7] J. Kraus and R. Marhefka, Antennas for All Applications. New York: McGraw Hill, 2002. [8] R. L. Haupt, Antenna Arrays: A Computational Approach. Hoboken, NJ: Wiley, Apr. 2010. [9] T. T. Wu and R. King, “The cylindrical antenna with nonreflecting resistive loading,” IEEE Trans. Antennas Propag., pp. 369–373, May 1965.

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[10] H. Nakano, “A meander spiral antenna,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2004, vol. 3, pp. 2243–2246. [11] D. Filipovic and J. Volakis, “Broadband meanderline slot spiral antenna,” in IEE Proc. Microwaves, Antennas and Propagation, Apr. 2002, vol. 149, no. 2, pp. 98–105. [12] K. Louertani, R. Guinvarc’h, N. Ribiere-Tharaud, and M. Darces, “Design of a spiral antenna with coplanar feeding solution,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 2008, pp. 1–4. [13] J. Duncan and V. Minerva, “100:1 bandwidth balun transformer,” Proc. IRE, vol. 48, no. 2, pp. 156–164, 1960. [14] R. Klopfenstein, “A transmission line taper of improved design,” Proc. IRE, vol. 44, no. 1, pp. 31–35, 1956.

Régis Guinvarc’h (S’02–M’04) received the B.S. and M.S. degrees in 2000 and the Ph.D. degree (with a Conventions Industrielles de Formation par la Recherche (CIFRE) grant) in 2003, all in electrical engineering from the Institut National des Sciences Appliques (INSA), Rennes, France. From 2000 to 2003, he was with the Etienne Lacroix company, France, as a Research Engineer, where he was engaged in research on microwave remote sensing through discrete random media. Since 2004, he is an Associate Professor at Supelec in the Supelec ONERA NUS DSTA Research Alliance (SONDRA), Gif-Sur-Yvette, France, where he is working on antennas and HF surface wave radar.

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Randy L. Haupt (M’82–SM’90–F’00) received the B.S.E.E. degree from the USAF Academy, CO, in 1978, the M.S. degree in engineering management from Western New England College, Springfield, MA, in 1982, the M.S.E.E. from Northeastern University, Boston, MA, in 1983, and the Ph.D. degree in electrical engineering from The University of Michigan, Ann Arbor, in 1987. He was Senior Scientist and Department Head at the Applied Research Laboratory, Pennsylvania State University, Professor and Department Head of ECE at Utah State, Professor and Chair of EE at the University of Nevada Reno, and Professor of EE at the USAF Academy. He was a Project Engineer for the OTH-B radar and a Research Antenna Engineer for Rome Air Development Center early in his career. He is coauthor of the books Practical Genetic Algorithms (2nd ed., Wiley, 2004), Genetic Algorithms in Electromagnetics (Wiley, 2007), and Introduction to Adaptive Antennas (SciTech, 2010), as well as author of Antenna Arrays a Computation Approach (Wiley, 2010). Dr. Haupt was the Federal Engineer of the Year in 1993 and is a Fellow of the Applied Computational Electromagnetics Society (ACES). He is a member of the IEEE Antenna Standards Committee, IEEE AP-S Fellows Committee, and ACES Fellows Committee. He serves as an Associate Editor for the “Ethically Speaking” column in the IEEE AP-S Magazine.

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Wideband Planar Microwave Lenses Using Sub-Wavelength Spatial Phase Shifters Mudar A. Al-Joumayly, Student Member, IEEE, and Nader Behdad, Member, IEEE

Abstract—We present a new technique for designing low-profile planar microwave lenses. The proposed lenses consist of numerous miniature spatial phase shifters distributed over a planar surface. The topology of each spatial phase shifter (SPS) is based on the design of a class of bandpass frequency selective surfaces composed entirely of sub-wavelength, non-resonant periodic structures. A procedure for designing the proposed lenses and their constituting spatial phase shifters is also presented in the paper. This design procedure is applied to two different planar lenses operating at X-band. Each lens is a low-profile structure with an overall thickness of 0 08 0 and uses sub-wavelength SPSs with dimensions of 0 2 0 0 2 0 , where 0 is the free-space wavelength at 10 GHz. These prototypes are fabricated and experimentally characterized using a free-space measurement system and the results are reported in the paper. The fabricated prototypes demonstrate relatively wide bandwidths of approximately 20%. Furthermore, the lenses demonstrate stable responses when illuminated under oblique angles of incidence. This feature is of practical importance if these lenses are to be used in beam-scanning antenna applications. Index Terms—Frequency selective surfaces, lens antennas, lenses, periodic structures.

I. INTRODUCTION ENSES are used extensively throughout the microwave and millimeter-wave frequency bands in applications such as imaging [1], radar systems [2], quasi-optical power combining [3], [4], quasi-optical measurement systems [5], and high-gain beam-steerable antenna arrays [6], [7]. Dielectric lenses were among the first structures to be investigated and are still used in various antenna and radar applications [8]–[10]. Dielectric lenses, similar to the one shown in Fig. 1(a), tend to operate over relatively wide bandwidths. However, they suffer from reflection losses caused by mismatch between the refractive index of the lens material and that of its surrounding environment. Moreover, dielectric lenses that operate at low

L

Manuscript received March 01, 2011; revised May 07, 2011; accepted June 21, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. This work was supported in part by a Young Investigator Program (YIP) Award from the Air Force Office of Scientific Research (Award No. FA9550-11-1-0050) and in part by the National Science Foundation (Awards ECCS-1101146 and ECCS-1052628). M. Al-Joumayly was with the Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, WI 53706 USA. He is now with TriQuint Semiconductor, Apopka, FL 32703 USA (e-mail: [email protected]). N. Behdad is with the Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, WI 53706 USA. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165515

Fig. 1. (a) Topology of a conventional double-convex dielectric lens. (b) A planar microwave lens composed of arrays of transmitting and receiving antennas connected to each other using transmission lines with variable lengths. (c) Topology of the proposed MEFSS-based planar lens.

microwave frequencies are generally bulky, heavy, and expensive to manufacture. This has limited their use primarily to millimeter-wave and sub-millimeter-wave frequency bands. In addition to dielectric lenses, a wide range of other techniques have also been used to design lenses that operate throughout the RF and microwave frequencies [11]–[23]. In particular, planar microwave lenses have received considerable attention and have been investigated by several groups of researchers over the years [3], [6], [13]–[17]. Typically, these types of microwave lenses are designed using planar arrays of transmit and receive antennas connected together using a phase shifting mechanism as depicted in Fig. 1(b). Different techniques have been used to achieve the required transmission phase between the arrays of receiving and transmitting antennas. Examples include using transmission lines with variable lengths between the two arrays of antennas [15] or using an array of coupling apertures between them [13]–[17]. In [16], two microstrip patch antennas coupled together using an aperture embedded in their common ground plane form the basic building block of a flat lens. A similar structure is studied in [17], where a resonant aperture (filter) is used to couple the arrays of receiving and transmitting microstrip patch antennas together. However, the relatively large inter-element spacing of such a planar array limits its performance under oblique angles of incidence. As the element size and the inter-element spacing increase, the phase responses of the lens’ spatial phase shifters become more sensitive to any change in the angle of incidence of the electromagnetic (EM) wave [24]. Therefore, the phase-delay profile of the lens and its focusing characteristics can change considerably as the angle of incidence changes. Recently, artificially engineered materials have also been used to design microwave lenses [18]–[20]. In [18], a lens

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AL-JOUMAYLY AND BEHDAD: WIDEBAND PLANAR MICROWAVE LENSES USING SUB-WAVELENGTH SPATIAL PHASE SHIFTERS

consisting of stacked parallel metal plates or arrays of parallel metal wires is reported. This structure behaves as an artificial dielectric with a permittivity less than one. Other artificially engineered materials, such as complementary split ring resonators [19] or negative-refractive-index transmission lines [20] have also been used to design three-dimensional microwave lenses operating at microwave frequencies. In another approach, Fresnel zone plates (FZP) are employed as focusing elements to design a variety of microwave lenses. The classical microwave FZP lens consists of circular concentric metal rings that lay over the odd or even Fresnel zones [21], [22]. Frequency selective surfaces (FSSs) have also been used in the design of planar lenses. In [23], a combination of a Fresnel zone plate lens and an FSS is employed to achieve a new compound diffractive FZP-FSS lens with enhanced focusing and frequency filtering characteristics. In FSS-based lenses, the phase shift required for beam collimation is achieved from the phase response of the FSS’s transfer function. The phase shift that can be achieved from a bandpass FSS is directly related to the type and order of its frequency response. Since many planar lenses require phase shifts in the range of 0 to 360 , FSSs with higher-order responses are generally required. To achieve a higher-order filter response from a conventional FSS, multiple first-order FSSs must be cascaded with a spacing of a quarterwavelength between each panel [24]. This increases the overall thickness of the FSS and enhances the sensitivity of its frequency response to the angle and polarization of incidence of the EM wave. Recently, the authors presented a new class of low-profile frequency selective surfaces that are composed entirely of non-resonant periodic structures [25], [26]. These structures, which are referred to as miniaturized element frequency selective surfaces (MEFSSs), are composed of multiple closely spaced periodic structures and can provide filter responses of any desired order [26]. This way, a high-order filter response can be achieved from a low-profile, light-weight FSS. Additionally, MEFSSs demonstrate stable phase and amplitude responses as a function of angle and polarization of the incidence of the EM wave [25]–[28]. In this paper, we use the generalized MEFSS reported in [26] to design low-profile, planar microwave lenses. In doing this, we treat the unit cells of a typical MEFSS as spatial phase shifters (SPSs). By populating a planar surface with SPSs that provide different phase shifts, planar lenses with a desired phase shift profile can be designed as depicted in Fig. 1(c). Because of their extremely thin profiles and sub-wavelength dimensions, the phase responses of the SPSs constituting the lens is extremely stable as a function of the angle of incidence of the EM wave. This way, the phase-delay profile provided by these lenses and their responses remain stable over a relatively wide angular range. Moreover, these planar lenses can operate over relatively wide bandwidths. In what follows, we will first present the principles of operation and the design procedure for the proposed planar lenses in Section II. In Section III, we discuss the measurement techniques used to characterize the proposed planar lenses and present the measurement results of two fabricated prototypes. Finally, in Section IV, we present a few concluding remarks.

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Fig. 2. Phase delay of a spherical wave at the input of the lens aperture (referenced to the phase at the center of the aperture) and the phase delay, which must be provided by the lens to achieve a planar wavefront at the output aperture of the lens. The results are calculated for a lens with aperture size of 23.4 cm 18.6 cm and focal distance of 30 cm.

2

II. LENS DESIGN AND PRINCIPLES OF OPERATION Fig. 1 shows a comparison between a double-convex dielectric lens, a traditional planar lens composed of arrays of receiving antennas connected to arrays of transmitting antennas using transmission lines with variable lengths, and the proposed MEFSS-based planar lens. Irrespective of the implementation technique, the planar microwave lenses shown in Fig. 1(b) and (c) attempt to mimic the transmission phase profile of the double-convex dielectric lens shown in Fig. 1(a). When illuminated by a point source located at its focal point, the double-convex lens converts an incident spherical wavefront at its input surface to a planar wavefront at its output. For a radiating point source located at the lens’ focal point, the radiated spherical wave arrives at different points on the lens’ aperture with different phase delays. In this case, the ray that passes through the center of the lens experiences the smallest phase delay in free space. Using this phase delay as a reference, the excess free-space phase delay that a ray arriving at an arbitrary point on the aperture acquires can be calculated. Fig. 2 shows this calculated excess phase delay profile for a lens with an cm having a focal length of aperture size of cm. From this curve, the phase delay profile that the lens must provide to achieve beam collimation can be calculated. This phase delay changes from zero at the edge of the aperture to 1 at its center, where is is the free space wavelength, the free space wavenumber, is the focal length, and is the aperture size of the lens. Fig. 2 also shows the phase delay that the lens must provide at every point on its aperture to collimate the input spherical wavefront. In a conventional convex lens (Fig. 1(a)), this phase shift is provided in a continuous fashion by the lens. In particular, the surface profiles of the two curved surfaces determine the lens’ phase transmission function. In planar lenses, however, there is no flexibility in choosing the surface profiles of the input and output apertures of the lens. Therefore, alternative phase shifting mechanisms are usually used. In the next two subsections, we discuss a method for achieving the desired phase

0

1Notice that ( (A=2) + f f ) is the difference between the path length of a ray that passes through the focal point and the edge of the lens and one that passes through the lens’ focal point and the center of the aperture.

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Fig. 4. Simulated transmission coefficients (magnitude and phase) of three MEFSSs having second-, third-, and fourth-order bandpass responses. Simulation results are obtained in CST Microwave Studio. As the order of the MEFSS response increases, the phase of the transmission coefficient changes over a wider range within the pass band of the MEFSS.

N

Fig. 3. (a) 3D topology of an th-order MEFSS reported in [26]. (b) 2D top view of the unit cell of a sub-wavelength capacitive patch layer (top) and an inductive wire grid layer (bottom). The two layers are used as building blocks for the th-order MEFSS. (c) Equivalent transmission line circuit model of the th-order MEFSS [26].

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shift by synthesizing sub-wavelength spatial phase shifters that accomplish this task. A. Miniaturized-Element Frequency Selective Surfaces as Spatial Phase Shifters Fig. 3(a) shows the three-dimensional (3-D) topology of a generalized MEFSS with an th-order bandpass response [26]. The MEFSS is a multi-layered structure composed of a number of closely spaced metallic layers separated from one another by very thin dielectric substrates. Each metallic layer is in the form of a two-dimensional (2-D) periodic arrangement of sub-wavelength capacitive patches or a 2-D wire grid with sub-wavelength periodicity. The unit cells of a sub-wavelength capacitive patch and a wire grid are shown in Fig. 3(b). An th-order bandpass MEFSS has capacitive layers and inductive layers. The first and last layers are always composed of sub-wavelength capacitive patches while the interior layers alternate between capacitive and inductive surfaces. Fig. 3(c) shows the equivalent circuit model of this structure. A comprehensive analytical design and synthesis procedure for this structure is also reported in [26]. Using this method, by specifying the center frequency of operation of the MEFSS, , its operational bandwidth, BW, and its response type and order, all of the element values of the equivalent circuit model shown in Fig. 3(c) can be determined and mapped to the geometrical parameters of the MEFSS shown in Fig. 3(a). Compared to regular FSSs with the same response type and order, MEFSSs have significantly lower overall thicknesses and smaller periodicities [26]. Moreover, they demonstrate very stable frequency responses as functions of the incidence angle and the polarization of the incident wave [26]. Within its pass band, a bandpass MEFSS allows the signal to pass with little attenuation. However, the transmitted signal experiences a phase shift, which can be determined by the phase

of the MEFSS’s transmission coefficient. Therefore, within its pass band, a bandpass MEFSS acts as a phase shifting surface (PSS). The unit cell of such an MEFSS is the smallest building block constituting the PSS. This way, each unit cell acts as a spatial phase shifter. SPSs providing different phase shifts can be used to populate the aperture of a lens and provide the phase shift profile shown in Fig. 1(c). In an MEFSS, different phase shifts can be achieved at a single frequency by de-tuning the center frequency of operation of the MEFSS to control the transmission phase. This can be seen by examining the magnitudes and phases of the transmission coefficients of several MEFSSs shown in Fig. 4. Within their pass bands, the transmission phases of all three MEFSSs change with frequency. The third-order at 9 GHz MEFSS, for example, provides a phase shift of at 11 GHz. If the center frequency and a phase shift of of operation of this MEFSS is tuned, both the magnitude and the phase responses shift with frequency. Therefore, for a fixed frequency that falls within the pass band of both structures, only a change in phase is observed. This method can be used to achieve MEFSSs that provide different transmission phase shifts at a single frequency. The maximum phase shift that can be achieved from an MEFSS depends on the order of its filter response. As seen from Fig. 4, the higher the order of the response is, the larger the phase shift range will be. For example, the second-order MEFSS whose response is shown in Fig. 4 provides a maximum phase variation of 180 within its pass band. This phase shift range increases to 270 and 360 for the third-order and fourth-order MEFSSs, respectively. Therefore, the desired phase shift profile of a planar lens determined from its aperture size, , and focal distance, , determines the order of the MEFSSs that need to be used to synthesize the required spatial phase shifters. B. Lens Design Procedure Let’s assume that the planar microwave lens, shown in Fig. 5, is located in the - plane and has a rectangular aperture with aperture dimensions of and . A point source located at the radiates a spherlens’ focal point at ical wavefront that impinges upon the surface of the lens and is transformed to a planar wavefront at the output aperture. At the

AL-JOUMAYLY AND BEHDAD: WIDEBAND PLANAR MICROWAVE LENSES USING SUB-WAVELENGTH SPATIAL PHASE SHIFTERS

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Fig. 6. Topology of the two lens prototypes described in Section II. The lens aperture is divided into a number of concentric zones populated with identical spatial phase shifters (a) first prototype with five concentric zones. (b) Second prototype with ten zones. Fig. 5. Top view of the proposed planar MEFSS-based lens. A spherical wave is launched from a point source located at the focal point of the lens, (x = 0; y = 0; z = f ). To transform this input spherical wavefront to an output planar one, k r + 8(x; y ) must be a constant for every point on the aperture of the lens. 8(x; y ) is the phase delay, which is provided by the lens.

0

lens’ input surface, the input spherical wave can be expressed as: (1) where is the amplitude of the electric field of plane and the incident spherical wave on the is the distance between an arbitrary point on the lens’ aperture specified by its coordinates and the focal point of the lens . The electric field distribution at the output aperture of the lens can be expressed as: (2) where is the amplitude of the electric field over is the phase delay the output aperture of the lens and . provided by the spatial phase shifters of the lens at point To ensure that the output aperture of the lens represents an must be a constant. equiphase surface, the term can be calculated as: Consequently, (3) where is a positive constant that represents a constant phase delay added to the response of every SPS on the aperture of the conlens. Let’s assume that the lens’ aperture is divided into centric zones populated with spatial phase shifters of the same type. If the coordinates of a point located at the center of zone are given by (where ), then the desired transmission phase required from SPSs that populate this zone can be calculated from:

(4)

Having determined the transmission phase for each zone, the lens design procedure can be summarized as follows. 1) Select the desired center frequency of operation of the lens, . 2) Select the desired aperture shape and its dimensions. For a and values rectangular aperture, choose appropriate and the focal distance, . Notice that these parameters can be chosen at the discretion of the designer. 3) Divide the lens aperture into concentric discrete regions is an arbitrary positive integer. or zones, where 4) Determine the transmission phase delay profile of the lens by calculating the phase delay required from SPSs populating each zone using (4). All calculations should be performed at the desired center frequency of operation of . over the 5) Depending on the maximum variation of lens’ aperture, determine the order of the MEFSS needed to implement the spatial phase shifters. For example, Fig. 4 shows that a fourth-order frequency response would be sufficient if a phase variation of more than 270 but less than 360 is needed. 6) Use the synthesis procedure described in [26] to design the spatial phase shifter that populates the central zone of the lens. This SPS should provide a phase delay larger than at the desired frequency of oper. From the phase response of this SPS at , ation, can be determined. is the difference the value of between the actual phase delay provided by the SPS and . 7) Use the synthesis procedure described in [26] to design the spatial phase shifters that populate zone . These SPSs should provide the required phase shift calculated from (4). In doing this, the MEFSS designed for Zone 0 can be used as a starting point and its frequency response can be shifted towards higher frequencies. This decreases the phase shift . provided by the structure at We applied the aforementioned procedure to design two planar lens prototypes shown in Fig. 6. The lenses are designed to operate at 10 GHz and both have focal lengths of cm and aperture dimensions of cm and

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TABLE I PHYSICAL AND ELECTRICAL PROPERTIES OF THE SPATIAL PHASE SHIFTERS THAT POPULATE EACH ZONE OF THE FIRST PROTOTYPE. INSERTION LOSS VALUES ARE w : mm AND h h h h : mm IN dB AND ALL PHYSICAL DIMENSIONS ARE IN m. FOR ALL OF THESE DESIGNS, w

=

=24

=

=

=

=05

TABLE II PHYSICAL AND ELECTRICAL PROPERTIES OF THE SPATIAL PHASE SHIFTERS THAT POPULATE EACH ZONE OF THE SECOND LENS PROTOTYPE. INSERTION LOSS VALUES ARE IN dB AND ALL PHYSICAL DIMENSIONS ARE IN m. FOR ALL OF THESE DESIGNS, w w : mm AND h h h h : mm

=

cm. With these specifications, the maximum phase variation required over the aperture of the lens is 169 along the direction and 264 along the direction. The desired phase profile of the lens, along the axis is shown in Fig. 2. As can be seen from Fig. 4, this phase shift range can be achieved from a third-order MEFSS. The unit cell of a third-order MEFSS consists of three capacitive and two inductive layers separated from one another by very thin dielectric substrates. Typical , where is the substrate thicknesses are approximately [26]. free space wavelength at The spatial phase shifters used in this lens are designed based on the assumption that they will operate in an infinite two-dimensional periodic structure. This assumption is also used in the design of reflectarrays and is referred to as the local periodicity condition [29]. However, the proposed lens is inherently non-periodic, since it uses a number of different SPSs over its aperture. Therefore, the frequency responses of the SPSs used in the design of the lens are expected to change when placed in this operational environment. In our first proof-of-concept experiments, we considered two lens prototypes with different number of zones. In the first prototype, the lens’ aperture is divided into five discrete zones, as shown in Fig. 6(a) and in the second prototype the aperture is divided into 10 zones as seen in Fig. 6(b). Fig. 7 shows the detailed layout of the two lens prototypes and the arrangement of the spatial phase shifters that populate different zones of each lens. Each square cell in these figures represents one SPS with physical dimensions of 6.1 mm 6.1 mm. The numbers on each rectangular cell represent the zone in which the SPS is located. For a lens with only a few discrete zones, the structure can be considered to be locally periodic in each zone. Therefore, the phase responses of SPSs that populate each zone are expected to be closer to the ideal case (infinite periodic structure). Such a lens, however, does not provide a smooth phase delay profile over its aperture. On the other

=24

=

=

=

=05

Fig. 7. Layouts of the two lens prototypes designed using the proposed design procedure discussed in Section II.B. The lens aperture is discretized into small area pixels. Each pixel is occupied by a spatial phase shifter with physical dimensions of 6.1 mm 6.1 mm designed using the design procedures discussed in Section II.B. (a) Layout of the first prototype, where the lens aperture is divided into five discrete zones. The zones are numbered from 0–4. The specifications of the spatial phase shifters that populate each zone are listed in Table I. (b) Layout of the second prototype, where the lens aperture is divided into ten discrete zones. The zones are numbered from 0–9. The specifications of the spatial phase shifters that populate each zone are listed in Table II.

2

hand, a lens with a higher number of zones can provide a smooth phase variation over its aperture. However, since fewer SPSs of the same type are used within each zone, the uncertainty in their actual frequency response could be higher. The two lens prototypes shown in Fig. 7(a) and (b) are representative examples of these two scenarios and were chosen to experimentally study these potential tradeoffs. For each lens, following the aforementioned design procedure, the required transmission phase shift is first calculated for each zone. Then, the spatial phase shifter (MEFSS unit cell) that populates Zone 0 of the lens (Z0-SPS) is designed. The structure is then simulated using full-wave EM simulations in CST Microwave Studio. In doing this, the MEFSS unit cell is simulated as part of an infinite periodic structure by placing it inside a waveguide with periodic boundary condition (PBC) walls.

AL-JOUMAYLY AND BEHDAD: WIDEBAND PLANAR MICROWAVE LENSES USING SUB-WAVELENGTH SPATIAL PHASE SHIFTERS

The structure is then excited by a plane wave and the magnitude and phase of its transmission coefficient are calculated. The physical and geometrical dimensions of the Z0-SPS are used as a reference for designing the SPSs that populate other zones of the lens. Since the SPSs of each zone must have different relative transmission phase, they are designed by slightly de-tuning the frequency response of the Z0-SPS. Doing this shifts both the magnitude and the phase of the transfer function of the MEFSS response in frequency. Therefore, for a given fixed frequency, a relative phase shift can be achieved. In a third-order MEFSS, this de-tuning can be achieved by changing the size of the three capacitive patch layers without changing the inductive layers. This facilitates the design of the lens by reducing the number of variables that must be changed from one SPS to the other. Having achieved the desired relative transmission phases, the spatial phase shifters are then populated over their corresponding zones. In general, the frequency and phase responses of each SPS are functions of angle and the polarization of incidence of the EM wave. In designing SPSs that populate each zone of the lens, these effects are taken into account. The SPSs designed for the two lens prototypes shown in Fig. 7 provide almost identical phase shifts (with less than 2 difference) under oblique incidence angles for the transverse electric (TE) and transverse magnetic (TM) polarizations. This is mostly due to the fact that all SPSs of these lenses operate over relatively small incidence angles (20 or less). Table I and II show the physical and geometrical parameters of SPSs, which populate different zones of each lens. In all of these designs, the desired response is achieved by only changing the dimensions of the three capacitive patches of , and in Fig. 3) while keeping a third-order MEFSS ( the dimensions of the two inductive grids the same ( and in Fig. 3). Fig. 8(a) and (b) show the magnitude and phase of the transmission coefficients of the spatial phase shifters that populate different zones of the two lens prototypes. The frequency ranges where the magnitudes of the responses overlap are highlighted in both figures. The overlap region spans a fractional bandwidth of 10% and 8% for the first and second prototypes respectively. This overlap range can be considered as a lower bound for the bandwidth of the lens.

III. EXPERIMENTAL VERIFICATION The two lens prototypes discussed in the previous section were fabricated using standard PCB fabrication techniques. Both prototypes use four 0.508-mm-thick dielectric substrates (RO4003C from Rogers Corporation). Different metal layers are patterned on one or both sides of each substrate and the substrates are bonded together using three 0.1-mm-thick bonding films (RO4450B from Rogers Corporation). The overall thickness of each lens, inclusive of the bonding films, is 2.3 mm or , where is the free space wavelength at equivalently 10 GHz. The dimensions of the spatial phase shifters are 6.1 . Fig. 9 shows the mm 6.1 mm or equivalently photograph of the fabricated prototype whose layout is shown in Fig. 7(b).

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Fig. 8. Frequency responses of the spatial phase shifters that populate the lens aperture of (a) first prototype and (b) second prototype. The spatial phase shifters are designed using the procedure described in Section II.

Fig. 9. Photograph of the fabricated lens prototype that consists of ten zones. In this figure, only one of the five metallic layers that constitute the lens is visible (the first capacitive layer). Notice that the sizes of the capacitive patches on the layer decrease as we move away from the center of the lens.

A. Measurement System To characterize the two fabricated prototypes, two different types of measurements are carried out using the measurement setup shown in Fig. 10. The measurement setup consists of a large metallic screen with dimensions of 1.2 m 1.0 m. The screen has an opening with dimensions of 19 cm 24 cm at its center to accommodate the fabricated lens prototypes. An X-band horn antenna with aperture dimensions of 7.5 cm 9 cm is placed 120 cm away from the test fixture and radiates a vertically polarized EM wave to illuminate the lens. At the other side of the lens, a probe is used to measure the received signal. The probe is also vertically polarized and is in the form

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axis of rotation at the center of the lens. The length of the arm can be adjusted as desired. The receiving probe is swept over the focal arc with 2 increments and the received power is measured. B. Measurement Results

Fig. 10. (a) Perspective view of the measurement setup used to experimentally characterize the performance of the two fabricated lens prototypes. (b) In a set of measurements, the lens is excited with a plane wave and a receiving probe is swept in a rectangular grid in the vicinity of its focal point (x = 0; y = 0; z = f ) to characterize its focusing properties as a function of frequency. The measurement grid’s area is 8 cm 8 cm and the resolution is 1 cm. (c) In another set of measurements, the lens is illuminated with plane waves arriving from various incidence angles and the received power pattern on the focal arc of the lens is measured. Here, a probe is swept over the focal arc with 2 increments to measure the received power.

0

2

of an open-ended, semi-rigid coaxial cable with its center conductor extended by 10 mm. Both the transmitting antenna and the receiving probe are connected to a vector network analyzer. Fig. 10(b) shows a top view of the measurement system and the scenario used for the first series of measurements. In this case, the fixture that holds the receiving probe is swept over a measurement grid with dimensions of 8 cm 8 cm in the - plane with 1 cm increments in the and directions. The measurement grid is centered at the expected focal point of the lens. The measurement of the fabricated lens prototypes are conducted in two steps. For each grid point, the transmission response of the fixture without the lens is measured first. Then, the transmission response of the system with the presence of the lens is measured. By normalizing the latter measured values to the former one, the focusing gain of the lens at each grid point can be calculated. These measurements are then repeated for every point in the 8 cm 8 cm grid to obtain a two-dimensional plot of the focusing gain of the lens in the vicinity of its expected focal cm). This measurement is carpoint ried out for both fabricated prototypes at different frequencies across the X-band. In the second series of measurements, we examined the performance of the fabricated prototypes under oblique incidence angles. This is done by illuminating the lens with a plane wave from various incidence angles ranging from normal to 60 and measuring the field distribution over the focal arc of the lens using the measurement setup shown in Fig. 10(c). Here, the receiving probe is mounted on a rotating arm with its

Fig. 11 shows the focusing gain of the first fabricated prototype measured in the frequency range of 8–12 GHz with 0.5 GHz increments. At each frequency, the location where the maximum value of the focusing gain is achieved is indicated with symbol. At 10 GHz, a maximum focusing gain of a cross cm), which is 9.6 dB is achieved at 2 cm away from the expected focal point of the lens. This can be attributed to the difference between the actual and simulated responses of the SPSs, numerical errors in full-wave simulations, and fabrication tolerances. Further examination of Fig. 11 reveals that at frequencies below 10 GHz, the focal point2 is located closer to the lens and it moves away from the lens as frequency is increased. This frequency-dependent movement of the focal point can be explained using the Fermat’s principle. It states that the path taken between two points by a ray of light is the path that can be traversed in the least time. In the context of a focusing system, such as a reflectarray or a microwave lens, Fermat’s principle requires that the net time delay acquired by any ray propagating from the focal point to the aperture must be the same. Therefore, to achieve a frequency-independent focal point, the spatial phase shifters of a planar lens (or a reflectarray) must act as true-time-delay (TTD) units that provide different time delay values based on their position of the lens’ aperture [30]. In frequency domain, a constant time delay corresponds to a phase response that changes linearly with frequency. Therefore, the phase responses of different time-delay units that populate the aperture will be linear functions of frequency with different slopes. As can be seen from Fig. 8(a) and (b), the phase responses of different SPSs used in the two lens prototypes can roughly be approximated with linear functions of frequency. However, all of these linear functions have approximately the same slope. This frequency-dependent focal point movement is not unique to this lens and is observed in any non-TTD microwave lens or reflectarray. Nonetheless, as is observed from Fig. 11, the focusing properties of the lens remain within acceptable margins in the vicinity of the desired frequency of operation. The same measurement is also conducted to characterize the response of the second prototype. Fig. 12 shows the measured focusing gain of the second lens prototype over the same 8 cm 8 cm rectangular grid in the frequency range of 8 GHz to 12 GHz with 0.5 GHz increments. Fig. 12(e) shows that a maximum focusing gain of 10 dB is achieved at cm) at 10 GHz, which is 1 cm away from the expected focal point of the lens. The slight increase in the focusing gain is attributed to the better approximation of the desired phase delay profile across the lens aperture compared to that of the first prototype. Similar to the previous case, a frequency-dependent movement of the focal point is also observed in this prototype. Fig. 13 shows the measured focusing gains 2Defined

as the location where the focusing gain attains its maximum value.

AL-JOUMAYLY AND BEHDAD: WIDEBAND PLANAR MICROWAVE LENSES USING SUB-WAVELENGTH SPATIAL PHASE SHIFTERS

Fig. 11. Measured focusing gain of the first fabricated planar lens prototype (with five zones) in a rectangular grid in the vicinity of its expected focal point cm, y cm, z cm). In all of the figures, the horizontal axis is (x the x axis with units of [cm] and the vertical axis is the z axis with units of [cm]. The color bar values are in dB. The x marker in all of these figures represents the exact coordinate where the focusing gain maxima occur. (a) 8.0 GHz, (b) 8.5 GHz, (c) 9.0 GHz, (d) 9.5 GHz, (e) 10.0 GHz, (f) 10.5 GHz, (g) 11.0 GHz, (h) 11.5 GHz, and (i) 12.0 GHz.

=0

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of the two prototypes, at the expected focal point of the lens cm), as a function of frequency. As can be seen, the gain of the first prototype does not vary by more than 3 dB in the frequency range of 9.4–11.4 GHz, which is

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Fig. 12. Measured focusing gain of the second fabricated planar lens prototype (with ten zones) in a rectangular grid in the vicinity of its expected focal point (x cm, y cm, z cm). In all of the figures, the horizontal axis is the x axis with units of [cm] and the vertical axis is the z axis with units of [cm]. The color bar values are in dB. The x marker in all of these figures represents the exact coordinate where the focusing gain maxima occur. (a) 8.0 GHz, (b) 8.5 GHz, (c) 9.0 GHz, (d) 9.5 GHz, (e) 10.0 GHz, (f) 10.5 GHz, (g) 11.0 GHz, (h) 11.5 GHz, and (i) 12.0 GHz.

=0

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equivalent to a fractional bandwidth of 19.2%. For the second prototype, this bandwidth is extended to 20% (9.4–11.5 GHz). The electrical dimensions of the fabricated lens prototypes are relatively large and the structures have very small minimum features that gradually change over their apertures. For example,

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Fig. 13. Calculated and measured focusing gains of the two lens prototypes at cm, y cm) as a function cm, z their expected focal point (x of frequency. The 3 dB gain bandwidths of the two prototypes are respectively 19.2% and 20% for the first and second prototypes.

=0

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= 030

the gaps between the capacitive patches of the lens shown in Fig. 9 have dimensions of 150 m at the center of the aperture and gradually increase to 300 m at its edges. Modeling this structure using full-wave numerical simulations requires using a very dense grid system that can resolve the small variations in the physical dimensions of the structure from one SPS to the other (see the dimensions reported in Tables I and II). Moreover, the overall electrical dimensions of the structure are significantly larger than the minimum feature of a given SPS. Consequently, conducting a full-wave EM simulation of these prototypes requires significant computational resources and will be very time consuming. A simple alternative method to model the proposed lens is to consider it as a two-dimensional antenna , where is the physical array with an element spacing of size of each spatial phase shifter of the lens (6.1 mm). Considering each element of this array to be an electrically-small Hertizan dipole with a given amplitude and phase, one can easily calculate the overall array gain of this structure and evaluate it in the vicinity of its expected focal point. The amplitude and phase of each radiating element of this array can be determined from the simulated values of insertion loss and transmission phase of each SPS provided in Tables I and II. This process is carried out for both lens prototypes and figures similar to Figs. 11 and 12 are generated. To keep this article concise, these figures are not shown here. However, the same frequency-dependent focal point movement is observed in both cases. More importantly, the focusing gain calculated through this procedure is found to be very close to the measured values for both prototypes. The cm), calculated focusing gains, at are shown in Fig. 13 along with the measured ones. As can be seen the calculated results match very well with the measured ones over the entire frequency band of operation of the lens. The small differences observed between the calculated and measured values of the focusing gain can be attributed to the Ohmic losses of the lens and the errors introduced by the local periodicity assumption described before. The performance of the two fabricated prototypes are also characterized under oblique incidence angles. Each lens is illuminated with vertically-polarized plane waves with incidence angles of 0 , 15 , 30 , 45 , and 60 in the - plane and the

Fig. 14. Received power pattern on the focal arc of the first fabricated prototype (with five zones) when excited with plane waves arriving from various incidence angles. (a) Calculated. (b) Measured. All results are reported at 10 GHz.

received power pattern on the focal arc is measured. Fig. 14(a) and (b) show respectively the simulated and measured power patterns on the focal arc of the first lens prototype. Fig. 15(a) and (b) show the same results for the second prototype. For each lens, the incident power density is maintained for all incidence angles. All measured power values are normalized to the maximum measured value, which occurs at the focal point for normal incidence. Therefore, in these figures, 0 dB refers to the power level received on the focal point of the lens when it is excited with a normally incident plane wave. These simulated and measured results show the expected and actual performance of the lens response under oblique incidence angles. As the angle of incidence changes, the power pattern on the focal arc is steered by an angle equal to that of the incident angle and its peak value decreases. For incidence angles up to 45 , however, this decrease is relatively small (1.3 dB for first prototype and 1.6 dB for second prototype at 45 ). Beyond this angle, however, the response of the lens starts to deteriorate quite rapidly and at 60 the maximum value of the measured power pattern of the first and second prototypes are reduced respectively by 7.8 dB and 7.1 dB compared to the peak values measured for normal incidence. Nonetheless, the measured results shown in Figs. 14(b) and 15(b) demonstrate a good performance for incidence angles up to 45 for both prototypes. Additionally, comparison of the calculated results with the measured ones shows a relatively good agreement between the two

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zones and thus providing only a rough approximation of the ideal phase profile. In the second prototype, the lens aperture is divided into ten zones and a smoother phase variation over the aperture is obtained. Both prototypes were fabricated and experimentally characterized using a free-space measurement setup. The two prototypes operate over a relatively wide band of operation with fractional bandwidths of respectively 19.2% and 20%. Additionally, under oblique angles of incidence, both prototypes demonstrate a good performance up to 45 . As a simple analysis technique, the two lenses were treated as two-dimensional antenna arrays composed of Hertzian dipole radiators and the expected performance of the lenses were also calculated. For both prototypes, this expected performance was found to be very close to the measurement results. This suggests that the responses of the spatial phase shifters of the lenses do not change significantly, from their ideal responses, when placed in this non-periodic environment. In general, the performance of the second prototype, which provides a smoother phase variation, is found to be better than that of the first one. This suggests that increasing the number of discrete phase shifting zones is advantageous in designing this type of lens. REFERENCES

Fig. 15. Received power pattern on the focal arc of the second fabricated prototype (having ten zones) when excited with plane waves arriving from various incidence angles. (a) Calculated. (b) Measured. All results are reported at 10 GHz.

despite the simplified method used to model the structure. Comparing the measurements shown in Figs. 14 and 15 reveals that the beam-width of the power pattern of the second prototype is narrower than that of the first one. Additionally, the performance of the second prototype at oblique angles of incidence is slightly better than that of the first one. These effects may be attributed to the smoother phase delay profile across the aperture of this prototype. This indicates that the performance of this type of lens can potentially be further enhanced by increasing the number of zones. In the limiting case, the width of each zone can be made as narrow as that of one spatial phase shifter (6.1 mm in this case). IV. CONCLUSION A new technique for designing low-profile planar microwave lenses is introduced in this paper. The proposed lenses use the unit cells of suitably designed miniaturized element frequency selective surfaces as their spatial phase shifters. A procedure for designing such planar lenses was also presented. The lens design procedure is based on designing each spatial phase shifter of the lens by treating it as part of an infinite 2-D periodic structure. Two prototypes that approximate the phase profile of a conventional convex dielectric lens were designed following this procedure. The first prototype accomplishes this by dividing the aperture of the lens into five phase-shifting

[1] C. Stephanis and G. Hampsas, “Imaging with microwave lens,” IEEE Trans. Antennas Propag., vol. 28, no. 1, pp. 49–52, Jan. 1980. [2] F. Gallee, G. Landrac, and M. M. Ney, “Artificial lens for third-generation automotive radar antenna at millimetre-wave frequencies,” in IEE Proc. on Microwaves Antennas Propag., Dec. 2003, vol. 150, no. 6, pp. 470–476. [3] S. Hollung, A. E. Cox, and Z. B. Popovic, “A bi-directional quasioptical lens amplifier,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2352–2357, 1997. [4] A. R. Perkons, Y. Qian, and T. Itoh, “TM surface-wave power combining by a planar active-lens amplifier,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 6, pp. 775–783, Jun. 1998. [5] 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, no. 2, pp. 333–336, Apr. 2003. [6] C. R. White, J. P. Ebling, and G. Rebeiz, “A wide-scan printed planar K-band microwave lens,” in Proc. IEEE AP-S Int. Symp., Washington, D.C., Jul. 3–8, 2005, vol. 4, pp. 313–316. [7] P. K. Singhal, R. D. Gupta, and P. C. Sharma, “Design and analysis of modified Rotman type multiple beam forming lens,” in Proc. IEEE Region 10 Int. Conf. on Global Connectivity in Energy, Computer, Communication and Control, New Delhi, India, Dec. 17–19, 1998, vol. 2, pp. 257–260. [8] B. Panzner, A. Joestingmeier, and A. Omar, “Ka-band dielectric lens antenna for resolution enhancement of a GPR,” in Proc. Int. Symp. on Antennas, Propagation and EM Theory, Kunming, China, Nov. 2–5, 2008, pp. 31–34. [9] S. Ravishankar, “Analysis of shaped beam dielectric lens antennas for mobile broadband applications,” in Proc. IEEE Int. Workshop on Antenna Technology: Small Antennas and Novel Metamaterials, Singapore, Mar. 7–9, 2005, pp. 539–542. [10] Z. X. Wang and W. B. Dou, “Dielectric lens antennas designed for millimeter wave application,” in Proc. Infrared Millimeter Waves and 14th Int. Conf. on Teraherz Electronics, Shanghai, China, 2006, pp. 376–376. [11] J. Ruze, “Wide-angle metal plate optics,” in Proc. IRE, 1950, vol. 38, pp. 53–59. [12] W. Rotman and R. F. Turner, “Wide-angle microwave lens for linesource applications,” IEEE Trans. Antennas Propag., vol. 11, no. 6, pp. 623–632, 1963. [13] D. T. McGrath, “Slot-coupled microstrip constrained lens,” in Proc. Antenna Applications Symp., Monticello, IL, Sep. 1987, pp. 139–168. [14] D. T. McGrath, “Planar three-dimensional constrained lenses,” IEEE Trans. Antennas Propag., vol. 34, no. 1, pp. 46–50, Jan. 1986.

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[15] Y. Zhou, S. Rondineau, D. Popovic, A. Sayeed, and Z. Popovic, “Virtual channel spacetime processing with dual-polarization discrete lens antenna arrays,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2444–2455, 2005. [16] D. M. Pozar, “Flat lens antenna concept using aperture coupled microstrip patches,” Electron. Lett., vol. 32, no. 23, pp. 2109–2111, Nov. 1996. [17] A. Abbaspour-Tamijani, K. Sarabandi, and G. M. Rebeiz, “A millimetre-wave bandpass filter lens array,” IET Microw. Antennas Propag., vol. 1, no. 2, pp. 388–395, 2007. [18] P. Ikonen, M. Karkkainen, C. Simovski, P. Belov, and S. Tretyakov, “Light-weight base-station antenna with artificial wire medium lens,” IEE Proc. Microw. Antennas Propag., vol. 153, no. 2, pp. 163–170, 2006. [19] R. Liu, X. M. Yang, J. G. Gollub, J. J. Mock, T. J. Cui, and D. R. Smith, “Gradient index circuit by waveguided metamaterials,” Appl. Phys. Lett., vol. 94, no. 7, Feb. 2009, Article no. 073506. [20] A. K. Iyer and G. V. Eleftheriades, “A multilayer negative-refractive-index transmission-line (NRI-TL) metamaterial free-space lens at X-band,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2746–2753, 2007. [21] D. N. Black and J. C. Wiltse, “Millimeter-wave characteristics of phase-correcting Fresnel zone plates,” IEEE Trans. Microw. Theory Tech., vol. 35, no. 12, pp. 1122–1128, 1987. [22] A. Petosa, S. Thirakoune, I. V. Minin, and O. V. Minin, “Array of hexagonal Fresnel zone plate lens antennas,” Electron. Lett., vol. 42, no. 15, pp. 834–836, 2006. [23] Y. Fan, B. Ooi, H. D. Hristov, and M. Leong, “Compound diffractive lens consisting of Fresnel zone plate and frequency selective screen,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 1842–1847, Jun. 2010. [24] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley-Interscience, 2000. [25] M. Al-Joumayly and N. Behdad, “A new technique for design of low-profile, second-order, bandpass frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 57, pp. 452–459, Feb. 2009. [26] M. Al-Joumayly and N. Behdad, “A generalized method for synthesizing low-profile, bandpass frequency selective surfaces with non resonant constituting elements,” IEEE Trans. Antennas Propag., vol. 58, no. 12, pp. 4033–4041, Dec. 2010. [27] N. Behdad, M. Al-Joumayly, and M. Salehi, “A low-profile third-order bandpass frequency selective surface,” IEEE Trans. Antennas Propag., vol. 57, pp. 460–466, Feb. 2009. [28] N. Behdad and M. Al-Joumayly, “A generalized synthesis procedure for low-profile frequency selective surfaces with odd-order bandpass responses,” IEEE Trans. Antenna Propag., vol. 58, no. 7, pp. 2460–2464, Jul. 2010. [29] D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarrays,” IEEE Trans. Antenna Propag., vol. 45, no. 2, pp. 287–296, Feb. 1997. [30] E. Carrasco, J. A. Encinar, and M. Barba, “Bandwidth improvement in large reflectarrays by using true-time delay,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2496–2503, Aug. 2008.

Mudar Al-Joumayly (S’11) received the B.S. degree in electrical engineering from Mutah University, Jordan, in 2003, the M.S. degree in computer engineering from the New York Institute of Technology, Jordan, in 2005, the M.S. degree in electrical engineering from University of Central Florida, Orlando, in 2009, and the Ph.D. degree in electrical engineering from the University of Wisconsin-Madison, in 2011. From 2006 to 2009, he worked as a Research Assistant in the Antennas, RF, and Microwave Integrated Systems (ARMI) Laboratory, University of Central Florida, and from 2009 to 2011, as a Research Assistant in the University of Wisconsin-Madison Applied Electromagnetics Lab. He is currently a Development Engineer at TriQuint Semiconductor, Apopka, FL. His research interests include antenna miniaturization, frequency selective surfaces, phased array antennas, microwave filters, and bulk acoustic wave devices. Dr. Al-Joumayly received the third prize in the Best Student Paper Competition at the Antenna Applications Symposium in Sep. 2010.

Nader Behdad (M’06) received the B.S. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 2000 and the M.S. and Ph.D. degrees in electrical engineering from University of Michigan—Ann Arbor in 2003 and 2006, respectively. Currently, he is an Assistant Professor in the Electrical and Computer Engineering Department, University of Wisconsin-Madison. From 2006 to 2008, he was as an Assistant Professor in the Department of Electrical Engineering and Computer Science, University of Central Florida, Orlando. His research expertise is in the area of applied electromagnetics. In particular, his research interests span the fields of electrically small antennas, biomimetics and biologically inspired systems in electromagnetics, periodic structures, passive high-power microwave devices, frequency selective surfaces, and phased array antennas. Dr. Behdad is the recipient of the 2011 CAREER award from the National Science Foundation, the 2011 Young Investigator Award from Air Force Office of Scientific Research, and the 2011 Young Investigator Award from the Office of Naval Research. He received the Office of Naval Research Senior Faculty Fellowship in 2009, the Young Scientist Award from the International Union of Radio Science (URSI) in 2008, the Horace H. Rackham Predoctoral Fellowship from the University of Michigan in 2005–2006, the best paper awards in the Antenna Applications Symposium in Sep. 2003, and the second prize in the paper competition of the USNC/URSI National Radio Science Meeting, Boulder, CO, in January 2004. His graduate students were the recipients of the first and third place awards in the student paper competition of Antenna Applications Symposium respectively in 2008 and 2010. He is currently serving as an Associate Editor for IEEE Antennas and Wireless Propagation Letters and the Co-Chair of the technical program committee of the 2012 IEEE International Symposium on Antennas and Propagation and USNC/URSI National Radio Science Meeting.

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Light Weight and Conformal 2-Bit, 1 4 Phased-Array Antenna With CNT-TFT-Based Phase Shifter on a Flexible Substrate Daniel T. Pham, Harish Subbaraman, Maggie Yihong Chen, Xiaochuan Xu, and Ray T. Chen

Abstract—This paper presents the development and characterization of an ink-jet printed 2-bit, 1 2 4 phased-array antenna (PAA) system on a flexible Kapton polyimide substrate utilizing carbon nanotube thin-film transistors (CNT-TFTs) as switching elements in the phase-shifting network. A multilayer metal interconnection strategy is used to fabricate a complete PAA system with control lines. By appropriately controlling the ON and OFF states of various switches, a 4.99 GHz signal is steered from 0 to 027 , and the measured far-field radiation patterns agree very well with the simulated data. Bending tests performed on the fabricated system at bending radii of 6.5 cm, 9.5 cm, 12 cm, and 24 cm also demonstrate good agreement with the simulations. Index Terms—Beam steering, carbon nanotube, flexible antenna, phased-array antenna, phase shifter, thin-film transistor.

I. INTRODUCTION

F

LEXIBLE antenna has become very attractive in the last decade due to the development of several interesting flexible circuit components that can be integrated into one system on a light weight, conformal flexible platform. Such flexible antenna components are being considered for several applications including communication, sensing, RFID, etc. For example, proximity surface activity applications for use on robotic devices or on human clothes, mandate a small size, light weight, and low power antenna system that can be used in a desired frequency band for certain data services. Due to the low profile of conformal antenna designs, the local networks for flexible antennas are expected to provide coverage for short m) to medium range ( – km) operations [1]. ( An important system utilizing such flexible antenna elements, such as a conformal phased array antenna (PAA), is formed by combining the flexible antenna elements and

Manuscript received October 18, 2010; revised April 05, 2011; accepted May 06, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. This work was supported by NASA under Contract NNX09CA37C, Air Force Office of Scientific Research (AFOSR) under contract FA9550-09-C-0212, and Office of Naval Research under contract N00014-10-M-0317. D. T. Pham and R. T. Chen are with the Microelectronics Research Center, Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX 78758 USA (e-mail: [email protected]). H. Subbaraman is with Omega Optics, Inc., Austin, TX 78759 USA. M. Y. Chen is with the Ingram School of Engineering, Texas State University, San Marcos, TX 78666 USA. X. Xu is with the Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX 78758 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165490

electronics involved in transmit/receive (T/R) modules, on a single flexible substrate. The PAA plays an important role in the modern radar systems since large angle electronic beam steering can be achieved without mechanically rotating the antenna array [2]. Although, it is possible to develop flexible antenna elements using ink-jet printing, embedding the phase shifting chip for the T/R modules on a flexible substrate is not an easy task. Due to the fabrication resolution limit of the printer, line widths smaller than 50 m for the metal lines are difficult to achieve. Additionally, the material and physical properties of the available flexible substrates lead to large feature sizes on the phase shifter in order to achieve 50 Ohm impedance matching with the printed antenna elements. Thus, the phase shifter and associated electronics take up a lot of space due to their large size. The assembly of these components on flexible substrates, herefore, is prone to reliability issues. Recently, CNT transistor has shown tremendous progress due to their excellent mobility characteristics [3]. CNT TFT-based devices on flexible substrates have achieved high carrier mobilities using ultrapure electronics-grade CNT solutions [4], [5] by ink-jet printing technique. Other techniques such as dielectrophoresis (DEP) [6], spin-coating [7], and spray-coating [8] to form CNT thin-film transistors have also been demonstrated. All of these techniques yield a random network of CNTs on the substrate. Utilizing ink-jet printing technique, we previously developed and demonstrated the working of a 2-element phased array antenna system that utilizes CNT-TFT containing random network of TFTs [9]. Further improvement in the CNT transistor device performance has been demonstrated through the use of aligned CNTs [10]–[14]. In comparison with a random network of CNTs, aligned CNTs improve the drain current by decreasing the average carrier path length. A majority of the reported aligned CNT thin films utilize direct deposition on silicon or quartz substrates via chemical vapor deposition (CVD) technique [10]–[14]. This deposition technique is unsuitable for flexible substrates due to high deposition temperatures (400 C–1000 C) involved. Among the flexible substrate compatible methods for achieving self-aligned CNTs, dip-coat technique has shown good performance results [13]–[15]. In this paper, a combination of ink-jet printing and stamping technique is used to fabricate a fully functional 1 4 phasedarray antenna with 2-bit CNT-TFT based phase-shifter on a flexible Kapton polyimide substrate. The self-aligned CNTs are formed on a silicon substrate using a dip-coat method, and a stamping technique is used to form the CNT-TFTs in the phase shifter [14], [15].

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Fig. 2. Schematic of a 2-bit, 1 4 phased-array antenna. Printing transmission lines and antenna elements separately provides good printing coverage.

Fig. 1. Fujifilm Dimatix Materials Printer.

II. FABRICATION PROCESS A. Ink-Jet Printing The ink-jet printer used in this work is a Fujifilm Dimatix Materials Printer (DMP-2800) as shown in Fig. 1. The printer uses a piezoelectric printing cartridge (DMC-11610) consisting of 16 independently controllable nozzles. The ink droplets dispensed from the nozzles have a nominal volume of 10 pL. The machine is suitable for room temperature printing of circuits on any kind of substrate, including glass, plastic, rubber, textile, etc., since the noncontact printing is substrate topography independent. In order to print a complicated pattern consisting of small and large structures or closely spaced lines next to each other, such as in the PAA system, the orientation of the PAA system needs to be considered in order to achieve uniform deposition of material ink at all locations. Several orientations of the PAA system were investigated, and it was found that the PAA system in horizontal position, as shown in Fig. 2, provides the best result. While printing large areas, the nozzles operate continuously in a long endurance time mode. In this mode, the printing material wets the nozzle surface. This wetting causes a problem, especially while printing on small area or printing small lines. Therefore, in order to print small features, a single nozzle is utilized. Another problem which requires consideration is the deposition uniformity. For example, during the beginning of the printing cycle, the nozzles provide uniform deposition, and after printing a large area (long operating time), the performance slowly degrades. Therefore, breaking the printing area down to several small parts is a good way to have better uniform printing. Fig. 2 shows the PAA system architecture formed by connecting two parts. The first part contains narrow width ( micron) transmission lines. This part is printed using a single nozzle on the cartridge for better precision. The second part consisting of the large antenna elements, is printed using multiple nozzles. Using this type of combination printing, the printed product has better quality and is performed with an optimal printing time. Please note that alignment marks are used to precisely align the two printing steps, which are also shown in Fig. 2.

Fig. 3. (a) microstrip transmission line, co-planar waveguide and coupler design; (b) microstrip antenna design.

B. Principle of Operation of Phased-Array Antenna Subsystem Fig. 2 shows the layout of the 2-bit 1 4 element phased array antenna subsystem. In order to design the full PAA system, standard microstrip antenna design techniques are used [16]–[20]. Schematic drawings of RF coupler, microstrip transmission line, coplanar waveguide to microstrip line coupler, antenna elements designed for operation at 5 GHz are shown in Fig. 3. In order to achieve beam steering using the PAA, first, the 5 GHz RF signal is applied through the input RF port. The coupling section transitions the signal from the coplanar waveguides to the microstrip line. The signal is then split into two branches, with each branch split further into two more branches, thus giving 4 branches with equal length (for 4 elements). The 4 branches feed a 2-bit phase shifting network, schematically shown in Fig. 4. Each of the switches (numbered from 1 through 16) in the phase shifter is a CNT-TFT. By controlling the ON/OFF ratio of the ,0 , switch pairs as indicated in Table I, beam steering at 27 , and 45 is achieved. C. Formation of Carbon Nanotube Thin-Film Transistor (CNT-TFT) In this work, Carbon nanotube thin-film transistors (CNT-TFTs) are formed by a combination of printing and stamping techniques on a 127 m thick Kapton Polyimide substrate [14], [15]. Fig. 5 shows a schematic of the bottom gate integration process flow for the CNT-TFT. At first, the gate electrode is printed along with the transmission lines and

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Fig. 4. Schematic of a 2-bit, 1 4 phase shifter. CNT-TFTs are numbered from 1 through 16. Lengths indicated within boxes denote additional lengths of microstrip line.

TABLE I SWITCH PAIR SELECTION VERSUS STEERING ANGLE

Fig. 5. Schematic of bottom gate integration for our CNT-TFT which acts as a switch in the phase shifter.

D. Multilayer Metal Interconnection

antenna elements by utilizing Silver nanoparticle ink from Cabot Corporation. After annealing the printed lines at 160 C for 10 minutes, a spin-on glass dielectric material is printed on top of the gate electrode and cured. Then, the silver source and drain (same width as transmission line, which is 300 m) electrodes are printed with the gate length of 100 m. After annealing, wet silver droplets are printed on the source-drain regions in order to provide a good contact between source drain electrodes with CNT film. The CNT is aligned using dip-coat technique on sacrificial silicon substrate [12]–[14], [22], [23], and the film is lifted-off of the silicon substrate using a special thin Kapton substrate (25 m thick) with an adhesive on one side. This Kapton substrate containing the CNT film is bonded on top of the printed source-drain regions containing wet silver droplets and the entire structure is annealed to form the CNT-TFT. Detailed process integration, electrical data and bending test results of our CNT-TFT are reported elsewhere [14], [15].

Multilayer metal interconnect is also developed to provide connection to the gate electrodes from an external power supply and aid in the packaging of the system. A thin Kapton (25 m) substrate with adhesive coating on one side (same as the one used to develop the CNT-TFTs) is bonded on top of the first substrate containing the printed PAA subsystem. Contact vias are formed on the thin substrate prior to attaching. A pressurized annealing process at 100 C is used to bond these layers together. After bonding, silver ink is printed on the thin Kapton layer to form the metal connection lines. The liquid silver ink is printed one or multiple times to fill the contact vias, which makes contact between the bottom gate contact pads and top interconnection lines. Another anneal process is performed to complete the contact. Details of this multilayer metal interconnection are also reported in [15]. Fig. 6 shows a picture of a fully fabricated 2-bit, 1 4 PAA system on a Kapton substrate. Notice that a third thin Kapton layer is used on top of the second layer to protect the metal interconnection lines and vias. Probing pads are formed using double-sided copper tape. III. EXPERIMENTAL SET UP AND DATA The parameter of the printed thin film antenna is measured first to confirm the radiation of the patch antenna, as shown in Fig. 7. It can be seen from the figure that the antenna

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Fig. 6. Picture of a complete 2-bit 1 4 PAA system containing CNT-TFTs as switches in the phase shifting network. Multilayer metal interconnection produces a fully packaged system with metal interconnection lines.

Fig. 7. S parameter of the printed antenna element. The antenna shows good radiation around 4.99 GHz.

radiates well around 4.99 GHz, agreeing well with the designed frequency of 5 GHz. Figs. 8(a) and (b) show the entire measurement setup for far-field pattern measurement of the fabricated PAA system. The PAA system is spread out flat on a thick flexi glass substrate. We designed the stage such that the flexi glass can also be used to perform bending experiments and study the influence of bending on far-field patterns. The entire circuit is mounted vertically on a precision rotation stage along with the DC and RF probes. The RF signal from a HP8510C network analyzer is applied at the input of the PAA system. The CNT-TFT switch network is controlled using a mainframe computer with a switch control module. As shown in Table I, for each steering angle, 8 CNT-TFT switches are controlled corresponding to each desired steering angle. RF absorbers are arranged around the PAA setup in order to eliminate multipath effects. The radiated signal is received by a receiving horn antenna which is connected to a microwave spectrum analyzer (MSA). The received power is measured on the MSA as a function of the rotation angle, thus producing the far field patterns. The radiation pattern for a 4.99 GHz signal is measured using the above setup. We measure all the four azimuth steering angles. Fig. 9 shows the measured and simulated far field radiation patterns of the PAA system at 0 degree (black curves) and degree (red curves) steering angles. The measured points are indicated by data points whereas the simulated patterns are shown as smooth curves. It can be seen from the results that the mea-

Fig. 8. (a) Experimental setup to measure the far-field radiation pattern of the printed 1 4 PAA system; (b) close up picture showing the 1 4 PAA system laid flat on a flexi-glass substrate holder.

2

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Fig. 9. Measured (indicated by data points) and simulated (indicated by smooth curves) far-field radiation patterns of a 4.99 GHz signal at 0 degree (indicated by black curves) and 27 degrees (indicated by red curves).

0

sured and simulated far field patterns agree very well with each other. IV. BENDING EXPERIMENT We also performed an experiment to observe the effect of bending on the performance of the PAA system. Three PAA

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V. CONCLUSION A light weight, flexible 2-bit, 1 4 phased-array antenna system on Kapton substrate has been presented. CNT-TFT is used as a switch in the phase shifting network. Multilayer metal interconnection is implemented for packaging the full system. By controlling the ON/OFF states of the transistors, has been beam steering of a 4.99 GHz signal from 0 to demonstrated. The antenna system also shows good stability and tolerance under different bending radii of curvature. Compared to traditional antenna design, this light weight, flexible and conformal PAA is a good candidate for portable wireless system to meet challenging requirements. REFERENCES

Fig. 10. Theoretical (solid) and measured (dots) far-field radiation pattern for (a) 6.5 cm bending radius, (b) 9.5 cm bending radius, 12 cm bending radius, and (d) 24 cm bending radius.

Fig. 11. Theoretical (solid) and measured (crosses) 3 dB beamwidth of the farfield radiation pattern as a function of bending radius of curvature.

samples were fabricated to evaluate their bending characteristics. The PAA system is spread out on circular tube or on bent flexi glass substrate. The entire measuring setup is similar to that shown in Fig. 8. In order to evaluate the effect of bending on far-field radiation pattern, we first developed a code to simulate the PAA patterns. Using the results from this simulation code, we evaluated the bending performance of our system. Fig. 10 shows the measured (indicated by data points) and simulated (indicated by smooth curves) far-field radiation patterns for four different bending radii. A summary plot showing the calculated and meaas a function of bending radius sured 3 dB beamwidth of curvature is shown in Fig. 11 for all three PAA systems. It can be seen that the measured and calculated curves agree well with each other, thus confirming good operation for conformal operations.

[1] L.-R. Zheng, M. Nejad, S. Rodriguez, L. Zhang, C. Chen, and H. Tenhunen, “System-on-flexible substrates: Electronics for future smart-intelligent world,” in Proc. HDP, 2006, pp. 29–36. [2] R. J. Mailloux, Phased Array Antenna Handbook. Norwood, MA: Artech House, 1994. [3] C. Rutherglen, D. Jain, and P. Burke, “Nanotube electronics for radiofrequency applications,” Nature Nanotechnol., vol. 29, Nov. 2009. [4] J. Vaillancount, H. Zhang, P. Vasinajindakaw, H. Xia, X. Lu, X. Han, D. Janzen, W. Shih, C. Jones, M. Stroder, M. Y. Chen, H. Subbaraman, R. T. Ray, U. Berger, and M. Renn, “All ink-jet-printed carbon nanotube thin-film transistor on a polyimide substrate with an ultrahigh operating frequency over 5 GHz,” Appl. Phys. Lett., vol. 93, p. 243301, 2008. [5] L. Nougaret, H. Happy, G. Dambrine, V. Dercycke, J.-P. Bourgoin, A. A. Green, and M. C. Hersam, “80 GHz FET produced using high purity semiconducting SWCNT,” Appl. Phys. Lett., vol. 94, p. 243505, 2009. [6] S. Selvarasah, K. Anstey, S. Somu, A. Busnaina, and M. R. Dokmeci, “High flexible and biocompatible carbon nanotube thin-film trasistor,” in Proc. 9th IEEE Conf. Nanotechnology, 2009, pp. 29–32. [7] M. LeMieux, M. Roberts, S. Barman, and Y. W. Jin, “Self-sorted, aligned nanotube newworks for TFT,” Science, vol. 321, pp. 101–104, 2008. [8] E. Artkovic, M. Kaempgen, D. S. Hecht, S. Roth, and G. GrUner, “Transparent and flexible carbon nanotube transistors,” Nano Lett., vol. 5, no. 4, pp. 757–760, 2005. [9] M. Y. Chen, X. Lu, H. Subbaraman, and R. T. Chen, “Fully printed phased-array antenna for space communications,” Proc. SPIE, vol. 7318, p. 731814, 2009. [10] S. Kang, C. Kocabas, T. Ozel, M. Shim, N. Pimparkar, M. Alam, S. Rotkin, and J. Rogers, “High-performance electronics using dense, perfectly aligned arrays of single-walled carbon nanotubes,” Nature Nanotechnol., vol. 2, pp. 924–931, 2007. [11] S. Kim, S. Ju, J. Back, Y. Xuan, P. Ye, M. Shim, D. Janes, and S. Mohammadi, “Fully transparent thin-film transistors based on aligned carbon nanotube arrays and indium tin oxide electrodes,” Adv. Mater., vol. 21, no. 5, pp. 564–568, 2009. [12] F. Ishikawa, H. Chang, K. Ryu, P. Chen, A. Badmaev, L. Gomez, D. Arco, G. Chen, and C. Zhou, “Transparent electronics based on transfer printed aligned carbon nanotubes on rigid and flexible substrate,” ACS Nano, vol. 3, no. 1, pp. 73–79, 2009. [13] M. Engel, J. Small, M. Steiner, M. Freitag, A. Green, M. Hersam, and P. Avouris, “Thin film nanotube transistor based on self-assembled, aligned, semiconducting carbon nanotube arrays,” ACS Nano, vol. 2, no. 12, pp. 2445–2452, 2008. [14] D. Pham, H. Subbaraman, M. Y. Chen, X. Xu, and R. T. Chen, “Bending tests of carbon nanotube thin-film transistors on flexible substrate,” in Proc. SPIE Conf. Carbon Nanotube, Graphene, and Associated Devices III, 2010, vol. 7761, pp. 7761–7725. [15] D. Pham, H. Subbaraman, M. Y. Chen, X. Xu, and R. T. Chen, “Self-aligned carbon nanotube thin-film transistors on flexible substrates with novel source-drain contact and multi-layer interconnection,” IEEE Trans. Nanotechnol., to be published. [16] K. R. Carver and J. W. Mink, “Microstrip antenna technology,” IEEE Trans. Antennas Propagat., vol. AP-29, no. 1, pp. 2–24, Jan. 1981. [17] J. R. James, P. S. Hall, and C. Wood, Microstrip Antenna Theory and Design. London, U.K.: Peter Peregrinis, 1981, pp. 87–89. [18] P. Bhartia et al., Millimeter-Wave Microstrip and Printed Circuit Antennas. Norwood, MA: Artech House, 1991.

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[19] B. C. Wadell, Transmission Line Design Handbook. Norwood, MA: Artech House, 1991. [20] D. Pavlidis and H. L. Hartnagel, “The design and performance of threeline microstrip couplers,” IEEE Trans. Microw. Theory Tech., vol. 24, no. 10, pp. 631–640, Oct. 1976. [21] G. P. Gauthier et al., “W-band finite ground coplanar waveguide (FGCPW) to microstrip line transition,” IEEE MTT-S Dig., vol. TU2E-3, pp. 107–109, 1998. [22] A. Dimitrov and K. Nagayama, “Continuous convective assembling of fine particles into two-dimensional arrays on solid surfaces,” Langmuir, vol. 12, pp. 1303–1311, 1996. [23] A. Dimitrov and K. Nagayama, “Steady-state unidirectional convective assembling of fine particles into two-diemnsional arrays,” Chem. Phys. Lett., vol. 243, pp. 462–468, 1995.

Daniel T. Pham received the B.S. and M.S. degrees in chemical engineering from Texas A&M University, College Station, and Rice University, Houston, TX. He is currently pursuing the Ph.D. degree in Electrical and Computer Engineering Department, University of Texas at Austin. He worked in the semiconductor industry for more than ten years, focused on semiconductor processes, device integration, and novel device research. He has over 25 publications and eight awarded patents.

Harish Subbaraman received the B.E. degree in electronics and communication engineering from the Chaitanya Bharathi Institute of Technology, Hyderabad, India, in 2004, and the M.S. and Ph.D. degrees in electrical engineering from the University of Texas at Austin in 2006 and 2009, respectively. With a strong background in RF photonics and X-band phased array antennas, he has been working on true-time-delay feed networks for phased array antennas and carbon nanotube based thin-film transistors for the last four years. Throughout these years, he has laid a solid foundation in both theory and experimental skills. He has served as principal investigator for projects from National Aeronautics and Space Administration (NASA), Air Force Office of Scientific Research (AFOSR), and the Navy. He has over 15 publications in refereed journals and conferences.

Maggie Yihong Chen received the Ph.D. degree in electrical engineering in optoelectronic interconnects from the University of Texas at Austin in 2002. She is an Assistant Professor of electrical engineering at Texas State University, San Marcos. Her research work over the past ten years has been focused on optical true-time delay feeding networks for phased-array antennas, planar waveguide devices, optoelectronic interconnects, and diffractive optical elements. She has authored over 50 publications in refereed journals and conferences. Dr. Chen is a senior member of the OSA and SPIE.

Xiaochuan Xu received the B.S. and M.S. degrees in electrical engineering from the Harbin Institute of Technology, China. He is currently pursuing the Ph.D. degree in the Electrical and Computer Engineering Department, University of Texas at Austin.

Ray T. Chen received the B.S. degree in physics from the National Tsing-Hua University, Taiwan, R.O.C., in 1980, and the M.S. degree in physics in 1983 and the Ph.D. degree in electrical engineering in 1988, both from the University of California. He joined the University of Texas (UT) at Austin as a faculty to start the optical interconnect research program in the ECE Department in 1992. Prior to his UT professorship, he was a research scientist, manager, and Director of the Department of Electrooptic Engineering, Physical Optics Corporation, Torrance, CA, from 1988 to 1992. He also served as the CTO/founder and chairman of the board of Radiant Research from 2000 to 2001 where he raised 18 million dollars A-Round funding to commercialize polymer-based photonic devices involving over 20 patents, which was acquired by Finisar in silicon valley (NASDAQ: FNSR). He also serves as the founder and Chairman of the board of Omega Optics, Inc., since its initiation in 2001. Over five million dollars of research funds were raised for Omega Optics. His research work has been awarded with 99 research grants and contracts from sponsors such as DOD, NSF, DOE, NASA, the state of Texas, and private industry. The research topics are focused on three main subjects: nano-photonic passive and active devices for optical phased array and interconnect applications; thin film guided-wave optical interconnection and packaging for 2-D and 3-D laser beam routing and steering; and true time delay (TTD) wide band phased array antenna (PAA). Experiences garnered through these programs in polymeric material processing and device integration are pivotal elements for the research work conducted by his group. He currently holds the Cullen Trust for Higher Education Endowed Professorship at UT Austin. He is the Director of the Nanophotonics and Optical Interconnects Research Lab within the Microelectronics Research Center. He is also the Director of the newly formed AFOSR Multi-Disciplinary Research Initiative (MURI) Center for Silicon Nanomembrane involving faculty from Stanford University; the University of Illinois, Urbana-Champaign; Rutgers University; and UT Austin. His group at UT Austin has reported its research findings in more than 540 published papers including over 60 invited papers. He holds 20 issued patents. He has also served as a consultant for various federal agencies and private companies and delivered numerous invited talks to professional societies. Dr. Chen has chaired or been a program-committee member for more than 100 domestic and international conferences organized by the IEEE, SPIE (The International Society of Optical Engineering), OSA, and PSC. He has served as an editor or co-editor for 18 conference proceedings. He is a Fellow of OSA and SPIE. He was the recipient 1987 UC Regent’s dissertation fellowship and of 1999 UT Engineering Foundation Faculty Award for his contributions in research, teaching, and services. He is also the recipient of 2008 IEEE teaching award. During his undergraduate years at National Tsing-Hua University, he led a university debate team in 1979 which received the national championship of national debate contest in Taiwan. There are 33 students who have received the electrica engineering Ph.D. degree in Dr. Chen’s research group at UT Austin.

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High Gain Metal-Only Reflectarray Antenna Composed of Multiple Rectangular Grooves Yong Heui Cho, Woo Jin Byun, and Myung Sun Song

Abstract—Using an overlapping T-block method based on modematching technique and virtual current cancellation, the scattering formulations for a metal-only reflectarray antenna composed of multiple rectangular grooves are rigorously presented in fast convergent integrals. By matching the normal boundary conditions at boundaries, we get the simultaneous equations of the TE and TM modal coefficients for plane-wave incidence and Hertzian dipole excitation. A metallic-rectangular-grooves based reflectarray fed by a pyramidal horn antenna was fabricated and measured with near-field scanning, thus resulting in 42.3 [dB] antenna gain at f = 75 [GHz]. Our circular reflectarray has 30 [cm] diameter and 5 961 rectangular grooves on its metal surface. The simultaneous equations for a Hertzian dipole feed are solved to approximately obtain the radiation characteristics of a fabricated reflectarray. The measured results are compared with those of the overlapping T-block method and the FDTD simulation and they show favorable agreements in terms of radiation patterns and antenna gain. Index Terms—Electromagnetic diffraction, Green’s function methods, metal-only reflectarray, millimeter wave technology, mode matching methods.

I. INTRODUCTION N the millimeter-wave frequency bands, there have been a lot of interesting applications in the fields of broadband radio links for backhaul networking of cellular base stations, Gbps-class wireless personal area network (WPAN), and millimeter-wave imaging to detect concealed weapons and nonmetal objects [1]–[4]. Especially, 70 and 80 GHz communication systems within 1-mile distance will play an important role in the next-generation wireless networks, because the cell site connectivity will require more than 1 Gbps data rates. For these millimeter-wave bands, a well-designed antenna for narrow beamwidth and high antenna gain is essential to compensation for severe free-space path loss and to prevention against signal interference. Although a high gain antenna can be manufactured by the concept of a parabolic reflector antenna, a reflectarray antenna has been extensively studied in [5]–[8],

I

due to the fact that the reflectarray technology has several advantages such as low profile, low cost, easy manufacturing, low feeding loss, and simple controllability of mainbeam and polarization. The reflectarray antenna in [5] consists of rectangular waveguide arrays and a feed waveguide to reflect electromagnetic waves efficiently. The phases of reflected waves are controlled based on the variation of a surface impedance. In view of a transmission line theory, the variation of waveguide depth results in that of surface impedance at the end of each waveguide. This indicates that reradiated waves can be designed in the predefined way and then the high-gain antenna with very low loss can be easily implemented. Modern reflectarrays [6]–[8] have the same operational principle of the original reflectarray [5]. In addition, the metal-only high gain antennas in [8], [9] were proposed for the millimeter-wave bands, where the loss characteristics are very important to maintain good communication links. In this work, we propose a novel analytic approach suitable for a metal-only metallic-rectangular-grooves based reflectarray. A 2-D metal-only reflectarray antenna has already been analyzed and fabricated in [8]. In order to analyze the problem of a 3-D metal-only metallic-rectangular-grooves based reflectarray, we will introduce an overlapping T-block method [8], [10] based on mode-matching technique, virtual current cancellation, and superposition principle, thereby obtaining analytic scattering equations in rapidly convergent and numerically efficient integrals. In view of mode-matching and Green’s function, we can represent the discrete modal expansions for closed regions and the continuous wavenumber spectra for open regions. Using virtual current cancellation by means of virtual PEC covers related to closed regions, the vector potential formulations for open regions are easily evaluated. To apply the superposition principle, we divide multiple rectangular grooves into several simple T-blocks and a source block [8], [10]. II. FIELD REPRESENTATIONS FOR SINGLE GROOVE

Manuscript received July 16, 2010; revised February 05, 2011; accepted May 16, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. This work was supported by the IT R&D Program of KCC/KCA (2008-F-013-04, Development of Spectrum Engineering and Millimeterwave Utilizing Technology). Y. H. Cho is with the Department of Electrical and Computer Engineering, University of Massachusetts Amherst, Amherst, MA, 01003 USA and also with the School of Information and Communication Engineering, Mokwon University, Daejeon, 302-729, Korea. W. Jin Byun and M. S. Song are with the Millimeter-Wave Research Team, Electronics and Telecommunications Research Institute, Daejeon, 305-700, Korea. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165479

with Let’s consider the TE plane-wave incidence incident angles, and , shown in Fig. 1. The time convention is suppressed throughout. The incident magnetic field is (1) where the incident angles, and , are defined as in terms of the - and -axes

0018-926X/$26.00 © 2011 IEEE

and

(2) (3)

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Fig. 1. Geometry of a metallic rectangular groove in a perfectly conducting plane.

, and . The incident electric field for the TE plane-wave is also formulated as

Fig. 2. Artificial geometry for virtual current cancellation.

(4) In order to match boundary conditions efficiently, we define the reflected electromagnetic waves from a perfectly conducting as plane at

(14)

(5)

and are the unknown TE modal cowhere efficients for regions (I) and (II), respectively,

(6) where

at

is a unit-step function. Enforcing the yields

(7)

(15)

(8) Similar to the TE plane-wave, we obtain the TM incident and as, respectively reflected plane-waves

, and field continuity

where we presume that

(9) (10) and TM polarizations, the incident Considering the TE and reflected electric fields are represented as

(11)

(12) and are polarization constants for the TE and TM where modes, respectively. Since all electromagnetic fields can be formulated with corresponding electric and magnetic vector potentials, we introduce and (II) the electric vector potentials for regions (I) illustrated in Fig. 1 as

(13)

(16) and . Note that in (16) is formulated with virtual PEC covers placed at and shown in Fig. 2. Even though the virtual PEC covers in Fig. 2 are absent from the original geometry illustrated in Fig. 1, the PEC covers are artificially inserted to accommodate the field formulations which will be described in (17). In order to make the fields continuous for , we define a scattered component which is implicitly related . Adding the virtual PEC covers inevitably gento erates the unwanted surface current densities on and . By means of the Green’s function, is utilized to remove a current density produced by the -field discontinuities. Then

(17)

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where

is an outward normal unit vector denoted in Fig. 2 and indicates the -directional Green function excited by the -directed source in terms of a magnetic vector potential . Inserting (16) into (17) yields

(24) The integral (24) is numerically efficient for or where the Green function in (24) does not have any and . Using (24), singularity in the region of we obtain the asymptotic -potential in region (II) as

(18) where

(25) (19) (20)

. It should be noted that our preand is right because vious assumption such as . To avoid pole singularities at of and on the real axis, we deform the integral path as for [10]. By using this substitution, a radiation integral (18) can be transformed to that without singularities as

, and shown in Fig. 1. In the next step, the magnetic vector potentials for Fig. 1 are formulated as where

(26)

(27) and are the unknown TM modal where coefficients for regions (I) and (II), respectively. Applying the field continuity at , we get (21)

(28)

is presented where a precise and efficient evaluation of in Appendix A. For numerical computation, a path-deforming variable in (21) can be empirically truncated to as

where is an equivalent electric charge density which may be produced by the field discontinuities at bound, and aries, we assume that

(22) where

is the maximum value of

(29)

and (23)

Similar to the Green’s function relation already described in as (17), we get the formula for

. Applying the Green’s second integral identity, and we reduce (16) and (18) as (30)

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where . To remove the -field disconand the -field tinuities at and discontinuities at are given by, respectively

(31) Fig. 3. Geometry of multiple rectangular grooves in a perfectly conducting plane (P : the number of grooves in the x-axis and (T ; S ): a translation point of the pth groove in the x-y plane).

(32) (36) where

illustrated in Fig. 1. The expression for is obtained from (29) and (31), and written by

In the far-field, the represented as

-potential in region (II) is asymptotically

(33) and is formulated in Appendix A. where Similarly, integrating (32) with (16), we get

(37) where (38)

(34) where Note that

and

is shown in Appendix A. are satisfied due to

. Using the Green’s second integral identity, we simplify (29) and (31) as

III. FIELD MATCHING FOR MULTIPLE GROOVES Due to large number of metallic rectangular grooves illustrated in Fig. 3, scattering formulations and analyses are a little complicated. These difficulties can be easily overcome with field superposition principle [10], [11]. In the field superposirelated to tion, electromagnetic fields in open region those of each metallic rectangular groove ( , ) for Fig. 1 are additively superimposed to produce and the total electromagnetic fields for multiple grooves shown in Fig. 3. Based on the superposition principle, the total electric and magnetic vector potentials are represented as

(35) Similarly,

is reduced to

(39)

(40)

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where is the total number of grooves, and of the th groove are translation positions for the - and -axes, respectively, and

(41) (42)

and fields continuities By matching the , we can obtain simultaneous scattering equations for at arbitrarily polarized plane-wave incidence in (11). Multiplying continuity at with the for the th groove and integrating over and yields

Fig. 4. Behaviors of the backscattered copolarization RCS (= ) versus a plane-wave incident angle ( ) with P = 1; N = 2; 2a = 2:5 ; 2b = d = 0:25 ;  = 0; u = 0; v = 1;  =  =  ;  =  =  .

0

for the th groove and integrating with respect to and

gives

(44)

(43) is an equivalent magnetic charge where density generated by the field discontinuities between reis a parameter gions (I) and (II) placed at for the th groove, is are the Kronecker delta, and is defined in Appendix B. Note that a center point of the th groove for field matching and is an inevitable term which must be included in normal magnetic field matching. continuity with Similarly, multiplying the

where

are defined in Appendix B.

IV. NUMERICAL COMPUTATIONS AND MEASUREMENT Plane-wave scattering from a rectangular metallic groove in a perfectly conducting infinite plane is extensively studied with numerical [12], [13] and analytic [14] techniques. In order to validate our formulations, (43) and (44), we compared our numerical results for a wide groove ( and in Fig. 1) with other simulations [12], [14]. Fig. 4 illustrates the normalized backscattered copolarization radar cross section (RCS) for an incident angle of a plane-wave illustrated in Fig. 1. All simulated results in Fig. 4 are strongly consistent

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Fig. 5. Fabricated 3-D metal-only reflectarray composed of rectangular grooves with a whole diameter ( ) = 30 [cm] and the total number of grooves ( ) = 5 961.

P

D

for . In addition, Fig. 4 indicates the convergence behavior of our simultaneous equations, (43) and (44), where and denote the truncation numbers of modal coefficients with respect to and in (13) and (26) for numerical computation, respectively. As the number of modes for increases, the backscattered RCS converges very fast for any . A lower-mode solution ( and ) is very good approximation for . Fig. 5 shows a fabricated 3-D metal-only reflectarray with prime focus composed of multiple rectangular grooves. A thick circular metal plate with 30 [cm] diameter and 1 [cm] thickness contains 5 961 rectangular metallic grooves. A pyramidal horn antenna used as a feed has 7 [mm] 5 [mm] aperture size and 12 [mm] waveguide transition, and an input waveguide for a feed is WR-12 (3.1 [mm] 1.5 [mm]). For simple design, we assume that each rectangular groove illustrated in Figs. 3 and 5 has the same aperture size of and the same separations of for all in (39) and (40), where is the number of grooves in the -axis. A depth for the th groove is individually calculated with a formula based on the phase matching condition [8] as

(45) [GHz], where and are a focus and the maximum depth of a paraboloid, respectively, which is effectively formed by a metal-only flat reflectarray in Fig. 5. Because of the periodicity of reflected phase, the depth in (45) can be limited to half the guided wavelength [8] ( [mm] for 78.5 [GHz]). Considering the phase center of a pyramidal horn antenna denoted as [mm], the feed in Fig. 5 is placed at where is the focus of a paraboloid, [8]. Fig. 6 presents the H- and E-plane radiation patterns of a metal-only reflectarray in Fig. 5 for 78.5 [GHz] and RA

Fig. 6. Characteristics of the H- and E-plane antenna gain patterns versus an observation angle ( ) with = 78 5 [GHz], = 5961 = 2 = 1 2 = 3 2 [mm], 2 = 2 7 [mm], 2 = 2 = 0 5 [mm], = = 0 = 193 845 [mm], = = = = RA = 0 75 = 25 [mm], obtained from (45) (a) H-plane ( = 0 ), (b) E-plane ( = 90 ).

a x

y

:

: ;d 

;z



b

f : :

:

d

P T0 a  

;M ;N ; S0 b : ;  ; 

. In our computations, we used the simultaneous equations, (43) and (44), with Hertzian dipole excitation polarized in the -axis. This means that the right hand sides of (43) and (44) should be modified for a Hertzian dipole. Our formulations based on the overlapping T-block method are compared with planar near-field measurement [8], [16] and numerical simulation [15]. We obtained planar near-field measurement results with a WR-10 (2.54 [mm] 1.27 [mm]) OERW (Open-Ended Rectangular Waveguide) probe and the parameters such as distance between probe and [cm], [mm], and [cm]. In Fig. 6, the radiation behaviors of our method and FDTD simulation agree very well for all observation angle. The GEMS parallel FDTD simulation [15] for Figs. 5 and 6 requires the parameters such as the [GB], the number of , and hours. The FDTD simulation is performed for the geometry shown in Fig. 5 without three metallic struts to support a pyramidal horn feed. In contrast

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higher than others, due to the fact that our computation is based on Hertzian dipole excitation and thus cannot include the feed characteristics. The measured aperture efficiencies for 75, 77, 78.5 [GHz] are 30.2, 27.2, 23.3 [%], respectively. V. CONCLUSION

Fig. 7. Behaviors of the co- and cross-polarization excited gain patterns versus an observation angle () (The parameters are selected from those in Fig. 6).

Rigorous and analytic solutions for scattering from multiple rectangular grooves in a perfectly conducting plane are obtained with the overlapping T-block method based on superposition principle and Green’s function relation. The simultaneous scattering equations for Hertzian dipole excitation can be utilized to predict radiation characteristics of a metal-only reflectarray fed by a pyramidal horn antenna. Our simulations were compared with commercial FDTD computation and near-field measurement and all results show favorable agreements. The modematching and Green’s function approach for a metal-only reflectarray with rectangular grooves can be extended to that with nonrectangular grooves by using suitable modal expansions. In further work, we will investigate the general feed modeling in near-field region and the corresponding phase matching condition for a metal-only reflectarray.

APPENDIX A INTEGRALS FOR The definitions of by

, AND , and

are written

(46)

(47)

Fig. 8. Co-polarization excited antenna gain variations versus a frequency (The parameters are chosen from those in Fig. 6).

to the FDTD simulation, our calculation time with CPU 2 [GHz] and RAM 2 [GB] is 4.4 minutes. Fig. 6 also shows the discrepancy between simulations and measurement results in the side-lobe region. This noticeable difference is caused by our simple feed modeling such as Hertzian dipole excitation and finite measurement scan area (52.497 [cm] 52.497 [cm]) [16]. Fig. 7 shows characteristics of co- and cross-polarization excited gain patterns for 78.5 [GHz]. The co- and cross-polarized excitations were simulated with Hertzian dipoles polarized in the - and -axes, respectively, when all parameters of a metal-only reflectarray were fixed. In case of cross-polarized excitation, radiation patterns have the null point at and their side-lobe levels are approximately 15 [dB] higher than those of co-polarized excitation. Fig. 8 indicates a succinct comparison of gain behaviors in terms of our method, FDTD simulation, and near-field measurement. Although the discrepancy among simulated and measured results is maximally 3 [dB], overall tendency of gain behaviors is not significantly different among results. It should be noted that antenna gain of our method is

(48) where , and

(49) To evaluate efficiently, we utilize the residue calculus. Thus, (46) can be transformed in terms of pole and branch-cut contributions as

(50)

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is the th order Hankel function of the first kind where . Note that (55) is and very efficient for numerical computation when or . Similar to the evaluation of , we obtain the following integrals as

Fig. 9. Deformed integral path to remove singularities.

where and . However, the integral (50) has rapidly oscillating behaviors when . These oscillatory characteristics can be removed with proper path deforming. As such, we propose a novel intefor any as gral path shown in Fig. 9 which always

(56)

(51) (57) Based on the path parameter (51) and Fig. 9, we modify (50) to a fast convergent integral without singularities as

For large argument approximation ( ), (56) and (57) are also formulated as

or

(58)

(52) Since the integrand in (52) has complex exponential functions and the complex numbers, , and in (52) have positive imaginary parts, the integral (52) converges exponentially. This also converges means that the double integral (18) with very rapidly. For Gaussian quadrature technique, the integral as (52) can be empirically truncated to (53) where

(59)

APPENDIX B MATCHING INTEGRALS The matching integrals for simultaneous modal equations, (43) and (44), are defined as

is defined in (21) as and (54) (60)

When and , the integral (52) still has unwanted numerical oscillations with respect to . To avoid numerical oscillations of integrand in (52), we analytically reduce (52) to a finite integral as

(55)

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(61)

(67)

(68) (62)

(69) (63)

(64) Since the integrands from (60) to (64) are composed of simple elementary functions, we can easily evaluate the above integrals in closed form. When , (60) through (64) are approximately formulated as

where and are the parameters for the th and th grooves. It should be noted that the formulations in (65) through (69) are very useful to obtain modal matrixes of . the simultaneous scattering equations, (43) and (44) for This is because the simplified integrals in (65) through (69) are in closed form without double infinite integrals, whereas the original integrals, (60) through (64), still have infinite integrals. By using simplified integrals, (65) through (69), we can compute the modal matrixes for a very large metal-only reflectarray very efficiently. REFERENCES [1] W.-J. Byun, B.-S. Kim, K. S. Kim, K.-C. Eun, M.-S. Song, R. Kulke, O. Kersten, G. Möllenbeck, and M. Rittweger, “A 40 GHz vertical transition having a dual mode cavity for a low temperature co-fired ceramic transceiver module,” ETRI J., vol. 32, no. 2, pp. 195–203, Apr. 2010. [2] W. Byun, B.-S. Kim, K.-S. Kim, M.-S. Kang, and M.-S. Song, “60 GHz 2 4 low temperature co-fired ceramic cavity backed array antenna,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 2009, pp. 1–4. [3] D. Lockie and D. Peck, “High-data-rate millimeter-wave radios,” IEEE Microw. Mag., vol. 10, no. 5, pp. 75–83, Aug. 2009. [4] D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 9, pp. 1581–1592, Sep. 2001. [5] D. G. Berry, R. G. Malech, and W. A. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propag., vol. 11, no. 6, pp. 645–651, Nov. 1963. [6] J. Huang and J. A. Encinar, Reflectarray Antennas. Piscataway, NJ: Wiley-IEEE Press, 2007. [7] A. G. Roederer, “Reflectarray antennas,” in Proc. Eur. Conf. Antennas Propag., Mar. 2009, pp. 18–22. [8] Y. H. Cho, W. J. Byun, and M. S. Song, “Metallic-rectangular-grooves based 2D reflectarray antenna excited by an open-ended parallel-plate waveguide,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1788–1792, May 2010. [9] A. Lemons, R. Lewis, W. Milroy, R. Robertson, S. Coppedge, and T. Kastle, “W-band CTS planar array,” in IEEE Microw. Symp. Dig., 1999, vol. 2, pp. 651–654.

2

(65)

(66)

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[10] Y. H. Cho, “TM plane-wave scattering from finite rectangular grooves in a conducting plane using overlapping T-block method,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 746–749, Feb. 2006. [11] Y. H. Cho, “Transverse magnetic plane-wave scattering equations for infinite and semi-infinite rectangular grooves in a conducting plane,” IET Microw. Antennas Propag., vol. 2, no. 7, pp. 704–710, Oct. 2008. [12] K. Barkeshli and J. L. Volakis, “Electromagnetic scattering from an aperture formed by a rectangular cavity recessed in a ground plane,” J. Electromag. Waves Applicat., vol. 5, no. 7, pp. 715–734, 1991. [13] J. M. Jin and J. L. Volakis, “A finite element-boundary integral formulation for scattering by three-dimensional cavity-backed apertures,” IEEE Trans. Antennas Propag., vol. 39, no. 1, pp. 97–104, Jan. 1991. [14] H. H. Park and H. J. Eom, “Electromagnetic scattering from multiple rectangular apertures in a thick conducting screen,” IEEE Trans. Antennas Propag., vol. 47, no. 6, pp. 1056–1060, Jun. 1999. [15] GEMS Quick Start Guide, [Online]. Available: http://www.2comu.com [16] A. D. Yaghjian, “An overview of near-field antenna measurements,” IEEE Trans. Antennas Propag., vol. 34, no. 1, pp. 30–45, Jan. 1986.

Yong Heui Cho was born in Daegu, Korea, in 1972. He received the B.S. degree in electronics engineering from the Kyungpook National University, Daegu, Korea, in 1998, the M.S. and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2000 and 2002, respectively. From 2002 to 2003, he was a Senior Research Staff with the Electronics and Telecommunications Research Institute (ETRI), Daejeon, Korea. In 2003, he joined the School of Information and Communication Engineering, Mokwon University, Daejeon, Korea, where he is currently an Associate Professor. In 2011, he is on the sabbatical leave with the Department of Electrical and Computer Engineering, University of Massachusetts

Amherst, MA. His research interests include dispersion characteristics of waveguides, electromagnetic wave scattering, and design of reflectarrays.

Woo Jin Byun received the B.S. degree in electronic engineering from the Kyung Pook National University, Taegu, Korea, in 1992, and the M.S. and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology, (KAIST) Daejeon, Korea, in 1995 and 2000, respectively. In 1999, he joined Samsung Electro-Mechanics Company, Suwon, Korea, where he developed mobile communication devices such as power amplifiers and radio modules for five years. He is currently with the Future Radio Technology Research Team of ETRI as a project leader of engineering staff. His current research areas include RF/millimeter-wave/THz integrated circuits and system designs, planar and reflector antennas and electromagnetic scattering analysis.

Myung Sun Song received the B.S. and M.S. degrees in electronics engineering from Chungnam National University, Korea in 1984 and 1986, respectively. Since 1986, he works for the Radio Technology Research Department of ETRI as a principal member of engineering staff and team leader. He developed RF and millimeter-wave communication system and cognitive radio technologies as a project leader. His interests include system engineering for cognitive radio, RF and millimeter-wave communication system, especially standardization relating to cognitive radio-based communication system.

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Integral-Equation Analysis of Scattering From Doubly Periodic Array of 3-D Conducting Objects Fu-Gang Hu, Member, IEEE, and Jiming Song, Senior Member, IEEE

Abstract—The integral-equation (IE) method formulated in the spatial domain is employed to calculate the scattering from the doubly periodic array of three-dimensional (3-D) perfect electric conductor (PEC) objects. The special testing and basis functions are proposed to handle the problem with non-zero normal components of currents at the boundary of one period. Moreover, a relationship between the scattering from the PEC screen and its complementary structure is established. In order to efficiently compute the matrix elements from the IE approach, an acceleration technique with the exponential convergence rate is applied to evaluate the doubly periodic Green’s function. The formulations in this technique are appropriately modified so that the new form facilitates numerical calculation for the general cases. Index Terms—Basis functions, integral equations, method of moments, periodic structures.

I. INTRODUCTION

I

N the last several decades, periodic structures have gained intensive interests and attention of researchers in the field of electromagnetics. They can find a variety of applications for periodic structures in the area of electromagnetics [1]–[3], including microwave or quasi-optical filters, polarizers, radomes, artificial dielectrics, semiconductor lasers, and so on. Recently, researchers found that the material with negative permittivity and permeability can be realized by applying the periodic array of conducting objects [4]. The electromagnetic scattering from periodic structures has been investigated by many methods, including the mode-matching method [5]–[7], finite element method (FEM) [8], [9], boundary integral-modal (BI-modal) method [10], finite element-boundary integral (FE-BI) method [11], [12], and integral-equation (IE) method [2], [13]–[17]. The mode-matching method is suitable to the canonical geometry, such as rectangular or circular PEC patches or apertures perforated from PEC screens. The FEM, which is a full-wave approach, can deal with the arbitrary shape and inhomogeneous media. The FE-BI method takes advantage of BI on the top Manuscript received December 21, 2009; revised August 09, 2010; accepted January 29, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. This work was supported in part by the National Science Foundation CAREER Grant ECS-0547161. F.-G. Hu was with the Department of Electrical and Computer Engineering, Iowa State University, Ames, Iowa 50011 USA. He is now with Temasek Laboratories at National University of Singapore 117411, Singapore (e-mail: [email protected]). J. Song is with the Department of Electrical and Computer Engineering, Iowa State University, Ames, Iowa 50011 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165466

and/or bottom surfaces of one unit cell, instead of absorbing boundary condition (ABC) used in the FEM. The application of BI is more accurate than that of ABC at the cost of the partially full coefficient matrix. The IE approaches for periodic structures can be formulated in spectral and spatial domains. The former [2], [13], [14] is limited to planar periodic metallic structures. In addition, the expansion of electromagnetic (EM) quantities into Floquet’s modes has a slow convergence rate. The latter can treat the doubly periodic objects with arbitrary shape [15]–[17]. However, it calls for the efficient calculation of the periodic Green’s function (PGF). Fortunately, some acceleration techniques have been proposed to achieve the fast convergence for the PGF [18]–[21]. In this paper, the electric field integral equation (EFIE) formulation in the spatial domain is employed to calculate scattering from doubly periodic array of PEC objects. The following three issues about the IE approach are addressed. First, the special testing and basis functions are proposed to handle the problem with non-zero normal components of currents at the boundary of one period. As we know, the computational domain for periodic structures is restricted to one period. The objects with periodicity may be truncated by the four-side periodic boundaries (PB). In this case, the electric current flowing out of the boundary of the PEC surface may not be zero. If one adopts the Rao-Wilton-Glission (RWG) basis functions [22] to expand the current, and treat the boundary in the manner which is applied to single PEC plate, namely, one does not assign unknowns on the boundary, the solution to the current will be probably inaccurate or even wrong. This is because this procedure enforces the condition of zero outgoing current on the truncated boundary of PEC. Mittra, Chan, and Cwik in [2] and Cwik in [23] point out it is only necessary to sample the current along one of the opposing boundaries which are related by the periodic boundary condition (PBC). In [24], Simon applied the modified RWG basis functions to analyze the periodic structures. This paper fully addresses the issue about modeling of outgoing current on the PB. The special test and basis functions on the truncated boundary are proposed to handle this problem. In Appendix A, the calculation of the matrix elements of MoM is discussed. It is shown that the matrix elements of MoM only are of the double-surface integral involving the PGF rather than the gradient of PGF. Thus, the matrix elements have the similar form with the MoM approach for the PEC surface without periodicity. The application of the designed test and basis functions make the implementation of MoM straightforward. Second, the relationship is addressed between the scattering from the PEC screen with periodicity and its complementary

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structure. The Babinet’s principle for this dual problem has been proved in several different ways [25]–[27]. As shown in [25], the Babinet’s principle can come out from the equivalence principle. In [27], the equations of Babinet’s principle for single aperture were derived using the IE approach. However, the derivation is complex due to the application of the scalar equations. This paper shows a simple way for proof of the Babinet’s principle for periodic structures. In Appendix B, it will be shown the proof of Babinet’s principle involves the IE formulation with the compact form and the periodic Green’s function. In addition, the relationship for the reflection and transmission coefficients between the PEC screen with periodicity and its complementary structure will be derived using the IE approach. It should be mentioned that this relationship also can be obtained directly from Babinet’s principle [2], [6], [23]. In this paper, instead of starting from Babenit’s principle, it is first derived of the expressions for the reflection and transmission coefficients, which are the integral of (or ) with the Floquet’s modes. Then, the relationship between the PEC screen and its complementary structure is found, which will be shown in Appendix C. Furthermore, it is found that the obtained relationship has practical applications in the calculation of scattering from the PEC screen. For periodic apertures perforated from the screen, one can apply integral equations about the electric current on the PEC part of the screen. However, the unknown density near the aperture should be made large enough to achieve the accurate solution for the scattering. In contrast, it will be easier to achieve the convergence of solution if one solves the integral equation for the electric current on PEC patches in its dual problem. Then one may find the solution for scattering from the apertures using the relationship mentioned. Finally, the acceleration technique in [21] is applied to evaluate the PGF. It has an exponential convergence rate and can be implemented easily. One can take advantage of the intrinsic function in Fortran to evaluate the error function involved in [21] since its argument can be real number. The formulations in this technique are appropriately modified so that the new form facilitates numerical calculation for the general cases. Therefore, there will exist no obstacles to effectively evaluate the matrix element of the IE approach. For the first two issues above, the first two subsections in next section give a formal framework for the known techniques in [2], [23], [24], [27]. II. FORMULATION Fig. 1 shows the unit cell including the 3-D PEC object in a skew 2-D lattice. and are the primitive lattice vectors in . the -plane. Without loss of generality, let is the incident electric field. is the polarization angle. and . are incident angles. The wave vector is . The incident plane wave of -polarization and -polarization are the th-order TMz and TEz Floquet’s modes, respectively. Here TMz (TEz) indicates the magnetic (electric) field transverse to -direction.

Fig. 1. Unit cell including the 3-D PEC object in a skew 2-D lattice.

A. Basis and Testing Functions and EFIE Matrix Equation The EFIE is based on the boundary condition that the total tangential electric field on the PEC objects is zero. It is expressed as [28], [29]

(1) where is a doubly periodic Green’s function. The basis functions on triangular elements are employed to discretize the electric current. After using the method of moments (MoM), one can obtain the current on the PEC object in one unit cell. Then, the current in the other unit cells can be found by using the Floquet’s theorem. The PEC surface is discretized into triangular elements. As shown in Fig. 2, there are three types of edges on triangular elements: 1) inside the domain enclosed by the periodic boundary; and ; 3) on periodic boundary 2) on periodic boundary and . For the first type of edges, the RWG basis functions [22] are adopted to be both basis functions and testing functions (2) and . indicate the coorwhere , which are opposite to dinate of the vertices of the triangles the common edge. is the area of the triangle and is the length of the common edge. For the first type of edges, the testing function is applied to be the same as the basis function. For the other two types, it is assumed the edges exist in a dual are translated with pair. As shown in Fig. 2, the edges on to get its counterpart on . One dual pair the displacement of edges are associated with one unknown. The corresponding and testing function are given by basis function (3)

(4) are translated with the displaceSimilarly, the edges on to get its counterpart on . The corresponding basis ment and testing basis functions are given by functions

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Fig. 2. Three types of edges on triangular elements. The periodic boundary (PB) C is comprised by P ; P ; P , and P . The PB C is comprised of P ; P ; P , and P . The PB C is comprised of P ; P ; P , and P . The PB C is comprised of P ; P ; P , and P .

(5)

Fig. 3. One unit cell of the PEC screen and its complementary structure. (a) Structure I: One unit cell of the PEC screen. (b) Structure II: Complementary structure of (a).

(6) , and are the same as in (2). The definition of is the length of edges . The special basis functions for the edges on the periodic boundary can guarantee that the outgoing elecsatisfies the periodic boundary condition tric current . Because the basis function is applied to the edges on the periodic boundary, the condition of zero outgoing current is not enforced on the truncated boundary of PEC any more. With respect to this issue, Cwik proposed to only sample the current along one of the opposing boundaries which are related by the PBC in [23]. In addition, Simon designed the modified RWG basis functions for analysis of the periodic structures in [24]. In this paper, the testing functions are also formulated, which are different from the basis functions. After applying weighted residue method, one obtains the matrix equation (7)

Actually, there are two ways to generate the matrix equation from EFIE for the periodic array. The first way is to apply the RWG basis functions as the basis and testing functions to the whole array. Then, with the help of Floquet’s theory or periodic boundary condition (PBC), the resultant infinite matrix equations can be rearranged and reduced to finite matrix equations about the unknowns on one unit cell. In this way, the computational domain should first be defined as more than one unit cell. The second way, which is proposed in this paper, is to first restrict the computational domain to be one unit cell. Then, the three types of basis and testing functions are applied to the computational domain in one unit cell. Among them, two types of basis and testing functions are designed for the edges on the periodic boundary. The dimension of the resultant matrix equation is naturally finite. It is worth noting that the PBC is not applied by this approach, rather than guaranteed by the designed basis function. These two ways result in the same matrix equation. The difference is that the second way starts from the viewpoint of basis function for the edges on the periodic boundary. B. Relationship of Scattering From the PEC Screen With Periodicity and Its Complementary Structure

where (8) (9) In (8), . The derivation of (9) is given in Appendix A. It is worth (or noting that the phase factor of the testing function ) has a different sign from the basis function (or ). Therefore, as given in Appendix A, the line-surface integral is involves only the double cancelled and the matrix element surface integral. Furthermore, the difference of sign can cancel the phase shift of the testing and basis functions. Thus, the diagonal elements are dominant in the resultant coefficient matrix.

Fig. 3 shows two planar structures: the one unit cell of the PEC screen and its complementary structure. Here, assume the screens are located at . In Structure I, the PEC parts of the screen are denoted by , and the apertures in the screen by . In Structure II, the apertures are denoted by , and the PEC parts by . Babinet’s principle [27] describes the basic relationship of the scattering fields from the PEC screen and its complementary structures. In Appendix B, the proof of this principle is given in a simple way. It involves the integral equations. For any one of these two structures, there are two ways to calculate the reflection and transmission coefficients. One way is to apply the magnetic field integral equation (MFIE) on the aperdenote a unit surface in ture part of the screen. Let

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plane at which the reflection coefficient is computed. denotes plane at which the transmisa unit surface in sion coefficient is calculated. Assume the incident wave is the th-order Floquet’s mode. Its transverse component is given by

In conclusion, the relationship for the reflection and transmission coefficients between the PEC screen and its complementary structure can be given by

(10)

(23)

The details about the Floquet’s modes are given in Appendix C. For the th-order Floquet’s mode, the reflection and transmission coefficients are given by (11) (12) where and denotes the aperture surface in the unit cell. is the Kronecker delta function. The other way is to apply the EFIE on the PEC part of the screen to obtain

(22)

(24) By using the above relationship, the problem for seeking MFIE solution is transformed to that of the EFIE. It should be mentioned that this relationship can also be obtained on the basis of the Babinet’s principle, which was discussed by Mitrra et al. in [2], Chen in [6], and Cwik in [23]. C. Application of the PGF . can be efficiently evaluated by appliLet cation of the following acceleration technique discussed in [21]. When (25)

(13) where (14) Here, denotes the PEC surface in the unit cell. Derivation of (11)–(14) are given in Appendix C. If these two approaches are applied to the same periodic structure with the same incident wave, in principle, it should hold that (15) Now these two approaches are applied to the dual periodic structures. Assume MFIE is applied to Structure I (II) with the TEz incident wave, and EFIE is applied to Structure II (I) with the TMz incident wave. Using the IE approach in Appendix B, it is easy to find

The above expression is the spectral representation of the PGF, which is with the exponential convergence. When (26) , and are given as follows. and are where modified from [21] to facilitate the numerical implementation (27)

(16) (28)

In addition, the following conditions are satisfied

(17) As a result, the following relationship can be found (18) (19) Assume MFIE is applied to Structure I (II) with the TMz incident wave, and EFIE is applied to Structure II (I) with the TEz incident wave. In a similar manner, one can find

(29) where

(20) (21) because the following conditions are satisfied

In [21], involves the calculation of order to achieve the specified accuracy for

or . In , the number of

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terms of the summation is generally large. As a result, or often may be beyond the range of floating point numbers. is modified to To avoid this situation, the expression for the present form (27). when apIt is obvious that the limit exists for proaches to zero. If the plane wave is normally incident, at . There is no trouble to calculate for this case using (29). It should be noted that unlike the free-space Green’s function, the periodic Green’s function has the singular points in both and , the spatial and spectral domains. When and , respectively. In PGF encounters the singularity in the case of , one has to deal with the singularity [30]–[33] when evaluating the matrix elements obtained from the MoM method. III. NUMERICAL RESULTS The EFIE approach on PEC is implemented using Fortran language. The evaluation of matrix elements is time-consuming because of the very frequent direct calculation of the PGF. To improve the efficiency of this approach, the interpolation technique is applied to the tabulated PGF. To validate this approach, the reflection and transmission coefficients of several structures are calculated. The first example is the infinite PEC plate. The PEC plate cm and is infinitely thin. The 2-D lattice has cm. Assume the geometrical center of the unit cell . is at the origin. The PEC plate is located at The incident plane wave has the incident angles and . The incident electric field is along -direction. As shown in Appendix C, this incident plane wave is the th TEz Floquet’s mode. The special testing and basis function on PB are applied to this example. The total numbers of edges and unknowns are 630 and 600, respectively. The Fortran program is run on a PC machine with a 3.2 GHz Pentium IV processor. The CPU time is about 2.99 s per frequency point. This array of PEC plate is essentially the infinite PEC plate. Thus, the induced normalized electric current should be exactly . Fig. 4 shows the normalized current distribution at 9 GHz. In Fig. 4(a), the current varies with along the line at mm. In Fig. 4(b), the current varies with along the line at mm. As shown in this figure, there are good agreements between the numerical results and exact solution. Fig. 5 shows the reflection coefficient of the th-order TEz Floquet’s mode for this array. The exact solution of reflection coef. If the testing and basis functions (3)–(6) are ficient is applied to the edges on the PB, there are excellent agreements between the numerical results and the exact solution. However, if there is no testing and basis functions assigned to the edges on PB, the reflection coefficient is totally wrong. The second example is the same as the first one, except for a rectangular aperture perforated in the PEC plate. The rectancm and height gular aperture has the width cm. Its geometrical center is also located at the origin. The transmission coefficient is computed by using two approaches. When directly using (14), i.e., EFIE on the PEC part, the special testing and basis functions for the edge on PB is applied. The number

= ^1 + ^p3 3

= 9 GHz. a = x^2 cm and = 0 . (a) Along x-direction. (b)

Fig. 4. Normalized current distribution at f a x y = cm.  and  Along y -direction.

= 60

= ^1 + ^p3 3

Fig. 5. Reflection coefficient of the 0th-order TEz mode for the infinite PEC x cm and a x y = cm.  and  . plate. a

= ^2

= 60

=0

of unknowns is 3601. The CPU time is about 377 s for each frequency point. When using (24), the EFIE is first applied to calculate the transmission coefficient for the complementary structure. Because the PEC plate in its complementary structure is inside the unit cell, the special testing and basis functions for the edge on PB are not applied. For the second approach, the number of unknowns is 237. The CPU time is about 2.36 s for

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Fig. 6. Transmission coefficient of the 0th-order TEz Floquet’s mode for the x cm and a x doubly periodic array of rectangular apertures. a : cm and height y = cm. The rectangular aperture has the width l : cm.  and  . l

^p3 3 = 0 12

= 60

=0

= ^2 = 12

= ^1 +

= ^6

= ^6

Fig. 8. Two meshed cylinders in a unit cell. a x mm and a y mm. Each cylinder has the radius r : mm and length L mm. The space mm. between two cylinders is d

= 06 =6

Fig. 7. Reflection coefficient of the 0th-order TEz mode for the PEC strip. a x m and a y : m. The rectangular PEC plate has the width l : m and height l : m.  and  .

05

= ^1

= ^0 5 =05

=0

=0

=

each frequency point. Fig. 6 shows the transmission coefficient of the th-order TEz Floquet’s mode for this array. Good agreements are observed between the results of three approaches. It is worth noting that the first approach demands much more dense mesh to achieve the converged results than the second one. This is due to the singular current distribution near the edge of the

=6

plate. Thus, the second approach is suggested for the periodic array of apertures. In the third example, the PEC strip is simulated to demonstrate the convergence of reflection coefficients. The strip is inm and m. In finite long along -direction. m and height one unit cell, the PEC plate has the width m. The geometrical centers of the unit cell and the PEC plate coincide with each other. The top and bottom sides involve testing and basis functions for the edges on PB. The left and right sides are not associated with the unknowns. The inciand . dent plane wave has the incident angles The incident electric field is along -direction. For this case, the 3-D problem is reduced to the 2-D problem. Then the 2-D IE approach can be used to calculate the reflection coefficient [34], [35]. Fig. 7 shows the convergence of the reflection coefficient of the th-order TEz mode. As shown in this figure, the dense mesh is required to achieve the convergence. This is due to the edge effect of PEC. The last example is the doubly periodic array of two PEC cylinders. Fig. 8 shows one unit cell including the meshed cylinders. The axis of each cylinder is along -direction. mm and mm. Each cylinder has the radius mm and length mm. The space between two cylinders mm. Two cross sections of each cylinder touch two is sides of the periodic boundary. Thus, these two sides involve the testing and basis functions for the edges on PB. The total number of unknowns is 600. The average CPU time is about 3.9 s per frequency point. The reflection coefficients are shown in Fig. 9. As shown in this figure, there are good agreements between the results of the present approach and 2-D IE approach except at some dips. The discrepancy is due to the different mesh type and density for two approaches.

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(A-1) Applying the following identity (A-2) and the surface divergence theorem, one can change the integral of the first term in (A-1) into

(A-3) where denotes two triangular elements associated with , . denotes the unit and denotes two ones associated with bounding the two triangular elevector normal to contour and are the two elements denoted by ments . Assume , as shown in Fig. 2. Because holds at the four , (A-3) edges that do not correspond to the testing function will reduce to

(A-4)

Fig. 9. Reflection coefficient of the 0th-order TMz mode for the two-layer array of PEC cylinders shown in Fig. 8.  and  .

=0

=0

IV. CONCLUSION This paper addresses several issues in the integral-equation (IE) method for scattering from the doubly periodic array. First, the formulation in [21] is modified to facilitate the numerical implementation. Second, special testing and basis functions are proposed for the edges on the periodic boundary. The application of the special basis functions can model the currents flowing out the truncated boundary of PEC. Third, the relationship for the reflection and transmission coefficients between the PEC screen with periodicity and its complementary structure is given. Numerical results are provided to validate the proposed approach.

Here and denote the edges corresponding to on and , respectively. is a constant and can be moved outside the line-surface integral. For the first type of edge, at and , respectively. Thus, the two line-surface integrals are cancelled with each other. and For the second type of edge, at and , respectively. Then the term about line-surface integral is given by

(A-5) The coordinate transformation is used for the first term in the bracket of (A-5). In addition, according to the Floquet’s theorem, (A-6) Thus,

(A-7)

APPENDIX A DERIVATION OF MATRIX ELEMENTS In this section, the derivation of (9) is given. Multiplying the testing function on both sides of (1) and performing the integral over PEC surface give (8) and

Combination of (A-4) and (A-7) gives (A-8) In the similar manner, one can obtain (A-8) for the case of the third type of edges.

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APPENDIX B PROOF OF BABINET’S PRINCIPLE

TABLE I DESCRIPTION OF TMZ AND TEZ FLOQUET’S MODES

i) Let the EM field (A-9) on the PEC screen in the plane be incident in Let the total field in be . ii) Let the EM field

;

(A-10) on the complementary PEC screen be incident in in the plane ; Let the total field in be . Then, the Babinet’s principle asserts that [27]

(A-19)

(A-11)

(A-20)

In fact, one may have (A-12) (A-21) for Problem (i), and (A-22)

(A-13) for Problem (ii). Here and . As shown in satisfy the same integral equation (A-12) and (A-13), and on the same domain. Thus,

Table I shows the vector function for the TMz and TEz Floquet’s modes [16]. Here (A-23)

(A-14) The total field in

When

is given by

and (A-24)

(A-15)

for the TMz case and

for Problem (i), and

(A-25) (A-16)

. for the TEz case. Here The incident plane wave is a special case of (A-19)–(A-22), whose electric field is expressed as

(A-17)

(A-26)

Taking curl of both sides of (A-17) and then taking advantage gives of Maxwell equations and

, and . where It is easy to verify that the incident plane wave is the combination of th TMz and TEz Floquet’s modes. For -polarization

for Problem (ii). Adding (A-15) to (A-16) yields

(A-18)

(A-27) APPENDIX C REFLECTION AND TRANSMISSION COEFFICIENTS OF FLOQUET’S MODES The electric and magnetic fields can be expanded into the TMz and TEz Floquet’s modes

where -polarization

and

. For

(A-28) . where For the MFIE approach, one first can apply the equivalence principle and image theory [25], [26] to find the magnetic field

HU AND SONG: INTEGRAL-EQUATION ANALYSIS OF SCATTERING FROM DOUBLY PERIODIC ARRAY OF 3-D CONDUCTING OBJECTS

in terms of the tangential electric fields on the apertures , one may have Assuming the screen is located at

.

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The transmission coefficient can be found by (A-41)

(A-29)

Substitution of (A-40) into (A-41) yields

where (A-42) (A-30)

It does hold that

. is the reflected field by infinite PEC surface. From (A-29), the MFIE on the aperture can be established (A-43)

(A-31) The scattered field above the aperture can be found by

for the TEz case, and (A-32)

The transverse components of incident and reflected magnetic fields are given by, respectively (A-33)

(A-44) for the TMz case. Here denotes and TMz cases, one can obtain

or

. Thus, for both TEz

and (A-34) At panded as

, the transverse component of

(A-35) Using (A-35) and the orthogonality of Floquet modes yields (A-36) because . The reflection (transmission) coefficient is defined as the ratio of the Floquet’s modes coefficient of the reflected (transmitted) electric field to that of the incident electric field. From (A-32), (A-34), (A-19), and (A-21), the reflection coefficient can be found by (A-37) Substitution of (A-36) into (A-37) yields (A-38) At

(A-45)

can be ex-

can be expanded as (A-39)

Using (A-39) and the orthogonality of Floquet modes yields (A-40)

and are above and below the scatAssume the planes terers, respectively. Substitution of (A-45) into (A-38) and (A-42) yields (11) and (12). The derivation of (13)–(14) is similar to that in [16]. REFERENCES [1] T. K. Wu, Frequency Selective Surface and Grid Array. New York: Wiley, 1995. [2] 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. [3] E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett., vol. 58, pp. 2059–2062, May 1987. [4] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, pp. 4184–4187, May 2000. [5] C. C. Chen, “Transmission through a conducting screen perforated periodically with apertures,” IEEE Trans. Microwave Theory Tech., vol. 18, no. 9, pp. 627–632, Sep. 1970. [6] C. C. Chen, “Scattering by a two-dimensional periodic array of conducting plates,” IEEE Trans. Antennas Propag., vol. 18, no. 5, pp. 660–665, Sep. 1970. [7] C. C. Chen, “Diffraction of electromagnetic waves by a conducting screen perforated periodically with circular holes,” IEEE Trans. Microwave Theory Tech., vol. 19, no. 5, pp. 475–481, May 1971. [8] I. Bardi, R. Remski, D. Perry, and Z. Cendes, “Plane wave scattering from frequency-selective surfaces by the finite-element method,” IEEE Trans. Magn., vol. 38, no. 2, pp. 641–644, Mar. 2002. [9] G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 973–980, Jun. 2000.

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[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] S. D. Gedney, J. F. Lee, and R. Mittra, “A combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech., vol. 40, no. 2, pp. 363–370, Feb. 1992. [12] T. F. Eibert, J. L. Volakis, D. R. Wilton, and D. R. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 843–850, May 1999. [13] R. Pous and D. M. Pozar, “A frequency-selective surface using aperture-coupled microstrip patches,” IEEE Trans. Antennas Propag., vol. 39, no. 12, pp. 1763–1769, Dec. 1991. [14] C. Wan and J. A. Encinar, “Efficient computation of generalized scattering matrix for analyzing multilayered periodic structures,” IEEE Trans. Antennas Propag., vol. 43, no. 11, pp. 1233–1242, Nov. 1995. [15] A. W. Mathis and A. F. Peterson, “Efficient electromagnetic analysis of a doubly infinite array of rectangular apertures,” IEEE Trans. Antennas Propag., vol. 46, no. 1, pp. 46–54, Jan. 1998. [16] I. Stevanovic´, P. Crespo-Valero, K. Blagovic´, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 10, pp. 3688–3697, Oct. 2006. [17] X. Dardenne and C. Craeye, “Method of moments simulation of infinitely periodic structures combining metal with connected dielectric objects,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2372–2380, Aug. 2008. [18] N. Kinayman and M. I. Aksun, “Comparative study of acceleration techniques for integrals and series in electromagnetic problems,” Radio Sci., vol. 30, no. 6, pp. 1713–1722, 1995. [19] K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the Green’s function for the Helmholtz operator on periodic structures problems,” J. Comput. Phys., vol. 63, no. 6, pp. 222–235, 1986. [20] M. J. Park and S. Nam, “Efficient calculation of the Green’s function for multilayered planar periodic structures periodic structures,” IEEE Trans. Antennas Propag., vol. 46, no. 10, pp. 1582–1583, Oct. 1998. [21] M. G. Silveirinha and C. A. Fernandes, “A new acceleration technique with exponential convergence rate to evaluate periodic Green functions,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 347–355, Jan. 2005. [22] S. M. Rao, D. R. Wilton, and A. W. Glission, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 409–418, May 1982. [23] T. Cwik, “Scattering From General Periodic Screens,” Ph.D. dissertation, University of Illinois, Urbana, IL, 1986. [24] P. S. Simon, “Modified RWG basis functions for analysis of periodic structures,” in Proc. IEEE MTTS Int. Symp., Jun. 2002, pp. 2029–2032. [25] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [26] R. E. Collin, Field Theory of Guided Waves, 2nd ed. New York: Wiley-Interscience, 1991, ch. 1, pp. 1–54. [27] E. T. Copson, “An integral equation method for solving plane diffraction problems,” Proc. Roy. Soc. (London) Ser. A, vol. 186, no. 1, pp. 100–118, 1946. [28] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989, ch. 12, pp. 670–742. [29] W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithm in Computational Electromagnetics. Boston: Artech House, 2001, ch. 3, pp. 77–118.

[30] R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green’s function or its gradient on a plane triangle,” IEEE Trans. Antennas Propag., vol. 41, no. 10, pp. 1448–1455, Oct. 1993. [31] M. G. Duffy, “Quadrature over a pyramid or cube of integrands with a singularity at a vertex,” SIAM J. Numer. Anal., vol. 19, no. 6, pp. 1260–1262, Dec. 1982. [32] D. R. Wilton, S. M. Rao, A. W. Glission, D. H. Schaubert, O. M. Al-Bundak, and C. M. Bulter, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag., vol. 32, no. 3, pp. 276–281, Mar. 1984. [33] J. R. Mosig, “Integral-eqation technique,” in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, T. Itoh, Ed. New York: Wiley, 1989, ch. 3, pp. 133–213. [34] F. G. Hu and J. Song, “Integral equation analysis of scattering from multilayered periodic array using equivalence principle and connection scheme,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 848–856, Mar. 2010. [35] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. Oxford: Oxford Univ. Press, 1998, ch. 7, pp. 261–300.

Fu-Gang Hu (M’08) received the B.Eng. and M.Eng. degrees from Xidian University, Xi’an, China, in 1999 and 2002, respectively, and Ph.D. degree in electrical engineering from Iowa State University, Ames, in 2010. He was an Associate Scientist from 2002 to 2007, and is a Research Scientist, since 2010, with Temasek Laboratories, National University of Singapore, Singapore. His current research interest includes electromagnetic modeling using numerical techniques.

Jiming Song (SM’99) received the B.S. and M.S. degrees, both in physics, from Nanjing University, China, in 1983 and 1988, respectively, and the Ph.D. degree in electrical engineering from Michigan State University, East Lansing, in 1993. From 1993 to 2000, he worked as a Postdoctoral Research Associate, a Research Scientist and Visiting Assistant Professor at the University of Illinois at Urbana-Champaign. From 1996 to 2000, he worked as a Research Scientist at SAIC-DEMACO. He was the principal author of the Fast Illinois Solver Code (FISC), which has been distributed to more 400 government and industrial users. From 2000 to 2002, he was a Principal Staff Engineer/Scientist at Digital DNA Research Lab., Semiconductor Products Sector, Motorola, Tempe, AZ. In 2002, he joined the Department of Electrical and Computer Engineering, Iowa State University, Ames, as an Assistant Professor and is currently an Associate Professor. His research has dealt with modeling and simulations of interconnects on lossy silicon and RF components, the wave scattering using fast algorithms, the wave propagation in metamaterials, and transient electromagnetic field. He has co-edited one book and published seven book chapters, 44 journal papers and 125 conference papers. Dr. Song received the NSF Career Award in 2006 and the Excellent Academic Award from Michigan State University in 1992. He was selected as a National Research Council/Air Force Summer Faculty Fellow in 2004 and 2005.

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A Calderón Multiplicative Preconditioner for the PMCHWT Integral Equation Kristof Cools, Francesco P. Andriulli, Member, IEEE, and Eric Michielssen, Fellow, IEEE

Abstract—Electromagnetic scattering by penetrable bodies often is modelled by the Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) integral equation. Unfortunately the spectrum of the operator involved in this equation is bounded neither from above or below. This implies that the equation suffers from dense discretization breakdown; that is, the condition numbers of the matrix resulting upon discretizing the equation rise with the mesh density. The electric field integral equation, often used to model scattering by perfect electrically conducting bodies, is susceptible to a similar breakdown phenomenon. Recently, this breakdown was cured by leveraging the Calderón identities. In this paper, a Calderón preconditioned PMCHWT integral equation is introduced. By constructing a Calderón identity for the PMCHWT operator, it is shown that the new equation does not suffer from dense discretization breakdown. A consistent discretization scheme involving both Rao-Wilton-Glisson and Buffa-Christiansen functions is introduced. This scheme amounts to the application of a multiplicative matrix preconditioner to the classical PMCHWT system, and therefore is compatible with existing boundary element codes and acceleration schemes. The efficiency and accuracy of the algorithm are corroborated by numerical examples. Index Terms—Boundary integral equations, Calderon multiplicative preconditioning, dense discretization breakdown, penetrable objects, Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) equation.

I. INTRODUCTION HE Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) integral equation is widely used to model time-harmonic electromagnetic scattering by homogeneous penetrable objects [1]. The PMCHWT only requires surface discretizations, resulting in (comparatively) small interaction matrices that can be applied rapidly to arbitrary vectors by using fast multipole and related algorithms (see [2] and references therein), and yields solutions that automatically satisfy the radiation condition. That said, the PMCHWT integral equation has several drawbacks. Indeed, as will be shown in this work, the PMCHWT operator’s spectrum is neither bounded from above

T

Manuscript received November 11, 2009; revised December 23, 2010; accepted March 12, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. K. Cools is with the University of Gent, Antwerpen 9000, Belgium (e-mail: [email protected]). F. P. Andriulli is with the Politecnico di Torino, 10129 Torino, Italy (e-mail: [email protected]). E. Michielssen is with the University of Michigan, Ann Arbor, MI 48109 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165465

or below. This implies that the PMCHWT integral equation is susceptible to dense discretization breakdown: if the mesh edge length tends to zero, the condition number of the system matrix resulting upon discretizing the PMCHWT integral equation grows arbitrarily large. This renders the iterative solution of the system very difficult for densely meshed surfaces. When modelling scattering from densely meshed penetrable surfaces, it may appear tempting to use the Müller integral equation [3] instead of the PMCHWT integral equation. This is because the operator involved in the former is a compact perturbation of the identity, and as a result the equation does not suffer from dense discretization breakdown. Indeed, the resulting system matrices are well-conditioned regardless the mesh edge length. Lamentably, it is documented [4] that, upon discretization, the Müller integral equation yields numerical solutions that are far less accurate than those obtained by discretizing the PMCHWT integral equation on the same mesh. Thus the incentive for constructing a regular equation that challenges the PMCHWT integral equation’s accuracy remains. A dense discretization breakdown phenomenon, similar to the one intrinsic to the PMCHWT integral equation, plagues the electric field integral equation (EFIE) that is used to model scattering by perfect electrically conducting (PEC) bodies [5]. It is known, however, that the EFIE’s dense discretization breakdown can be remedied by leveraging Calderón identities [6], [7]. These identities state that the EFIE operator is self-regularizing, i.e., its square is a compact perturbation of the identity. Recently, this preconditioning strategy was extended to the combined field integral equation (CFIE) which contains the EFIE operator as one of its constituents [8]. In this contribution, the Calderón preconditioning scheme of [7] is extended into the realm of scattering by dielectric objects by constructing a Calderón identity for the PMCHWT operator. More specifically, it is shown that the PMCHWT integral equation exhibits a self-regularizing property, similar to that of the EFIE. The self-regularizing property of the PMCHWT operator, however, does not follow immediately from the Calderón identities. The operators whose spectrum needs to be assessed are significantly more intricate due to the presence of two sets of material parameters. It will nevertheless be shown, both by computations and numerical experiments that the PMCHWT has a self-regularizing property. Next, the newly obtained Calderón preconditioned PMCHWT integral equation is discretized. The choice of basis functions and testing functions is motivated. The proposed discretization scheme is shown to yield a multiplicative preconditioner for the classical PMCHWT system, which implies that the scheme can be easily implemented in existing PMCHWT

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integral equation solvers and that it is fully compatible with a wide range of matrix-vector product accelerators such as the MLFMA [2]. This paper is organized as follows. In Section II, the classical PMCHWT integral equation and its discretization are revisited. The source of the dense discretization breakdown is discussed and illustrated by a concrete example. In Section III, the Calderón preconditioned PMCHWT integral equation is introduced. It is proven that its operator indeed is both bounded and well-posed, and thus expected to yield well-conditioned system matrices upon discretization. A discretization scheme that results in a multiplicative preconditioner is detailed. Finally, in Section IV, numerical results are presented that testify to both the efficiency and accuracy of the proposed scheme. II. THE PMCHWT INTEGRAL EQUATION A. Background and Formulation Consider a bounded domain with boundary that is occupied by a scatterer with constant permittivity , permeability , wave number , and characteristic impedance and can be complex. The exterior domain is filled with a background medium with permittivity , permeability , wave number , and characand are assumed real. The teristic impedance external normal to is denoted . The incident electromagnetic , produced by sources in , illuminates the scatfield terer. Upon interaction of with the total fields result. These fields satisfy the following external and internal electric and magnetic field integral equations [9]

(1)

(2) where, the operators

and

are

(3)

(1), however, yields the Poggio-Miller-Chan-Harrington-WuTsai (PMCHWT) equation (5) where (6) (7) (8) This equation has a unique solution [10]. A classical scheme to discretize the PMCHWT integral equation is described next. A triangulation of is defined. . It is The number of edges in this triangulation is denoted well-known that belong to the space of divergence conforming functions such that

(9) and should be approximated by This implies that expansions of div-conforming basis functions, e.g., Rao-Wilton[11], as Glisson functions (RWG), (10) (11) A linear system of equations in the unknown coefficients is obtained by substituting (10)–(11) into (5) and computing the inner product of the resulting equation with a suitable set of testing functions. Since the range space of the PMCHWT operator consists of tangential traces of electric and magnetic fields, it also is con. Therefore, the equations should be tested tained in dual of . The by functions that reside in the dual space of is [12]. This implies that sets of curl-conforming functions such as the “rotated” Rao-Wilton[13] are good candidates to test Glisson (RWG) functions both equations. This discretization scheme yields the system of linear equations (12)

(4) where are given by (3) and (4) upon replacing The operators and by . The integral in the first term of the definition of contains a hypersingular integrand and should be interpreted in the Cauchy principal value sense. Neither (1) nor (2) can be solved for the trace of the total electromagnetic field on . The kernel of (1) consists of all radiating with resolutions to the homogeneous Maxwell equations in spect to the background medium. The kernel of (2) comprises the traces of solutions to the homogeneous Maxwell equations in with respect to the scatterer’s medium. Subtracting (2) from

(13) (14) (15) with

, and (16)

COOLS et al.: A CALDERÓN MULTIPLICATIVE PRECONDITIONER FOR THE PMCHWT INTEGRAL EQUATION

Similar definitions hold for . Furthermore,

, and

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is the (scalar) spherical harmonic. Indeed, where this basis as

(21) (22)

(17) (18) Operators whose singular values tend towards infinity are called unbounded or discontinuous. Operators that are not unbounded are called bounded or continuous. Bounded operators whose singular values cluster at zero are called ill-posed. If an operator is not ill-posed, it is called well-posed. The sum of a continuous and a discontinuous operator is a discontinuous operator. The sum of a well-posed and an ill-posed operator is a well-posed operator. Upon discretization, operators that are discontinuous or illposed will give rise to ill-conditioned system matrices. This is so because the singular values of the system matrix tend to approximate those of the operator, which either go to infinity or zero. In both situations, the system matrix’ condition number, i.e., the ratio between the matrix’ largest and smallest singular value, will tend to infinity with the discretization density. Upon discretization, operators that are continuous and wellposed will give rise to well-conditioned matrices. This is so because the singular values of the matrix tend to approximate those of the operator, which are bounded from above and below, giving rise to a bounded condition number as the discretization density increases. Compact operators are continuous and ill-posed operators. Identities are continuous and well-posed operators. Second kind operators are the sum of an identity and of a compact operator. As a consequence, second kind operators are continuous and well-posed and, upon discretization, give rise to well-conditioned system matrices. The operator is both discontinuous and ill-posed. The operator is compact. The considerations above suggest that the in the definition of the PMCHWT operpresence of and ator could negatively affect the conditioning of the discretized PMCHWT system matrix. The next section will show that this is indeed the case. From the considerations above, it is desirable to turn the PMCHWT operator into a second kind operator, and this will be done in Section III. B. Spectral Analysis To gain further qualitative insights into the behavior of the singular value spectrum of the operator (5), and hence indirectly of the system matrix (12), the action of the operator on the traces of vector spherical harmonics on a sphere is considered. The arguments here are not unlike those in [14]. For a spherical geometry, the operators and are (skew-)diagonal in a basis of the vector spherical harmonics

acts on

while for

it holds that (23) (24)

The functions and Hankel functions defined as

denote the Riccati Bessel and (25) (26)

where are the spherical Hankel functions [15]. Functions and derivatives functions of and Similar expressions hold for [16]

Bessel and first kind and denote the with respect to . . Note that for large

(27) (28) (29) (30) The Galerkin discretization of the PMCHWT operator in the basis of spherical harmonics yields

(31)

where denotes the inner product. By using (27)–(30) together with the Gershgorin disk theorem [17], it can be concluded that the condition number of the matrix in (31) will grow , when . This suggests that the numerical soluas tion of (12) will be increasingly difficult for meshes with decreasing element sizes (dense discretization breakdown). III. THE CALDERÓN PRECONDITIONED PMCHWT INTEGRAL EQUATION

(19)

For PEC objects, the PMCHWT integral equation reduces to the EFIE [18]

(20)

(32)

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Just as its PMCHWT extension, the EFIE suffers from dense discretization breakdown. The EFIE, however, can be regularized by leveraging the Calderón identities (33) (34) is a compact perturbation of the idenEquation (33) says that therefore is expected to yield a welltity. Discretization of conditioned system matrix [7]. Not surprisingly, the Calderónpreconditioned EFIE (35) overcomes the conditioning problems of the standard EFIE and has been studied extensively [13], [14], [19], [7]. Since the EFIE constitutes the high conductivity limit of the PMCHWT integral equation, the question arises whether this regularization procedure can be extended to the latter as well. To this end, the Calderón preconditioned PMCHWT integral equation (36) is considered and its spectral properties are studied. In what folwill be investilows, the spectral properties of the operator gated.

+

+

Fig. 1. Eigenvalue distribution of T; T ; T T ; KT T K; KT T K at  , (b)). low frequencies (ka  , (a)) and at high frequencies (ka

=

=4

A. Spectral Analysis of the Calderón Preconditioned PMCHWT Integral Equation Consider the operator

(37)

and in the high frequency regime . The eigenvalues are plotted in the complex plane, together with the eigenvalues of for comparison (Fig. 1(a)–(b)). As expected, the spectrum of comprises two branches, one reaching to infinity and the other accumulating at zero. In the absence of material contrast, the spectrum accumulates at 1/4, as predicted by the Calderón accumulate at two different but identity. The eigenvalues of and . still finite and non zero values: To understand this, consider the expansion of into its hypersingular and weakly singular terms. The tildes remind that the main frequency dependency has been extracted expands to explicitly from these terms. The product

1) Diagonal Blocks: The diagonal block operator satisfies (39)

(38) In the derivation of (38), (33) was used. The terms and are obviously compact due to the compactness of and . Even though no Calderón identities such as (33) exist for and , they are well-behaved [14]. the mixed products To establish this, an eigenvalue analysis of both and has m composed of been performed for a sphere of unit radius material in the low frequency regime a

The first term disappears because the image of comprises surface curls of scalar functions on the boundary, which are divergence free and thus in the kernel of . The last term is a compact contribution and does not affect the position of the accumulation points in the spectrum. In the absence of contrast, the coefficients of the second and third terms both are 1. From this, it can be concluded that the accumulation points of the spectra of and (wich are compact perturbations of both and , respectively), both are 1/4. Thus, the accumulation points of the spectrum of are at and . This analysis implies that the method will break down when applied to very high contrast media.

COOLS et al.: A CALDERÓN MULTIPLICATIVE PRECONDITIONER FOR THE PMCHWT INTEGRAL EQUATION

In the high frequency case, the eigenvalues corresponding to low index values are scattered over a wider region. This however, does not affect the position of the accumulation points. In other words, the condition number can grow near resonant frequencies, i.e., when one of the eigenvalues approaches the complex plane’s origin. Nevertheless, dense grid breakdown is absent, even at high frequencies. To conclude, the diagonal blocks are continuous and wellposed operators, corresponding to well-conditioned blocks in the system’s matrix, especially for moderate material contrasts. 2) Off-Diagonal Blocks: The upper right off-diagonal block operator satisfies

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well-posed. Therefore, in contrast with the standard PMCHWT integral equation, the Calderón-preconditioned PMCHWT integral equation (36) is expected to yield well-conditioned system matrices upon discretization. B. Discretization of the Calderón Preconditioned PMCHWT Integral Equation As in the case of the Calderón-preconditioned EFIE [7], the discretization of the Calderón-preconditioned PMCHWT integral equation will necessitate the use of both RWG basis funcand Buffa-Christiansen (BC) basis functions . For tions a detailed description of BC functions and their properties, see [21]. The discretized Calderón-preconditioned PMCHWT integral equation reads (43) where , and spectively, and

have been defined in (13), (14), and (15), re-

(40) (44) where the second transition is obtained by using (34). Note that (45) and (46)

(41) so that the integral operator has a weakly singular kernel and therefore is compact. In fact is clearly weakly singular, while (42) is weakly singular as so that well. It is well known that integral operators with weakly singular kernels are compact [9]. It follows that is compact. Note that (42) contains the same singularity cancellation effect occurring when defining the Muller equation [20]. In a similar way it is shown that the other three terms appearing in (40) are compact. It can thus be concluded that the off-diagonal block (40) is compact. Similar arguments can be invoked to show the compactness of the lower left off-diagonal block in (37). These theoretical considerations agree with numerical results and (Fig. 1(a)–(b)). The eigenvalues of both accumulate at zero. Analogously to and , the eigenvalues scatter over a wider region in the high frequency regime, but this does not affect the position of the accumulation points. Numerical examples will show that it is not necessary to form explicitly. This significantly the regular operator simplifies the implementation of the method. From the considerations in Subsections III.A.1 and III.A.2 is both bounded and it can be concluded that the operator

The definition of the matrices , and parallels those of , and in (13). Note that in the linear system (43), the matrix will act as a preconditioner for the standard PMCHWT linear system and matrix . Moreover the choice of the basis functions is justified by the fact that both matrices and have to be discretizations of the PMCHWT operator such that • The domain of the operator is discretized with div-conforming basis functions. • The range of the operator is discretized with curl-conforming basis functions. • The Gram matrix linking the range of the rightmost operator to the domain of the leftmost is well-conditioned. IV. NUMERICAL RESULTS This section presents numerical results that corroborate the theory and demonstrate the effectiveness of the proposed preconditioning scheme as well as its applicability to complex problems. Remark: To provide a fair comparison with the Calderón-preconditioned PMCHWT integral equation, the system (12) was reordered as (47) It is known that iterative solution of (47) is faster than that of (12) because the dominant contributions (that arise from discretizations of ) are on the diagonal. The Calderón-preconditioned PMCHWT integral equation was tested on a sphere of unit radius, residing in a background

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Fig. 3. The far-field radiated by a sphere of relative permittivity 2.25 and permeability 1.0 when illuminated by a plane wave of wavelength 2.0 meter, as computed with the Mie series, the PMCHWT integral equation and the Calderón preconditioned PMCHWT integral equation.

Fig. 2. The condition numbers and iterations counts of the PMCHWT and Calderón preconditioned PMCHWT systems for scattering by a sphere (a) and a cube (b) as a function of the mesh parameter.

medium with and the incident electric field

. The sphere is illuminated by (48)

meter, and has and with and . The simulation is carried out for five uniform discretizations with the mesh parameter (the maximum edge length in the triangulation) ranging from 0.6 to 0.2 meter. For each mesh, the condition number of the system matrix was computed, and compared to the condition numbers of the classical PMCHWT integral equation system matrix on the same meshes (Fig. 2(a)). Clearly, the condition numbers of the systems resulting upon discretization of the classical , PMCHWT integral equation tend towards infinity as whereas the condition numbers of the systems resulting upon discretization of the Calderón-preconditioned PMCHWT integral equation remain virtually constant. Even for moderately dense meshes, the difference is significant. The high condition numbers are reflected in the number of iterations needed to solve the systems using the TFQMR iterative method [22]. Next, the correctness of the results obtained using the Calderón-preconditioned PMCHWT integral equation is verified. The far-field scattered by the sphere is evaluated by solving the classical PMCHWT and the Calderón-preconditioned PMCHWT

Fig. 4. Magnitude of the tangential trace of the magnetic field on the surface of the sphere computed using the Mie-series, the PMCHWT integral equation, and the Calderón preconditioned PMCHWT integral equation.

with

integral equations, and by using the analytical Mie series solution. The sphere was modeled by a mesh comprising 514 triangles, resulting in a total number of 1542 unknowns. The standard PMCHWT integral equation converged within a tolerance in 371 iterations while the Calderón-precondiof tioned PMCHWT integral equation converged in 9 iterations. The far-field is computed along the meridian in the plane (Fig. 3). Finally, the accuracy of the proposed solution obtained is demonstrated by comparing the solutions for from the classical PMCHWT integral equation, the Calderónpreconditioned PMCHWT integral equation, and the Mie-seerror on both the PMCHWT ries (Fig. 4). The relative and CP-PMCHWT solutions is 5 percent. To discuss the effective speed-up, one needs to take into account extra computational costs in the setup and matrix-vector product stages of the solution process. Concerning the setup stage, it is interesting to note that the preconditioning matrix (obtained through discretization with BC functions) can be computed with very low accuracy without loosing its preconditioning properties. More

COOLS et al.: A CALDERÓN MULTIPLICATIVE PRECONDITIONER FOR THE PMCHWT INTEGRAL EQUATION

Fig. 5. The far-field radiated by a cube of relative permittivity 1.0 and permeability 2.25 when illuminated by a plane wave of wavelength 2 meter, as computed with the PMCHWT integral equation and the Calderón preconditioned PMCHWT integral equation.

explicitly, the total number of quadrature points required to evaluate the preconditioning matrix and the preconditioned matrix is more or less the same. In the context of accelerating methods such as the fast multipole method, a much more careful analysis is required. This analysis is out of the scope of the work presented in this manuscript. Concerning the matrix-vector product stage, the extra cost consist of the multiplication with the inverse of a well-conditioned Gram matrix, and the multiplication with a BC discretized system matrix that has the same dimensions as the original system matrix. Neglecting the time to invert the sparse and well-conditioned Gram matrix, an iteration of the Calderón-preconditioned system takes twice as long as one of the classical system. The effective speedup in this example thus is about 20. To demonstrate that the preconditioner is sufficiently robust when applied to non smooth surfaces, a similar analysis is performed on a cube of unit side. The cube is illuminated by the incident electric field in (48) with meter, and has and . The simulation is carried out for nine uniform discretizations with the mesh parameter (the maximum edge length in the triangulation) ranging from 0.28 to 0.11 meter. For each mesh, the condition number was computed, and compared to of the system matrix the condition numbers of the classical PMCHWT integral equation system matrix on the same meshes (Fig. 2(b)). It can be concluded that also for non smooth surfaces the condition numbers of the systems resulting upon discretization of the classical , PMCHWT integral equation tend towards infinity as whereas the condition numbers of the systems resulting upon discretization of the Calderón-preconditioned PMCHWT integral equation remain virtually constant. Next, the correctness of the results obtained using the Calderón-preconditioned PMCHWT integral equation is verified. The mesh parameter is held at 0.16 m. The far-field scattered by the cube is computed by solving the classical PMCHWT and the Calderón-preconditioned PMCHWT integral equations. The cube was modeled by a mesh comprising 972 triangles, resulting in a total number of 2916 unknowns. The standard PMCHWT integral equation converged within a tolerance of in 250 iterations while

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Fig. 6. Triangulation of Thunderbird model.

Fig. 7. Convergence history for the TFQMR solution of the Thunderbird scattering problem.

the Calderón-preconditioned PMCHWT integral equation converged in 7 iterations; an effective speedup of 18. The far-field plane (Fig. 5). is computed along the meridian in the Finally the Calderón-preconditioned PMCHWT integral equation was tested on a Thunderbird model (Fig. 6). The Thunderbird is illuminated by the incident electric field in meter, and has and . The (48) with mesh parameter is 0.025 m. The far-field scattered by the Thunderbird is computed by solving the classical PMCHWT and the Calderón-preconditioned PMCHWT integral equations. The Thunderbird was modeled by a mesh comprising 3040 triangles, resulting in a total number of 9120 unknowns. Convergence history as a function of the requested tolerance can be found in Fig. 7. The classic PMCHWT system could not using less than 2500 be solved within a tolerance of TFQMR iterations. The Calderón-preconditioned PMCHWT in 373 iterations. The system was solved up to far-field (Fig. 8) is computed along the meridian in the plane using the solution of the classical PMCHWT integral ) and equation after 2500 iterations (correct up to

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where (54) (55) The definition of the matrices is analogous to the definitions of in (13) and (46).

, and , and

REFERENCES

Fig. 8. The far-field radiated by a thunderbird of relative permittivity 1.0 and permeability 2.25 when illuminated by a plane wave of wavelength 2 meter, as computed with the PMCHWT integral equation and the Calderón preconditioned PMCHWT integral equation.

using the solution of the Calderón-preconditioned PMCHWT ). integral equation (correct up to V. CONCLUSION A Calderón preconditioned PMCHWT integral equation that leverages a Calderón identity for the PMCHWT operator was introduced. It was shown that the new equation does not suffer from dense discretization breakdown. A consistent discretization scheme involving both Rao-Wilton-Glisson and Buffa-Christiansen functions that leads to a multiplicative matrix preconditioner to the classical PMCHWT system compatible with existing boundary element codes and acceleration schemes, was presented. Numerical examples demonstrated the robustness of the proposed approach.

APPENDIX CALDERÓN PRECONDITIONING OF THE CHEN-WILTON DISCRETIZATION OF THE PMCHWT In [23] the following alternative discretization of the PMCHWT was given (49) where (50) (51) (52) are the functions introduced in [23]. Here the functions This discretization also suffers from ill-conditioning for decreasing mesh parameters. A valid discretion of (36) can be found that results in a preconditioner for the linear system (49): (53)

[1] A. Poggio and E. Miller, Computer Techniques for Electromagnetics. Oxford, UK: Pergamon Press, 1973, ch. IV, Integral Equation Solutions of Three-dimensional Scattering Problems. [2] W. C. Chew, J. M. Jin, C. C. Lu, E. Michielssen, and J. M. Song, “Fast solution methods in electromagnetics (Invited),” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 420–431, 1997. [3] P. Ylä-Oijala and M. Taskinen, “Well-conditioned Müller formulation for electromagnetic scattering by dielectric objects,” IEEE Trans. Antennas Propag., vol. 10, pp. 3316–3323, 2005. [4] P. Ylä-Oijala, M. Taskinen, and S. Järven, “Analysis of surface integral equations in electromagnetic scattering and radiation problems,” Engrg. Analys. Bound. Elem., vol. 32, pp. 196–209, 2008. [5] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. New York: Wiley-IEEE Press, 1997. [6] A. Buffa and S. H. Christiansen, “A dual finite element complex on the barycentric refinement,” Math. Comp., vol. 260, pp. 1743–1769, 2007. [7] F. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative Calderon preconditioner for the electric field integral equation,” IEEE Trans. Antennas Propag., vol. 56, pp. 2398–2412, Aug. 2008. [8] H. Bagci, F. Andriulli, K. Cools, F. Olyslager, and E. Michielssen, “A Calderón multiplicative preconditioner for the combined field integral equation,” IEEE Trans. Antennas Propag., vol. 57, pp. 3387–3392, Oct. 2009. [9] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory. New York: Wiley, 1983. [10] A. Buffa and R. Hiptmair, “Galerkin boundary element methods for electromagnetic scattering,” in Topics in Computational Wave Propagation. Berlin: Springer, 2003, vol. 31, Lect. Notes Comput. Sci. Eng., pp. 83–124. [11] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, pp. 409–418, May 1982. [12] M. Cessenat, Mathematical Methods in Electromagnetism, ser. Series on Advances in Mathematics for Applied Sciences. River Edge, NJ: World Scientific, 1996, vol. 41, Linear theory and applications. [13] R. J. Adams, “Physical and analytical properties of a stabilized electric field integral equation,” IEEE Trans. Antennas Propag., vol. 52, pp. 362–372, Feb. 2004. [14] H. Contopanagos, B. Dembart, M. Epton, J. Ottusch, V. Rokhlin, J. Visher, and S. M. Wandzura, “Well-conditioned boundary integral equations for three-dimensional electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 50, pp. 1824–1930, Dec. 2002. [15] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. Mineola, NY: Dover, 1965. [16] G. C. Hsiao and R. E. Kleinman, “Mathematical foundation for error estimation in numerical solution of integral equations in electromagnetics,” IEEE Trans. Antennas Propag., vol. 45, pp. 316–328, Mar. 1997. [17] R. S. Varga, Ger˘sgorin and His Circles, ser. Springer Series in Computational Mathematics. Berlin: Springer-Verlag, 2004, vol. 36. [18] A. Bendali, “Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method. I. The continuous problem,” Math. Comp., vol. 43, no. 167, pp. 29–46, 1984. [19] K. Cools, F. P. Andriulli, and E. Michielssen, “Time-domain Calderón identities and preconditioning of the time-domain EFIE,” in Proc. IEEE Antennas Propag. Society Int. Symp., Jul. 2006, pp. 2975–2978. [20] C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves. Berlin: Springer-Verlag, 1969.

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[21] A. Buffa and S. H. Christiansen, “A dual finite element complex on the barycentric refinement,” C. R. Math. Acad. Sci. Paris, vol. 340, no. 6, pp. 461–464, 2005. [22] R. Freund, “A transpose-free quasi-minimal residual algorithm for nonhermitian linear systems,” SIAM J. Sci. Comput., vol. 14, pp. 470–482, 1993. [23] Q. Chen and D. Wilton, “Electromagnetic scattering by three-dimensional arbitrary complex material/conducting bodies,” in IEEE Antennas and Propagation Society Int. Symp.: Merging Technologies for the 90’s Digest, May 1990, vol. 2, pp. 590–593. Kristof Cools was born in Belgium, in 1981. He received the M.S. degree in physical engineering from Ghent University, Belgium, in 2004. His master’s dissertation dealt with the full wave simulation of metamaterials using the low frequency multilevel fast multipole method. He received the Ph.D. degree from the University of Ghent, in 2008, under the advisership of Prof. Femke Olyslager and Prof. Eric Michielssen. In August 2004, he joined the Electromagnetics Group, Department of Information Technology (INTEC), Ghent University. His research focuses on the spectral properties of the boundary integral operators of electromagnetics. Dr. Cools was awarded the Young Scientist Best Paper Award at the International Conference on Electromagnetics and Advanced Applications, in 2008.

Francesco P. Andriulli (S’05–M’09) received the Laurea degree in electrical engineering from the Politecnico di Torino, Italy, in 2004, the M.S. degree in electrical engineering and computer science from the University of Illinois at Chicago, in 2004, and the Ph.D. degree in electrical engineering from the University of Michigan at Ann Arbor, in 2008. From 2008 to 2010, he was a Research Associate with the Politecnico di Torino. Since 2010, he has been with the Microwave Department, École nationale supérieure des télécommunications de Bretagne (TELECOM Bretagne), Brest, France, where he is currently a Maître de conférences. His research interests are in computational electromagnetics with focus on preconditioning and fast solution of frequency and time domain integral equations, integral equation theory, hierarchical techniques, and single source integral equations. Dr. Andriulli was awarded the University of Michigan International Student Fellowship and the University of Michigan Horace H. Rackham Predoctoral

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Fellowship. He was the recipient of the Best Student Paper Award at the 2007 URSI North American Radio Science Meeting. He received the first place prize of the student paper contest of the 2008 IEEE Antennas and Propagation Society International Symposium, where he authored and coauthored two other finalist papers. He was the recipient of the 2009 RMTG Award for junior researchers and was awarded a URSI Young Scientist Award at the 2010 International Symposium on Electromagnetic Theory.

Eric Michielssen (M’95–SM’99–F’02) received the M.S. degree in electrical engineering (summa cum laude) from the Katholieke Universiteit Leuven (KUL, Belgium) in 1987 and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign (UIUC), in 1992. He joined the faculty of the UIUC Department of Electrical and Computer Engineering in 1993, reaching the rank of Full Professor in 2002. In 2005, he joined the University of Michigan as a Professor of electrical engineering and computer science where, since 2009, he has been the Director of the University of Michigan Computational Science Certificate Program. He authored or coauthored over one 160 journal papers and book chapters and over 280 papers in conference proceedings. His research interests include all aspects of theoretical and applied computational electromagnetics. His research focuses on the development of fast frequency and time domain integral-equation-based techniques for analyzing electromagnetic phenomena, and the development of robust optimizers for the synthesis of electromagnetic/optical devices. Dr. Michielssen is a Fellow of the IEEE and a member of URSI Commission B. He received a Belgian American Educational Foundation Fellowship in 1988 and a Schlumberger Fellowship in 1990. He was the recipient of a 1994 International Union of Radio Scientists (URSI) Young Scientist Fellowship, a 1995 National Science Foundation CAREER Award, and the 1998 Applied Computational Electromagnetics Society (ACES) Valued Service Award. He was named the 1999 URSI United States National Committee Henry G. Booker Fellow and selected as the recipient of the 1999 URSI Koga Gold Medal. He was also awarded the UIUC’s 2001 Xerox Award for Faculty Research, appointed 2002 Beckman Fellow in the UIUC Center for Advanced Studies, named 2003 Scholar in the Tel Aviv University Sackler Center for Advanced Studies, and selected as UIUC 2003 University and Sony Scholar. He served as the Technical Chairman of the 1997 Applied Computational Electromagnetics Society (ACES) Symposium (Review of Progress in Applied Computational Electromagnetics, March 1997, Monterrey, CA), and served on the ACES Board of Directors (1998–2001 and 2002–2003) and as ACES Vice-President (1998–2001). From 1997 to 1999, he was as an Associate Editor for Radio Science, and from 1998 to 2008 he served as Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

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Fast-Factorization Acceleration of MoM Compressive Domain-Decomposition Angelo Freni, Senior Member, IEEE, Paolo De Vita, Paola Pirinoli, Member, IEEE, Ladislau Matekovits, Senior Member, IEEE, and Giuseppe Vecchi, Fellow, IEEE

Abstract—Domain-decomposition (DD) for Integral Equation can be achieved by aggregating standard basis functions into specialized basis functions on each sub-domain; this results in a strong compression of the MoM matrix, which allows an iteration-free (e.g., LU decomposition) solution also for electrically large problems. Fast matrix-vector product algorithms can be used in the matrix filling and compression process of the employed aggregate-functions approach: this hybrid approach has received considerable attention in recent literature. In order to quantitatively assess the performance, advantages and limitations of this class of methods, we start by proposing and demonstrating the use of the Adaptive Integral Method (AIM) fast factorization to accelerate the Synthetic Function eXpansion (SFX) DD approach. The method remains iteration free, with a significant boost in memory and time performances, with analytical predictions of complexity scalings confirmed by numerical results. Then, we address the complexity scaling of both stand-alone DD and its combined use with fast MoM; this is done analytically and discussed with respect to known literature accounts of various implementations of the DD paradigm, with nonobvious results that highlight needs and limitations, and yielding practical indications. Index Terms—Adaptive integral method, aggregate functions, domain decomposition, fast methods, moment method, numerical methods.

I. INTRODUCTION

O

NE of the most successful approaches to the Method of Moments (MoM) analysis of large scattering and antenna problems is that adopted by the so called “fast methods;” they are based on the use of iterative solvers, and essentially aim at the reduction of the cost and memory occupation needed at each step of the iterative algorithm. In a somewhat dual approach, other works (e.g., [1]–[3]) address the memory and time reduction without necessarily requiring an iterative solution. In this alternative approach, the Manuscript received June 13, 2010; revised February 23, 2011; accepted May 09, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. This work was supported in part by the European Space Agency within the European Antenna Modeling Library (EAML) initiative (Contract no. 18802/04/NL/JD). L. Matekovits was supported by a Marie Curie International Outgoing Fellowship within the 7th European Community Framework Programme. A. Freni is with the Department of Electronics and Telecommunications, University of Florence, I-50139 Firenze, Italy. P. De Vita was with the Department of Electronics and Telecommunications, University of Florence, I-50139 Firenze, Italy. He is now with IDS Ingegneria dei Sistemi, 1 56010 Pisa, Italy. P. Pirinoli, L. Matekovits, and G. Vecchi are with the LACE, Department of Electronics, Politecnico di Torino, I-10129 Torino, Italy. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165474

overall large problem is addressed by domain-decomposition (DD); the actual size of the final Method-of-Moment (MoM) algebraic problem is reduced by grouping basis functions into “aggregate functions” defined over sub-domains, i.e., portions of the structure that are significantly larger than one cell of the initial mesh. Because of the reduced size of the final linear system, an LU decomposition solver can be used and so approaches of this kind can be classified as “iteration-free”, and this class of approaches will be referred to here as “compressive DD”. These approaches drastically reduce the memory occupation and the solution time, but they do not act on the matrix filling time, at least in their basic version. Because of this, methods scaling like the fast methods will eventually perform better when the number of unknowns is sufficiently large, provided an efficient preconditioner is applied; this latter issue is of crucial importance in antennas, and in general from multiscale problems (e.g., see [4]). However, fast iterative methods require a complete solution of the linear system when excitation changes; this is a relevant drawback in large array problems (when the full impedance/S matrix is desired, and/or embedded patterns), and/or when many incidence directions are required in scattering problems. In these situations, iteration-free approaches have an obvious advantage (LU decomposition storage), that shifts the point where fast iterative methods become better performing upwards. The dual approaches of fast iterative methods and aggregatefunctions methods (that can be iteration-free) are not mutually exclusive, and indeed they can be combined into a hybrid “fastaggregate” method to increase the efficiency of both: this has become an active research area in recent years. In [6], the fast multipole (FMM) decomposition was employed to reduce the matrix filling time of an aggregate function technique [7], called Macro Basis Functions (MBF), applied to the solution of medium size array problems. While it resulted efficient only for electrically small radiating elements on a periodic (regular) lattice, that seminal work pioneered the hybrid fast-aggregate approach. Application of the Adaptive Integral Method (AIM) [8] to the Synthetic Functions aggregation paradigm [1], [2] was first presented in the conference paper [10]; the Characteristic Basis Function (CBF) [3] aggregation method was combined in [12] with the Adaptive Cross Approximation (ACA) low-frequency fast factorization [16], [17]. In both cases, arrays were the primary emphasis of the methods. At variant with that, the work in [5] introduced the CBF aggregation in the multilevel fast multipole algorithm (MLFMA)

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FRENI et al.: FAST-FACTORIZATION ACCELERATION OF MOM COMPRESSIVE DOMAIN-DECOMPOSITION

retaining the iterative solution scheme; while the approach can be considered more general, the emphasis there was on scattering problems from objects with large smooth portions, and the special case of NURBS patches was considered for the initial MoM discretization; the CBF aggregate functions [3] were used to reduce the overall number of unknowns. Thus, although the CBF generation process increases the preprocessing CPU time of about a factor of three, the computation time required to solve the linear system for each excitation is reduced. Finally, the multilevel application of the MBF aggregation process was recently investigated [13]; although the results in those papers reported a marginal advantage over single-level implementation, we understand that the payoff rests on the ability to analyze larger problems in an iterative-free manner. At last, we remark that a different recent line of research has addressed the issue of noniterative solution via matrix compression [14]; this recent addition is better described after the overall complexity discussion in Section V. Starting from the above-described state of the art, the aim of this paper is two-fold. We investigate on the complexity scaling of the fast-accelerated compressive DD method, adopting a general format applicable to all endeavors of this class. Along this line, we first employ the fast matrix-vector properties of the AIM to directly compute on-the-fly the compressed MoM matrix resulting from application of the aggregate functions approach to the initial MoM problem. For quasi-planar structures, the AIM has an complexity per iteration, and for fully volume problems; seen in terms of a general fast factorization algorithms, AIM summarizes in these two situations the performance of single-level FMM and of FFT-based approaches and that of the more performing and more complex ). multilevel version of the FMM (MLFMA, We will prove the effectiveness of the AIM acceleration approach by providing both analytic estimates of the complexity scaling properties and numerical results. The proposed method is more general than the one in [6] and can be applied to both scattering and antenna problems, without any limitation on the size and location of the radiating elements. Furthermore, it can be applied in a simple manner also to multilayer structures when the formulation in [9] is used instead of classical AIM. As alluded above, a prototypical version of this endeavor was reported in the conference paper [10]. In carrying out the above, we will specifically refer to the Synthetic Functions (SFs) method [1], [2]; similar considerations can be applied for instance to the approach described in [3]. The general results of the scaling analysis will be validated and substantiated by SFX-AIM results and literature data for the other approaches. It is apparent that the hybrid class of methods of present interest are based on fast factorizations, and therefore, this class as a whole should not be contrasted to the class of fast methods. Instead, the key question for anyone having a fast factorization code available is the following: is it more convenient to use the fast factorization “alone” in a conventional iterative solver, or is it more convenient to employ it in a compressive DD method to

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fill in the compressed matrix and then exploiting its direct (e.g., LU decomposition) solution? As it can be easily perceived, the answer to this question depends on: a) the comparative complexity of compressed matrix filling versus complexity of one iteration; b) the speed of convergence of the iterative solution, i.e., the conditioning of the problem and the efficiency versus complexity of a preconditioner; and: c) the number of RHS (e.g., incidence directions, array ports). It is apparent that the answer cannot be but problem-dependent. In this work, we will especially concentrate on deriving a complexity scaling for the fast-DD methods. Some specific examples of this comparison will be outlined in Section V-G. To the best of our knowledge, both general and specific issues addressed here have never been addressed in the existing literature.

II. SF APPROACH TO DD The Synthetic Functions (SF) method is described in detail in [2]; therefore, we will recall only those aspects that are relevant to the present approach. On the other hand, we will carry out a detailed analysis of the associated numerical complexity scaling, an issue that has been only cursorily addressed in previous works, and that plays a key role in the present endeavor. The SF approach essentially consists in dividing the overall structure to be analyzed into portions, recognizing that the degrees of freedom of the field coupling between the solution on these portions are limited, and building basis functions that exploit this property. The SF generation procedure starts with the separation of the overall geometry into geometric sub-structures, that are called “blocks”. In general all the blocks have a different size, of basis functions; in oder and include different numbers to simplify notation and ease understanding the subsequent complexity analysis, we will drop the explicit indication of referring to the average number the block-dependent size of functions per block, . The SFs are then generated (separately) on each block and the necessary number of them will be selected, as detailed in [2]. Again, we assume an average the average number of such funcsituation, and we call tions; they are subsequently collected into the matrices of ; in the version described in [2] are average size real, but that is not a limitation in general. Because the SF are represented as linear combinations of the original (say, RWG) basis functions, expressing the MoM problem for the SF basis amounts to a linear algebraic transinto its formation from the original MoM problem counterpart for the SF basis. Since each standard basis function belongs to one block only, the transformation is block-wise; it is, therefore, convenient to visualize the transformation process by accordingly numbering the original basis functions with the same block grouping. In this way the

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total number of basis functions defined on the whole strucand the initial ture is divided in groups with an average size MoM linear system can be re-organized as

.. .

..

.

.. . .. .

.. .

(1)

.. .

When a solid metal is “torn up” into disconnected pieces by the geometrical subdivisions, the RWG defined on the boundary edges between two blocks remain as independent (and nonredundant) degrees of freedom. These “loose” functions are conveniently grouped all in one block, typically placed at the end of the sequence. All this is totally transparent to the procedure described in the following. , we comIn the standard SFs method, for pute the block-wise compressive matrix transformations (2) (3) (4) that transform the MoM system for the original basis (e.g., RWG) into the compressed linear system in the SFs basis ; with transparent notation, globally we write (5) where

is the block-diagonal matrix .. . .. .

..

(6)

.

The SF problem to solve has size erage reduction of a factor

with an av(7)

with respect to the original one in (1). The numerical complexity required to obtain the compressed linear system is given by the sum of the time required to evaluate the matrices , plus the matrix filling time, and the compression time needed to compute (2)–(4). In the standard SFX-MoM scheme, the matrix filling time is that of the stan, where is a constant that depends dard MoM, i.e., on the (average) complexity of computing one reaction integral. Each compressed matrix (2) is obtained with a computational ; since the entire consists of cost blocks, the total computational complexity for the compression results to be (8) Thus, the overall complexity to build the

matrix is (9)

Concerning the required dynamic memory, it is essentially storage to which we have to add a small amount due to the equal to that necessary to store the largest compression matrix plus the largest matrix block, i.e., . As seen above, and well known (e.g., [2]), the cost of computing the compressed matrix in all DD methods (like SFX, CBF, etc.) is not lower than with the standard MoM; the advantage is in the (dramatic) reduction of solve time, as well as (dynamic) memory allocation. The reduction of the filling time can instead be addressed with a fast factorization to perform matrix-vector products intrinsic in the compressed matrix: this is described in the next sections.

III. ON-THE-FLY DD COMPRESSION WITH FAST FACTORIZATION As well known, fast factorizations allow a reduced-cost computation of the MoM matrix-vector product . In the following we will employ a fast factorization method to directly compute the compressed MoM matrix resulting from application of a compressive DD (e.g., SFX, CBF); in particular, we will diblocks in (2), employing a fast factorrectly compute the ization for both avoiding the complete filling and storage of the and for performing the compressive original matrix blocks products in (2) at a reduced computational cost. To this end, note that to effect the compressive product in (2) one has to perform times a matrix-vector product between and each of the columns of the compression matrix ; using the fast factorization, we can efficiently perform these products at a reduced cost, and without needing the computation and storage of the . entire In particular, we will employ here AIM fast factorization; as and as an already discussed AIM behaves both as an algorithm (for nearly planar or 3-D structures), thus representing both classes of fast factorization methods. As well known, AIM acts by mapping the basis functions on a general mesh onto a regular (volume) grid, and then performing via FFTs. By considering the clasthe matrix-vector product sical AIM approach [8], the impedance matrix is written as , where the near-field matrix is a sparse matrix , with average number of nonzero elements per of size gives the far-field contribution, matrix row, while with a (three-level) block Toeplitz matrix and an extremely sparse matrix. It is well known [8] that with the AIM the MoM for 3-D matrix filling time is of the order of for planar or surface problems, and reduces to quasi-planar structures, as typical for printed antennas and/or planar arrays. Note that is again a constant that depends on the numerical implementation. We observe that AIM complexity for 3-D and planar surfaces also parallels those of the single level and multilevel implementations of the (3-D) FMM. Therefore, the scaling derived here for AIM are of general applicability; the constant may be different in the two cases, of course. Initially describing the process at global level for ease of notavia repeated mation, we construct the compressed matrix trix-vector products of the kind , where is the column

FRENI et al.: FAST-FACTORIZATION ACCELERATION OF MOM COMPRESSIVE DOMAIN-DECOMPOSITION

of corresponding to the th column of interactions, we have

; separating near-far

(10) The near-field part products are effected essentially as in the standard SFs method, the numerical advantages obviously arising from the proper handling of the “far” interactions. In this near-field part, in order to take advantage of the sparsity of matrix and reduce the storage requirements, we have the block by block and, only to build the near-field matrix immediately after, compress it. Acting in this way we obtain a and a memory allocation requirements complexity of . The far-field contribution will be computed via AIM. Because of convolutional character of the Toeplitz kernel, the can be computed by using a fast Fourier vector transform as

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have escaped the investigations of the now sizeable community working on compressive DD; in fact, to the best of the authors’ knowledge, the only attempt to draw some general guidelines are those in [15]. While it is intuitive that dense-mesh structures lend to an obvious compressive gain, as discussed in [15], the compression rate in smooth structures depends on the global geometry, and ultimately on the ability of the specific DD method to catch the essential degrees of freedom of the problem. However, it is the experience of several groups involved in this research field that the methods described in this paper typically have a lower compression rate in structures with large smooth parts. Therefore, in Section V-F we will evaluate the impact of this factor on overall performances of SFX-AIM (of which we have a complete information) for different values of , also beyond the values typical of arrays and of large smooth . structures IV. FAST-ACCELERATED DD: RESULTS

(11) For surface problems, the computation complexity to calculate is , where is a constant that each vector depends on the AIM grid. Note that for quasi-planar structure . this complexity reduces to Finally, when we have calculated the vector , the th column of the th block of the compressed matrix can be evaluated by performing the product , without storing the , at a computational cost of . entire matrix Since this operation has to be done for all the columns of , the total numerical complexity to obtain is . Therefore, the overall computation complexity for obtaining matrix with the AIM is given by the compressed (12) . with Moreover, the memory allocation requirements are less than (13) where is the number of nonzero elements per row of the . matrix , which is It is worth nothing that for quasi-planar structures, the term in the RHS of (12) and (13) is not present. We observe that the asymptotic complexity scaling is still at best, i.e., for quasi-planar AIM, or MLFMA; this might appear surprising, and — to the best of our knowledge — never reported in the relevant literature. However, the regime kicks in at a very large number of unknowns, as discussed later in Sections IV and V, and the use of a fast method for the filling of the compressed matrix remains a significant boost of efficiency for a very large class of problems of practical interest. Finally, it is apparent that the effectiveness of the compressive DD methods (SFX or anyone else) depends on the compression factor . However, general bounds on this factor appear to

We will consider three array antennas, with rather different properties. For the already discussed reasons of complete availability of data we will consider the AIM (MLayAIM) fast factorization, and the SFX embodiment of compressive DD methods; for the presented structures the AIM factorization complexity, and thus repreis largely within the sentative of this class (e.g., of MLFMA on non quasi-planar structures). Finally, we observe that we will devote a specific section (Section V) to the discussion of various other options documented in the literature. Initially, we consider an array of bowtie dipoles on a regular Cartesian layout; the overall layout is as reported in [10] and not replicated here for the sake of conciseness. In this initial case we have chosen a Cartesian grid to the only purpose of expediting generation of the several different arrays to obtain complexity scaling results. The next examples will consider a general layout. We have considered two different grouping schemes: a) all blocks are composed of a single bowtie, with SFs on each block; b) all blocks include each two adSFs per block; the choice jacent bowties, keeping now of has been done so as to have the same compression ratio for both. In what follows we will obtain the values of the constant by fitting the appropriate scaling law to the observed CPU times for the considered range of number of un. Next, Tables I knowns . Furthermore, we can estimate and II show the total computational cost (computation of vectors compression) for obtaining the system in the SF basis, and the storage occupation. Fourth and fifth columns show the CPU time relevant to two different numerical implementations of the SFX-AIM scheme. In particular, the fifth column labeled “SFX-AIM blk” is pertinent to the case where the near-field mais calculated block by block, and each block is distrix charged (deallocated) when it is no more necessary to build the block. Instead, the fourth column is related to the pertinent is stored as a standard case where the near-field matrix AIM sparse matrix. We can easily notice that in the latter case the CPU time is slightly smaller than the one relevant to the “SFX-AIM blk” scheme (due to the implicit simpler numerical

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TABLE I COMPARISON OF THE SFX-MOM AND SFX-AIM SCHEMES ( = 17 = 0 08) FOR ARRAYS OF BOWTIES (EACH SF BLOCK MATCHES ONE BOWTIES): MATRIX; (COLUMNS 5–7) DYNAMIC MEMORY REQUIREMENTS (COLUMNS 3–5) CPU TIME ON A CENTRINO 1.8 GHZ, 2 GB RAM FOR FILLING THE

TABLE II COMPARISON OF THE SFX-MOM AND SFX-AIM SCHEMES ( = 34 = 0 08) FOR ARRAYS OF BOWTIES (EACH SF BLOCK MATCHES TWO BOWTIES): MATRIX; (COLUMNS 5–7) DYNAMIC MEMORY REQUIREMENTS (COLUMNS 3–5) CPU TIME ON A CENTRINO 1.8 GHZ, 2 GB RAM FOR FILLING THE

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Fig. 1. CPU time on a Centrino 1.8 GHz for filling the SFX-MoM or SFX-AIM schemes are used ( = 0 08).

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implementation), but a larger amount of dynamic memory is required. However, in both cases it appears that the total numerical complexity of the SFX-AIM scheme is always smaller than that of the SFX-MoM, and that the advantage of using AIM increases with the number of unknowns. Fig. 1 graphically represents the main content of Tables I and II, and shows the adcomvantage in using the proposed schemes for filling the pressed matrix. First, it could be observed that the regression curve crossing (specithe triangular symbols relative to the case fied as “one dipole” in the figure legend) shows that the comwhile that plexity of the “SFX-AIM blk” scheme is one crossing the square symbols shows that the complexity of . This behavior can be exthe “SFX-AIM” scheme is plained by the fact that the practical arrays that have been anunknowns) but not yet alyzed are quite large (in excess of enough to exhibit the asymptotic quadratic behavior; since the

scalings are not specific to the SFX and AIM methods employed here, this issue will be addressed at a more general level in Section V. A second consideration concerns instead the fact that, increasing the size of the blocks the numerical complexity of the “SFX-AIM blk” scheme becomes similar to that of SFXAIM, even though it preserves a reduced RAM occupation, as shown by columns seven and eight in Table II. Having assessed the complexity scaling, we now pass to consider a general case. We consider the antenna in Fig. 2(a), consisting in an array of 31 identical radiating elements on a nonregular layout. The geometry of each element is sketched in Fig. 2(b): it consists of a dual-pol circular patch element capacitively coupled with four square patches each of which is fed through a metallic strip. The overall array therefore has a total , and thus, it is an example number of ports equal to of structure for the analysis of which the use of aggregate functions is particular convenient (the need to consider two ports for each polarization derives from the fact that the four-to-two port reduction is effected by the beam-forming network for which this information is relevant). The antenna has been designed to aperture distribution, with normalapproximate the ized radius [18]. The entire structure has been discretized with an edge mesh with 36456 RWG functions; this number of unknowns is reduced to 1302 in the SFX formulation. In order to check the accuracy of the SFX-AIM procedure, the field radiated by the antenna has been computed and compared with that obtained with the SFX-MoM; Fig. 3 shows the isoflux directivity pattern for the right hand circular polarization (RHCP) component obtained in the symmetry plane containing five radiating elements: the curves relevant to the two methods are practically indistinguishable. Almost identical results have been obtained for all the other planes. Table III compares the SFX-AIM performances with both the SFX-MoM and the classical AIM ones. The second column shows the CPU time required for filling all the matrices and vectors, while the third one is relevant to the

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TABLE III COMPARISON OF THE SFX-MOM AND SFX-AIM SCHEMES ( = 42 = 0 035) FOR THE ARRAY OF FIG. 2. CPU TIME IS RELEVAT TO A CENTRINO 1.8 GHZ, 2 GB RAM

Fig. 4. Sub-array of the large SAR antenna [11].

Fig. 2. Geometry of the entire array (a) and of a single element (b).

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Fig. 3. Directivity pattern of the array shown in Fig. 2 when fed in order to obtain an isoflux pattern (RHCP polarization, = 1 35 GHz): SFX-MoM (solid), SFX-AIM (dot dashed).

CPU time needed to solve the linear system. An LU decomposition and a conjugate gradient (BiCGstab) method have been used for SFX and AIM formulations, respectively. Total CPU time is then reported in column four and the dynamic memory occupation in column five. It can be noted that, even if the array does not require a very large number of unknowns, the use of the SFX-AIM formulation yields an acceleration factor of three with respect to the SFX-MoM. On the other hand, we have a slight increase of the dynamic storage which, however, remains quite moderate, especially for the SFX-AIM blk scheme. The second row of Table III is relevant to the classical AIM method. It is worth noting that here the filling time is very close to that obtained with the SFX-AIM procedure. However, a higher effort is required for the solution of the final linear

system due to the presence of a larger number of unknowns (36456 RWG) with regard to the SFX scheme (1302 SFs). The data reported in Table III are relative to the single-RHS case (all array elements are fed simultaneously) which is the worst case for comparison with a fast (iterative) method; indeed, as already pointed out in Section I, the advantage of using a DD approach (like SFX) with respect to the intrinsically iterative fast methods (like AIM) increases with the number of RHSs. In the present array case, the multiple RHS case corresponds to computing the embedded patterns or the scattering matrix. The computational effort to do this with a fast method (AIM) alone is equal to the CPU time reported in the fourth column of Table III multiplied times the number of array elements; instead, resorting to SFX (or a like DD method) it is at most the filling time in the second column plus the solution time multiplied by the number of elements (as a matter of fact, using an LU decomposition and back-substitution it is still less). Finally, as an example of application to an existing antenna, two very large planar arrays for SAR applications [11] have been analyzed; the structure of the sub-array is sketched in Fig. 4. The first array is made by 16 sub-arrays while the other one by 48. Each sub-array is composed by the eight dual-polarization interconnected stacked patches in Fig. 4, discretized by 4756 triangular elements; the 16-tile array has a grand total of 88 416 unknowns, and the 48-tile array 265 248 unknowns; Table IV shows the computational effort and the storage required for this analysis. As the results show, the proposed SFX-AIM formulation allows performing the analysis of the first (second) array 33 (56) times faster than the standard SFX-MoM method; in agreement with the predicted scaling, even for a large array (in excess of 250 000 unknowns) the reduction in filling time is still dramatic. For the sake of clarity, we would like to point out that the results in Table IV are relative to the case in which geometrical symmetries of the structure have been considered. While this choice was mandatory to analyze the structure with the standard SFX-MoM, the same symmetries have been exploited with the SFX-AIM only to compare the two methods. The more general case, where no symmetries are considered, will be addressed in Section V-G.

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TABLE IV COMPARISON OF MOM, SFX-MOM AND SFX-AIM SCHEMES FOR THE ARRAY IN [11]. CPU TIME IS RELEVAT TO A 3.0 GHZ QUAD-CORE PROCESSOR, 128 GBYTE RAM, 1 THREAD

V. OVERVIEW OF DD METHODS ACCELERATED BY FAST FACTORIZATIONS We have already recalled that the scaling results obtained here are largely independent of the use of SFX for effecting the DD, and of AIM for computing on-the-fly the compressed MoM matrix; also, we have seen in Section III that the asymptotic complexity scaling does not seem to free the combined algorithm complexity. However, our results in Section IV for from unknowns did not show onset of problem well in excess of this unfavorable regime, and none of the results in the literature did either. Since none of the presently known results goes significantly beyond the number of unknowns employed here, it is of paramount importance to investigate at what level a demise of the DD-fast factorization could appear. This will be the main objective of this section. To this end, we will consider the most widely employed approaches found in the recent literature. In particular, we will consider: the algorithm described by Craeye in [6], based on the application of the (single-level) fast multipole method, that we will refer to as MBF-FMM; the formulation shown in [12] where the Characteristic Basis Function (CBF) Method is used in combination with the Adaptive Cross Approximation Algorithm (CBF-ACA); and the multilevel implementation of the latter (ML-CBFM) as described in [13]. For the complexity scaling we will employ numeric data about these formulations as directly taken from the cited works. To appreciate the actual benefit of the accelerated formulation over the basic DD approaches, it is convenient to introduce the ratio between the complexity of the accelerated algorithm and the standard DD-MoM formulation: this will allow a quantitative appreciation of the reduced effort in computing the compressed MoM matrix. A. SFX-AIM Method By using (12) and (9) we easily obtain (14) where (15) (16) (17) , (14) shows Note that, apart from a small constant term two different trends with respect to the number of unknowns :

Fig. 5. Ratio between the numerical complexity for the matrix compression in the standard SFX-MoM and in the SFX-AIM, as given by (14) (a = 2:5 10 ; a = 3000; a = 2:5 10 ).

the first decreases proportionally to while the other one increases logarithmically for quasi-planar structures, and proporfor surface 3-D objects. This term tionally to scaling, which is well known derives from the overall to be a worst-case; therefore, actual scalings for specific cases might exhibit a lower complexity. As already mentioned in Section III, surprising as it may seem, it is apparent that in the asymptotic limit of very large the SFX-AIM formulation looses any advantage over the implementation with conventional SFX-MoM. However, while the is about a few thousands, is very value of the constant small (usually an order of magnitude smaller than the constant ) and, when typical values of the constants are considered, we obtain the behavior shown in Fig. 5. It is worth noting that for the analysis of surface 3-D obgrows rapidly and the complexity jects the term ratio is less that unity (i.e., SFX-AIM is a favorable technique) . On the contrary, when we consider only until about a quasi-planar structure, since the term is absent, the resulting curve goes up very slowly and the SFX-AIM is very fa, that is about the size of the largest vorable also for presently addressable problems on supercomputers [19]. These scalings can be readily extended to using single- and multiple-level FMM as a fast factorization, since AIM scales , like single-level FMM for 3-D surface structures for quasi-planar and like multiple-level FMM structures. In order to correctly understand the scaling analysis here and in Section III, we recall, however, that the multiplicative constant may be different and dependent on block size implementation.

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B. MBF-FMM Method Since the method shown in [6] is based in the application of the fast multipole method (FMM) we can identify two contributions to the numerical complexity required to make the commatrix: the first is due to the near interaction terms pressed of the impedance matrix , while the other is relevant to the far interaction terms. If we extend the formulation in [6] to the analysis of objects of arbitrary geometry and position, the complexities due to the near and far parts result and , respectively. In the derivation of the previous complexities we have considered the optimum condition where the number of FMM groups, of unknowns in each group, . From the and of points in the angular domain, are equal to latter hypothesis, it follows: (18) where (19) (20) (21) . It is worth noting that for asympwith totic large the ratio (18) goes to . For the sake of simplicity, we have assumed that each synthetic functions block exactly overlap an FMM group. For arbitrary geometry structures this is not always possible. Hence, in addition to a higher intricacy of the numerical algorithm, a slight larger numerical complexity is expected. C. CBF-ACA Method According to the adaptive cross approximation algorithm described in [12] the numerical complexity required to build the matrix is , where is the average effective rank of the submatrices relevant to all synthetic functions blocks. As a consequence the ratio between the numerical complexity of the CBF-ACA method and the standard CBF-MoM results

(22) where the assumptions have been made. The ratio in (22) depends on the structure itself, the adopted discretization, and the specific subdivision in blocks. Hence, it is not easy to give it a numerical evaluation a priori. In absence of an a-priori estimate, we have extracted the relevant data from matrix Fig. 5 of [12]; for a relative error in calculating the elements of 1%, we can estimate the latter ratio of the order of . The complexity ratio for this method will be reported in the following (e.g., in Fig. 6) as constant, since the number of unknowns does not appear explicitly in the ACA rank , as this is not mentioned in [12]. The above appears an optimistic estimate, though; the ACA rank is approximately independent of at zero-frequency, i.e.,

Fig. 6. Ratio between the numerical complexity of the standard DD-MoM (SFX-MoM or CBF-MoM) and that relevant to the accelerating formulations. Markers indicate computed complexities (see text); black dots: SFX-AIM, red triangles: CBF-ACA, blue diamonds: ML-CBF.

for the Laplace problem for which it has been devised [17], and means increasing discretization denin which increase of sity; for dynamic problems with a Helmholtz kernel, the MoM (sub)matrices are increasingly less and less rank-deficient. As a result, if the increase of derives from an increase of electrical size of the problem (as opposed to an increase in discretization density) the effective rank of ACA is bound to increase up to full rank, where the ACA becomes a full pivoted LU decomposition. can be Keeping this into account, a scaling of obtained that remains valid up to moderate electrical sizes and [16]. Inserting this scaling sampling rate of approximately in the derivation leading to (14) in Section V-A one obtains the following alternative expression (23) that, for the sake of completeness, will also be reported graphically in the following. D. ML-CBFM Method Concerning the multilevel CBF method described in [13], the numerical complexity relevant to the compression of the matrix is dependent on the number of levels, and on the number of characteristic or synthetic functions in each level. However, it is possible to give an upper bound to this complexity , where is the average number of characas teristic functions associated to the level that presents the minimum number of characteristic functions. Then, we have to add the numerical complexity relevant to the computation of the en. Thus tries of the impedance matrix , i.e., (24) assuming

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E. Discussion In order to better compare the accelerating methods considered above, we have summarized the associated scaling com-

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plexity ratios in Fig. 6; we remark that using ratios one can meaningfully compare results obtained on different machines. For a better readability we show now the inverse of the ratio in (14), (18), (22), and (24); the above gives a gain figure of employing a given fast factorization with respect to using conventional MoM for the filling and compression. (Note that Fig. 6 reports the inverse of the ratio in Fig. 5). Fig. 6 shows both the analytically derived scalings — reported as curves — and actual values of complexity. The latter have been obtained for the arrays analyzed in Section IV for SFX-AIM (black dots), and using the data present in [12] (red triangles) and in [13] (blue diamonds), for the CBF-ACA and the ML-CBFM, respectively. The theoretical scalings for methods involving ACA have been reported both as as constants and as in (23) for the reasons discussed above in Section V-C. It is worth noting that for the SFX-AIM case the computed gain ratio increases with increasing dimension of the structure (i.e., the number of unknowns), even if it is always a slightly lower than the theoretical estimate; this is quite understandable, and simply accounts for obvious programming overheads. The curve relative to MBF-FMM has been obtained emand deriving from the data for ploying the values of SFX-AIM, since the data in [6] are all relative to a symmetric case implying a different (lower) complexity scaling than for the general case of present interest. The gain of the CBF-FMM unknowns, with exhibits an increasing regime up to about a subsequent decay; that is a consequence of the scaling of the single-level FMM, and would likewise appear for a non-quasi-planar structure and AIM, albeit at a possibly different value of . For the CBF-ACA, under the assumption of constant-rank scaling, and with the data in [12], one finds the constant line in Fig. 6; it shows that CBF-ACA method is ten times faster than the standard CBF-MoM as a maximum. The red triangles show the complexity ratio obtained directly from the results reported in [12]. As a comparison—and with the caveat mentioned above—we have also reported the theoretical scaling deriving from [12], that appears to correctly fit the decreasing part of the trend. As the results show, the undoubted simplicity of ACA is paid in terms of effectiveness, due to its less performing frequency scaling with electrical size. Likewise, the data in [13] indicate that the gain of ML-CBFM is never greater than three, and it is even less than one for the largest structure; we understand, though, that the sought advantage here is the ability to address problems large enough not to be amenable to a direct solution with a single-level CBF. A recent work [14] has demonstrated a direct solution using ACA matrix compression that has computational complexity of order and asymptotic memory requirements of order ; this appears lower than the asymptotic complexity of compressive DD methods but the multiplicative constant is higher, leading to computation times of 12 h for 500 000 unknowns, as reported in [14]. We understand that the frequency range of applicability of this interesting approach is the same as of the underlying ACA [16]. F. Impact of Compression Factor Finally, we investigate on the effectiveness of the fast-accelerated compressive DD with specific regard to the effect of the

Fig. 7. Effectiveness of fast-acceleration in compressive DD: ratio between the numerical complexity of the standard SFX-MoM and the SFX-AIM as a function of the number of unknowns, and for different values of the compression factor 1=C (solid curves). The dashed line connects the maxima of the acceleration ratio attained for a specific 1=C (and the number of unknowns at which the maximum is obtained); overlaid, the 1=C scale for ease of reference. Blue dots denote the total dynamic memory required with the DD approach for a few specific compression ratios (i.e., for a few specific products M = CN ).

compression ratio on the onset of the asymptotic complexity scaling discussed in Section III. In order to have all data necessary to a full scaling, we employ data deriving from our SFX-AIM implementation. The relevant results are collected in Fig. 7. It clearly shows the gain that even for very unfavorable compression factors over standard SFX-MoM remains evident; in particular, it is noted that while a decay after a maximum value exists, in practice this simply shows up as a saturation, i.e., the gain over standard SFX-MoM reaches a value that remains independent of the number of unknowns, and dependent on the compression ratio . Indeed we have found that this saturated gain depends lin. Hence, despite the asymptotic limit, one early on can safely conclude that up to 100 M unknowns there is always a net advantage in the use of fast factorizations to accelerate the matrix setup of compressive DD methods. The gain, and ultimately the performance of the accelerated DD methods will . depend on the compression factor G. Accelerated DD Versus Iterative Fast-Factorization In the previous sections fast-accelerated DD approaches have been compared with their baseline MoM based version; in this section we will conclude our discussion comparing the numerical complexity of fast-accelerated DD with that of a plain fast solver. The aim of this comparison is to answer the question posed in the Introduction (“is it more convenient to use the fast factorization alone in a conventional iterative solver, or is it more convenient to employ it in a compressive DD method?”); we recall the discussion there about the practical limitations inherent in attempting to answer that question before proceeding. In order to have a meaningful comparison we are forced to rely on data deriving from our own computations, i.e., using SFX as DD and MLayAIM [9] as fast acceleration; we point out that the results are valid — at least as a trend — for all other

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TABLE V COMPARISON OF MLAYAIM, SFX-AIM AND SFX-MOM SCHEMES FOR THE ARRAY IN [11]

Fig. 8. CPU time for the analysis of the planar array in [11] with preconditioned (iterative) MlayAIM and the the SFX-AIM or the SFX-MoM schemes versus the number of array feeding configurations (RHS).

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leads to a computation time the compression factor , and resulting of about 2 times as much as with in a break-even for about 32 RHS, which remains significantly smaller than the total number of ports (192). In order to put in a more general perspective the number of RHS, it is convenient to think in terms of the degrees of freedom (DOF) of the scattered or radiated field; the total area occupied by the array is about 390 , and considering roughly one [21] one finds 1560 DOF; since the (microstrip) DOF per array is backed by a ground plane it is a conservative measure to halve this number, i.e., considering 780 DOF. This would be the minimum number of RHS to completely characterize the scattering of the object, irrespective of its texture (i.e., it would not be different for a plate and the presently considered microstrip array). In this light, the break-even for employing the fast factorization in a compressive DD (rather than in the conventional way) appears for a number of RHS significantly smaller than the DOF of the problem. As a final remark, we observe that for many array problems, as in this case, the geometry of the radiating part consists of a replicated basic cell (in our case the tile in Fig. 4); when this is the case, the symmetry may be employed to perform the generation of the DD basis functions (e.g., SF or CBF) only once, and at a few other steps of the overall DD algorithm. This may make the complexity of the standard implementation of DD (MoM with no acceleration) significantly lower, as also reported in Fig. 7. Of course, this would be lost in a general case. VI. CONCLUSION

DD methods and for -complexity fast factorizations. We take as a case study the largest case for which we have all necessary pieces of information, i.e., the large (microstrip) array for SAR application [11] already considered in Section IV. Table V serves to set the stage comparing the performances of the SFX-AIM and of the standalone AIM (MLayAIM in this case) for the case of one RHS — i.e., the worst case for DD methods. Since (see Introduction) the performance comparison depends heavily on the effectiveness of the preconditioner used in the standalone (iterative) fast method, we have used here the preconditioner described in [20], that in our experience has nearly-optimum performance for this type of problems. Next, Fig. 8 reports the total time required for the simulation as a function of the number of RHS, i.e., of the number of different array feeding configurations (e.g., number of ports, if excited one at a time). The data in Fig. 8 and Table V refer to . As expected, the solution a compression ratio of time for the standalone (iterative) AIM increases linearly with the number of RHS; the solution time with DD is dominated by the matrix filling and LU factorization, so that the linear dependence on top of that is barely visible, and practically a constant with the number of RHS. As result, there is a break-even point at 16 RHS. Otherwise said, for a very efficient preconditioner, the AIM-accelerated DD becomes more convenient than the iterative version of AIM for 16 RHS. one can interpolate the For a compression factor of data in Fig. 7, finding a minor reduction in effectiveness; the solve time (LU) is a fraction of the total time in this range, so

We have addressed the acceleration of compressive Domain-Decomposition (DD) methods for the Integral-Equation, Method of Moments approach. In this context, DD is achieved by constructing special basis functions on each domain via aggregation of the standard basis functions; the aggregation is done by suitable processing of the solution for each isolated domain. These DD methods result in a strong compression of the MoM matrix, which allows an iteration-free (e.g., LU decomposition) solution, and an essential annihilation of the solve time; in their basic setting, they do no reduce the computational cost of generating the MoM matrix and compressing it. Fast factorizations, devised for the timeand memory-reduction of MoM matrix-vector product, can be considered as an effective means for reducing the complexity of DD approaches. In the above framework, we have shown how to hybridize the fast matrix-vector product of the Adaptive Integral Method (AIM) and the Synthetic Function (SFX) Domain-Decomposition approach. We have shown the advantages of the combined method over both the stand-alone SFX and AIM, both analytically and with numerical examples. Due to the significant advantage of AIM and its extensions for quasi-planar and layered media, our application examples have focused on arrays, with a nonregular element layout. At a more general level, we have found that the complexity scaling of all the fast-DD hybrids shows a scaling which is in the limit of . This is to some exasymptotic to tent surprising, and never discussed in the existing literature. We have therefore investigated the onset of the unfavorable regime,

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employing our data and those present in published accounts of similar hybrids. It has been found that for fast matrix-vector products exscaling the hybrid method starts loosing hibiting an unknowns, i.e., it is always some efficiency only beyond convenient; this is the case of AIM for quasi-planar structures (like array antennas), and of the multilevel FMM. A signifiunknowns) in concantly earlier cutoff point appears (about fast methods, like the AIM for 3-D strucnection with tures and the single-level FMM. The Adaptive Cross Approximation (ACA) has been applied in the literature in connection with the Characteristic Basis Function (CBF) DD approach; analysis of published data for this method shows a lower effectiveness than those mentioned above; it is conjectured that this be due to the fact that ACA is effective in over-sampled problems, less so when the electrical size of the structure increases. In summary, efficiency comparison between iterative fast methods and fast methods applied to compressive DD involves several factors, notably because of the profoundly different nature of the solution, i.e., iterative versus noniterative (as in DD approaches). Eventually, this involves iteration count and number of RHS; in turn, convergence speed depends on the characteristics of the problem at hand and the efficiency of the employed preconditioner. As a final comment, independent of their efficiency in a serial implementation, compressive DD methods are easily parallelizable. In fact, the overall compressed matrix can be partitioned in blocks that can be assembled separately, and this does not change when a fast-factorization method is employed (as examined here) to accelerate matrix filling. This property is likely to play a key role, since high-performance algorithms are being scrutinized under a different perspective in view of their parallel scalability. REFERENCES [1] L. Matekovits, G. Vecchi, G. Dassano, and M. Orefice, “Synthetic function analysis of large printed structures: The solution space sampling approach,” in Proc. Dig. IEEE Antennas and Propagation Soc. Int. Symp. (in ), Boston, MA, 2001, pp. 568–571. [2] L. Matekovits, V. A. Laza, and G. Vecchi, “Analysis of large complex structures with the synthetic-functions approach,” IEEE Trans. Antennas Propagat., vol. 55, no. 9, pp. 2509–2521, Sep. 2007. [3] V. V. S. Prakash and R. Mittra, “Characteristic basis function method: A new technique for fast solution of integral equations,” Microw. Opt. Technol. Lett., vol. 36-2, pp. 95–100, Jan. 2003. [4] P. De Vita, A. Freni, F. Vipiana, P. Pirinoli, and G. Vecchi, “Fast analysis of large finite arrays with a combined multiresolution—SM/AIM approach,” IEEE Trans. Antennas Propagat., vol. 54, no. 12, pp. 3827–3832, Dec. 2006. [5] E. Garcia, C. Delgado, I. G. Diego, and M. F. Catedra, “An iterative solution for electrically large problems combining the characteristic basis function method and the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propagat., vol. 56, no. 8, pp. 2363–2371, Aug. 2008. [6] C. Craeye, “A fast impedance and pattern computation scheme for finite antenna arrays,” IEEE Trans. Antennas Propagat., vol. 54, no. 10, pp. 3030–3034, Oct. 2006. [7] E. Suter and J. R. Mosig, “A subdomain multilevel approach for the efficient MoM analysis of large planar antennas,” Microw. Opt. Technol. Lett., vol. 26, pp. 270–277, Mar. 2000. [8] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci., vol. 5, pp. 1225–1251, 1996. [9] P. De Vita, F. De Vita, A. Di Maria, and A. Freni, “An efficient technique for the analysis of large multilayered printed arrays,” IEEE Antennas Wireless Propagat. Lett., vol. 8, pp. 104–107, 2009.

[10] P. De Vita, A. Freni, L. Matekovits, P. Pirinoli, and G. Vecchi, “A combined AIM-SFX approach for large complex arrays,” in Proc. Dig. IEEE Antennas and Propagation Soc. Int. Symp., Honolulu, HI, 2007, pp. 3452–3455. [11] G. Gheri and J. R. Mosig, “The multilevel interactive objects approach to model and mesh large planar antennas and arrays: A full-JAVA Tool,” in High-Performance Computing and Networking. Berlin/Heidelberg, Germany: Springer, 1998, vol. 1401, pp. 969–971. [12] R. Maaskant, R. Mittra, and A. Tijhuis, “Fast analysis of large antenna arrays using the characteristic basis function method and the adaptive cross approximation algorithm,” IEEE Trans. Antennas Propagat., vol. 56, no. 11, pp. 3440–3451, Nov. 2008. [13] J. Laviada, F. Las-Heras, M. R. Pino, and R. Mittra, “Solution of electrically large problems with multilevel characteristic basis functions,” IEEE Trans. Antennas Propagat., vol. 57, no. 10, pp. 3189–3198, Oct. 2009. [14] A. Heldring, J. M. Rius, J. M. Tamayo, J. Parrón, and E. Ubeda}, “Multiscale compressed block decomposition for fast direct solution of method of moments linear system,” IEEE Trans. Antennas Propagat., vol. 59, no. 2, pp. 526–536, Feb. 2011. [15] L. Matekovits, G. Vecchi, and F. Vico, “Physics-based aggregate-functions approaches to large MoM problems,” ACES J., vol. 24, pp. 143–160, Apr. 2009. [16] K. Zhao, M. N. Vouvakis, and J. F. Lee, “The adaptive cross approximation algorithm for accelerated method of moments computations of EMC problems,” IEEE Trans. EMC, vol. EMC-47, no. 11, pp. 763–773, Nov. 2005. [17] M. Bebendorf, “Approximation of boundary element matrices,” Numer. Math., vol. 86, no. 4, pp. 565–589, Oct. 2000. [18] C. Brumbaugh, A. Love, G. Randall, D. Waineo, and S. Wong, “Shaped beam antenna for the global positioning satellite system,” in Proc. Dig. IEEE Antennas and Propagation Soc. Int. Symp., Oct. 1976, vol. 14, pp. 117–120. [19] J. M. Taboada, L. Landesa, F. Obelleiro, J. L. Rodriguez, M. G. Araujo, J. M. Bertolo, J. C. Mourino, and A. Gomez, “Supercomputing challenges in electromagnetics,” in Proc. EuCAP, Barcelona, Spain, 2010, vol. 1, pp. 1–3. [20] F. Vipiana, M. A. Francavilla, and G. Vecchi, “EFIE modeling of highdefinition multiscale structures,” IEEE Trans. Antennas Propagat., vol. 58, no. 7, pp. 2362–2374, Jul. 2010. [21] O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propagat., vol. 37, no. 7, pp. 918–926, Jui. 1989. Angelo Freni (S’90–M’91–SM’03) received the Laurea (Doctors) degree in electronics engineering from the University of Florence, Italy, in 1987. Since 1990, he has been with the Department of Electronic Engineering, University of Florence, first as Assistant Professor, and since 2002, as Associate Professor of electromagnetism. From 1995 to 1999, he was also an Adjunct Professor at the University of Pisa, Italy, and in 2010 a Visiting Professor at the TU Delft University of Technology, Delft, The Netherlands. During 1994, he was involved in research in the Engineering Department, University of Cambridge, U.K., concerning the extension and the application of the finite element method to the electromagnetic scattering from periodic structures. Between 2009 and 2010, he also spent one year as a researcher at the TNO Defence, Security, and Safety, The Hague, The Netherlands, working on the electromagnetic modeling of kinetic inductance devices and their coupling with array of slots in THz range. His research interests include meteorological radar systems, radiowave propagation, numerical and asymptotic methods in electromagnetic scattering and antenna problems, electromagnetic interaction with moving media, and remote sensing. In particular, part of his research concerned numerical techniques based on the integral-equation, with a focus on domain-decomposition and fast solution methods.

Paolo De Vita received the M.S. degree in electronic engineering and the Ph.D. degree in information and telecommunication engineering from the University of Florence, Florence, Italy, in 1999 and 2004, respectively. From 1999 to 2007, he served as a Research Assistant in the Department of Electronics and Telecommunication, University of Florence. Since 2008, he has been with IDS Ingegneria dei Sistemi, Pisa, Italy. His main research interests are in numerical techniques for electromagnetic radiation and scattering problems.

FRENI et al.: FAST-FACTORIZATION ACCELERATION OF MOM COMPRESSIVE DOMAIN-DECOMPOSITION

Paola Pirinoli (M’96) received the Laurea and Ph.D. (Dottorato di Ricerca) degrees in electronic engineering, from the Politecnico di Torino, Italy, in 1989 and 1993, respectively. In October 1994, she joined the Department of Electronics of the Politecnico di Torino as an Assistant Professor; since December 2003, she has been an Associate Professor with the same department. From November 1996 to February 1997, she was a Visiting Research Fellow at the Laboratoire d’Electronique, Antennes et Telcomunication of the University of Nice (F), through a Fellowship of the Communaut de Travail des Alpes Occidentales (COTRAO). Her main research activities include the development of analytically based numerical techniques, essentially devoted to the fast analysis of printed structures on planar or curved substrates, the modelling of non conventional substrates as bi-isotropic ones, the design and analysis of antennas for wireless communications, and the development of efficient optimization techniques for the design of microwave devices. Dr. Pirinoli received the Young Scientist Award in 1998, the “Barzilai” prize for the best paper at the National Italian Congress of Electromagnetic (XII RiNEm) in 1998, and the prize for the best oral paper on antennas at the Millennium Conference on Antennas and Propagation (AP2000) in 2000. She is member of the program committees of several conferences and serves as reviewer for different journals.

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Ladislau Matekovits (M’94–SM’11) was born in Arad, Romania, on November 19, 1967. He received the degree in electronic engineering from the Institutul Politehnic din Bucures¸ti, Bucures¸ti, Romania, and the Ph.D. degree (Dottorato di Ricerca) in electronic engineering from Politecnico di Torino, Torino, Italy, in 1992 and 1995, respectively. Since 1995, he has been with the Electronics Department of the Politecnico di Torino, first with a postdoctoral fellowship, then as a Research Assistant. He joined the same department as Assistant Professor in 2001 and was appointed as Senior Assistant Professor in 2005. In late 2005, he was a Visiting Scientist at the Antennas and Scattering Department of the FGAN-FHR, Wachtberg, Germany. Since July 1, 2009, he has been a Marie Curie Fellow at Macquarie University, Sydney, NSW, Australia. His main research activities concern numerical analysis of printed antennas and in particular development of new, numerically efficient full-wave techniques to analyze large arrays, optimization techniques, and active and passive metamaterials. Dr. Matekovits is a recipient of many awards in international conferences and is a member of various conferences program committees. He was an Assistant Chairman and Publication Chairman of the European Microwave Week 2002 (Milan, Italy). He serves as a reviewer for different journals, including the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS.

Giuseppe Vecchi (M’90–SM’07–F’10) received the Laurea and Ph.D. (Dottorato di Ricerca) degrees in electronic engineering from the Politecnico di Torino, Torino, Italy, in 1985 and 1989, respectively, with doctoral research partly carried out at the Polytechnic University, Farmingdale, NY. He was a Visiting Scientist at Polytechnic University from 1989 to 1990. In 1990, he joined the Department of Electronics, Politecnico di Torino, as an Assistant Professor (Ricercatore) where, from 1992 to 2000, he was an Associate Professor and, since 2000, he has been a Professor. He was a Visiting Scientist at the University of Helsinki, Finland, in 1992, and has been an Adjunct Faculty in the Department of Electrical and Computer Engineering, University of Illinois at Chicago, since 1997. His current research activities concern analytical and numerical techniques for analysis, design and diagnostics of antennas and devices, RF plasma heating, electromagnetic compatibility, and imaging.

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Multilevel Adaptive Cross Approximation (MLACA) José M. Tamayo, Alexander Heldring, and Juan M. Rius

Abstract—The Multilevel Adaptive Cross Approximation (MLACA) is proposed as a fast method to accelerate the matrix-vector products in the iterative solution of the linear system that results from the discretization of electromagnetic Integral Equations (IE) with the Method of Moments (MoM). The single level ACA, already described in the literature, is extended with a multilevel recursive algorithm in order to improve the asymptotical complexity of both the computational cost and the memory requirements. The main qualities of ACA are maintained: it is purely algebraic and independent of the integral equation kernel Green’s function as long as it produces pseudo-rank-deficient matrix blocks. The algorithm is presented in such a way that it can be easily implemented on top of an existing MoM code with most commonly used Green’s functions. Index Terms—Adaptive cross approximation, fast integral equation methods, impedance matrix compression, method of moments, numerical simulation.

I. INTRODUCTION N recent years, a wide range of fast methods [1] have been developed for accelerating the iterative solution of the electromagnetic integral equations [2] discretized by method of moments (MoM) [3]. Most of them are based on multilevel subdomain decomposition and require a computational cost per itor . One of these methods eration of order is the multilevel fast multipole algorithm (MLFMA) [4]. The MLFMA has been widely used in the last years to solve very large electromagnetic problems [5], [6] due to its excellent computational efficiency. The main drawback of the MLFMA is the dependence of its formulation on the problem Green’s function. Notwithstanding, other general purpose methods have been developed. For instance, the multilevel matrix decomposition algorithm (MLMDA) [7]–[9] or its renewed version, the matrix decomposition algorithm - singular value decomposition (MDA-SVD) [10] exploit the compressibility of MoM submatrices corresponding to well separated subscatterers by using equivalent surface basis/testing functions cleverly distributed that radiate

I

Manuscript received July 01, 2010; revised March 09, 2011; accepted June 02, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. This work was supported in part by the Spanish Interministerial Commission on Science and Technology (CICYT) under projects TEC2009-13897-C03-01, TEC2009-13897-C03-02 and TEC2010-20841-C04-02 and CONSOLIDER CSD2008-00068 and in part by the Ministerio de Educación y Ciencia through the FPU fellowship program. The authors are with the AntennaLab, Department of Signal Processing and Telecommunications, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165476

and receive the same fields as the original basis/testing functions. Another fast solver, the adaptive integral method (AIM) [11] replaces the actual basis and testing functions with a regular volumetrical grid, that again radiates and receives the same fields as the original discretization, in order to efficiently compute the integral equations convolution using the FFT algorithm. However, all the aforementioned methods rely on the appropriate selection of elements with an equivalent physical behavior, either multipoles or equivalent surface or volume basis/testing functions. Hence the interest of purely algebraic methods, whose formulation is still independent of the problem Green’s function and operate solely with some of the impedance matrix elements, such as the IE-QR Algorithm [12], the IES3 [13], [14] and the adaptive cross approximation (ACA) [15]. Unfortunately, these algebraic methods present an asymptotic computational time and memory requirement not as good as that of the above-mentioned methods. The ACA, which is the basis of the multilevel algorithm proposed here, was developed in 2000 by Bebendorf [15] and since then has been widely used to solve large magnetostatic problems [16]. Recently, the ACA has been applied also to antenna radiation and RCS computation [17], [18]. For very large problems, the compression of well separated subscatterers matrix blocks with ACA has an asymptotical computational for matrix compression and for complexity of storage, as proved in [19], where is the number of unknowns of each subscatterer. In practice, these asymptotical behaviors is of the order of several millions, are not reached unless and therefore for moderately large values of one usually observes a much lower computational complexity. The aim of this paper is to present a multilevel version of the ACA method, first introduced by the authors in [20] at a preliminary state, using physical concepts similar to those presented in [7] for the MLMDA. Maintaining the purely algebraic formulation and in comparison with the single level ACA, the computaand tional cost and memory requirements are reduced to , respectively. The memory, and therefore the matrix-vector product time, are comparable to that of the MLFMA. II. DESCRIPTION OF THE ALGORITHM A. Tree Domain Decomposition Based on the Solid Angle Hereupon and up to Section V we address the interaction between two boxes or sets of samples contained within two spheres that do not intersect each other. These boxes come from a domain decomposition of the object that will be introduced in Section V. We shall call the two separated boxes, source and observation boxes, respectively. The degrees of freedom (DoF) of this far-interaction basically depend on the solid angle, or the angular portion, that one of the

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Fig. 1. Representation of the subdivision of the blocks of samples based on the solid angle seen from the other block. Coplanar on the left and perpendicular on the right.

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A SVD recompression of the obtained ACA decomposition can be performed utilizing two QR factorizations [21], so that either the or matrices can be done orthonormal, with an extra saving in memory. We have chosen the ACA threshold ten times lower than the SVD recompression threshold . The ACA-SVD has a computational cost or number of opwhereas the final amount of erations scaling with necessary memory scales with . The last quantity coincides also with the number of matrix elements evaluations in to decompose allowing us not to compute the whole matrix. Although this number of evaluations is asymptotically smaller than the computational cost of the ACA, for medium size problems it can dominate the total computational time as Green’s function evaluations can be highly time consuming. C. Degrees of Freedom (DoF)

boxes occupies from the view point of the other box, i.e., the solid angle when the coordinates origin is placed at the center of the other box. One box is subdivided into children boxes having all of them the same solid angle and therefore approximately the same number of DoF in their interaction with the other box. Two examples of this sort of subdivision are shown in Fig. 1. This subdivision must be applied to both the source and observation boxes placing the origin at the center of the other. The use of a decomposition based on the solid angle rather than just on the electrical size of the boxes is crucial to have a real gain in computational complexity and memory already for moderately electrically large problems. In the matrix representations throughout the article, the basis and testing functions will be reordered following the tree decomposition described, in order to obtain block-like representation. It means that functions in the same box of the tree are represented by contiguous rows/columns of the matrix. Furthermore, although the tree decomposition is not necessarily binary (see Fig. 1 on the right), it has been considered binary without loss of generality in the description of the algorithm. B. Adaptive Cross Approximation-Singular Value Decomposition (ACA-SVD) The ACA [15] is a purely algebraic technique for matrix compression. It accurately decomposes a low -rank matrix arising from an asymptotically smooth function into the product of two new matrices in a compressed form (1) being a matrix, a matrix and a matrix with , where is the -rank or number of DoF. The error in this approximation is controlled by a threshold which determines when to stop looking for more columns and rows of and , respectively. Most of the open Green’s functions in electromagnetism lead to an integral equation function (EFIE, MFIE, CFIE, ) which, when dealing with the interaction between two well-separated blocks in space, lead in turn to an interaction matrix with a deficient pseudo-rank that can be efficiently compressed with the ACA technique.

As stated in [7], a far interaction not only have a reduced number of DoF, fact exploited by the ACA, but also the DoF accomplish the following two properties, necessary for the MLACA. The number of DoF remains asymptotically constant for large boxes when the two interacting boxes are inversely changed in size, fixing everything else. The DoF of the interaction between a box at the finest level in one object, source or observation, and the whole other object remains also constant if the finest level is kept at a certain fixed electrical size, easily done varying the actual number of levels of the tree subdivision. D. Multilevel Adaptive Cross Approximation (MLACA) Applying the method of moments (MoM) it is possible to express the two boxes far-interaction with an impedance matrix with dimensions , where and are the number of basis functions in which the observation and source boxes have been discretized, respectively. In the first place, we must subdivide each box recursively into smaller domains utilizing the tree decomposition described in Section II-A. If is the number of levels, at the finest level we have in each subset or elements, considering subdomain approximately . is transformed into the product of In step 0 [Fig. 2(a)] two new matrices and . The procedure to obtain them is to split into strips, each corresponding to the interaction of one subset of basis functions at the finest level in the source box with the whole observation box (see first row of pictures in Fig. 3). Each of those strips only has DoF. They can be compressed with the ACA-SVD algorithm and regrouped [see Fig. 2(a)] concatenating horizontally the left-hand side matrices forming the new matrix and diagonally the right-hand side . In that matrices forming the new block-diagonal matrix way, the product of the two new matrices gives us an approximatrix. Then only the “ ” matrices mation of the original need to be permanently stored and the “ ” matrices will be subdivided and recompressed in the next steps of the algorithm. At step 1 , the matrix from level 0 is transformed into four new matrices [Fig. 2(b)] as follows. Due to the proper reordering of columns according to the tree decomposition of the source box, consecutive couples of strips in matrix correspond to the children of each box

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Fig. 4. Multilevel decomposition matrices for zero elements whereas gray blocks are filled.

Fig. 2. Graphical representation of the matrix transformations in the MLACA algorithm described in Section II. First step or step 0 (a) and step i (b) with i 1.



L

= 4 levels. White parts are

are transformed into four new matrices [Fig. 2(b)], where we join the contributions of the source box in the next level of the tree decomposition while splitting the observation object in the boxes from the previous level (each row of Fig. 3 represents a step of the algorithm). Each interaction has again DoF, and therefore is recompressible. After the we have the whole set of matrices which correctly step . combined represent a compressed version of the matrix ” from the last step need Only the “ ” matrices and the “ to be permanently stored. In this procedure, all the “ ” matrices are orthonormal to maintain the information (singular values) of each interaction in the “ ” matrices, which are the ones to recompress at each step. decomposed as a In order to have our original matrix product of very sparse matrices (see Fig. 4) we need to regroup the obtained matrices as follows: (2) For each

:

..

. (3)

Fig. 3. Schematic representation of the described multilevel procedure. Each row of figures represents each step on the algorithm. Gray blocks are the ones considered at each step. It starts with the interaction between all the finest level boxes of the source box “S” and the total observation box “O.” In the next steps, the source blocks are sequentially grouped whilst the observation boxes are split.

in the next decomposition level. Similarly, due to the proper reordering of rows according to the tree decomposition of the observation box, the matrix can be horizontally split obtaining that each subblock corresponds to each one of the two halves of the observation box (second row of pictures in Fig. 3). According to Section II-C, the new strips have again the same DoF. As the actual number of columns of the strips is , they can be recompressed with the ACA-SVD and regrouped for each matrix in the same manner as in step 0. The obtained and are sent to the next step in order to new matrices equally proceed recursively. Generalizing this procedure in a recursive manner, at step , the set of “ ” matrices from step ,

..

..

.

.

(4)

The previous procedure has a computational cost of and a final storage of (see Section III). However, during the algorithm the amount of necessary memory at some (for instance, at step 0, the steps can be temporarily of matrix has elements). Nevertheless, this temporary storage can be reduced by simply reordering the operations. It means that we do exactly the same operations and the result is exactly the same but the order of the operations is different. A representation of that reordering is shown in Fig. 5 for two levels. Each row in the figure represents all the matrix blocks

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Similarly, the total temporary necessary memory at the worst part of the algorithm can be proved to equal the following expression, development omitted for the sake of brevity and readability

(7) Only remains to obtain the computational cost of the construction of the compressed matrix. At step 0 of the algorithm ACA decompositions of matrices with we need to perform DoF and dimensions . Furthermore, at each step with we need to compute ACA decompositions of . matrices with again DoF but with dimensions Therefore, the total computational cost becomes

(8) Fig. 5. Representation of the order to follow (numbers) showing schematically which are the elements of the matrix involved. The arrows display which steps are necessary from the previous row in order to compute the matrices below. In levels. this case there are L

=2

involved at each step of the algorithm. The numbers above each matrix determine the order in which the operations are computed. It can be understood as an under demand algorithm, in the sense that each computation is only done when it is necessary, starting from the final goal, i.e., the interaction of the complete source box with each finest level observation box. III. COMPUTATIONAL COST AND STORAGE We remind that this section deals with time and storage of the MLACA algorithm applied to the interaction between two separated sets of samples and not to the global algorithm applied to the entire problem under study. Here is the constant number of DoF of each submatrix interaction inside the multilevel procedure defined in Section II-D. The final storage, which coincides with the computational cost of a matrix-vector product, corresponds with the whole set matrices (see Fig. 4). The maof matrices and the contains small block-matrices each one with ditrix . The matrices conmensions submatrices with dimensions . Finally, tain each one the matrix contains submatrices with dimensions . Therefore, the final storage is: (5) (6) where it has been used that and and that and are independent of . So finally, the total necessary amount of memory to store the matrix and the com. putational cost of a matrix-vector product is

(9) Summarizing, the computational cost of computing the matrix is , and the computational cost of a matrix-vector . product as well as the storage is IV. SVD THRESHOLD Similarly to what happens with real numbers, if we have a product of matrices and we want to assure a certain relative error in the resulting matrix, we have to consider the worst case where all the relative errors of each matrix are summed. So, to have a better error than with the direct application of the ACA-SVD ), we need to use a (equivalent to the MLACA with instead of because the matrix is decomthreshold matrices. The influence of that posed into the product of change of threshold in the computational costs and storage is analyzed below. The only thing which is affected by the threshold is the and number of DoF at each level that now depends on therefore on . Let us consider that the singular values decay exponentially up to some point (10) Then we must accomplish (11) and substituting (10) for finally obtain

and isolating we

(12) (13)

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As the coefficients of the order in the last section were proportional to , what we need to add is a factor to the computational cost and storage. In practice the new factor is negligible against the others. V. GLOBAL ALGORITHM (PUTTING IT TOGETHER) In this section, we are going to consider a complete radiation or scattering problem. It is necessary to subdivide the whole set of interactions in near and far interactions, so that we can apply the multilevel MLACA algorithm described above to the far parts. Commonly in the literature, a hierarchical octal tree is used to split the object or the related matrix and this approach can also be applied in our case if desired. Notwithstanding, we propose the utilization of a binary tree for this hierarchical subdivision. The differences in the results should not be too high in the vast majority of the cases. The only gain comes when the object discretization is very nonuniformly distributed or the object is much larger in one dimension than the rest, because in the proposed decomposition the boxes are adapted to the object at each subdivision. Once the matrix is divided in far and near interactions, the near parts are directly computed whereas the far parts are computed with the presented MLACA algorithm. Then we have a routine to perform a matrix-vector product which can be introduced in any existing iterative method, like a GMRES for factor to the aforemeninstance. This procedure adds a tioned asymptotical costs, obtaining finally a computational cost and a storage and matrix-vector growing with . product with VI. NUMERICAL RESULTS All the numerical experiments reported in this section have been performed on a PC with 64 GB of RAM and a Dual Intel Xeon X5460 processor at 3.16 GHz (eight cores). The computations were done in MATLAB© 7.8.0, always forcing to use only one CPU. We present two different experiments where we consider only the interaction between two sets of samples separated in space (Section VI-A). They are the most important in order to evaluate the computational cost and the necessary memory of the algorithm. They could well be a part of a major object or electromagnetic problem. Then, three complete object examples are presented (Section VI-B). In all the presented numerical experiments, when reference is made to a relative error in a computed (vector) parameter , this error has been calculated according to (14) is the 2-norm and is an independently created where reference solution. The proper reference solution is usually hard to obtain when dealing with extremely large-scale problems due to the limitations in the available resources. Therefore, the obtained error is usually an estimated error which has only into account the error introduced by the accelerated solver rather than the actual error, which would include the effect of the finite discretization among others.

Fig. 6. Storage requirement and CPU time versus the number of unknowns, with a fixed electrical discretization size, in the interaction between two separated parallel square plates for different number of levels in the MLACA. The threshold  equals 10 . (a) Storage. (b) CPU time.

A. Space Separated Blocks 1) Parallel Square Plates: In this experiment, a couple of 1 m 1 m square plates are placed in the same plane, separated 2m. Varying the frequency, and accordingly the number of unknowns of each square plate to have a fixed discretization of /10, we perform the computation of the interaction MoM-EFIE matrix between the two plates with the presented MLACA, for different number of used levels. Fig. 6 shows the required memory and the computational is used. time of this simulation when a threshold Clearly, the memory requirement decreases significantly with the number of levels, mainly when the electrical size of the plates grows. With respect to the CPU time, the optimum for small elec. Nonetheless, there trical sizes is the single level ACA or a plate side is a certain point, approximately for of , where the time for levels starts to be equal or even better when the plates size increase. The same happens for at a further point, approximately for or

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TABLE I STORAGE REQUIREMENT AND CPU TIME IN THE INTERACTION BETWEEN TWO SEPARATED PARALLEL SQUARE PLATES WITH  = 2428200 SAMPLES PER PLATE FOR DIFFERENT NUMBER OF LEVELS IN THE MLACA. THE THRESHOLD EQUALS 10

N



a plate side of . Exactly the same will happen for , not represented in the figure as it is beyond the maximum computed value, and so on and so forth. Therefore, an appropriate utilization to have a good trade-off between necessary memory and CPU time is the following: using the maximum number of levels but assuring that the CPU time is at least equal to the CPU . The number of levels to use is then chosen detime for pending on the solid angles between the two blocks of samples. For larger solid angles, larger number of levels. To stress the gain that we are actually obtaining, we have extracted the results for the largest computed parallel square plates or square side ) and included them in Table I. ( , whose comA good trade-off in this case would be to use putation takes 8 minutes less than with the single level ACA with 41% of the memory (from 16.47 GB to 6.79 GB). 2) Opposite Square Plates: As will be shown in the present section, the gain is much greater when the two square plates are placed face to face instead of in the same plane. The two 1 m 1 m square plates are oppositely placed and separated 2m. Again, the computation of the MoM-EFIE compressed interaction matrix is computed for different frequencies and number of and the disMLACA levels. The used threshold is cretized elements have an average length of . Fig. 7 shows the required memory and the computational time of this simulation. For electrically large blocks, the memory requirement is much smaller when the number of levels grows. For small blocks, the size with more levels is larger because the threshold is reduced proportionally to the number of levels (see Section IV). is faster As for the CPU time, the single level ACA until blocks with about samples or a side size of . becomes the reference until blocks with From this point, or a side size of . Afterwards, is the fastest and it follows successively for larger electrical sizes and number of MLACA levels. In this case, the better behavior of the multilevel, even in terms of CPU time is much more evident. Similarly to the parallel square plates case, we have included the results for the electrically largest simulated plates into samTable II. It corresponds to square plates with . If we use levels, it only takes ples or a side size of a 37% of the time with respect to the ACA and it only needs a , although it takes 15% of the memory. Furthermore, with a 46% of the ACA time which is a bit slower than with , it only needs a 7.2% of the memory. Therefore, the number of MLACA levels to use will depend on our resources. If we are more restricted by memory, we can use a larger number of levels. If CPU time is our main constraint there is always an optimum number of levels in terms of time, with still an excellent compression.

Fig. 7. Storage requirement and CPU time versus the number of unknowns, with a fixed electrical discretization size, in the interaction between two separated opposite square plates for different number of levels in the MLACA. The threshold equals 10 . (a) Storage, (b) CPU time.



TABLE II STORAGE REQUIREMENT AND CPU TIME IN THE INTERACTION BETWEEN TWO SEPARATED OPPOSITE SQUARE PLATES WITH  = 606600 SAMPLES PER PLATE FOR DIFFERENT NUMBER OF LEVELS IN THE MLACA. THE THRESHOLD EQUALS 10

N



B. Complete Problem Solutions 1) Large PEC Square Plate: The first object under study is . The threshold a PEC square plate with a length side of for the SVD recompressions is and the size of the finest level in the subdivision of the object to split in far and near in. The square plate has RWG teractions is unknowns with an average edge length of the discretized ele. In this case we have used the EFIE formulation ments of and as iterative method we have used a GMRES with a residual . Note that in this case all the far interthreshold to stop of actions are of the kind where there are square plates in the same plane, which as proved before, is the case where the MLACA needs electrically larger blocks to be worth using.

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TABLE III MLACA PERFORMANCE ON CURRENTS COMPUTATION OF 59:7 SIDE PEC SQUARE PLATE UTILIZING EFIE FORMULATION. N = 1385840 UNKNOWNS AND THRESHOLD  = 10

Fig. 8. Bistatic RCS of a 59:7 side PEC square plate with unknowns.

N

= 1385840

The reference current vector to compute the error has been obtained solving the problem with the MLACA with a threshold . We tried to obtain it with the ACA to be fairer but it was then beyond the memory limit with the correspondent slowing down of the simulation when the swapping to disk started, making it unfeasible in a reasonable time. Table III shows the performance of this simulation with the single level ACA and the MLACA. With the MLACA method we obtain a further compression of 5.5 GB with respect to the ACA with the same total solution time. It represents a reduction of an 11% of the total necessary memory. The estimated error is also lower for the MLACA than for the ACA, and in particular is of the same order as the selected threshold . Something worth mentioning, which will appear in all the simulation from here on, is that the iterations time is not improved in the same amount as the used memory. This is because the GMRES has an additional cost apart from a matrix-vector product which increases with each iteration. Fig. 8 shows the bistatic RCS of this simulation for the different methods. As expected from the errors in the surface currents, an excellent agreement can be observed for the RCS. Note that we only use multiple levels for the far interaction of electrically large blocks. Therefore, in this case only a few and 1006 interactions are improved (78 interactions with ), which do not represent a considerable percentage with

TABLE IV MLACA PERFORMANCE ON CURRENTS COMPUTATION OF 26:4 DIAMETER PEC SPHERE UTILIZING CFIE FORMULATION. N = 786432 UNKNOWNS AND THRESHOLD  = 10

of the total matrix size. However, this compression will increase considerably when the object is larger. This behavior will be better analyzed for the PEC missile in Section VI-B.3. 2) Large PEC Sphere: In order to avoid conditioning problems, which are not the issue of this manuscript, and having a good convergence, in the next two examples we will only analyze closed geometries using the CFIE formulation. A drawback of this approach is that the matrix is not symmetric. Therefore, all the interactions need to be computed. The next object under study is a PEC sphere with a diameter with unknowns. The threshold for the of , the size of the finest level in the SVD recompressions is subdivision of the object to split in far and near interactions is and the threshold for the GMRES residual to stop is again . The discretization has an average length of the elements . of The reference current vector to compute the error has been obtained solving the problem with the ACA with a threshold . Therefore, in any case it should benefit the ACA simulation, but as expected (see Section IV), the error will be slightly smaller for the MLACA. Table IV shows the results of this simulation. In this case, the MLACA has only been used at the first level (largest blocks) of the object subdivision where there are far interactions (1946 in), even though it should not be applied yet teractions with if we wanted to optimize the CPU time. The memory requirement is 6.77 GB lower for the MLACA than for the ACA with CPU time 21 minutes higher. It represents a gain in memory of a 18.6% with an increasing of 6.6% of CPU time. When the object becomes electrically larger, the gain in memory will be much more important together with a time improvement. The estimated relative error is slightly smaller for the MLACA and of the same order as the used threshold . Fig. 9 shows the bistatic RCS of the sphere computed from the surface currents obtained with the different methods. We have also included the reference available for the sphere from the Mie series development. There are slight differences which are normal considering the errors in the surface currents. These errors can be easily reduced by decreasing the threshold , which . has been set very high for this case 3) PEC Missile: Finally, we analyze a PEC missile presented . The shape and dimensions of in [22] which has a length of

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TABLE V MLACA PERFORMANCE ON CURRENTS COMPUTATION OF 75 PEC MISSILE UTILIZING CFIE FORMULATION. N = 1222419 UNKNOWNS AND THRESHOLD  = 10

Fig. 9. Bistatic RCS of a PEC sphere with diameter 26:4 and N = 786432 unknowns. TABLE VI MLACA PERFORMANCE ON THE MATRIX COMPUTATION OF 75 PEC MISSILE UTILIZING CFIE FORMULATION. THE CONTRIBUTION TO SUBBLOCKS OF THE ORIGINAL OBJECT OF DIFFERENT SIZES IS CONTEMPLATED. N = 1222419 UNKNOWNS AND THRESHOLD  = 10

Fig. 10. Bistatic RCS of a 75 length PEC missile with unknowns.

N

= 1222419

the missile can be extracted from the referenced paper. The misunknowns and an sile has been discretized with average length of the elements of as in the previous cases. As in the sphere, we use the CFIE formulation with a threshold and the size of the finest level in the subdivision of the . However, the object to split in far and near interactions of residual tolerance to stop the GMRES algorithm is set to due to the worse convergence rate. We were not able to obtain a good reference solution as we are very close to the RAM memory constraint of our computer. However, a comparison of the bistatic RCS with our results (see Fig. 10) and the one included in [22], calculated with a very different approach, shows a good agreement. Table V shows the results for this simulation. In this case, there are two columns for the MLACA. We use MLACA 1 when multiple levels are only applied to the block interactions at the subdivision levels 2 and 3 of the object which correspond with the largest far-interaction blocks (16 interactions with and 38 with ). In the MLACA 2, also the forth level far-interactions are calculated with multiple levels (2110 interactions ). With the inclusion of the forth level, the build with time is a bit higher but the gain in compression is considerable. In this case, the number of iterations is higher for the ACA than

the others. Often the one which takes longer to converge is the one with a higher relative error in the matrix elements but this cannot be concluded in the absence of a good reference. In any case, the total time of the MLACA is smaller than the ACA and even the build time, without considering the iterations is very close. The gain in memory is 10.4 GB with respect to the ACA which corresponds with a 18% of the total. To evaluate the contribution of the MLACA to the different far-interaction blocks, we have included Table VI. When the MLACA is applied to the relatively electrically small blocks at level 4 of the object subdivision we save a 21% of the memory but with an increase of a 14% of CPU time. In the next larger blocks level 3, the gain in memory is already a 55% with only 4 minutes more of time. And finally, the gain at the largest level 2, with only 6 far-interactions, is a 75% with just 3 extra minutes. Therefore, if the object is electrically larger, the compression of the larger block interactions will be much better, and also the contribution of the largest blocks interactions will represent a good proportion of the total memory. Fig. 10 shows the bistatic RCS computed with the different methods. There are some small differences which are normal considering the high threshold we are using. The main difference is in the single level ACA. VII. CONCLUSION The MLACA has been presented as a novel multilevel version of the well-known ACA algorithm, for the compression of matrices appearing in the Method of Moments discretization of radiation and scattering electromagnetic problems. Equally to ACA, MLACA applies when the source and field domains corresponding to the impedance matrix block being compressed are well-separated in space, which needs a previous hierarchical domain decomposition of the object under study

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and a different treatment of compressible far field blocks from noncompressible near field blocks. The storage and the matrix-vector product (or iteration) time and the matrix compresscale asymptotically as . The accuracy is expected to sion time scales as improve by reducing a threshold . Several numerical experiments have corroborated the gain in terms of memory requirement and CPU time with respect to the single level version ACA, already for objects of around one million of unknowns. It has been proved that the MLACA will be much superior for problems that are one or two orders of magnitude larger. Therefore, the author expects that the MLACA will find its place in the framework of supercomputers and parallelization, which are becoming a reality nowadays even at very low prices and will allow the solution of much larger objects. REFERENCES [1] W. C. Chew, J.-M. Jin, C.-C. Lu, E. Michielssen, and J. Song, “Fast solution methods in electromagnetics,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 533–543, Mar. 1997. [2] N. Morita, N. Kumagai, and J. Mautz, Integral Equation Methods for Electromagnetics. Boston, MA: Artech House, 1990. [3] R. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968. [4] J. Song, C.-C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag., vol. 45, no. 10, pp. 1488–1493, Oct. 1997. [5] S. Velamparambil and W. Chew, “Analysis and performance of a distributed memory multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2719–2727, Aug. 2005. [6] L. Gurel and O. Ergul, “Fast and accurate solutions of extremely large integral-equation problems discretised with tens of millions of unknowns,” Electron. Lett., vol. 43, no. 9, pp. 499–500, Apr. 2007. [7] E. Michielssen and A. Boag, “A multilevel matrix decomposition algorithm for analyzing scattering from large structures,” IEEE Trans. Antennas Propag., vol. 44, no. 8, pp. 1086–1093, Aug. 1996. [8] J. M. Rius, J. Parrón, E. Úbeda, and J. R. Mosig, “Multilevel matrix decomposition algorithm for analysis of electrically large electromagnetic problems in 3-d,” Microw. Opt. Technol. Lett., vol. 22, no. 3, pp. 177–182, Aug. 1999. [9] J. Parron, J. Rius, and J. Mosig, “Application of the multilevel matrix decomposition algorithm to the frequency analysis of large microstrip antenna arrays,” IEEE Trans. Magn., vol. 38, no. 2, pp. 721–724, Mar. 2002. [10] J. M. Rius, J. Parrón, A. Heldring, J. M. Tamayo, and E. Úbeda, “Fast iterative solution of integral equations with method of moments and matrix decomposition algorithm - singular value decomposition,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2314–2324, Aug. 1998. [11] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci., vol. 31, no. 5, pp. 1225–1251, Sep./Oct. 1996. [12] S. M. Seo and J.-F. Lee, “A single-level low rank IE-QR algorithm for PEC scattering problems using EFIE formulation,” IEEE Trans. Antennas Propag., vol. 52, no. 8, pp. 2141–2146, Aug. 2004. [13] S. Kapur and D. Long, “IES3: A fast integral equation solver for efficient 3-dimensional extraction,” in Proc. ICCAD, 1997, pp. 448–455. [14] S. Kapur and D. Long, “IES3: Efficient electrostatic and electromagnetic simulation,” IEEE Comput. Sci. Eng., pp. 60–66, Oct.-Dec. 1998. [15] M. Bebendorf, “Approximation of boundary element matrices,” Numer. Math., vol. 86, no. 4, pp. 565–589, 2000. [16] S. Kurz, O. Rain, and S. Rjasanow, “The adaptive cross-approximation technique for the 3D boundary-element method,” IEEE Trans. Magn., vol. 38, no. 2, pp. 421–424, Mar. 2002. [17] K. Zhao, M. Vouvakis, and J.-F. Lee, “The adaptive cross approximation algorithm for accelerated method of moments computations of EMC problems,” IEEE Trans. Electromagn. Compat., vol. 47, no. 4, pp. 763–773, Nov. 2005.

[18] J. Shaeffer, “Lu factorization and solve of low rank electrically large MOM problems for monostatic scattering using the adaptive cross approximation for problem sizes to 1 025 101 unknowns on a PC workstation,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Sep. 9–15, 2007, pp. 1273–1276. [19] A. Heldring, J. Tamayo, and J. Rius, “On the degrees of freedom in the interaction between sets of elementary scatterers,” in Proc. 3rd Eur. Conf. Antennas Propag., 23–27, 2009, pp. 2511–2514. [20] J. M. Tamayo, A. Heldring, and J. M. Rius, “Multilevel adaptive cross approximation (MLACA),” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 2009, pp. 1–4. [21] M. Bebendorf and S. Kunis, “Recompression techniques for adaptive cross approximation,” J. Integ. Equat. Appl., vol. 21, no. 3, pp. 331–357, 2009. [22] W.-D. Li, W. Hong, and H.-X. Zhou, “An IE-ODDM-MLFMA scheme with DILU preconditioner for analysis of electromagnetic scattering from large complex objects,” IEEE Trans. Antennas Propag., vol. 56, no. 5, pp. 1368–1380, May 2008. José M. Tamayo was born in Barcelona, Spain, on October 23, 1982. He received the degree in mathematics and the degree in telecommunications engineering from the Universitat Politècnica de Catalunya (UPC), Barcelona, both in 2006, and the Ph.D. degree in telecommunications engineering from the Universitat Politècnica de Catalunya (UPC), Barcelona, in 2011. In 2004, he joined the Telecommunications Department, Universitat Politècnica de Catalunya (UPC), Barcelona. His current research interests include accelerated numerical methods for solving electromagnetic problems.

Alex Heldring was born in Amsterdam, The Netherlands, on December 12, 1966. He received the M.S. degree in applied physics and the Ph.D. degree in electrical engineering from the Delft University of Technology, Delft, The Netherlands, in 1993 and 2002, respectively. He is presently an Assistant Professor with the Telecommunications Department, Universitat Politècnica de Catalunya, Barcelona, Spain. His research interests include integral equation methods for electromagnetic problems and wire antenna analysis.

Juan M. Rius received the “Ingeniero de Telecomunicación” degree in 1987 and the “Doctor Ingeniero” degree in 1991, both from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain. In 1985 he joined the Electromagnetic and Photonic Engineering group at UPC, Department of Signal Theory and Communications (TSC), where he currently holds a position of “Catedrático” (equivalent to Full Professor). From 1985 to 1988 he developed a new inverse scattering algorithm for microwave tomography in cylindrical geometry systems. Since 1989 he has been engaged in the research for new and efficient methods for numerical computation of electromagnetic scattering and radiation. He is the developer of the Graphical Electromagnetic Computation (GRECO) approach for high-frequency RCS computation, the Integral Equation formulation of the Measured Equation of Invariance (IE-MEI) and the Multilevel Matrix Decomposition Algorithm (MLMDA) in 3D. Current interests are the numerical simulation of electrically large antennas and scatterers. He has held positions of “Visiting Professor” at EPFL (Lausanne) from May 1, 1996 to October 31, 1996; “Visiting Fellow” at City University of Hong Kong from January 3, 1997 to February 4, 1997; “CLUSTER Chair” at EPFL from December 1, 1997 to January 31, 1998; and “Visiting Professor” at EPFL from April 1, 2001 to June 30, 2001. He has more than 50 papers published or accepted in refereed international journals (28 in IEEE TRANS.) and more than 150 in international conference proceedings.

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Efficient Analyzing EM Scattering of Objects Above a Lossy Half-Space by the Combined MLQR/MLSSM Zhaoneng Jiang, Yuan Xu, Yijun Sheng, and Maomao Zhu

Abstract—A large dense complex linear system is obtained when solving an electromagnetic scattering of arbitrary shaped object located above a lossy half-space with the surface integral equation approach. To analyze the large dense complex linear system efficiently, the multilevel QR (MLQR) is used to accelerate the matrix–vector multiplication when the corresponding matrix equation is solved by a Krylov-subspace iterative method. Although the MLQR is more efficient than the direct solution, this paper presents a novel recompression technique to further reduce computation time and storage memory. The technique applies the multilevel simply sparse method (MLSSM) to the matrices of MLQR. Using the MLSSM, a sparser representation of the impedance matrix is obtained, and a more efficient matrix–vector multiplication is implemented. The combined MLQR/MLSSM is comparable to the MLQR and the adaptive cross-approximation/singular value decomposition (ACA-SVD) in terms of computation time and memory requirement. Remarkably, the new formulation can reduce the computational time and memory by about one order of magnitude, with excellent accuracy. Index Terms—Compression techniques, half-space, multilevel QR (MLQR), Multilevel simply sparse method (MLSSM).

I. INTRODUCTION

D

IFFERENT electromagnetic scattering problems have been studied in recent years. They include, but are not limited to, radar cross-section (RCS) computations, antenna analysis, remote sensing, biomedicine, electromagnetic interference (EMI), and electromagnetic compatibility (EMC). In this paper, the scattering properties of arbitrary shaped objects located above a lossy half-space are analyzed. Simulating these problems is very time demanding, and an efficient numerical method is required. The method of moments (MoM) [1], [2] is one of the most widely used techniques for solving electromagnetic problems. For a large electromagnetic problem, the number of unknowns will be large and it will be difficult to solve the matrix equation due to the memory requirement of and computational complexity of . This difficulty can be circumvented by using the Krylov iterative method, which can reduce the operation complexity to .

Manuscript received November 13, 2010; revised June 08, 2011; accepted June 13, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. This work was supported in part by the Major State Basic Research Development Program of China (973 Program: 2009CB320201), the Natural Science Foundation of China under Grants 60871013 and 60701004, and the Jiangsu Natural Science Foundation under Grant BK2008048. The authors are with the Department of Communication Engineering, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2165493

Many fast methods have been developed for accelerating the iterative solution of the electromagnetic integral equations discretized by MoM. Most of them are based on multilevel subdomain decomposition and require a computational cost of order . One of the most popular techniques is multilevel fast multipole algorithm (MLFMA) [3], [4], which has been widely used to solve very large electromagnetic problems due to its excellent computational efficiency. However, in the MLFMA, the formulation, implementation, and occasionally performance depend on a priori knowledge of Green’s function. Hence, it is inconvenient for the complex form of special domain Green’s functions of half-space. In [5] and [6], fast multipole method (FMM) has been extended for general targets in the presence of a lossy half-space. The “far” terms are evaluated via an approximation to dyadic Green’s function using a single appropriately weighted image in real space [7]. However, this real image representation of Green’s function is accurate when the source and observation points are separated by a wavelength or more [8]. This limits the minimum group size of the FMM, and leads to a low efficiency. QR is another popular technique used to analyze the scattering/radiation [9]–[11], which exploits the well-known fact that the submatrices for well-separated subscatterers are rank deficient and can be compressed [12], [13]. In contrast to MLFMA, the QR is purely algebraic, and therefore, independent of the problem of Green’s function. The multilevel QR (MLQR) has been successfully applied in [14] and [15] to electromagnetic problems. The aim of this paper is to present combined MLQR/MLSSM for analyzing the electromagnetic problems of objects located above a lossy half-space. It utilizes the multilevel simply sparse method (MLSSM) [16]–[21] to recompress the matrices of MLQR. Simulation results show that the combined MLQR/MLSSM is more computationally efficient than the MLQR and the adaptive cross-approximation/singular value decomposition (ACA-SVD) [22]. This paper is organized as follows. Section II gives a brief review of the electric field integral equation (EFIE). Section III describes the combined MLQR/MLSSM in detail. Section IV provides numerical examples to demonstrate accuracy and efficiency of the proposed method. Conclusions are provided in Section V. II. FORMULATIONS In this paper, the analysis is based on the electric field integral equation (EFIE) [23]

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(1)

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is the interaction matrix between observation and where stands for the number of the test funcsource boxes, and tions in the observation box and stands for the number of the basis functions in the source box, and denotes the rank of , which is much smaller than and . Therefore, the matrix–vector product operation of the two matrices is much easier than the operation of the direct multiplication. The impedance matrix can be expressed by MLQR as (6) is the number of nonempty groups at level and, denotes the number of far interaction groups of the th at level . The nonempty group for each observation group is associated with the interaction between product and the source group . For a given the observation group observation group , it is needed to store the matrix for , increasing the memory requirement. different source group The MLSSM is utilized to recompress the matrices of MLQR in this paper. It provides a sparser impedance matrix of the QR than the conventional MLQR in solving 2-D and 3-D electromagnetic problems. where

Fig. 1. The sketch of the oct-tree structure.

where is spatial domain dyadic Green’s function and can be expressed as follows: (2) Utilization of Galerkin’s method results in a matrix equation (3) where is the column vector containing the unknown coefficients of the surface current expansion with RWG basis functions [2]. Using the fast algorithm, the impedance matrix can be written as (4) is the near part of and it is computed directly. is Here the far part of and it is compressed by MLSSM or ACA-SVD. are not explicitly computed and stored. Those elements in

B. Combined MLQR/MLSSM The impedance matrix filled by MLSSM is carried out based on the same multilevel spatial decomposition of the underlying geometry. The single level of SSM is presented in [24] and the MLSSM is shown in [16]–[18]. The structure of the SSM representation is expressed in a multilevel recursion manner as (7)

Take 3-D problems into account; MLQR is based on the data structure of the oct-tree [14], [15]. In Fig. 1, the box enclosing the object is subdivided into smaller boxes at multiple levels, in the form of an octal tree. The largest boxes not touching each other are at level 3, while the smallest boxes are at level . The subdivision process runs recursively until the finest level . The boxes which are nontouching each other are analyzed by QR. Consider any two boxes: one is a source box consisting of a list of basis functions and the other is an observation box consisting of a list of test functions. The impedance matrix between two well-separated boxes can be expressed through low-rank representations [9]–[11]. QR utilizes this low-rank nature of the interaction between two well-separated boxes. In QR implementation, the impedance matrix of two well-separated boxes can be expressed as two small matrices [9]–[11]

where is the reduced order impedance matrix and it consists , which will be compressed of only far interactions at level in the coarser levels recursively up to level 3. There is no level near interactions at the finest level . Thus, is the impedance matrix . In (7), is a sparse matrix containing all near-neighbor interactions at level of the oct-tree which are and are the not represented at finer level of the oct-tree. new testing and basis function matrices, respectively, which are block diagonal unitary matrices that represent interaction between source and observation groups in nontouching groups at level . Note that the operation of MLSSM can be done with operations at level , and denotes the number of unknowns. is the rank of far-field matrix, which is much lower than . Therefore, the operation of MLSSM is at level . Summing over all approximate to levels, the complexity of this multilevel algorithm is approximate to . 1) The Single Level of the Combined MLQR/MLSSM: For simplicity, the procedure of the single level of the combined MLQR/MLSSM is first described in the following. The impedance matrix can be expressed as

(5)

(8)

III. COMBINED MLQR/MLSSM A. Multilevel QR

JIANG et al.: EFFICIENT ANALYZING EM SCATTERING OF OBJECTS ABOVE A LOSSY HALF-SPACE BY THE COMBINED MLQR/MLSSM

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The following procedure is used to form matrices , and at level 3. 1) At finest level 3, compute the near interaction matrix directly. 2) Fill the far-field submatrix using QR. 3) Form the new testing function matrix , the row transformation matrix at level 3. For a given observation group , loop over source group , which are not near-neigh. Then, concatenate all mabors of observation group , which are expressed by (5), and obtain matrix trices

(9) Maintaining the admissible error , use QR and SVD to compress

Fig. 2. Level 3 SSM matrices: (a) U ; (b) Z ; and (c) V

.

where

(10)

(15)

(11)

and are a little different from The procedures to form that of the single-level case. The main differences are in the proand . In the multilevel procedures of forming matrices for cedure, it needs to concatenate not only all matrices at level 4, but also the corresponding matrices matrix at level 3. The form of is expressed as

which gives the final expression of

is the th diagonal block of . Implementing the procedure as above for all observation groups, we can build . The residual matrix is multiplied with matrix which is expressed by (5). 4) Form the new basis function matrix , which is the column transformation matrix at level 3. For a given , loop over observation group , which source group . Concatenate are not near-neighbors of source group all matrices and obtain matrix .. . (12) Maintaining the admissible error , use QR and SVD to compress (13) is the th diagonal block of . Implementing the procedure as the above for all source groups, we can build . forms matrix The residual matrix The essential part of the above procedure is to postcompress the far interaction of impedance matrix. The major usage of the , and at novel form of QR is to store the matrices are very sparse, where and are level . Matrices is both unitary and block diagonal. The dimension of matrix , and are shown in very small. The forms of matrices Fig. 2 at level 3. 2) The Multilevel of the Combined MLQR/MLSSM: The procedure of multilevel is a little more complicated than the single-level combined MLQR/MLSSM. For simplicity, suppose that the object is decomposed in four-level oct-tree, then the impedance matrix can be expressed as (14)

(16) where denotes the near interaction groups of the th at level . To nonempty group for each observation group form matrix , it also needs to concatenate corresponding matrices at level 3. , and are obtained, matrices and are When , far interaction part of at level 3. obtained from matrix and . Repeat steps 3 and 4 at level 3 to obtain The function of the above procedure is to postcompress the far interaction of impedance matrix . The major usage of the , and at all levels. MLSSM memory is to store matrices The truncating tolerances of (11) and (13) do not affect the complexity of the MLSSM; they only affect the coefficient of the scaling. Using the MLSSM for finding the impedance matrix , a fast matrix–vector multiplication algorithm can be obtained as follows. Subroutine MVM Begin

Call MVM (y2, y3, l-1);

End

The above matrix–vector multiplication algorithm is very similar to the MLFMA in the manner. Because of recompression of far interactions, the matrix–vector multiplication of the MLSSM is more efficient than that of the MLQR.

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Fig. 4. Bistatic scattering cross section of a two-cylinder geometry. Fig. 3. Bistatic scattering cross section of a missile geometry. TABLE I THE FAR-FIELD MEMORY AND ONE MATRIX–VECTOR MULTIPLICATION TIME FOR MISSILE GEOMETRY

TABLE II THE FAR-FIELD MEMORY AND ONE MATRIX–VECTOR MULTIPLICATION TIME FOR TWO CYLINDERS

B. Two Cylinders Above Half-Space Interface IV. NUMERICAL RESULTS To validate and demonstrate the accuracy and efficiency of the proposed method, some numerical results are presented in this section. In the implementation of the proposed method, the restarted version of the GMRES algorithm [25], [26] is used as the iterative method. The restart number of the GMRES is set to be 30 and the stop precision of the restarted GMRES is . The truncating tolerances of the MLSSM, denoted to be relative to the largest the ACA-SVD, and the MLQR are all singular value. All experiments are performed on a Core-2 6300 with 1.86-GHz CPU and 1.96-GB RAM in single precision. A. A Missile Above Half-Space Interface The first half-space example is a missile geometry, the bottom of which is located at 5 m above the half-space interface. The height of the cylinder is 8.2 m, and the radius of the cylinder is 0.6 m. The rotation axis of missile geometry is -axis, and the frequency is 100 MHz. It consists of the missile geometry with 0 0 18 789 unknowns. The incident direction is and the scattered angles vary from 0 to 89 in the azimuthal direction when the pitch angle is fixed at 0 . Fig. 3 shows that the result of the proposed method agrees very well with that of the MLQR and the ACA-SVD. Table I summarizes the MVP time and the far-field memory storages of the proposed method, the ACA-SVD, and the MLQR. “MVP time” in the table indicates the time of one matrix–vector production. When compared with the MLQR algorithm, it decreases the far-field memory storage by a factor of 6 in this example, while it reduces the MVP time by a factor of 5 in this example.

The second example is the scattering of two cylinders, the bottom of which is located at 0.2 m above the half-space interface. The height of the large cylinder is 1.02 m, and the radius of the large cylinder is 0.2 m. The height and the radius of the small cylinder are 0.42 and 0.12 m, respectively. The rotation axis of this geometry is -axis, and the frequency is 900 MHz. It consists of the cylinder geometry with 24 489 0 0 and the unknowns. The incident direction is scattered angles vary from 0 to 89 in the azimuthal direction when the pitch angle is fixed at 0 . The bistatic RCS by use of the proposed method is shown in Fig. 4, and agrees well with that of the MLQR and the ACA-SVD. Table II shows the far-field memory storages and MVP time of the proposed method, the ACA-SVD, and the MLQR. When compared in terms of the far-field memory storage and the MVP time, the proposed method has a gain over the MLQR algorithm by a factor of 7 in the far-field memory storage, while it has gain a factor of 6 in the MVP time. C. A Sphere Above Half-Space Interface The last example is a sphere above the half-space interface, as shown in Fig. 5. The radius of the sphere is 0.5 m, and the bottom of the geometry is situated 0.4 m above the soil interface. The frequency is 300 MHz. It consists of the sphere geometry with 11 028 unknowns. The bistatic RCS of the proposed method is given in Fig. 5. It can be seen that there is an excellent agreement among them. Figs. 6 and 7 show the memory requirement and CPU time per iteration as a function of the number of unknowns for the sphere geometry. This experiment was done on an Intel(R) Core(TM) 2 Quad CPU at 2.83 GHz and 8 GB of

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Fig. 5. Bistatic scattering cross section of a sphere geometry. Fig. 8. Far-field memory for two truncating tolerances needed for the sphere geometry.

Fig. 6. Far-field memory needed for the sphere geometry.

Fig. 9. The CPU time per iteration of sphere geometry for two truncating tolerances.

of unknowns for two truncating tolerances and are given in Figs. 8 and 9. With reference to Figs. 8 and 9, the scalings of the memory requirement and the MVP time are the same for different tolerances. However, the truncating tolerance does not affect the scalings of the memory requirement and the MVP time; it only affects the coefficient of the scaling. V. CONCLUSION

Fig. 7. CPU time for a matrix–vector operation as a function of the number of unknowns for the sphere geometry.

RAM. With reference to Fig. 6, the memory requirement of the proposed method is much lower than that of the ACA-SVD and the MLQR. Also, it is much lower than that of the PILOT [14] implementations. With reference to Fig. 7, the MVP time of the proposed method is also much lower than that of the ACA-SVD, the MLQR and the PILOT implementations. It can be found that the memory requirement and the MVP time of the proposed method scale with the unknowns linearly. The scalings of the memory requirement and the MVP time per iteration of the proposed method as a function of the number

In this paper, a new combined method is applied to efficiently solve the EM scattering of an arbitrary shaped object located above a lossy half-space. In contrast to the MLFMA, the new combined method is purely algebraic and, therefore, does not depend on the problem of Green’s function. It is more convenient than the MLFMA for solving complex Green’s functions’ problems. Since recompression of the matrices of the MLQR, the matrix–vector multiplication of the combined MLQR/MLSSM is much more efficient than that of the MLQR, while the memory required for the combined MLQR/MLSSM is also much lower than that of the MLQR. Moreover, the numerical results demonstrate that the combined MLQR/MLSSM is also much more efficient than the ACA-SVD and the PILOT. It can be used to compute monostatic RCSs of complex objects efficiently.

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REFERENCES [1] R. F. Harrington, Field Computations by Moment Methods. New York: MacMillan, 1968. [2] 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. 5, pp. 409–418, May 1982. [3] R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: A pedestrian prescription,” IEEE Antennas Propag. Mag., vol. 35, no. 6, pp. 7–12, Jun. 1993. [4] W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast Efficient Algorithms in Computational Electromagnetics. Boston, MA: Artech House, 2001. [5] X. D. Wang, D. H. Werner, L.-W. Li, and Y.-B. Gan, “Interaction of electromagnetic waves with 3-D arbitrarily shaped homogeneous chiral targets in the presence of a lossy half space,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3647–3655, Dec. 2007. [6] R. S. Chen, Y. Q. Hu, Z. H. Fan, D. Z. Ding, D. X. Wang, and E. Yung, “An efficient surface integral equation solution to EM scattering by chiral objects above a lossy half space,” IEEE Trans. Antennas Propag., vol. 57, no. 11, pp. 3586–3593, Nov. 2009. [7] Z. Liu, J. He, Y. Xie, A. Sullivan, and L. Carin, “Multilevel fast multipole algorithm for general targets on a half-space interface,” IEEE Trans. Antennas Propag., vol. 50, no. 12, pp. 1838–1849, Dec. 2002. [8] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Piscataway, NJ: IEEE Press, 1996, ch. 4. [9] N. A. Ozdemir and J. F. Lee, “A low-rank IE-QR algorithm for matrix compression in volume integral equations,” IEEE Trans. Magn., vol. 40, no. 2, pp. 1017–1020, Mar. 2004. [10] S. M. Seo and J. F. Lee, “A single-level low rank IE-QR algorithm for PEC scattering problems using EFIE formulation,” IEEE Trans. Antennas Propag., vol. 52, no. 8, pp. 2141–2146, Aug. 2004. [11] K. Z. Zhao and J. F. Lee, “A single-level dual rank IE-QR algorithm to model large microstrip antenna arrays,” IEEE Trans. Antennas Propag., vol. 52, no. 10, pp. 2580–2585, Oct. 2004. [12] S. Kapur and D. E. Long, “IES3: Efficient electrostatic and electromagnetic simulation,” IEEE Computer Sci. Eng., vol. 5, no. 4, pp. 60–67, Oct.–Dec. 1998. [13] S. Kapur, D. Long, and J. Zhao, “Efficient fullwave simulation in layered lossy medium,” in Proc. IEEE Custom Integr. Circuits Conf., May 1998, pp. 211–214. [14] D. Gope and V. Jandhyala, “Oct-Tree-Based multilevel low-rank decomposition algorithm for rapid 3-D parasitic extraction,” IEEE Trans. Comput.-Aided Design, vol. 23, no. 4, pp. 1575–1580, Nov. 2004. [15] D. Gope and V. Jandhyala, “Efficient solution of EFIE via low-rank compression of multilevel predetermined interactions,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3324–3333, Oct. 2005. [16] F. X. Canning and K. Rogovin, “Simply sparse, a general compression/ solution method for MoM programs,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 2002, vol. 2, pp. 234–237. [17] A. Zhu, R. J. Adams, and F. X. Canning, “Modified simply sparse method for electromagnetic scattering by PEC,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Washington, DC, 2005, vol. 4, pp. 427–430. [18] J. Cheng, S. A. Maloney, R. J. Adams, and F. X. Canning, “Efficient fill of a nested representation of the EFIE at low frequencies,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 2008, pp. 1–4. [19] R. J. Adams, A. Zhu, and F. X. Canning, “Sparse factorization of the TMz impedance matrix in an overlapped localizing basis,” Progr. Electromagn. Res., vol. 61, pp. 291–322, 2006. [20] Y. Xu, X. Xu, and R. J. Adams, “A sparse factorization for fast computation of localizing modes,” IEEE Trans. Antennas Propag., vol. 58, no. 9, pp. 3044–3049, Sep. 2010. [21] X. Xu, “Modular fast direct analysis using non-radiating local-global solution modes,” Ph.D. dissertation, Electr. Comput. Eng. Dept., Univ. Kentucky, Louisville, KY, 2009. [22] M. Bebendorf and S. Kunis, “Recompression techniques for adaptive cross approximation,” J. Integral Equations Appl., vol. 21, no. 3, pp. 331–357, 2009. [23] K. A. Michalski and D. Zheng, “Electromagnetie scattering and radiation by surfaces of arbitrary surfaces of arbitrary shape in layered media—Part I and Part II,” IEEE Trans. Antennas Propag., vol. 38, no. 3, pp. 335–352, Aug. 1990.

[24] F. X. Canning and K. Rogovin, “A universal matrix solver for integral equation-based problems,” IEEE Antennas Propag. Mag., vol. 45, no. 1, pp. 19–26, Feb. 2003. [25] Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput, vol. 7, pp. 856–869, Jul. 1986. [26] Y. Saad, Iterative Methods for Sparse Linear System. Boston, MA: PWS, 1996. [27] Z. J. Liu, J. Q. He, Y. J. Xie, A. Sullivan, and L. Carin, “Multilevel fast multipole algorithm for general targets on a half-space interface,” IEEE Trans. Antennas Propag., vol. 50, no. 12, pp. 1838–1849, Dec. 2002.

Zhaoneng Jiang was born in Jiangsu Province, China. He received the B.S. degree in physics from Huaiyin Normal College, Huai’an, Jiangsu, China, in 2007. He is currently working towards the Ph.D. degree in electromagnetic field and microwave technique at Nanjing University of Science and Technology (NJUST), Nanjing, China. His current research interests include computational electromagnetics, antennas and electromagnetic scattering and propagation, and electromagnetic modeling of microwave integrated circuits.

Yuan Xu received the B.S. and M.S. degrees in mathematics from Suzhou University, Suzhou, China, in 1988 and 1991, respectively, and the Ph.D. degree in electrical engineering from Nanjing University of Science and Technology (NJUST), Nanjing, China, in 2001. He worked as a Postdoctoral Scholar at the University of Kentucky, Lexington, in 2006 and 2010. He is currently an Associated Professor with the Electronic Engineering Department, NJUST. His research interests include computational electromagnetics, integral equation methods, and the analysis of large-scale electromagnetics problems.

Yijun Sheng was born in Jiangsu Province, China. He received the M.S. degree in physics from Nanjing University of Science and Technology (NJUST), Nanjing, China, in 2003, where he is currently working towards the Ph.D. degree in electromagnetic field and microwave technique. His current research interests include computational electromagnetics, antennas and electromagnetic scattering and propagation, and electromagnetic modeling of microwave integrated circuits.

Maomao Zhu was born in Anhui Province, China. She received the B.S. degree in physics from Anhui University, Hefei, China, in 2009. She is currently working towards the M.S. degree in electromagnetic field and microwave technique at Nanjing University of Science and Technology (NJUST), Nanjing, China. Her current research interests include computational electromagnetics, antennas and electromagnetic scattering and propagation, and electromagnetic modeling of microwave integrated circuits.

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Method-of-Moments Analysis of Resonant Circular Arrays of Cylindrical Dipoles George Fikioris, Senior Member, IEEE, Spyridon Lygkouris, and Panagiotis J. Papakanellos, Member, IEEE

Abstract—We perform a method-of-moments (MoM) analysis of a circular array of cylindrical dipoles. The array is known from earlier theoretical and experimental studies to possess very narrow resonances. The earlier theoretical studies were carried out using the “two-term theory.” The present paper is a direct continuation of a recent work showing that the problem possesses unique and particular difficulties. The main difficulties are overcome herein using a set of improved kernels in the usual Hallén-type integral equations (these kernels had been developed in previous works, and were successfully incorporated into the aforementioned two-term theory analyses). We make a detailed comparison of our MoM results to two-term theory results and, also, to the earlier experimental results. Index Terms—Antenna arrays, Galerkin method, method of methods, resonance.

I. INTRODUCTION

T

HIS paper deals with circular arrays of parallel, non-staggered cylindrical dipoles consisting of a large of elements, with only element no. 1 driven and number with the remaining elements being parasitic and unloaded. Recent theoretical and experimental studies [1]–[7] have demonstrated that these arrays can exhibit a series of very narrow resonances, called “phase-sequence resonances.” Each such resonance is a narrow peak of the driving-point —or, equivalently, of the conductance , see (1) —as a function of phase-sequence conductance , where is the current the frequency at the center of the driven dipole, is the single driving voltage, m/sec is the velocity of light in and the vacuum. At the resonance, the driving-point susceptance is zero (to be more precise, is very small at the resonance, while it exactly vanishes at a frequency very close to the resonant frequency). Resonances can occur only if we properly select the dipoles’ radius and length , as well as the inter-element spacing . An integer parameter is used to designate each resonance, with successive and increasing values of corresponding to increasing resonant frequencies, Manuscript received June 28, 2010; revised May 12, 2011; accepted June 02, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. The work of G. Fikioris and P. J. Papakanellos was supported by the PEBE 2009 basic research program. The work of S. Lygkouris was derived from his Senior Thesis at the Hellenic Air Force Academy. G. Fikioris is with the School of Electrical and Computer Engineering, National Technical University, GR 157-73 Zografou, Athens, Greece (e-mail: [email protected]). S. Lygkouris and P. J. Papakanellos are with the Hellenic Air Force Academy, GR 1010 Dekelia, Athens, Greece (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2165475

and with the last and most narrow resonance corresponding (for simplicity, we assume throughout that is to case of even). For lossless dipoles, and for the are theoretically the array of [2]–[5], the values of and predicted to be extremely large. These large values, however, greatly decrease for dipoles with finite conductivity [3], [4, Ch. 12]. Discussions related to array efficiencies can be found in [8]. The theoretical results in [1]–[7] were obtained using the “two-term theory” (TTT), which is an approximate solution coupled integral equations for the currents along the to the cylindrical dipoles. This theory was originally developed by R. W. P. King in 1959 [9]. Additional early references are discussed in [4]. The fact that the resonances are extremely narrow indicates that analysis of resonant circular arrays is a demanding problem. array was selected as a test case The aforementioned during the inaugural meeting of the EuRAAP working group on software [10], and was recently simulated using the array scanning method applied with the method of moments (MoM) [11]. The resonant values shown in [11] (see [11, Fig. 5]) are much smaller than the aforementioned TTT values. Also, they differ from the experimental values of [3]–[5]. The authors of [11] attribute their differences with the experiment to their use of the delta-function generator instead of a coaxial feed. Another recent work illustrating the unique difficulties of the problem is [12], to which the present paper is a direct continuation. Reference [12] is a first step toward an accurate analysis of resonant circular arrays using MoM applied to the well-known Hallén-type integral equations. The main conclusion of [12] is that, at least as far as resonances are concerned, it is of crucial importance to use the “modified” kernels in place of the “original” ones. The original kernels, however, are adequate for conventional arrays. The modified kernels were proposed in the early 1990s after detailed studies by D. K. Freeman and T. T. Wu [13], [14], and were successfully incorporated in the TTT analyses of [2]–[7] (note, however that they had not been incorporated in the early paper [1], which used the original kernels). More specifically, it is shown in [12] (see also [15]) that use of ; the original kernels can yield negative values of and on the other hand, no such negative values were found when the modified kernels were used. Since equals the radiated power, must always—for any array with a single driven element, and at any frequency—be a positive quantity. This is also true for , as explained in [12]. Thus, the aforementioned negative values of and are clearly meaningless results. The introduction of [12] discusses in detail the considerations motivating the investigations therein; it also discusses previous related works and potential applications of these arrays.

0018-926X/$26.00 © 2011 IEEE

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Fig. 1. Phase-sequence conductance , and frequency for (2b) and (2c).

N = 20; m = 10

G

G obtained by MoM as function of M = 26. The two curves correspond to

M

Fig. 2. Resonant conductance obtained by MoM as function of for and . The three sets of points correspond to (2b), (2c), and (2c) together with the use of the CAM of [20, Eq. (5)].

N = 20

m = 10

Although the use of the modified kernels does seem to elim, the results of [12] inate the problem of negative and are certainly not conclusive because the resonant (peak) values of do not appear convergent as the number of basis functions increases; this problem is so serious that even seems very difficult, see [12, Fig. 7], a rough estimation of as well as Fig. 2. On the other hand, a rough estimation of the is possible, see [12, Fig. 6] and Fig. 3. resonant frequency It is conjectured in [12, Sect. IV] that the problem will no longer occur if one uses the “refined modified kernels,” where our terminology is consistent with [4, Sect. 11.8]. The first purpose of the present paper is to demonstrate that this is indeed true and to compare the converged results with corresponding TTT results. For the purpose of MoM analysis, the “refined modified kernels” thus greatly improve upon the modified kernels; for this reason, we will use the term “improved kernels” instead of “refined modified kernels.” The latter term, however, is appropriate for the TTT. As further discussed in Section VI, the fact that a change in kernels greatly improves one method (MoM) but not another (TTT) is due to the aforementioned difficulty in obtaining MoM results for large values of , and should not be in considered paradoxical: There is no parameter similar to the TTT. In this sense, the TTT is an explicit approximate solution. Even though the TTT does include integrals that must be

Fig. 3. Like Fig. 2, but for the resonant frequency

G

.

evaluated numerically, it is still possible to use the TTT formulas to obtain a large number of general conclusions about resonant circular arrays [4, Ch. 11–12]. This is not possible with a purely numerical solution. The discussions above concern lossless dipoles. For the case of lossy (but highly conducting, e.g., brass) dipoles, proper integral equations were developed in [4] and [5], where they were solved approximately by means of an appropriately modified version of the TTT. The second purpose of the present paper is to apply MoM to those integral equations, and to compare with the corresponding TTT results. Once again, the integral equations involve the improved kernels rather than the modified or the original ones. Highly conducting dipoles in free space are equivalent to the physically unrealizable model of highly conducting monopoles over a perfectly conducting ground plane. The third purpose of this paper is to perform a MoM analysis for the more realistic case where the ground plane is also highly conducting. Specifically, we apply MoM to certain approximate integral equations developed in [4] and [5], and compare to the TTT results therein. We also compare both sets of results to the experimental results of [2]–[5]. In the experiment, the dipoles are brass and the ground plane is aluminium. cm, mm, and Throughout this paper, cm, as in [12]; these values come from the experimental circular array of [2]–[5].1 We use various values of ( in [12], while in [2]–[5]) and . All resonances occur near 2.5 GHz, so that the dipoles are unusually short and thick: , while is nearly 0.03. Parts from Sections II and III of this work have been presented as the conference paper [16]. II. PHASE SEQUENCES AND INTEGRAL EQUATIONS As in [12], denotes the current along the element no. when only the element no. 1 is driven by a voltage . is a superposition of the “phase-sequence currents” , is the current on element no. 1 when the array where is driven at its th phase-sequence, in which the driving-point admittance of any array element is 1To be more precise, the above values are the same as in [12], but have been rounded off from the theoretical values used in [2]–[5]; this explains the slight differences between the tables of [4] and the TTT values in Tables II–V.

FIKIORIS et al.: MoM ANALYSIS OF RESONANT CIRCULAR ARRAYS OF CYLINDRICAL DIPOLES

. The relation between

and

is [12] (1)

Both and must always (for any circular array, at any frequency) be a positive quantity. Therefore, at a resonance, is large because is large for some . In this paper, we apply MoM to the “ th phase-sequence integral equation” [12, Eq. (2)], which involves the “ th phase.2 This kernel is, in turn, a superposisequence kernel” according to [12, Eq. (4)]. Depending tion of the kernels on whether the original, modified, or improved kernels are used, is given by (2), shown at the the self-interaction kernel bottom of the page, [4, Sect. 11.8], [12]. For all three cases, are given by [4, the mutual-interaction kernels Sect. 11.8], [12]

(3) is the distance bewhere tween the axes of dipoles 1 and —the assumption of axis-toaxis interaction is logical because the dipoles are relatively thin. For the case of MoM analysis of resonant circular arrays, the crucial differences between (2a) and (2b) were the subject of [12]; here we mainly focus on the differences between (2b) and (2c). The real parts in (2b) and (2c) are the real parts of what are commonly called the “approximate” and “exact” kernels in Hallén’s and Pocklington’s equations for an isolated dipole [18]. The real part in (2c), which is singular, is calculated by extracting the singularity as in [5, Sect. 9.1.D]; this is adequate for our purposes, but we note that more efficient schemes exist [19]. pulse Our specific MoM is Galerkin’s method with basis functions [12]. The version of the TTT that we use includes the so-called “refinements for numerical calculations” [4, Sect. 11.6]; the specific TTT formulas are [4, Eqs. (12.38)–(12.68)] and [4, Eq. (10.34)]. Note that the real part in (2c) must be used in this last equation. We also use the techniques of [4, Sect. 13.5] 2Another method of taking advantage of the circular geometry would be to apply MoM to the usual uncoupled integral equations, and to reduce computer time by using the circulant-block characteristics of the matrix [17]. A detailed comparison between the two versions of MoM does not seem straightforward and is beyond the scope of this paper.

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to easily and accurately calculate the TTT values of .

and

III. RESULTS FOR LOSSLESS DIPOLES for and as Fig. 1 shows a function of the frequency , as calculated by MoM with 53 . The two curves are obpulse basis functions, so that tained by using (2b) and (2c). For both kernels, a narrow peak is seen to occur. It is seen that the peak corresponding to (2c) is shifted toward the right, is less narrow, and less in height. In both and the corresponding curves, we have marked the values maximum of . Both and , of course, depend on . The marked improvement in the convergence of is illustrated in Fig. 2, in which the abscissa is . Whereas the points corresponding to (2b) fail to converge, the ones corresponding to (2c) do appear convergent (horizontal), albeit somewhat slowly. We refer to [12] on further difficulties one encounters if one uses yet larger values of . We accelerated the convergence by applying the convergence acceleration method (CAM) of [20, Eq. (5)]; this gives the third set of points in Fig. 2, which is seen to converge much more rapidly. We stress that CAM is applied of peak values, each of which occurs at to the sequence a (slightly) different frequency. This is much more appropriate in which the frequency than applying CAM to a sequence is the same for all sequence values. It is natural to use this latter method in applications that do not involve narrow peaks [20], [21]. instead of . The Fig. 3 is like Fig. 2, but shows points corresponding to (2b) behave much better than their counterparts in Fig. 2; still, the ones corresponding to (2c) converge faster, while the ones obtained by further application of the CAM appear, at the scale of the figure, to be nearly horizontal. In what follows, we will obtain “final” MoM values and by applying CAMs as in Figs. 2 and 3, of and denote such final values by MoM-CAM. Thus, roughly . speaking, our MoM-CAM values correspond to using Table I compares the MoM-CAM values of and for and to TTT values. For , the differences are of the order of a fraction of a percent and increase as increases. The differences between the values of are within a few percent; here, the differences decrease as increases. remain small even for larger The differences between values of : Fig. 4 shows the logarithms of for , and for . For , the MoM-CAM value is the largest of the two and the difference is 5%; for , the MoM-CAM value is the smallest of the two and the

(2a) (2b) (2c)

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TTT AND MOM-CAM VALUES OF RESONANT FREQUENCIES f

TABLE I AND RESONANT PHASE-SEQUENCE CONDUCTANCES G

FOR

N

= 20

TABLE II . FOR m ; ; ; WE ONLY GIVE A SINGLE SIGNIFICANT DIGIT FOR G AND, FOR m , WE PROVIDE AN LIKE TABLE I, BUT FOR N INTERVAL RATHER THAN A SINGLE VALUE. WE DO THIS BECAUSE OF THE DIFFICULTIES OF APPLYING THE CAM WHEN THE VALUES ARE VERY LARGE

= 90

= 41 42 43 44

ln 2

Fig. 4. Logarithm of resonant phase-sequence conductance G (the meais Siemens) as function of N , for m N= and using TTT sure unit of G (dotted line) and MoM-CAM (solid line).

=

difference is between 28% and 40%. These percentage differences are between the values of , not the values of its log. For , arithm. The two values are closest when we give a range of percentage difference rather than a single value because it is difficult to obtain a reliable value of when that value is extremely large. The underlying reason for are the difficulties is illustrated in Fig. 5, where the values shown as a function of frequency. Note that the horizontal scale is very fine. The values are noisy, making it hard to determine a precise maximum value. Since the CAM is applied to the sefor various values of quence of maximum values of ( in Fig. 5), and since CAM results are highly susceptible to noise in the original sequence [20], [22], there is even

= 45

= 90 = 45 = 2 701742546546

= 80

Fig. 5. Like Fig. 1, but for, N ; results obtained ;m , and M by MoM with (2c). The scale on the horizontal axis is the frequency difference : GHz. Thus, the horizontal scale is very f f , where f fine and the peak is extremely narrow.

0

more noise in the transformed sequence. Thus, the 28%–40% range represents our reasonable effort to obtain a reliable value of (see also Table II). It may be possible to obtain more reliable values by reducing the noise in the original sequences (e.g., by using better numerical integration routines). We have not further investigated this possibility. Besides the closeness between TTT and MoM-CAM, Fig. 4 also shows the extremely rapid increase of the resonant values with . In fact, the increase is seen to be—to an excellent degree of accuracy—exponential, as predicted by the theoretical discussions based on the TTT in [4, Sect. 11.3].

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TABLE III LIKE TABLE II, BUT FOR BRASS DIPOLES

so-called exact kernel (4) should not be used in conjunction with the usual mutual kernels (3) for the purpose of resonant circular array analysis. This point is further discussed in Section VI. IV. BRASS DIPOLES: INTEGRAL EQUATIONS AND RESULTS Appropriate integral equations for dipoles with a finite—but large—conductivity have been proposed in [4] and [5]: The inof (2c) tegral equation is the same, but with the kernel replaced by [4, Sect. 12.2]

Fig. 6. Like Fig. 1, but with (4) in place of (2c). Here, we have a “negative peak.”

Table II is like Table I, but for . Here, we observe an extremely rapid increase of with . For , the differences between the results of the two methods are similar to the differences in Table I. For between 41 and 44, Table II only provides a single significant digit for the MoM-CAM value , the table contains a range for ; we do while, for this because of the difficulties discussed above. Fig. 6 is like Fig. 1, but with the self-term

(4) in place of (2c). This is precisely the exact kernel in Hallén’s equation for an isolated dipole. A negative peak is seen, just like the meaningless results in [12, Fig. 2] and [15]; larger values of , as well as other values of , still produce negative peaks. Fig. 6 thus leads to the important conclusion that, at least for , and used in the present paper, the the parameter values

(5) where [4, Eq. (12.14)] (6) in which is the dipoles’ conductivity. For brass, S/m. The aforementioned integral equation, which is approximate, seems to have first appeared in [5]; related equations can be found in [23]–[25]. The notation is in accordance with the literature [26], [27]. In [4] and [5] a version of the TTT is proposed that is appropriate for highly conducting dipoles, and Table III shows the results thus obtained. Note that the resonant frequencies are, to all significant digits shown in the table, the same as in the lossless case of Table II; for the case of the TTT, this coincidence is explained in [4, Sect. 12.4]. Once again, the values of exhibit good agreement for . However, the . differences increase to 18.5% for Qualitatively, the results differ from those in Table II in a manner that one might expect: The effect of ohmic losses is negligible in the first few cases, but drastic in the last few cases. eventually beFurthermore, as increases, the values of come constant. A detailed explanation of this behavior via the

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TABLE IV VALUES OF EFFECTIVE MONOPOLE CONDUCTIVITY  [4, TABLE 12.2]; RESONANT FREQUENCIES f OBTAINED BY TTT (cf. [4, TABLE 12.2]) AND MOM-CAM; EXPERIMENTAL RESONANT FREQUENCIES [4, TABLE 12.4]

TTT can be found in [4, Sect. 12.4]—the discussions there lead to an illuminating equivalent circuit for the resonant array. V. RESULTS FOR BRASS MONOPOLES OVER AN ALUMINUM GROUND PLANE As already mentioned, highly conducting dipoles are equivalent to highly conducting monopoles over a perfectly conducting ground plane of infinite extent. One obtains results for the latter case by multiplying the former results by a factor of two. For the more realistic case of highly conducting monopoles over a highly conducting ground plane (still assumed to be of infinite extent), appropriate integral equations have been proposed in [4] and [5]: For the case of a single driven element, one uses the integral equations described in Section IV but, for each , one replaces in (6) by an “effective monopole conductivity” , where . One then multiplies by two to get results valid for monopoles. The details of the calculation of will not be of concern here; rather, we will simply utilize the values determined in [4] and [5], and solve the respective integral equations using the MoM methods discussed previously. taken from [4, Table Table IV shows the values of 12.2], together with the resulting resonant frequencies obtained by the two theories (for the TTT values, cf. [4, Table 12.2]). Also shown are corresponding experimental resonant frequencies (taken from [4, Table 12.4]), and the percent differences between the experimental and the two theoretical values. coincide with those of Note that the theoretical values of Tables II and III; for the case of the TTT, this is explained in [4, Ch. 12]. In any row of Table IV, the three values are seen to be very close to each other; the differences between the MoM-CAM and the experimental values change sign when and are, generally, a bit smaller than the corresponding differences between TTT and experiment.

in Table IV comes from a single phase-sequence Each integral equation, the th equation. When the final quantity one should, strictly speaking, apply a CAM of interest is to all phase-sequence integral equations. Here, we do so only to the dominant equation, that is, to the th equation, where indicates the phase-sequence resonance we are interested in. Having thus determined and as in Table IV we use in the remaining equations rather than the “large” value using a CAM to obtain the remaining . Finally we sum all according to (1) to obtain . This saves a great deal of the computer time, with only a slight loss in accuracy. Note that the methods herein can only give resonant values of , not values of between the resonances. Fig. 7(a) and (b) show the MoM-CAM values of thus obtained together with a continuous curve of TTT values and another continuous curve of experimental values—both continuous curves appear in [4, Figs. 12.10 and 12.11]. The most noticeable differences between the three sets of data in Fig. 7 are frequency shifts, which are not constant from resonance to resonance. As we observed in Table IV, the shifts are very small in relative terms in the sense that the per cent differences between corresponding resonant frequencies are very small. The MoM-CAM and TTT resonant (peak) values of are also provided in Table V, together with the resonant experimental values taken from [4, Table 12.5] or [3, Table 1].3 From Table V and Fig. 7, it is seen that the TTT values of resonant are closer to the experimental values for larger values of , while the MoM-CAM values are closer to the experimental values for smaller values of . Usually, the agreement is within a few per cent. Let us stress that the experiment has difficulties of its own, as described in [3] and, in more detail, [5]. 3There are slight differences between the experimental values in Table V and the peak values of Fig. 7(a) and (b): The former set of values involves an averaging process and should be considered more precise than the latter—see [3], where the two sets appear as the third and sixth columns of [3, Table 1].

FIKIORIS et al.: MoM ANALYSIS OF RESONANT CIRCULAR ARRAYS OF CYLINDRICAL DIPOLES

G

f L. (b) Global and local coordinate system. Here, for any orientation of the local frame described by  and '; L will be greater than the maximum dimension of the source region V in that direction.

ated by a point source located at the origin depends only on the distance of the observation point from the origin. At a deeper level, we may take this symmetry condition as an integral trait of the underlying space–time structure upon which the electromagnetic field is defined.14 What is relevant to our present discussion, which is concerned with the nature of the antenna near field, is that the observation frame of reference can be rotated in an arbitrary manner around a fixed origin. Let us start then by fixing the choice for the origin of the source frame , and . Next, we define a global frame of reference and label its axes by , and . Without loss of generality, we assume that the source frame is coincident with the global frame. We then introduce another coordinate system with the same origin of the both the global and source frames and label its coordinates by , and . This last frame will act as our local observation frame. It can be orientated in an arbitrary manner as is evident from the freedom of choice of the coordinate system in the Weyl expansion (4). We allow the -axis of our local observation frame 14This observation can be further formalized in the following way. The field concept is defined at the most primitive level as a function of space and time. Now, what is called space and time is described mathematically as a manifold, which is nothing but the precise way of saying that space and time are topological spaces that admit differentiable coordinate charts (frames of references). We find then that the electromagnetic fields are functions defined on manifolds. The manifold itself may possess certain symmetry properties, which in the case of our Euclidean space are a rotational and translational symmetry. Although only the rotational symmetry is evident in the form of Weyl expansion given by (4), the reader should bear in mind that the translational invariance of the radiated fields has been already used implicitly in moving from (4) to expressions like (13) and (14), where the source is located at instead of the origin.

r

MIKKI AND ANTAR: THEORY OF ANTENNA ELECTROMAGNETIC NEAR FIELD II

to be directed at an arbitrary direction specified by the spherical angles and , i.e., the -axis will coincide with the unit vector in terms of the global frame. The situation is geometrically described in Fig. 1(b). There, the Weyl expansion will be , and with region written in terms of the local frame of validity given by , where is chosen such that it will be greater than the maximum size of the antenna in the direction specified by and . It can be seen then that our radiated electric fields written in terms of the global frame but spectrally expanded using the (rotating) local frame are given15

(17) where the new spectral vector is given by (18) The cartesian coordinates in (17) represent the source coordinates after being transformed into the language of the new frame .16 In terms of this notation, (17) is rewritten in the more compact form

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It should be immediately stated that this rotation matrix will also -plane around the -axis with some angle. We rotate the can further control this additional rotation by multiplying (20) by the following matrix: (23) where here represents some angle through which we rotate -plane around the -axis. However, as will be shown the in Section II-D, a remarkable characteristic of the field decomposition based on Weyl expansion is that it does not depend on the angle if we restrict our attention to the total propagating part and the total evanescent part of the electromagnetic field radiated by the antenna. , and From (18) and (22), it is found that therefore , where denotes matrix transpose operation. Moreover, it is easy to . Using these two relation, show that (19) can be put in the form

(24)

(19)

Therefore, from the definition of the spatial Fourier transform of the antenna current as given by (16), (24) can be reduced into the form

To proceed further, we need to write down the local frame coordinates explicitly in terms of the global frame. To do this, the following rotation matrix is employed:17 (25) (20)

Separating this integral into nonpropagating (evanescent) and propagating parts, we obtain, respectively

where the elements are given by

(26)

(27)

(21) In terms of this matrix, we can express the local frame coordinates in terms of the global frame’s using the following relations: (22) 15That is, we expand the dyadic Greens function (2) in terms of the local frame and then substitute the result into (1). 16These are required only in the argument of the dyadic Greens function. 17See

Appendix D for the derivation of the matrix elements (21).

We will refer to the expansions (26) and (27) as the general decomposition theorem of the antenna fields. They express the decomposition of the field at location into total evanescent and propagating parts measured along the direction specified by the unit vector , i.e., when the -axis of the local observation frame is oriented along the direction of . Moreover, since it can be proved that this decomposition is independent of an arbitrary rotation of the local frame around (see Section II-D), it follows that the quantities appearing in (26) and (27) are unique. However, it must be noticed that the expansions (26) and (27) are valid only in an exterior region, for example , where here is taken as the maximum dimension of the antenna current distribution. Using the explicit form of the rotation matrix (20) given in (21),

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we find that the general decomposition theorem is valid in the region exterior to the infinite slab enclosed between the two planes

(28) This infinite slab will be refereed to in this paper as the antenna horizon, meaning the horizontal range inside which the simple expressions in (26) and (27) are not valid. We immediately notice that the antenna horizon is changing in orientation with every angle and . This will restrict the usefulness of the expansions (26) and (27) in many problems in field theory as we will see later. However, a particularly attractive field structure, the radial streamline concept, will not suffer from this restriction. Toward this form we now turn. C. Concept of Antenna Near-Field Radial Streamlines We focus our attention on the description of the radiated field surrounding the antenna physical body using spherical coordinates. In particular, notice that by inserting into (22), and using the form of the rotation matrix given by (20) and (21), one can easily calculate .18 Therefore, the expansion (25) becomes

(29) where . Since the observation is of the field propagating or nonpropagating away from the antenna, we are always on the branch . Furthermore, by dividing the expansion (29) into propagating and nonpropagating parts, it is finally obtained

(30)

(31) where we have introduced the spectral polarization dyad defined as19 (32) We notice that in this way the general decomposition theorems (26) and (27) are alaways satisfied since for each direction specified by and , the slab enclosed between the two planes given by (28) will also rotate such that the observation point is always in the exterior region. This desirable fact is behind the great utility of the radial streamline concept (to be defined 18This computation can be considered as an alternative derivation of the rotation matrix compared to the one presented in Appendix D. 19For a discussion of the physical meaning of this dyad, and hence a justification of the proposed name, see Section V.

momentarily) in the antenna theory we are proposing in this paper. The expansions (30) and (31) can be interpreted as the decomposition of the electromagnetic fields into propagating and nonpropagating waves in the radial directions described by the spherical angles and . That is, we do not obtain a plane-wave spectrum in this formulation, but instead what we prefer to call radial streamlines emanating from the origin of the coordinate system (conveniently chosen at the center of the actual radiating structure). The physical meaning of “streamlines” here is analogous to the situation encountered in hydrodynamics, where material particles move in trajectories embedded within continuous fluids. In the case considered here, streamlines have for a propagating mode with the mathematical form constant phase speed , and hence are defined completely in terms of fields. As explained earlier, it is only such solutions that represent a genuine propagating mode; the remaining part, the evanescent mode in the electromagnetic problem, represents clearly the nonpropagating part of the radiated field. The concept of “electromagnetic-field streamlines” developed above is a logical deduction from a peculiarity in the Weyl expansion, namely the symmetry breaking of the rotational invariance of the scalar Greens function, a mathematical trait we propose to elevate to the level of a genuine physical process at the heart of the dynamics of the antenna near fields.20 It is this form of radial streamlines that appears to the authors to be the most natural representation of the inner structure of the antenna near fields since it is viewed from the perspective of the far fields, which in turn is most conventionally expressed in terms of spherical coordinates. Since antenna engineers almost always describe the antenna in the far-field zone (among other measures like the input impedance), and since such mathematical description necessitates a choice of a spherical coordinate system, we take our global frame introduced in the previous parts to coincide with the spherical coordinate system employed by engineers in the characterization of antennas. Therefore, our near-field picture, although it starts from a given current distribution in the antenna region, still partially reflects the perspective of the far field. In Section IV, we will develop the near-field picture completely from the far-field perspective by employing the Wilcox expansion. D. Independence of the Spectral Expansion From Arbitrary Rotations Around the Main Axis of Propagation/Nonpropagation We now turn to the issue of the effect of rotation around the main axis chosen to perform the spectral expansion. As we have already seen, the major idea behind the near-field theory is the interpretation of the rotational invariance of the scalar Greens function in terms of its Weyl expansion. It turned out that with respect to a given antenna current distribution, as long as one is concerned with the exterior region, the observation frame of reference can be arbitrarily chosen in order to enact a Weyl expansion with respect to this frame. It is our opinion that such 20The generalized concept of propagation in the near zone that goes beyond the streamline picture will be developed by the authors in separate publications. For example, see [3].

MIKKI AND ANTAR: THEORY OF ANTENNA ELECTROMAGNETIC NEAR FIELD II

freedom of choice is not an arbitrary consequence of the mathematical identity per se, but rather the deeper expression of the being of electromagnetic radiation as such. Indeed, the very essence of how antennas work is the scientific explication of a definite mechanism through which the near field genetically gives rise to the far field—in other words, the genesis of electromagnetic radiation out from the near-field shell. Although the full analysis of this problem will be addressed in future publications by the authors, we have introduced so far the concept of radial streamlines to describe the conversion mechanism above mentioned in precise terms. It was found that we can orient the -axis of the observation frame along the unit radial vector of the global frame in order to obtain a decomposition of the total fields propagating and nonpropagating away from the antenna origin along the direction of .21 It remains to see how our spectral expansion is affected by a rotation of the local frame -plane around the radial direction axis. More precisely, the problem is stated in the following manner. Consider a point in space described by the position vector in the language of the global frame of observation. Assume further that the expansion of the electric field into propagating and nonpropagating modes along the direction of the -axis of this frame was achieved, with values and giving the evanescent and propagating parts, respectively. Now, keeping the direction of the -axis fixed, we merely rotate the -plane by an angle around the -axis. The electric field is now expanded into evanescent and propagating modes again along the same -axis, and the results are and , respectively. The question we now investigate is the relation between these two sets of fields. To accomplish this, let us start from the original expansion (24), but replace by a rotation around the -axis through an angle , which can be used to obtain the transformed spatial and spectral variables through the equations and , where is given by (23). By direct calculation, we obtain and . These results suggest introducing the substitutions and , which are effectively a rotation of the -plane by and angle around the origin. Being a rotation, the Jacobian of this transformation is one, i.e., , where denotes the Jacobian of the transformation matrix applied to its argument. Also, it is evident that . Moreover, this implies that the two regions and transform into the regions and , respectively. After dividing (24) into evanescent and propagating part, then rotating the -plane and changing the spectral variables from and to and , we find

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Fig. 2. Regions in the spectral pq -plane in terms of which the decomposition of the electromagnetic field into propagating and nonpropagating modes is conducted. The circle p q marks the boundary between the so-called q > and the visible region p q < (a circular invisible region p disk). In general, the mathematical description of the field can be accomplished with any region in the spectral plane, not necessarily the total regions inside and q . In particular, we show an arbitrary region D outside the circle p located inside the propagating modes disk p q < . In general, D need not q < , but may include arbitrary portions be a proper subset of the region p of both this disk and its complement in the plane.

+

+

+

=1 1

+

=1

+

1

+

1

1

(34) Applying the results of the paragraph preceding the two prior equations, we conclude that (35) Therefore, the total evanescent and total propagating parts of the antenna radiated fields are invariant to rotation around the -axis of the observation frame. This result is true only when we are interested in field decomposition into regions in the spectral -plane that do not change through rotation. For example, if we are interested in studying part of the radiated field such that it contains the modes propagating along the -direction, but with spectral content in the -plane inside, say, a square, then since not every rotation is a symmetry operation for a square, we conclude that the quantity of interest above does vary with rotation of the observation frame around the -axis for this special case. In this paper, however, our interest will focus on the total propagating and nonpropagating parts since these are the quantities that help rationalize the overall behavior of antennas in general. However, it should be kept in mind that for more general and sophisticated understanding of near-field interactions, it is better to retain a general region in the -plane as the basis for a broad spectral analysis of the electromagnetic fields (see Fig. 2). E. Propagating and Nonpropagating Magnetic Fields The frequency-domain Maxwell’s equations in source-free homogenous space described by electric permittivity and magnetic permeability are given by

(33) 21Cf.

(30) and (31).

(36)

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The first curl equation in (36) can be used to compute the magnetic field if the electric field is known. We assume that the latter can be expressed by the general decomposition theorem as stated in (26) and (27). By direct calculation, we easily obtain22

(45) In other words, whatever is the direction of decomposition, the resultant fields are always Maxwellian.

(37) where the curl operator was brought inside the spectral integral. , we write Next, from the dyadic identity

(38) This allows us to conclude that (39) Therefore, after separating the integral into propagating and evanescent parts, (37) becomes

(40)

(41) The radial streamline magnetic fields corresponding to those given for the electric field in (30) and (31) are

(42)

(43) It is evident from the original (37) that the evanescent (propagating) magnetic field is found by applying the curl operator to the evanescent (propagating) part of the electric field. That is

F. Summary and Interpretation The expansion of the electromagnetic field into propagating and nonpropagating modes along a changing direction is well justified by the result of Section II-D, namely that such expansion along a given direction is independent of an arbitrary rotation of the local observation frame around this direction. This important conclusion simplifies significantly the analysis of the antenna near fields. The reason is that the full rotation group requires three independent parameters in order to specify an arbitrary 3-D orientation of the rotated observation frame. Instead, our formulation depends only on two independent parameters, namely and , which are the same parameters used to characterize the degrees of freedom of the antenna far field. This step then indicates an intimate connection between the antenna near and far fields, a connection which, relatively speaking, is not quite obvious on a priori ground. However, our knowledge of the structure of the near field, as can be discerned from the expansions (30) and (31), is enhanced by the record of the exact manner, as we progress away from the antenna along the radial direction , in which the evanescent field, i.e., the nonpropagating part, is being continually converted into propagating modes. As we reach the far-field zone, most of the field contents reduce to propagating modes, although the evanescent part still contributes asymptotically to the far field. For each direction and , the functional form of the integrands in (30) and (31) will be different, indicating the “how” of the conversion mechanism with which we are concerned. Since close to the antenna most of the near-field content is nonpropagating, we focus now our attention on the evanescentmode expansion of the electric magnetic field as given by (30).24 Let us introduce the cylindrical variables and such that and . Therefore, in the region (46) The integral (30) then becomes

(44) Moreover, the divergence of the evanescent and propagating parts of both the electric and magnetic fields is identically zero.23 We conclude from this together with (44) that

22We use the vector identity r2 ( A) = r 2 A + r1 A and the relation r exp(A 1 r) = A exp(A 1 r), which are true in particular for constant vector A and a scalar field (r). 23This can be seen by direct calculation and using the fact that the rotation  1R  = I. matrix is orthogonal, i.e., R

(47) where

(48) 24The subsequent formulation in this section can be also developed for the evanescent part of the magnetic field (42). Details are omitted for brevity.

MIKKI AND ANTAR: THEORY OF ANTENNA ELECTROMAGNETIC NEAR FIELD II

Next, perform another substitution . Since , it follows that the integral (47) reduces to (49) where

(50) Therefore, for a fixed radial direction and , the functional form of the evanescent part of the field along this direction takes the expression of a Laplace transform in which the radial position plays the role of frequency. This fact is interesting and suggests that certain economy in the representation of the field decomposition along the radial direction has been already achieved by the expansions (30) and (31). To better appreciate this point, we notice that since is a rotation matrix, it satisfies . In light of this, the change in the integrands of (30) and (31) with the orientation of the decomposition axis given by and can be viewed as, first, a rotation of the spatial Fourier transform of the current by the inverse rotation originally applied to the local observation frame and, second, as applying a similarity transformation to transform the spectral to , that polarization dyad is, the spectral matrix is undergoing a similarity transformation under the transformation , the inverse rotation. These results indicate that there is a simple geometrical transformation at the core of the change of the spectral content of the electromagnetic fields,25 which enacts the decomposition of the electromagnetic fields into nonpropagating and propagating modes. These transformations are simple to understand and easy to visualize. We summarize the entire process in the following manner. 1) Calculate the spatial Fourier transform of the antenna current distribution in a the global observation frame. 2) Rotate this Fourier transform by the inverse rotation . 3) Transform the spectral polarization dyad by the similarity transformation generated by the inverse rotation . 4) Multiply the rotated Fourier transform by the transformed spectral polarization dyad. Convert the result from cartesian coordinates and to cylindrical coordinates and and evaluate the angular (finite) integral with respect to . That is, average out the angular variations . 5) Transform as and compute the Laplace transform of the remanning function of . This will give the functional dependence of the antenna evanescent field on the radial position where will play the role of frequency in the Laplace transform. The significance of this picture is that it provides us with a detailed explication of the actual route to the far field. Indeed, since the radiation observed away from the antenna emerges from the concrete way in which the nonpropagating part is being transformed into propagating modes that escape to the far-field zone, it follows that all of the radiation characteristics of antennas, 25The

functional form of the integrands of (30) and (31)

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like the formation of single beams, multiple beams and nulls, polarization, etc., can be traced back into the particular functional form of the spectral function appearing in the Laplace transform expression (49). Moreover, we now see that the generators of the variation of this key functional form are basically geometrical transformations associated with the rotation through which we orient the local observation matrix frame of reference. In Section V, the theoretical narrative of the far-field formation started here will be further illuminated. III. CONCEPT OF LOCALIZED AND STORED ENERGIES IN THE ANTENNA ELECTROMAGNETIC FIELD A. Introduction Armed with the concrete but general results of the previous parts of this paper, we now turn our attention to a systematic investigation of the phenomena usually associated with the energy stored in the antenna surrounding field. We have already encountered the term “energy” in our general investigation of the antenna circuit model in [1], where an effective reactive energy was defined in conjunction with the circuit interpretation of the complex Poynting theorem. We have seen that this concept is not adequate when attempts to extend it beyond the confines of the circuit approach are made, pointing to the need to develop a deeper general understanding of antenna near fields before turning to an examination of various candidates for a physically meaningful definition of stored energy. In this section, we employ the understanding of the near-field structure attained in terms of the Weyl expansion of the free-space Greens function in order to build a solid foundation for the phenomenon of energy localization in general antenna systems. The upshot of this argument will be our proposal that there is a subtle distinction between localization energy and stored energy. The former is within the reach of the time-harmonic theory developed in this paper, while the latter may require in general an extension to transient phenomena. B. Generalization of the Complex Poynting Theorem Since we know at this stage how to decompose a given electromagnetic field into propagating and nonpropagating parts, the natural next step is to examine the power flow in a closed region. Our investigation will lead to a form of the Poynting theorem that is more general than the customary one (where the latter results from treating only the total fields). Start by expanding both the electric and magnetic fields as

(51) The complex Poynting vector is given by [7] (52) Substituting (51) into (52), we find

(53)

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Since it has been proved in Section II-E that each of the propagating and nonpropagating parts of the electromagnetic field is Maxwellian, it follows immediately that the first and the second terms of the RHS of (53) can be identified with complex Poynting vectors

in the original time-dependent form. By repeating the procedure that led to (64) but now in the time domain, it is possible to derive the following continuity equation:27

(54)

Here, we match the time-dependent “interaction” Poynting vector

(69)

(55) From the complex Poynting theorem [7] applied to a source-free region, we also find

(70) with the time-dependent electric and magnetic energy densities (71)

(56) (57) with electric and magnetic energy densities defined as (58) (59) It remains to deal with the cross terms (third and fourth terms) appearing in the right-hand side (RHS) of (53). To achieve this, we need to derive additional Poynting-like theorems. From (45), we obtain (60) and (61) Subtracting (61) and (60) and using a vector identity,26 we reach (62) By exactly the same procedure, the following dual equation can also be derived: (63) Adding (62) and (63), the following result is obtained: (64) where we defined the complex interaction Poynting vector by (65) and the time-averaged interaction electric and magnetic energy densities by

(66) respectively. It is immediate that

The justification for calling the quantities appearing in (66) energy densities is the following. Maxwell’s equations for the evanescent and propagating parts, namely (45), can be rewritten

r 1 (A 2 B) = B 1 (r 2 A) 0 A 1 (r 2 B).

C. Multifarious Aspects of the Energy Flux in the Near-Field Zone According to the fundamental expansion given in the general decomposition theorem of (26) and (27), at each spatial location , the field can be split into total nonpropagating and propagating parts along a direction given by the unit vector .29 Most generally, this indicates that if the near-field stored energy is to be associated with that portion of the total electromagnetic field that is not propagating, then it follows immediately that the definition of stored energy in this way cannot be unique. The reason, obviously, is that along different directions , the evanescent part will have different expansions, giving rise to different total energies. Summarizing this mathematically, we find that the energy of the evanescent part of the fields is given by (72) where denotes a volume exterior to the antenna (and possibly the power supply). In writing down this expression, we 27See

(67) (68)

26I.e.,

where and stand for the time-dependent (real) fields. We follow in this treatment the convention of electromagnetic theory in interpreting the quantities (71) as energy densities. It is easy now to verify that the expressions (66) give the time-average of the corresponding densities appearing in (71). Moreover, it follows that the time-average of the instantaneous . Poynting vector (70) is given by Therefore, the complex Poynting theorem can be generalized in the following manner. In each source-free space region, the total power flow outside the volume can be separated into , and . Each term individually is interthree parts, preted as a Poynting vector for the corresponding field. The evidence for this interpretation is the fact that a continuity-type equation Poynting theorem can be proved for each individual Poynting (flux) vector with the appropriate corresponding energy density.28

Appendix E for the derivation of (69). example, consider the energy theorem (69). This result states the following. Inside any source-free region of space, the amount of the interaction power flowing outside the surface enclosing the region is equal to negative the time rate decrease of the interaction energy located inside the surface. This interaction energy itself can be either positive or negative, but its “quantity” is always conserved as stated by (64) or (69). 28For

29Although the particular mathematical expressions given in (26) and (27) are not valid if the point at which this decomposition is considered lies within the antenna horizon, the separation into propagating and nonpropagating remains correct in principle, but the appropriate expression is more complicated.

MIKKI AND ANTAR: THEORY OF ANTENNA ELECTROMAGNETIC NEAR FIELD II

made the assumption that the directions along which the general decomposition theorem (26) is applied form a vector field .30 The first problem we encounter with the expression (72) is is infinite. This can that it need not converge if the volume is taken as the be most easily seen when the vector field constant vector . That is, we fix the observation frame for all points in space, separate the evanescent part, and integrate the amplitude square of this quantity throughout all space points exterior to the antenna current distribution. It is readily seen that since the field decays exponentially only in one direction (away from the antenna current along ), then the resulting expression will diverge along the perpendicular directions. D. Concept of Localized Energy in the Electromagnetic Field We now define the localized energy as the energy that is not propagating along certain directions of space. Notice that the term “localized energy” is: 1) not necessarily isomorphic to “stored energy”; and 2) is dependent on certain vector field . The first observation will be discussed in detail later.31 The second observation is related to the fundamental insight gained from the freedom of choosing the observation frame in the Weyl expansion. It seems then that the mathematical description of the wave structure of the electromagnetic field radiated by an antenna cannot be attained without reference to a particular local observation frame. We have now learned that only the orientation of the -axis of this local frame is necessary, reducing the additional degrees of freedom needed in explicating the wave structure of the near field into two parameters, e.g., the spherical angles and . This insight can be generalized by extending it to the energy concept. “Localization” here literally means to restrict or confine something into a limited volume. The electromagnetic near field possesses a rich and complex structure in the sense that it represents a latent potential of localization into various forms depending on the local observation frame chosen to enact the mathematical description of the problem. It is clear then that the localized energy will be a function of such directions and hence inherently not unique.32 The overall picture boils down to this: To localize or confine the electromagnetic energy around the antenna, you first separate the nonpropagating field along the directions in which the potential localization is to be actualized, and then the amplitude 30It is possible to find in [10, pp. 146–150] calculations for the propagating and nonpropagating complex power and momentum flux radiated across a plane z z > z . This case was developed there with the near-field scanning setting particularly in mind, where the “source” here is taken as the near-field measured data collected on a plane z z . The energy expression in the present work, as given by (72), provides a very different perspective since it is concerned with energy and not flux, but for a generally varying direction of decomposition into propagating and nonpropagating field. The calculation in [10] is related to our work in the sense that they apply most naturally to the waveguide port feeding the antenna system, where a preferred fixed direction of propagation is imposed in the longitudinal direction of the guiding structure. Such development was removed from our presentation and can be found in [4] and [10]. 31Cf. Section III-G. 32The reader should compare this to the definition of quantities like potential and kinetic energies in mechanics. These quantities will vary according to the frame of reference chosen for the problem. This does not invalidate the physical aspect of these energies since, relative to any coordinate system, the total energy must remain fixed in a (conservative) closed system. Similarly, relative to any local observation frame, the sum of the total propagating and nonpropagating fields yields the same actually observed electromagnetic field.

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square of this field is taken as a measure of the energy density of the localized field in question. By integrating the resulted energy density along the volume of interest, the total localized energy is obtained. The uncritical approach to the energy of the antenna fields confuses the stored energy with the localized energy, and then postulates—without justification—that this energy must be independent of the observation frame.33 It is because the far-field perspective involves an integration operation that the rich subwavelength effects of the antenna spatial current distribution on its generated field tend to be smoothed out when viewed from the vantage point of the antenna radiation pattern. In the refined approach of this paper, the crucial information buried in the antenna near-zone fields corresponds to the short-wavelength components, i.e., the , which are responsible for spectral components giving the field its intricate terrain of fine details. These components dominate the field as we approach the antenna current distribution and may be taken as the main object of physical interest at this localized level. E. Radial Evanescent Field Energy in the Near-Field Shell We now reexamine the concept of the near-field shell at a greater depth. The idea was introduced in Part I [1] in the context of the reactive energy, i.e., the energy associated with the circuit model of the antenna input impedance. As it has been concluded there, this circuit concept was not devised based on the field vantage point, but mainly to fit the circuit perspective related to the input impedance expressed in terms of the antenna fields as explicated by the complex Poynting theorem. We now have the refined model of the radial evanescent field developed in Section II-C. We define the localized energy in the near-field spherical shell as the self-energy of the nonpropagating modes . along the radial streamlines enclosed in the region The total local energy then is the limit of the previous expression . when To derive an expression for the localized electric34 radial energy defined this way, substitute (30) to (72) with the identifica. It is obtained35 tion

=

=

(73) 33One may hope that although the energy density of the evanescent part is not unique, the total energy, i.e., the volume integral of the density, may turn out to be unique. Unfortunately, this is not true in general. The total convergent evanescent energy in a given volume depends in general on the orientation of the decomposition axis u. 34For reasons of economy, throughout this section we give only the expressions of the electric energy. The magnetic energy is obtained in the same way.

^

35Throughout this paper, the conversion of the multiplication of two integrals into a double integral, interchange of order of integration, and similar operations are all justified by the results of the appendixes concerning the convergence of the Weyl expansion.

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By converting the space integral in (73) into spherical coordinates and using an integration identity36 to evaluate the radial , we end up with the following integral in the region expression:

F. Electromagnetic Interactions Between Propagating and Nonpropagating Fields We turn our attention now to a closer examination of the interaction electromagnetic field energy in the near-field shell of a general antenna system. The electric field will again be decomposed into propagating and evanescent parts as . The energy density becomes then

(77)

(75) where and . In particular, by taking the limit , it is found that the total radial energy is finite and is given by

(76) This final expression shows that the total energy of the radial evanescent energy in the entire space outside the exclusion sphere is convergent. Moreover, it was possible to analytically evaluate the infinite radial space integral. Indeed, expression (76) contains only finite space integrals along the angular dependence of the spectral expansion of the radial evanescent-mode field energy density. It appears to the authors that the radial evanescent-mode expansion is the simplest type of near-field decomposition that will give finite total energy. The conclusion encroached by (76) strongly suggests that the concept of radial streamlines introduced in Section II-C is the most natural way to mathematically describe the near field of antennas in general, especially from the engineering point of view. 36I.e.,

the identity

x e dx = e where c is a nonzero constant.

x c

0 2cx + c2

(74)

The first term is identified with the self-energy density of the evanescent field, the second with the self-energy of the pure propagating part. The third term is a new event in the near-field shell: It represents a measure of interaction between the propagating and nonpropagating parts of the antenna electromagnetic fields. While it is relatively easy to interpret the first two terms as energies, the third term, that which we dubbed the interaction link between the first two types of fields, presents some problems. We first notice that contrary to the two self-energies, it can be either positive or negative. Hence, this term cannot be understood as a representative of an entity standing alone by itself like the self-energy, but instead it must be viewed as a relative energy, a relational component in the description of the total energy of the electromagnetic system. To better understand this point, we imagine that the two positive energies standing for the self-interaction of both the propagating and nonpropagating parts subsist individually as physically existing energies associated with the corresponding field in the way usually depicted in Maxwell’s theory. The third term, however, is a mutual interaction that relates the two self-energies to each other such that the total energy will be either larger than the sum of the two self-subsisting energies (positive interaction term) or smaller than this sum (negative interaction term). In other words, although we imagine the self-energy density to be a reflection of an actually existing physical entity, i.e., the corresponding field, the two fields nevertheless exist in a state of mutual interdependence on each other in a way that affects the actual total energy of the system. Consider now the total energy in the near-field shell. This will be given by the volume integral of the terms of (77). In particular, we have for the interaction term the following total interaction energy:

(78) For a particular spherical shell, expressions corresponding to (75) and (76) can be easily obtained. Again, the total interaction energy (78) may be negative. Notice that from the Weyl expansion, most of the field very close to the antenna current

MIKKI AND ANTAR: THEORY OF ANTENNA ELECTROMAGNETIC NEAR FIELD II

distribution is evanescent. On the other hand, most of the field in the far-field zone is propagating. It turns out that the interaction density is very small in those two limiting cases. Therefore, most of the contribution to the total interaction energy in (78) comes from the intermediate-field zone, i.e., the crucial zone in any theory striving to describe the formation of the antenna radiated fields. It is the opinion of the present authors that the existence of the interaction term in (77) is not an accidental or side phenomenon, but instead lies at the heart of the genesis of electromagnetic radiation out of the near-field shell. The theoretical treatment we have been developing so far is based on the fact that the antenna near field consists of streamlines along which the field “flows” not in a metaphorical sense, but in the mathematically precise manner through which the evanescent modes are being converted to propagating modes, and vice versa. The two types of modes transform into each other according to the direction of the streamlines under consideration. This indicates that effectively there is an energy exchange between the propagating and nonpropagating parts within the near-field shell. Expression (78) is nothing but an evaluation of the net interaction energy transfer in the case of radial streamlines. Since this quantity is a single number, it only represents the overall average of an otherwise extremely complex process. A detailed theory analyzing the exact interaction mechanism is beyond the scope of this paper and will be addressed elsewhere. G. Notorious Concept of Stored Energy There exists a long history of investigations in the antenna theory literature concerning the topic of “stored energy” in radiating systems, both for concrete particular antennas and general electromagnetic systems.37 The quality factor is the most widely cited quantity of interest in the characterization of antennas. As we have already seen in [1], all these calculations circuit of are essentially those related to an equivalent model for the antenna input impedance. In such a simple case, the stored energy can be immediately understood as the energy stored in the inductor and capacitor appearing in the circuit representation. In the case of resonance, both are equal, so one type of energy is usually required. Mathematically speaking, undercircuit, there is a second-order ordinary differenlying the tial equation that is formally identical to the governing equation of a harmonic oscillator with damping term. It is well-known that a mechanical analogy exists for the electrical circuit model in which the mechanical kinetic and potential energies will correspond to the magnetic and electric energies. The stored mechanical energy can be shown to be the sum of the two mechanical energies mentioned above, while the friction term will then correspond to the resistive loss in the oscillator [12]. Now, when attempting to extend this basic understanding beyond the circuit model toward the antenna as a field oscillator, we immediately face the difficult task of identifying what stands for the stored energy in the field problem. is The first observation we make is that the concept of well defined and clearly understood in the context of harmonic oscillators, which are mainly physical systems governed by 37For a comprehensive view on the topic of antenna reactive energy and the associated quantities like quality factor and input impedance, see [4].

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ordinary differential equations. The antenna problem, on the other hand, is most generally governed by partial differential equations. This implies that the number of degrees of freedom in the field problem is infinitely larger than the number of degrees of freedom in the circuit case. While it is enough to characterize the circuit problem by only measuring or computing the input impedance as seen when looking into the antenna terminals, the field oscillator problem requires generally the determination of the spatio-temporal variation of six field components throughout the entire domain of interest. In order to bring this enormous complexity into the simple level of second-order oscillatory systems, we need to search for ordinary differential equations that summarily encapsulate the most relevant parameters of interest. We will not attempt such an approach here, but instead endeavor to clarify the general requirements for such a study. We start from the following quote by Feynman made as preparation for his introduction of the concept of quality factor [12]: Now, when an oscillator is very efficient the stored energy is very high—we can get a large stored energy from a relatively small force. The force does a great deal of work in getting the oscillator going, but then to keep it steady, all it has to do is to fight the friction. The oscillator can have a great deal of energy if the friction is very low, and even though it is oscillating strongly, not much energy is being lost. The efficiency of an oscillator can be measured by how much energy is stored, compared with how much work the force does per oscillation. The “efficiency” of the oscillator is what Feynman will immediately identify as the conventional quality factor. Although his discussion focused mainly on mechanical and electric (circuit) oscillators, i.e., simple systems that can be described accurately enough by second-order ordinary differential equations, we notice that the above quote is a fine elucidation of the general phenomenon of stored energy in oscillatory systems. To see this, let us jump directly to our main object of study, the antenna as a field oscillator. Here, we are working in the time-harmonic regime, which means that the problem is an oscillatory one. Moreover, we can identify mechanical friction with radiation loss, or the power of the radiation escaping into the far-field zone. In such a case, the antenna system can be viewed as an oscillator driven by external force, which is nothing but the power supplied to the antenna through its input terminal, such that a constant amount of energy per cycle is being injected in order to keep the oscillator “running.” Now, this oscillator, our antenna, will generate a near-field shell, i.e., a localized field surrounding the source, which will persist in existence as long as the antenna is “running,” an operation that we can insure by continuing to supply the input terminal with steady power. The oscillator function, as is well known, is inverted: In antenna systems, the radiation loss is the main object of interest that has to be maximized, while the stored energy (whatever that may be) has to be minimized. The stored energy in the field oscillator problem represents then an inevitable side-effect of the system: A nonpropagating field has to exist in the near field. We say nonpropagating because anything that is propagating is associated automatically with the oscillator loss; what we are left with

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belongs only to the energy stored in the fields and which averages to zero in the long run. The next step then is to find a means to calculate this stored energy. In the harmonic oscillator problem, this is an extremely easy task. However, in our case, where we are not in possession of such a simple second-order differential equation governing the problem, one has to resort to an indirect method. We suggest that the quantitative determination of the antenna stored energy must revert back to the basic definition of energy as such. We define the energy stored in the antenna surrounding fields as the latent capacity to perform work when the power supply of the system is switched off. To understand the motivation behind this definition, let us make another comparison to the time evolution of damped oscillators. Transient phenomena can be viewed as a discharge of initial energy stored in the system.38 When the antenna power supply is switched on, the radiation loss is completely compensated for by the power removed by the antenna terminals from the source generator, while the antenna stored energy remains the same. Now, when the power supply is switched off, the radiation loss can no longer be accounted for by the energy flux through the antenna port. The question here is about what happens to the stored energy. In order to answer this question, we need to be more specific about the description of the problem. It will be assumed that a load is immediately connected across the antenna input terminals after switching off the generator. The new problem is still governed by Maxwell’s equations and hence can be solved under the appropriate initial and boundary conditions. It is expected that a complicated process will occur, in which part of the stored energy will be converted to electromagnetic radiation, while another portion will be absorbed by the load. We define then the actual stored energy as the total amount of radiated power and the power supplied to the load after switching off the source generator. In this case, the answer to the question about the quantity of the stored energy can in principle be supplied. Based on this formulation of the problem, we find that our near-field theory cannot definitely answer the quantitative question concerning the amount of energy stored in the near field since it is essentially a time-harmonic theory. A transient solution of the problem is possible but very complicated. However, our derivations have demonstrated a phenomenon that is closely connected with the current problem. This is the energy exchange between the evanescent and propagating modes. As could be seen from (78), the two parts of the electromagnetic field interact with each other. Moreover, by examining the field expression of the interaction energy density, we discover that this “function over space” extends in a localized fashion in a way similar to the localization of the self-evanescent field energy. This strongly suggests that the interaction energy density is part of the “nonmoving” field energy, and hence should be included with the self-evanescent field energy as one of the main constituents of the total energy stored in the antenna surrounding fields. Unfortunately, such a proposal faces the difficulty that this total sum of the two energies may very well turn out to be negative, in which its physical interpretation becomes problematic. One way out of this difficulty is to put things in their appropriate level: The time-harmonic theory is incapable of giving the fine details of 38“By a transient is meant a solution of the differential equation when there is no force present, but when the system is not simply at rest.” [12].

the temporal evolution of the system; instead, it only gives averaged steady-state quantities. The interaction between the propagating and nonpropagating field, however, is a genuine electromagnetic process and is an expression of the essence of the antenna as a device that helps convert a nonpropagating energy into a propagating one. In this sense, the interaction energy term predicted by the time-harmonic theory measures the net average energy exchange process that occurs between propagating and nonpropagating modes while the antenna is running, i.e., supplied by steady power through its input terminals. The existence of this time-averaged harmonic interaction indicates the possibility of energy conversion between the two modes in general. When the generator is switched off, another energy conversion process (the transient process) will take place, which might not be related in a simple manner to the steady-state quantity.39 H. Dependence of the Radial Localized Energy on the Choice of the Origin In this section, we investigate the effect of changing the location of the origin of the local observation frame used to compute the radial localized energy in antenna systems. In (73), we presented the expression of such energy in terms of a local coordinate system with an origin fixed in advance. If the location of this origin is shifted to the position , then it follows from (16) that the only effect will be to multiply the spatial Fourier . transform of the antenna current distribution by Therefore, the new total radial localized energy will become

(79) , that is, the It is obvious that in general new localized energy corresponding to the shifted origin with respect to the antenna is not unique. This nonuniqueness, however, has nothing alarming or even peculiar about it. It is a logical consequence from the Weyl expansion. To see this, consider Fig. 3, where we show the old origin , the new origin located at , and an arbitrary observation point outside the antenna current region. With respect to the frame , the actually computed field at the location is the evanescent part along the unit vector . On the other hand, for the computation of the contribution at the very same point but with respect to the frame 39The reader may observe that the situation in circuit theory is extremely simple compared to the field problem. There, the transient question of the circuit can be answered by parameters from the time-harmonic theory itself. For example, in an RLC circuit, the Q-factor is a simple function of the capacitance, inductance, and resistance—all are basic parameters appearing throughout the steady state and the transient equations. It is not obvious that such simple parallelism will remain the case in the transient field problem.

MIKKI AND ANTAR: THEORY OF ANTENNA ELECTROMAGNETIC NEAR FIELD II

Fig. 3. Geometric illustration for the process of forming the radial localized energy with respect to different origins.

at , the field added there is the evanescent part along the direc. Clearly then the tion of the unit vector two localized energies cannot be exactly the same in general. The reader is invited to reflect on this conclusion in order to remove any potential misunderstanding. If two different coordinate systems are used to describe the radial energy localized around the same origin, i.e., an origin with the same relative position compared to the antenna, then the two results will be exactly the same. The situation illustrated in Fig. 3 does not refer to two coordinate systems per se, but to two different choices of the origin of the radial directions utilized in computing the localized energy of the antenna under consideration. There is no known law of physics necessitating that the localized energy has to be the same regardless to the observation frame. The very term “localization” is a purely spatial concept, which must make use of a particular frame of reference in order to draw a mathematically specific conclusion. In our particular example, by changing the relative position of the origin with respect to the antenna, what is meant by the expression “radial localization” also has to undergo certain change. Equation (79) gives the exact quantitative modification of this meaning.40 IV. NEAR-FIELD RADIAL STREAMLINES FROM THE FAR-FIELD POINT OF VIEW A. Introduction In this section, we synthesize the knowledge that has been achieved in [1], concerning the near field in the spatial domain, and Section II, which focused mainly on the concept of radial streamlines developed from the spectral domain perspective. The main mathematical device utilized in probing the spatial structure of the near field was the Wilcox expansion

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look at the problem. The view of the antenna presented in [1] is essentially an exterior region description. Indeed, inside the , which encloses the antenna physical body, there sphere is an infinite number of current distributions that can be compatible with the Wilcox expansion in the exterior region. Put differently, we are actually describing the antenna system from the far-field point of view. Indeed, as was already shown by Wilcox [5], it is possible to recursively compute all the higher-order terms in the expansion (80) starting from a given far field. Now, the approach presented in Section II is different essentially for the opposite reason. There, the mathematical description of the problem starts from an actual antenna current distribution using the dyadic Greens function as shown in (1). This means that even when inquiring about the fields radiated outside some sphere enclosing the antenna body, the fields themselves are determined uniquely by the current distribution. It is for this reason that the analysis in line with Section II is inevitably more difficult than [1]. Our purpose in the present section is to reach for a kind of compromise between the two approaches. From the engineering point of view, the Wilcox series approach is more convenient since it relates directly to familiar antenna measures like far field and minimum . On the other hand, as we have already demonstrated in detail, the reactive energy concept is inadequate when extensions beyond the antenna circuit models are attempted. The Weyl expansion supplied us with a much deeper understanding of the near-field structure by decomposing electromagnetic radiation into propagating and nonpropagating parts. What is required is an approach that directly combines the Wilcox series with the deeper perspective of the Weyl expansion. This we proceed now to achieve in the present section. We first generalize the classical Weyl expansion to handle the special form appearing in the Wilcox series. This allows us then to derive new Wilcox–Weyl expansion, a hybrid series that combines the best of the two approaches. The final result is a sequence of higher-order terms explicating how the radial streamlines split into propagating and nonpropagating modes as we progressively approach the antenna physical body, all computed starting from a given far-field pattern, B. Generalization of the Weyl Expansion We start by observing the following from the product rule: (81)

(80)

which is valid for . We will be interested in deriving since it is precisly this a spectral representation for factor that appears in the Wilcox expansion (80). From (81), write

On the other hand, the Weyl expansion (4) represented the major mathematical tool used to analyze the near field into its constitutive spectral components. There is, however, a deeper way to

(82)

40An example illustrating this relativity can be found in the area of rigid-body dynamics. There, the fundamental equations of motion involve the moment of inertia around certain axes of rotation. It is a well-known fact that this moment of inertia, which plays a role similar to mass in translational motion, does depend on the choice of the axis of rotation and varies even if the new axis is parallel to the original one.

The Weyl expansion (4) written in spherical coordinates reduces to (83)

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where

where

(84) (85)

(94)

By bringing the differentiation inside the integral, it is possible to achieve (95) (86)

Also, we have

Substituting (83) and (86) into (82), it is found that (96) (87) (97) Iterating, the following general expansion is attained: where (88) Observing the repeated pattern, we arrive to the generalized Weyl expansion41

(98)

(89)

(99)

In reaching into this result, the differentiation and integration were freely interchanged. The justification for this is very close to the argument in Appendix B and will not be repeated here. (i.e., ) On a different notice, the singularity is avoided in this derivation because our main interest is in the antenna exterior region.

The expansion electric and magnetic functions (94) and (95) can be interpreted in the following manner. The appearing in factor has an attenuating part . Therefore, the field described here consists of evanescent modes along the radial direction specified by the spherical angles and . Similarly, the expansion electric and magnetic functions (98) and (99) are pure propagating modes along the same radial direction. Thus, we have achieved a mathematical description similar to the radial streamline in Section II-C, mainly (30) and (31). In the new expansion, the rich information encompassing the near-field spectral structure is given by the funcand tions for the electric and magnetic fields, respectively. We immediately notice that this spectral function consists of direct multiplication of two easily identified contributions: The first is the Wilcox-type and , and expansion given by the angular functions the second is a common Weyl-type spectral factor given by . This latter is function of both the spectral variables and and the spherical angles and . We can now understand the structure of the antenna near field from the point of view of the far field in the following manner. Start from a given far-field pattern for a class of antennas of interest. Strictly speaking, an infinite number of actually realized antennas can be built such that they all agree on the supposed far field. Mathematically, this is equivalent to stating that the hybrid Wilcox–Weyl expansions above are valid

C. Hybrid Wilcox–Weyl Expansion We now substitute the generalized Weyl expansion (89) into the Wilcox expansion (80) to obtain

(90)

(91) By separating the spectral integral into propagating and evanescent parts, we finally arrive to our main results (92) (93) 41This result can be rigourously proved by applying the principle of mathematical induction.

MIKKI AND ANTAR: THEORY OF ANTENNA ELECTROMAGNETIC NEAR FIELD II

only in the exterior region . We then proceed by computing (recursively as in [5] or directly as in [1]) all the vecand starting from the radiation torial angular functions pattern. With respect to this basic step, a radial streamline spectral description of the near-field structure can be constructed and by just multiplying the obtained angular vector fields by . This will generate the dependence of the spectral content of the near field on the radial streamline orientation specified by and . The actual spatial dependence of the propagating and nonpropagating fields can be recovered by integrating the outcome of multiplying the above obtained spectrum with the radial streamline over the regions functions and , respectively. A striking feature in this picture is its simplicity. For arbitrary antennas, it seems that the spectral effect of including higherorder terms in the hybrid Wilcox–Weyl expansion is nothing but , and ,42 multiplication by higher-order polynomials of with coefficients directly determined universally by the direction cosines of the radial vector along which a near-field streamline is considered. On the other hand, antenna-specific details of the radial streamline description seem to be supplied directly by and , which are functions of the the angular vector fields (far-field) radiation pattern. It appears then that the expansions (92), (93), (96), and (97) provide further information about the antenna, namely the im(inside portance of size. Indeed, the smaller the sphere where the antenna is located), the more terms in those expansions are needed in order to converge to accurate values of the electromagnetic fields. Taking into consideration that the anand are functions of the far-field ragular vector fields diation pattern, we can see now how the hybrid Wilcox–Weyl expansion actually relates many parameters of interest in a unified whole picture: the far-field radiation pattern, the near-field structure as given by the radial streamlines, the size of the antenna, and the minimum (for matching bandwidth consideration).43 D. General Remarks We end this section with a few remarks on the Wilcox–Weyl expansion. Notice first that the reactive energy, as defined in [1], is the form of the total energy expressed through the Wilcox seterm excluded. It is very clear from the results ries with the of this section that this reactive energy includes both nonpropagating and propagating modes. This may provide an insight into the explanations and analysis normally attached to the relationship between reactive energy, localized energy, and stored energy.44 The second remark is about the nature of the new streamline here. Notice that although we ended up in the hybrid 42This is intuitively clear since, as we have found in Part I [1], higher-order terms in the Wilcox-type expansion correspond to more complex near-field radial structure as we descend from the far zone toward the source region, which in turns necessities the need to include significant short-wavelength components (i.e., large p and q components). 43It is for these reasons that the authors believe the results of this paper to be of direct interest to the antenna engineering community. 44Cf. Section III and [1].

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Wilcox–Weyl expansion with a radial streamline picture of the near field, there is still a marked difference between this particular streamline and those introduced in Section II-C from the source point of view. The difference is that the nonpropagating fields in (94) and (95) are damped sinusoidal functions, while those appearing, for example, in (30), are pure evanescent modes. This is related to a deeper difference between the two approaches of Section II-C and the present one. In using the Wilcox expansion for the mathematical description of the antenna electromagnetic fields, we are asserting a far-field point of view, and hence our obtained near-field insight is already biased. This appears behind the fact that the generalized Weyl integral (89), when separated into the two regions inside and , will not give a decomposition into outside the circle propagating and nonpropagating modes in general. The reason is that there exists in the integrand spatial variables, mainly the spherical angles and . Only when these two angles are fixed can we interpret the resulting quantity as propagating and nonpropagating modes with respect to the remaining spatial variable, namely . It follows then that from the far-field point of view, the only possible meaningful decomposition of the near field into propagating and nonpropagating parts is the radial streamline picture. V. MECHANISM OF FAR-FIELD FORMATION We are now in a position to put together the theory developed throughout this paper into a more concrete presentation by employing it to explain the structural formation of the far-field radiation. This we aim to achieve by relying on the insight into the spectral composition of the near field provided by the Weyl expansion. In the remaining parts of this section, our focus will be on applying the source point of view developed in Section II. The theory of Section IV, i.e., the far-field point of view, may be taken up in separate work. Let us assume that the current distribution on the antenna physical body was obtained by a numerical solution of Maxwell’s equations, ideally using an accurate, preferably higher-order, method of moment.45 We will now explicate the details of how the far-field pattern is created starting from this information. We focus on the electric field. Since the far-field pattern is a function of the angular variables and , the most natural choice of the appropriate mathematical tool for studying this problem is the concept of radial streamlines as developed in Section II-C. A glance at (30) and (31) shows that the quantity pertinent to the antenna current distribution is the spatial Fourier as defined in (16). Now, to start transform of this current . Relawith, we choose a global cartesian frame of reference tive to this frame, we fix the spherical angles and used in the description of both the far-field pattern and the radial streamline picture of the near field. The global frame is chosen such that the -axis points in the direction of the broadside radiation. For example, if we are analyzing a linear wire antenna or a planner 45It is evident that the problem formulated this way is not exact. However, since the integral operator of the problem is bounded, the approximate finite-dimensional matrix representation of this operator will approach the correct exact solution in the limit when N .

!1

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patch, the global frame is chosen such that the -axis is perpendicular to the wire in the former case and to the plane containing the patch in the latter case. Although we do not prove this here, it can be shown that, under these conditions, the Fourier transform of the current distribution in the previous two special cases, as a function of the spectral variables and , has its maximum value around the origin of the -plane as shown in Fig. 2. Since the majority of the contribution to the far field comes from the propagating modes appearing in (31), the rest being attenuated exponentially as shown in (30), we can picture the antenna operation as a two-dimensional low-pass spatial filer in the following manner. All spectral components within the unit circle (the visible domain) will pass to the far field, while components outside this region will be filtered out. Let us call this filter the visible domain filter.46 Now, the fact that when the global frame is chosen such that its -axis is oriented in the direction along which the spatial Fourier transform of the current , as a function of and , will have most of its distribution immediately values concentrated around the region explains why some antennas, such as linear wires and planner patches, have broadside radiation pattern to begin with. We unpack this point by first noticing how the near field splits into propagating and nonpropagating streamlines. The mechanism here, as derived in (30) and (31), is purely geometrical. To is maximum see this, let us call the region around which ; e.g., in the case of planner patch, this region will be . What happens is that for varying centered around spherical angles and , we have to rotate the spatial Fourier by the matrix . This will translate into transform the introduction of new nonlinear transformation of and as .47 The region is now transgiven by formed into . Since we are viewing the antenna operation of producing the far-field pattern as a global two-dimensional spatial filter, we must transform back into the language of will the global frame. The newly transformed region be written in the old language as . Therefore, varying the observation angles and is effectively equivalent to a nongiven by linear stretching of the original domain (100) This implies that a reshaping of the domain is the main cause for the formation of the far-field pattern. Indeed, by relocating points within the -plane, the effect of the visible domain filter will generate the far-field pattern. However, there is also a universal part of the filtering process that does not depend on the antenna current distribution. This defined by (32). is the spectral polarization dyad The multiplication of this dyad with , i.e., the spectral quantity , is the outcome of the fact that the electromagnetic field has polarization, or that the problem is vector in nature.48 It is common to all radiation processes. We now see that the overall effect of varying the observation angles can be summarized in the tertiary process. 46Similar

q

construction of this filter exists in optics. transformation is nonlinear because depends nonlinearly on via the relation . 47This 48Cf.

m = p1 0 p + q

Section II-C.

m

p and

• Rotate the spatial Fourier transform by . • Multiply (filter) the rotated Fourier transform by the specafter applying to the latter a tral polarization dyad similarity transformation. • Filter the result by the visible domain filter of the antenna. This process fully explicates the formation of the far-field pattern of any antenna from the source point of view. As it can be seen, our theoretical narrative utilizes only two types of simple operations : 1) geometrical transformations (rotation, stretching, similarity transformation), and 2) spatial filtering (spectral polarization filtering, visible domain filtering). VI. CONCLUSION It seems from the overall consideration of this work that there exists a deep connection between the near and far fields. Indeed, the results of Section II-D suggested that only two degrees of freedom are needed to describe the splitting of the electromagnetic field into propagating and nonpropagating parts, which supplied the theoretical motivation to investigate the radial streamline structure of the near field. Furthermore, the results of Section IV showed that the only near-field decomposition into propagating and nonpropagating modes that is possible from the far-field point of view is the radial streamline picture introduced previously from the source point of view. This shows that there exists an intimate relation between the far- and near-field structures, and we suggest that further research in this direction is needed in order to understand the deep implications of this connection for electromagnetic radiation in general. The spectral theory, which decomposes the fields into evanescent and propagating modes together with a fundamental understanding of their mutual interrelation, can be related to the ongoing research in nanooptics, imaging, and other areas relevant to nanostructures and artificial materials. Indeed, the crux of this new devolvement is the manipulation of the intricate way in which the electromagnetic fields interact with subwavelength (e.g., nano-) objects. Mathematically and physically, the resonance of such subwavelength structures occurs upon interaction with evanescent modes because the latter correspond to the high-wavenumber -components. Therefore, our work in Part II regarding the fine details of the process in which the total field is being continually split into propagating and evanescent modes appears as a natural approach for studying the interaction of a nanoantenna or any radiating structure embedded within a complex surrounding environment. What is even more interesting is to see how such kinds of applications (interaction with complex environments) can be studied by the same mathematical formalism used to understand how the far field of any antenna (in free space) is formed, as suggested particularly in Section V. The advantage of having one coherent formalism that can deal with a wide variety of both theoretical and applied issues is one of main incentives that stimulated us in carrying out this program of antenna near-field theory research. On the more conventional side, for the design and devolvement of antennas radiating in free space and in complex media, where in the latter short-range interactions are dominant, we have tried to illuminate the near-field structure from both the source point of view and the far-field perspective at the same time. Both views are important in the actual design process.

MIKKI AND ANTAR: THEORY OF ANTENNA ELECTROMAGNETIC NEAR FIELD II

For the source point of view,49 our analysis in Part II, especially Section II-C, relates in a fundamental way the exact variation in the antenna current distribution to the details of how the near field converts continually from evanescent to propagating modes. This can help antenna engineers in devising clues about how to modify the antenna current distribution in order to meet some desirable design or performance goals. The advantage gained from such an outcome is reducing the dependence on educated guess, random trial and error, and expensive optimization tasks, by providing a solid foundation for carrying the antenna devolvement process in a systematic fashion.

APPENDIX A ABSOLUTE AND UNIFORM CONVERGENCE OF THE WEYL EXPANSION We work on the integral representation (9). It is not difficult to for sufficiently large show that , say . Notice that this is valid for any and , which is the case here because we are for any working in the exterior region of the antenna system. We now apply the Weierstrass- [14] test for uniform convergence. It follows then that the integral is absolutely convergent and uniformly convergent in all its variables.

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APPENDIX D DERIVATION OF THE ROTATION MATRIX We know that the matrix describing 3-D rotation by an angle around an axis described by the unit vector is given by

with

, and . In order to rotate the -axis into the location described by the radial vector , we imagine the equivalent process of rotating the original coordinate system by an angle around an axis perpendicular to the unit vector and contained within the -plane. Such an axis of rotation is . Substituting described by the unit vector these values to the rotation matrix above, the form given by (20) and (21) follows readily.

APPENDIX E TIME-DEPENDENT INTERACTION POYNTING THEOREM Taking the inverse Fourier transform of equations (45), the following set is obtained:

APPENDIX B INTERCHANGE OF INTEGRATION AND DIFFERENTIATION IN WEYL EXPANSION We know from advanced calculus that if the integral under consideration is uniformly convergent, and the integral of the partial derivative of the original integrand is also uniformly convergent, then it is possible to interchange the order of differentiation and integration [14]. We use exactly the same ideas of Appendix A together with the recurrence relation of the derivative of the Bessel function. It follows that the two conditions mentioned above can be met.

(101) From (101) and the vector identity, , the following two dual equations can be obtained:

(102)

(103) APPENDIX C EXCHANGE OF ORDER OF INTEGRATIONS IN THE RADIATED FIELD FORMULA VIA THE SPECTRAL REPRESENTATION OF THE DYADIC GREENS FUNCTION We can exchange the order of integrations by using the folis continlowing theorem from real analysis [14]: If , and , and if uous for is uniformly convergent for , we conclude that . Now, we already proved that the Weyl expansion converges uniformly. In addition, since the antenna current distribution is confined to a finite region, it immediately follows by repeated application of the theorem above that we can bring the integration with respect to the source elements inside the spectral integral. 49For the far-field perspective, the reader may refer to the conclusion section in Part I [1].

Adding (102) and (103), and observing the Leibniz product rule in handling contributions of the RHS, (69) immediately follows. REFERENCES [1] S. M. Mikki and Y. M. M. Antar, “A theory of antenna electromagnetic near field—Part I,” IEEE Trans. Antennas Propag., vol. 59, no. 12, pp. 4691–4705, Dec. 2011. [2] S. M. Mikki and Y. M. M. Antar, “Critique of antenna fundamental limitations,” in Proc. URSI-EMTS Int. Conf., Berlin, Germany, Aug. 16–19, 2010, pp. 122–125. [3] S. M. Mikki and Y. M. M. Antar, “Morphogenesis of electromagnetic radiation in the near-field zone,” IEEE Trans. Antennas Propag., 2011, (in three parts) to be submitted, unpublished. [4] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298–1324, Apr. 2005. [5] C. H. Wilcox, “An expansion theorem for the electromagnetic fields,” Commun. Pure Appl. Math., vol. 9, pp. 115–134, 1956. [6] H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys., vol. 60, pp. 481–500, 1919. [7] D. J. Jackson, Classical Electrodynamics. New York: Wiley, 1999.

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[8] R. Knobel, An Introduction to the Mathematical Theory of Waves. Providence, RI: AMS, 2000. [9] W. C. Chew, Waves and Fields in Inhomogenous Media. New York: Van Nostrand Reinhold, 1990. [10] T. B. Hansen and A. D. Yaghjin, Plane-Wave Theory of Time-Domain Fields. New York: IEEE Press, 1999. [11] D. M. Kerns, “Plane-wave scattering-matrix theory of antennas and antenna-antenna interactions: Formulation and applications,” J. Res. Nat. Bureau Stand. B, Math. Sci., vol. 80B, no. 1, pp. 5–51, Jan.–Mar. 1976. [12] R. P. Feynman, Lectures on Physics I. Reading, MA: Addison-Wesley, 1963. [13] D. Bressoud, A Radical Approach to Real Analysis. Providence, RI: AMS, 1994. [14] S. Lang, Undergraduate Analysis. New York: Spinger-Verlag, 1983. [15] S. M. Mikki and Y. M. M. Antar, “Analysis of electromagnetic interactions in linear wire antennas using the antenna current Greens function method,” 2011, unpublished. [16] S. M. Mikki and A. Kishk, “Electromagnetic wave propagation in nonlocal media: Negative group velocity and beyond,” Prog. Electromagn. Res. B, vol. 14, pp. 149–174, 2009. [17] S. M. Mikki and A. A. Kishk, “Nonlocal electromagnetic media: A paradigm for material engineering,” in Handbook of Microwave and Millimeter Wave Technologies. Rijeka, Croatia: INTECH, 2010.

Said M. Mikki (M’08) received the Bachelor’s and Master’s degrees from Jordan University of Science & Technology, Irbid, Jordan, in 2001 and 2004, respectively, and the Ph.D. degree from the University of Mississippi, University, in 2008, all in electrical engineering. He is currently a Research Fellow with the Electrical and Computer Engineering Department, Royal Military College of Canada, Kingston, ON, Canada. He worked in the areas of computational techniques in electromagnetics, evolutionary computing, nanoelectrodynamics, and the development of artificial materials for electromagnetic applications. His present research interest is focused on foundational aspects in electromagnetic theory.

Yahia M. M. Antar (S’73–M’76–SM’85–F’00) received the B.Sc. (Hons.) degree from Alexandria University, Alexandria, Egypt, in 1966, and the M.Sc. and Ph.D. degrees from the University of Manitoba, Winnipeg, MB, Canada, in 1971 and 1975, respectively, all in electrical engineering. In 1977, he was awarded a Government of Canada Visiting Fellowship with the Communications Research Centre, Ottawa, ON, Canada, where he worked with the Space Technology Directorate on communications antennas for satellite systems. In May 1979, he joined the Division of Electrical Engineering, National Research Council of Canada, Ottawa, ON, Canada, where he worked on polarization radar applications in remote sensing of precipitation, radio wave propagation, electromagnetic scattering, and radar cross section investigations. In November 1987, he joined the staff of the Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, ON, Canada, where he has held the position of Professor since 1990. He has authored or coauthored over 170 journal papers and 300 refereed conference papers, holds several patents, chaired several national and international conferences, and given plenary talks at conferences in many countries. He has supervised or co-supervised over 80 Ph.D. and M.Sc. theses at the Royal Military College and Queen’s University, Kingston, ON, Canada, of which several have received the Governor General of Canada Gold Medal, the outstanding Ph.D. thesis of the Division of Applied Science, as well as many best paper awards in major symposia. He was elected and served as the Chairman of the Canadian National Commission for Radio Science (CNC, URSI,1999–2008), Commission B National Chair (1993–1999), holds an adjunct appointment at the University of Manitoba, and has a cross appointment at Queen’s University. He also serves, since November 2008, as Associate Director of the Defence and Security Research Institute (DSRI). Dr. Antar is a Fellow of the Engineering Institute of Canada (FEIC) and the Electromagnetic Academy. He was elected by the Council of the International Union of Radio Science (URSI) to the Board as Vice President in August 2008, and to the IEEE Antennas and Propagation Society Administration Committee in December 2009. On January 31 2011, he was appointed Member of the Defence Science Advisory Board Coordinating Committee (DSAB). He is an Associate Editor (Features) of the IEEE Antennas and Propagation Magazine and has served as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS. He is a member of the Editorial Board of the International Journal of RF and Microwave Computer-Aided Engineering. He served on NSERC grants selection and strategic grants committees, Ontario Early Research Awards (ERA) panels, and the National Science Foundation Electrical, Communications, and Cyber Systems review panel. In May 2002, he was awarded a Tier 1 Canada Research Chair in Electromagnetic Engineering, which was renewed in 2009. In 2003, he was awarded the 2003 Royal Military College “Excellence in Research” Prize.

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Bridging the Gap Between the Babinet Principle and the Physical Optics Approximation: Scalar Problem Gildas Kubické, Yacine Ait Yahia, Christophe Bourlier, Member, IEEE, Nicolas Pinel, Associate Member, IEEE, and Philippe Pouliguen

Abstract—For a two-dimensional (2-D) problem, this paper shows that the Babinet Principle (BP) can be derived from the physical optics (PO) approximation. Indeed, following the same idea as Ufimtsev, from the PO approximation and in far-field zone, the field scattered by an object can be split up into a field that mainly contributes around the specular direction (illuminated zone) and a field that mainly contributes around the forward direction (shadowed zone), which is strongly related to the scattered field obtained from the BP. The only difference relies on the integration surface. We also show mathematically that the involved integral does not depend on the shape of the object, which then corresponds to the BP. Simulations are provided to illustrate the link between the BP and PO. Index Terms—Babinet principle, forward scattering, physical optics, shadow radiation.

I. INTRODUCTION

T

HE ELECTROMAGNETIC wave scattering from a target in the forward scattering (FS) region (when the target lies on the transmitter–receiver baseline) [1] is a very interesting phenomenon and was first reported by Mie in 1908, when he discovered that the forward-scattered energy produced by a sphere was larger than the backscattered energy [2] in highfrequency domain. This configuration, which corresponds to a bistatic angle ( - ) near 180 (see Fig. 1), is a potential solution to detect stealth targets. Indeed, in high-frequency domain, the forward-scattering radar cross section (RCS) is mainly determined by the silhouette of the target seen by the transmitter and is almost unaffected by stealth absorbing coatings or shapings. This phenomenon can be physically explained by the fact that the scattered field in the forward direction represents the perturbation to the incident wave as a blocking effect that creates a shadowed zone behind the target. In this region, while the total field vanishes, the scattered field tends to the incident field (in amplitude) but in opposite phase. A simple explanation can be given using the Babinet principle (BP) [1], which states that Manuscript received December 08, 2010; revised March 10, 2011; accepted May 01, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. G. Kubické is with the CGN1 Division, Direction Générale de l’Armement/Direction Technique/Maîtrise de l’Information (DGA/DT/MI), 35170 Bruz, France (e-mail: [email protected]). Y. A. Yahia, C. Bourlier, and N. Pinel are with the Institut de Recherche en Electrotechnique et Electronique de Nantes Atlantique (IREENA) Laboratory, Université de Nantes, 44306 Nantes, France. P. Pouliguen is with the Direction Générale de l’Armement/Direction de la Stratégie/Mission pour la Recherche et l’Innovation Scientifique (DGA/DS/MRIS), 92221 Bagneux, France. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165498

Fig. 1. Geometry of the problem: The target is illuminated by an incident field, and a shadowed zone is produced behind the object. The projection of the object onto the plane orthogonal to the incident direction illuminated surface . The plane orthogonal to the incident and centered on the phase origin is and the direction splits the space into two subdomains: the illuminated zone . shadowed zone

the diffraction pattern (in forward direction) of an opaque body is identical to that of a hole (in a perfectly conducting screen) having the same shape as its silhouette. Nevertheless, the physical optics (PO) approximation is sometimes used instead of the BP [2]–[4] and provides good results near the forward direction. Ufimtsev [5]–[8] studied the shadow radiation and demonstrated that the PO approximation can be split up into two components [5], [8]: one that mainly contributes in the backward direction and thus corresponds to a reflected component, and the other one that mainly contributes in the forward direction and thus corresponds to a shadowed component. In this paper, the link between the BP and the PO approximations is provided. First, a theoretical study is presented, and then numerical results compare theses two asymptotic approaches. The time convention is omitted throughout the paper. II. THEORETICAL STUDY A. PO Approximation Induced currents on the object surface can be estimated by using the PO approximation. For a 3-D (vectorial) problem, PO currents are given by the well-known expressions

0018-926X/$26.00 © 2011 IEEE

(1)

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where

is the unitary normal vector to the surface, is a position vector on the surface, and are the electric and magnetic currents, respectively, and and are the reflection matrices given from the Fresnel reflection coefficients (2) By assuming that the scene is invariant according -direction, then the problem becomes scalar. For the TE polarization (electric field in the -direction), the unknown is the transverse component of the electric field . Then, one can show that (3) (4)

Fig. 2. Babinet principle. (a) Forward scattering from an arbitrary obstacle. (b) Forward scattering from the associated opaque screen. (c) Diffraction from a hole in the infinite plane. the complementary screen

This leads to an equivalence between 3-D and 2-D problems for the TE polarization (5) From a similar way for the TM polarization (magnetic field in the -direction), one can define the equivalence between 3-D and 2-D problems

(9)

(6)

is related to the where far-field 2-D Green function (the incident field is assumed to be unitary on the target). The last two lines of (9) correspond to the decomposition proposed by Ufimtsev [5], [8]. Ufimtsev then showed that mainly contributes in the specular direction and thus corresponds to a “reflected” component that we call “PO reflection.” Moreover, mainly contributes in the forward direction and thus corresponds to a “shadowed” component that we call “PO forward.”

For a 2-D (scalar) problem, assuming an incident plane wave on the target, PO currents are given by (7) where face,

is a vector on the surthe total field on the surface, and its normal derivative. The latter two quantities are the unknowns of the problem. Moreover, is the unitary normal vector to the surface, in defines the orientation of the normal vector which and is the surface slope. is the incident wave vector, is the Fresnel reflection coefficient, which depends on ; for a perfectly conducting object, for TM and TE polarizations, respectively. Lastly, is the target illuminated surface (see Fig. 1). Contrary to the Babinet induced currents (see below), PO currents have physical meaning and tend to the tangential fields measured at the object surface. The radiation of these currents is computed from the Huygens principle [10], (8) is the 2-D (scalar) Green function, with the observation vector. Thus, assuming the target in far field from the receiver, , substituting (7) into (8), one has where

B. BP The BP is an optical principle [11] (generalized to electromagnetics [12], [13]) that states that the diffraction pattern of an opaque body is identical to that of a hole having the same shape as its silhouette (see Fig. 2). Thus, according to this principle, the FS phenomenon is independent of the shape of the object; the scattering is only due to the target area projected onto the plane orthogonal to the incident direction (see Fig. 2): the silhouette of the target. The equivalent induced currents on the aperture are only due to the presence of the incident field (10) As depicted in Fig. 1, the plane orthogonal to the incident direction splits the space into two subdomains: the illuminated zone and the shadowed zone . is the target area projected onto the plane orthogonal to the incident direction and centered on the phase origin, thus in the case of normal incidence ( being the normal of here). It must be noted that the normal incidence case can be considered with no assumption. Indeed, by a rotation of the problem, one can make variable changes to always consider a local system of coordinates in which the incident wave vector is in the sense of negative .

KUBICKÉ et al.: BRIDGING THE GAP BETWEEN BABINET PRINCIPLE AND PHYSICAL OPTICS APPROXIMATION

Moving the emitter with a fixed target is equivalent to rotating the target with a fixed emitter. In the far field (the receiver is in far field from the screen), the substitution of (10) into (8) leads, for any polarization and , to

and are the lower and upper values of the where abscissa of the illuminated surface , respectively. , (16) becomes Since

(11) Comparing (11) to (9), if

, we can write

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(17) where

(12) so

(18) and (13)

In other words, equality (12) is satisfied if the integral is independent of the surface integration . The proof of this statement is given next. C. Proof: “Shadowed” Component By setting

(14)

and since

(normal incidence), in (13) becomes

, and

(15) with the length of the screen. Since (the normal vector on is assumed to be oriented in the sense of positive ), , and in (13) becomes

(19) and for an object of length It must be noted that (along -direction) centered on the phase origin. Then, the PO forward component is expressed from a sinc function and does not depend on the object shape. This is consistent with the “Shadow Contour Theorem” [5], [8], which states that, “The shadow radiation does not depend on the whole shape of the scattering object, and is completely determined only by the size and the geometry of the shadow contour.” Thus, equality between the PO forward component and BP in (13) holds if equality is obtained between (15) and (17). This holds for either of the following: • and : the two limit points of are the , same as those of the complementary Babinet screen i.e., and ; or • implying , which corresponds to the FS direction, for which and are collinear. , if , then It must be noted that, even with (except for : the FS direction) and and : There is only a shift of the PO surface in direction, which implies a constant phase shift (described by the term ) of the scattered field. Thus, it must be highlighted that even for , equality between the two approaches remains in terms of RCS. D. “Reflected” Component Using the same way for the “reflected” component can be shown from (9) and for a perfectly conducting object that

(16)

, it

(20)

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and for

(21) Fig. 3. Forward scattering from a triangularly shaped target. The two suband can be decomposed into domains and .

where

(22) Unlike the field “shadowed component,” (21) shows that the field in the illuminated zone depends on the surface profile a priori because in the illuminated zone.

This equation clearly shows that the component strongly contributes in the forward direction and depends on , which is related to the length of the object. On the contrary, the component vanishes in the forward direction. As the main conclusion of Section II, the BP is a good approximation of the PO near the FS direction and when for . The BP is then a particular case of the PO approach.

E. Discussion (corresponding In the reflected direction defined by to the specular direction for an horizontal plate), from (14), one has

(23)

Equations (17) and (20) then become

III. NUMERICAL RESULTS A. Combining PO and BP for a Triangular Target Let us consider the scene given in Fig. 3, in which a triangularly shaped target is illuminated by an incident plane wave. The lengths of the three elementary planar surfaces , and are and , respectively. and . Edge diffracAs can be seen, tion is neglected because the work is focused here on the FS phenomenon. Since , under the PO approximacan be written as tion,

(24) Since we consider normal incidence , the above equation clearly shows that the component vanishes in the spec, then . On the ular direction. Moreover, if contrary, the component strongly contributes in the specular direction. In the forward direction defined as , from (14), one has

(27) in which, for

(28) which simplifies as

(25) (29)

Equations (17) and (20) then become (26)

in which respectively.

, and

for

or ,

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The BP states that for a receiver in far field located inside , the scattered field is given from (15), with . This result is relevant for the subdomain for which the receiver is in the shadowed zone of the target, but results can be wrong in subdomains and in which the receiver is not and , respectively. That is to say, in the shadowed zone of contributes in reflection in . Indeed and more generally, it can be observed that the “shadowed” zone of is , is . Thus, whereas the “shadowed” zone of a means to obtain the scattered field is to compute the PO for the reflection from and the BP for forward scattering from [14]. Then, for each subdomain, one obtains

(30)

Fig. 4. RCS of the triangularly shaped target of Fig. 3 with , and for TE polarization, computed from the MoM, the PO, and the for PO combined with the BP.

in which, for and is obtained by using the Babinet induced currents on the surface . This corresponds to (11), but on the surface . It can be noticed that the integrand in (11) is exactly the same as the one in the term of (9), which leads to . Then, (30) can be written in terms of as

(31) Using PO combined with BP on each elementary surface imcomponent is neglected in the shadowed plies that the zone of the surface . According to PO, is much lower than in the FS direction of , but this can induce slight discontinuities in the RCS at the subdomain frontiers. Thus, can be seen as an approximation of PO. B. RCS of the Triangular Target The RCS is defined in the 2-D case as (32) Thus, the RCS computed from the PO approach is given from (32), in which given from (27) and (29). The RCS for the PO combined with the BP is given from (32), in which is given from (29) and (31). C. First Case:

, and TE Polarization

The RCS of these two asymptotic approaches are compared to the RCS computed from a benchmark method: the well-known method of moments (MoM). The comparison is depicted in Fig. 4 versus the scattering angle for and for TE polarization. The triangularly shaped target (see Fig. 3) is defined from , which implies . As can be seen, results from the two asymptotic approaches agree well with that of the benchmark method, and in particular around the specular direction of surface

Fig. 5. Enlarged detail of Fig. 4 around the frontier between

and

.

and around the FS direction . Slight differences between the classical PO and the PO combined with the BP can be observed for . Indeed, is set to zero in , and both and are set to zero in for the computation of , and as the observation . angle increases, these two contributions decrease in Moreover, a slight discontinuity in the RCS computed from can be observed at the frontier between and for (an enlarged detail is depicted in Fig. 5). Indeed, from this angle, is set to zero in PO combined with BP. This discontinuity does not appear with classical PO method. Fig. 6 compares the RCS of and the RCS of its two components: the reflected one and the FS one . As can be seen, the scattered field from PO is mainly due to the reflected component . In other words, the reflected component mainly contributes to . For increasing from the scattering process 120 , the reflected component decreases strongly, and the FS component becomes the main contribution to the scattered field; being negligible in the shadow region . It must be noted that this phenomenon begins to occur in subdomain , which is in the reflected zone of . Fig. 7 compares the RCS of the FS component of PO computed from (27), in which , and the BP given from (32), in which computed from (2) and (15).

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Fig. 6. Same simulation parameters as in Fig. 4, but computed from PO, PO in reflection, and PO in forward scattering.

Fig. 7. Same simulation parameters as in Fig. 4, but computed from PO in forward scattering and BP.

Fig. 8. RCS of the triangularly shaped target of Fig. 3 with , and for TE polarization, computed from the MoM, the PO, and for the PO combined with the BP.

Fig. 9. Enlarged detail of Fig. 8 around the frontier between

and

.

A perfect agreement is obtained. This illustrates the proof of equalities (12) and (13). Indeed, here and since the two limit points of are the same as those of the com( plementary Babinet screen and ). As theoretically demonstrated in Section II, in this case, the Babinet principle is included in PO approach since . Interestingly, it can be noted that these results perfectly match the results obtained with other values of : Even if the target is different, the same FS component is obtained. The shape of the illuminated surface does not play a role, which is consistent with the “Shadow Contour Theorem” [5], [8]. D. Second Case: Polarization

, and TE

The RCS of the PO and the PO combined with the BP approaches are compared to the RCS computed from the MoM in Fig. 8 versus the scattering angle for and for TE polarization. The triangularly shaped target (see Fig. 3) is now defined with , and . Here again, the results from the two asymptotic approaches agree well with that of the benchmark method, and in particular around the FS direction . Some differences between the classical PO and the PO combined with the BP can be observed from (and higher). Moreover, two can be discontinuities in the RCS computed from

Fig. 10. Enlarged detail of Fig. 8 around the frontier between

and

.

observed. The first one occurs at the frontier between and for (an enlarged detail is depicted in Fig. 9). Indeed, from this angle (and higher), is set to zero in PO combined with BP. The second discontinuity occurs at the frontier between and for (an enlarged detail is depicted in Fig. 10), from which is set to zero in PO combined with BP. Of course, these discontinuities do not appear with the classical PO method. Fig. 11 compares the RCS of and the RCS of its two and the FS one . components: the reflected one

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IV. CONCLUSION

Fig. 11. Same simulation parameters as in Fig. 8, but computed from PO, PO in reflection, and PO in forward scattering.

For a 2-D problem, this paper shows that the BP can be derived from the PO approximation. Indeed, following the same idea as Ufimtsev, from the PO approximation and in far-field zone, the field scattered by an object can be split up into a field that mainly contributes around the specular direction (illuminated zone) and a field that mainly contributes around the forward direction (shadowed zone), which is strongly related to the scattered field obtained from the BP. The only difference relies on the integration surface. A theoretical study has provided the mathematical proof that the involved integral in FS component of PO does not depend on the global shape of the object. Then, when the two limit points of are the same as those of the complementary Babinet , then BP exactly corresponds to the FS component screen of PO. Thus, BP is included in the PO approximation. When the two limit points are not the same, BP can be seen as an approximation of the PO approach, and BP provides exactly the same results as PO in the FS direction. These theoretical conclusions were illustrated with the scattering from a triangularly shaped target to better investigate the link between BP and PO. In order to complete the study, the new PO approach, recently published by Cátedra et al. [9], was investigated for a scalar problem in the Appendix. This enables us to demonstrate that, for a scalar problem, the Cátedra currents exactly correspond to the classical PO approximation.

APPENDIX A SCALAR CÁTEDRA APPROACH Fig. 12. Same simulation parameters as in Fig. 8, but computed from PO in forward scattering and BP.

Like for the first case, the reflected component decreases in the shadow region ( being negligible in ), and the FS component becomes the main contribution to the scattered field. Fig. 12 compares the RCS of the FS component, , and the RCS of the BP, . Here, the two approaches do not match. Indeed, and since the two limit points of are not the same as those of the complementary Babinet screen ( and ) due to the equivalent rotation of the target. As already said, moving the emitter with a fixed target is equivalent to rotating the target with a fixed emitter. This third case is equivalent to that of a normal incidence (like for the first and second cases), but with a rotated triangularly shaped target with an angle 25 . Thus, equalities (12) and (13) are only satisfied , implying , which corresponds to when the FS direction, for which and are collinear. As can be seen in Fig. 12, a perfect agreement is obtained in the FS direction . The Babinet principle can be seen as an approximation of the PO approach, which provides exactly the same results as the PO in the FS direction. Moreover, it is shown in the Appendix that, for a scalar problem, the Cátedra currents exactly correspond to the classical PO approximation.

In 2008, Cátedra et al. [9] proposed new induced PO currents to improve the FS computation. These currents extend over the whole body, including lit or shadowed parts. On the illuminated surface, the currents provide the reflected field , whereas on the shadowed surface, they provide the FS phenomenon . Applied to a scalar problem, for a perfectly conducting object, it can be shown that the Cátedra currents are given in TE polarization by

(33) and in TM polarization by

(34) Since target and

on the illuminated surface on the shadowed surface , (34) becomes

of the

(35)

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Applying Huygens’ principle (8) on these induced currents provides the scattered field in TM polarization and

(36) The second integral in the right-hand side corresponds to the scattering of the Babinet currents (10) applied on . We have shown above that this integral does not depend on the surface contour, and by considering the normal oriented in the sense of positive for both integrals (change of sign in the second integral), (36) can be written as

(37) which corresponds to the use of the classical PO approximation. Thus, for a scalar problem, the Cátedra currents provide exactly the same results as PO. The same conclusion can be drawn in TE polarization.

Gildas Kubické was born in Longjumeau, France, in 1982. He received the Engineering degree and M.S. degree in electronics and electrical engineering from Polytech’Nantes (Ecole polytechnique de l’université de Nantes), Nantes, France, in 2005, and the Ph.D. degree in electronics from the University of Nantes, Nantes, France, in 2008. He is currently working in the field of radar signatures at the Direction Générale de l’Armement (DGA), DGA-Maîtrise de l’Information, Bruz, France. He is also in charge of the Expertise and electoMagnetism Computation (EMC) Laboratory of DGA-Maîtrise de l’Information. His research interests include electromagnetic scattering and radar cross-section modeling.

Yacine Ait Yahia, photograph and biography not available at the time of publication.

Christophe Bourlier (M’99) was born in La Flèche, France, in 1971. He received the M.S. degree in electronics from the University of Rennes, Rennes, France, in 1995, and the Ph.D. degree in electronics from the Polytech’Nantes (University of Nantes, Nantes, France), in 1999. While at the University of Rennes, he was with the Laboratory of Radiocommunication, where he worked on antennas coupling in the VHF-HF band. Currently, he is with the Institut de Recherche en Electrotechnique et Electronique de Nantes Atlantique (IREENA) Laboratory in the SEC team at Polytech’Nantes. He works as an Assistant Researcher with the National Center for Scientific Research on electromagnetic wave scattering from rough surfaces and objects for remote sensing applications. He is the author of more than 120 journal articles and conference papers.

REFERENCES [1] K. M. Siegel, “Bistatic radars and forward scattering,” in Proc. Aero Electron. Nat. Conf., Dayton, OH, 1958, pp. 286–290. [2] J. I. Glaser, “Bistatic RCS of complex objects near forward scatter,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-21, no. 1, pp. 70–78, Jan. 1985. [3] J. I. Glaser, “Some results in the bistatic radar cross section (RCS) of complex targets,” Proc. IEEE, vol. 77, no. 5, pp. 639–648, May 1989. [4] P. S. Kildal, A. A. Kishk, and A. Tengs, “Reduction of forward scattering from cylindrical objects using hard surfaces,” IEEE Trans. Antennas Propag., vol. 44, no. 11, pp. 1509–1520, Nov. 1996. [5] P. Y. Ufimtsev, Fundamentals of the Physical Theory of Diffraction. Hoboken, NJ: Wiley, 2007. [6] P. Y. Ufimtsev, “Blackbodies and shadow radiation,” Sov. J. Commun., Technol. Electron., vol. 35, no. 5, pp. 108–116, 1990. [7] P. Y. Ufimtsev, “Blackbodies and the problem of invisible objects,” in Proc. JINA, 1992. [8] P. Y. Ufimtsev, “New insight into the classical Macdonald physical optics approximation,” IEEE Antennas Propag. Mag., vol. 50, no. 3, pp. 11–20, Jun. 2008. [9] M. F. Cátedra, C. Delgado, and I. G. Diego, “New physical optics approach for an efficient treatment of multiple bounces in curved bodies defined by an impedance boundary condition,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 728–736, Mar. 2008. [10] B. B. Bakker and E. T. Copson, The Mathematical Theory of Huygens Principle. Oxford, U.K.: Oxford Univ. Press, 1939. [11] M. Born and E. Wolf, Principles of Optics. New York: Pergamon, 1959. [12] H. G. Booker, “Slot aerials and their relation to complementary wire aerials (Babinet’s principle),” Proc. Inst. Elect. Eng., vol. 93, pp. 620–626, 1946. [13] P. Poincelot, “Sur le théorème de Babinet au sens de la théorie électromagnétique,” Ann. Télécommun., vol. 12, no. 12, pp. 410–414, Dec. 1957. [14] P. Pouliguen, J. F. Damiens, and R. Moulinet, “Radar signatures of helicopter rotors in great bistatism,” in Proc. APS/URSI, Jun. 2003, pp. 536–539.

Nicolas Pinel (A’09) was born in Saint-Brieuc, France, in 1980. He received the Engineering degree and M.S. degree in electronics and electrical engineering from Polytech’Nantes (Ecole polytechnique de l’université de Nantes), Nantes, France, in 2003, and the Ph.D. degree in electronics from the University of Nantes, Nantes, France, in 2006. He is currently a Research Engineer with the Institut de Recherche en Électrotechnique et Électronique de Nantes Atlantique (IREENA) Laboratory, Nantes, France. His research interests are in the areas of radar, optical remote sensing, and propagation. In particular, he works on asymptotic methods of electromagnetic wave scattering from rough surfaces and layers.

Philippe Pouliguen was born in Rennes, France, in 1963. He received the M.S. degree in signal processing and telecommunications and the Ph.D. degree in electronics from the University of Rennes 1, Rennes, France, in 1986 and 1990, respectively. In 1990, he joined the Direction Générale de l’Armement (DGA) at the Centre d’Electronique de l’Armement (CELAR), Bruz, France, where he was a DGA expert in electromagnetic radiation and radar signatures analysis. He was also in charge of the Expertise and electoMagnetism Computation (EMC) laboratory of CELAR. Now, he is the Head of acoustic and radio-electric waves domain at the Office for Advanced Research and Innovation of the Strategy Directorate, DGA, France. His research interests include electromagnetic scattering and diffraction, radar cross section (RCS) measurement and modeling, asymptotic high frequency methods, radar signal processing and analysis, antenna scattering problems, and electronic band-gap materials. In these fields, he has published more than 28 articles in refereed journals and more than 80 conference papers.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 12, DECEMBER 2011

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Pulsed Radiation From a Line Electric Current Near a Planar Interface: A Novel Technique Leonid A. Pazynin

Abstract—A novel technique has been suggested for the analysis of a transient electromagnetic field generated by a pulsed line current that is located near a planar interface between two dielectric nonabsorbing and nondispersive media. As distinct from the Cagniard-de Hoop method, which is widely used for the study of transient fields both in electrodynamics and in the theory of acoustic and seismic waves, our approach is based on the transformation of the domain of integration in the integral expression for the field in the space of two complex variables. As a result, it will suffice to use the standard procedure of finding of roots of the algebraic equation rather than construct auxiliary Carniard’s contours. A fresh type of the representation for the field has been derived in the form of an integral along a finite contour. The algorithm based on the representation of this kind may work as the most efficient tool for calculating fields in multilayered media. The method suggested allows extension to the case of arbitrary dipole sources. Index Terms—Electromagnetic radiation, modified Cagniard technique, planar interface, pulsed line source.

I. INTRODUCTION RANSIENT electromagnetic fields generated by pulsed currents located near a planar boundary between layered media are the subject of constant theoretical research, as from the B. van der Pol paper [1]. The approach based on the classical Cagniard method [2], [3] is the most efficient tool in this study. De Hoop [4] has suggested a modification of Cagniard’s method with the help of which exact solutions have been obtained for a number of problems about a dipole or a line source near an interface [5]–[9]. Various modifications of Cagniard’s technique have found wide application in the study of nonstationary acoustic and seismic wave propagation. Following paper [4], modifications of de Hoop’s technique [10], [11] as well as the alternative approaches free from some drawbacks to this method [12], [13] have been suggested. In the present paper, the approach alternative to Cagniard’s technique is used to study the nonstationary field generated by line sources located in flat-layered media. The approach suggested is applied to the already solved problem, namely, determination of the electromagnetic field generated by a pulsed line source located near a planar interface between two nonabsorbing and nondispersive media. The corresponding results have been discussed in considerable detail in [9]. In this paper, a one-sided Laplace transform with respect to time and two-sided

T

Manuscript received November 29, 2010; manuscript revised March 17, 2011; accepted April 30, 2011. Date of publication August 12, 2011; date of current version December 02, 2011. The author is with the Department of Mathematical Physics, Institute of Radio Physics and Electronics of the National Academy of Sciences of Ukraine, Kharkov 61085, Ukraine (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2164212

Fig. 1. Pulsed line source near the interface between two semi-infinite media.

Laplace transform with respect to a horizontal spatial variable have been applied and, as a consequence, the electromagnetic field has been represented in the form of some double integral. This integral can be calculated efficiently by the Cagniard-de Hoop method (CHM). The essence of the method is as follows. The original path of integration for one of two integrals forming the double integral is deformed into a so-called modified Cagniard contour. It is chosen such that upon the corresponding change of the integration variable in the integral along the modified contour, the original double integral turns into a composition of the direct and inverse Laplace transform for the known function. The central problem with this method is finding, generally speaking, numerically, the modified Cagniard contour whose shape changes as the observation point changes. The key point of the approach proposed in the present paper includes the following. To calculate the double integral efficiently, we suggest deforming its domain of integration (the real -space of two complex variables rather than to plane) in the -plane, as has been done deform one contour in the complex in CHM. It is shown that in this case the integral reduces to a sum of residues. The use of powerful apparatus of the residue theory instead of somewhat artificial way used in CHM is reason to hope that our approach can be efficient in the situations where the CHM fails, such as for anisotropic media. The method presented in the paper can be extended to multilayered media and arbitrary dipole sources. II. PROBLEM FORMULATION The field generated by the pulsed line electric current (1) located near a planar interface (Fig. 1) is to be found. The source of this kind excites the -polarized field

(2) where

0018-926X/$26.00 © 2011 IEEE

for

and

for

.

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The function

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 12, DECEMBER 2011

is the solution to the wave equation

that satisfies the conditions of continuity of - and ponents on the interface and the causality principle. The Fourier transform in time

or (3)

(11)

-com-

(12) Thus, we have the required field in the form of the following double integrals over the plane of real variables and (13)

(4)

applied to the boundary-value problem in (3), results in the following problem

(14)

(5) (15)

with the boundary conditions on (6) where . The solution of the equations in (5) is conveniently represented in the form [7]

(7) where

where

,

(

, 2).

III. TRANSFORMATION TO SINGLE INTEGRALS

(8)

(9)

(10) are the unknown functions, , boundary conditions in (6), we have: 1) 2)

(16)

. From the

In formulas (14)–(16), the integrands allow analytic contininto uation from the real plane -space of two complex variables and the . As previous analysis has shown, there is no need . To calculate efto operate with the whole of real 4-D space ficiently the integrals in (14)–(16), it is sufficient to consider a containing . In , 3-D space the single-valued branches of two square roots in the integrands should be chosen. In Appendix, it is shown that for a loss-free media the cut surface ensuring a choice of the branch for which in is (Fig. 4) a double we have sector that lies in the plane , contains the -axis, ( , 2). and is bounded by the branch lines The root is positive on the upper side of the right-hand sector and on the bottom side of the left-hand sector ; while it is negative on the other sides. Since the integrands in (14)–(16) are uniquely defined in -space with the specified cuts, the Cauchy-Poincare thethe orem [14] can be used to deform the surface of integration in . In accordance with the causality principle, the cut surfaces and have to adjoin the real plane from the bottom

PAZYNIN: PULSED RADIATION FROM A LINE ELECTRIC CURRENT NEAR A PLANAR INTERFACE

. Then, the integrands have no singularities in the half, and we have for all , space according the mentioned theorem. For positive values of the time variable , the -plane can . Then we have for an be deformed to a half-space , while for we have integrals integral over the surface . Here stands for the closed over the surface surface enveloping the cut . as an example, let us demonUsing the function strate how the integrals describing the secondary field in (15), (16) can be simplified. Denoting the integrand in (15) by , consider the following integral over the surface (17) . Let and be the rightwhere and the left-hand cavities of the surface hand ; is the closed contour generated by the intersection of with the coordinate plane . the surface Then we have

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where (20) and the contour envelopes the segment in the plane of the complex variable . Let us introduce an accessory for the sake of convergence acceleration; then parameter (19) can be rewritten in the form of (21), shown at the bottom of the page. in (17), we obtain a represenFor the second integral replaced by , where tation similar to (21) with is the contour enveloping the segment . Thus, for the function given by (15), which determines the secondary field in the first medium (see (13)), we arrive at the following expression:

(22) , is the contour enveloping the segment , . The root branches are determined by the inequalities with zero value of the argument on the bottom side of the cut along . the segment describing the field in the Similarly, for the function second medium, we obtain from (16) where

(18) In the second integral here, the change of variables has been carried out. By taking into account the evenness of the chosen branches of square roots with respect to this change of varientering the function , ables and by performing another change of variables we arrive at the following expression for the integral in (15)

(23)

(19)

where . The integrands in (22) and (23) are analytical in the plane of complex variable with the specified cut and decrease at infinity as . Therefore, these integrals

(21)

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can be reduced to the residues determined by zero values of the denominators in the square brackets

(24)

(31)

(25)

The wave reflected from the interface comes at the given point . For the time interval in the first medium at time , in view of (31), we obtain

IV. FIELD IN THE FIRST MEDIUM The roots of the (24) are readily determined and can be written as (26) (32)

and (27) for (24a) and (24b) respectively, where the square root

. For

where For the time interval

. , we have

, the same branch in the complex

plane of variable has been determined as for in the -plane. By calculating the corresponding residues, we have from (22) (33)

(28) Here we have used the equality

(29) It is easy to verify that the following relationships hold for the chosen branches of the square roots:

where . The behavior of the secondary field in the first medium for the times essentially depends on the relation and the second between the refractive indices for the first media. For an arbitrary point in the first medium, both of the roots are real (see (29)) if . Consequently, entering in , and the secondary field given by (32) we have is zero up to the moment of arrival of the reflected wave. In the case that , a more detailed analysis of the is required. Let us use the following notafunction , , , tions: stands for the angle of total internal reflection [7], where , where [15]. Let also introduce a parameter of this function has been chosen, the principal branch . Then we arrive at through the formula

(30) Therefore

(34)

PAZYNIN: PULSED RADIATION FROM A LINE ELECTRIC CURRENT NEAR A PLANAR INTERFACE

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V. FIELD IN THE SECOND MEDIUM Denote the roots of the (25a) and (25b) by and tively. Then the integral in (23) takes the form

, respec-

(36) Fig. 2. Wave fronts of the field generated by the line pulsed current located near a planar interface for n > n : primary (I), reflected (II), transmitted (III), and “side” (IV) waves; z x x A is the trajectory determining the time of arrival of the “side” wave at the point A,  is the angle of total internal reflection.

where (37)

Since for the space-time domain considered we have , then the arguments of the sine functions . Therein (34) find themselves within the interval fore, the function given by (34) has two roots and corresponding to the points of time and . There is no in Fig. 2) that difficulty to show (the trajectory , where is the time of arrival of the so-called ‘side’ [7] (or ‘diffraction’ [15]) wave at the observation point located in the first medium in the region . For , the variable goes to the unphysical sheet of the function , and the ‘side’ wave does not occur in this region. By virtue of the causality principle, for the times , there is no is of no importance secondary field and so the other zero . for Let us find the value of in the region . Here, the following relationships for the arguments of the sine functions in (32) are valid:

(38) The expressions for the roots can be written explicitly as the solutions of the associated algebraic quartic equations. However, they are too cumbersome because of six parameters entering (25) and are not used in the present paper. In view of the for , where is the causality principle, time of arrival of the transmitted wave at the observation point in , the roots are complex and, the second medium. For as evident from (25), in terms of (30), we have . Therefore, having regard to (31), we obtain for

(39)

which means that

. Considering that

, we have . and , the “side” wave Thus, for given by the function (32) is generated in the region . , From (32), (33), through the use of the substitutions , we arrive at the following expression for the characterizing the primary field: function

(35)

where is the time of arrival of the primary wave at the observation point in the first medium.

where

. VI. DISCUSSION AND CONCLUSION

Formulas (13) and (35) for the primary field, formulas (32) and (33) for the secondary field in the first medium, as well as formula (39) for the secondary field in the second medium coincide with the relevant expressions derived in [9] by CHM. The principal result of the work is a new representation for the field generated by a pulsed line current in a two-media configuration in the form of the integrals over a finite contour (20), (21). This method, like the CHM, is applicable to the problems of pulsed electromagnetic radiation from linear sources in media formed by an arbitrary finite number of homogeneous parallel layers with permittivity and permeability

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. In this case, for the field in the layers, the inte, grals over the contour enveloping the interval , are similar to represenwhere tations (20), (21). Two methods for calculating these integrals are possible. The first way is to reduce them by the Cauchy theorem to a sum of residues at the poles of the integrand. These poles are determined by the roots of algebraic equations that coincide with the equations for the modified Cagniard contours [9]. Therefore, this technique being alternative to the CHM in an analytical sense is equivalent to it in a calculating sense. Another way is to estimate numerically the integrals in (20), (21). It is easy to show that they can be reduced to the integrals . For example, the field in the first over the interval , in the medium (20) can be represented for following form:

Fig. 3. Sign distribution for Re and Im in the plane  = 0. Straight lines indicate the lines of intersection with the plane  = 0: a) bold line—for the cone Re = 0; b) dash line—for the planes Im = 0. Symbols ( ) specify a sign of Re , while [ ] specify a sign of Im ; sin  = n = n , l is the cone axis (A1).

6

0

6 jj

1. A surface

(A1) where

We can use a standard integrating procedure of any mathematby this formula. Comparison of ical package to calculate the data obtained by this way with the explicit expression given by (33) has demonstrated high efficiency and accuracy of this approach. The key point of the CHM is the solution of the algebraic equation determining the modified Cagniard contour. To do this, iterative numerical methods are used. The greatest difficulty inherent in these methods is to choose the starting value that is close enough to the required zero of the equation [18]. In the paper [9], such an initial approximation has been proposed for isotropic layers. For , the the medium consisting of efficiency of the iterative method has been shown. For more complex structures containing anisotropic layers, the initial approximation of this kind is unknown. (The CHM allows us to study as yet the simplest situation where the source and the observation point are located on the boundary of an anisotropic medium [19].) The method proposed in the paper being free from the complications of this kind reduces the calculation of the field generated by a line dipole in a multilayered medium to a standard procedure of numerical integration over a finite interval. APPENDIX Consider a function that the refractive index valued.

in

assuming is complex-

, , has the following invariants [16]: , , . Therefore, it represents a two-pole elliptic with its vertex cone symmetrical with respect to the plane at the origin of coordinates. Let us locate the axis of the cone. The lines of intersection of the cone with the symmetry plane are two mutually orthogonal straight lines with the bisecting lines and . Consequently, the cone axis is determined and . by the equations 2. A surface (A2) , , has the following invariants: . Therefore, it represents two mutually orthogonal planes intersecting along the -axis and determined by the equations: and . The first plane contains the axis of the cone (A1) being its another symmetry plane. From (A1) and (A2), we derive the following equations for the branch lines of : (A3) In Fig. 3, the distribution of signs for and in is shown. In (14)–(16), a single-valued branch of the function , , has been determined on the real for which . The above mentioned inequality is hold plane everywhere in if the following condition is satisfied: . In other words, the cut in that separates this branch should be determined by the conditions , . As is seen from Fig. 3, this takes place for a double sector formed by the intersection of the inner part of the cone (A1) with its symmetry plane . In with the . cut of this kind (Fig. 4) we have

PAZYNIN: PULSED RADIATION FROM A LINE ELECTRIC CURRENT NEAR A PLANAR INTERFACE

Fig. 4. Location of the branch lines l and the cut surface S ensuring a choice of the branch for which Im(!;  ) 0 in -space; l is the cone axis (A1).



R

A similar approach to choosing a branch of the square root is given in [17] for the case of a single variable. When passing to , the cut surface is shifted into the a lossless medium representing a double sector which contains the plane -axis and is bounded by straight branch lines . REFERENCES [1] B. van der Pol, “On discontinuous electromagnetic waves and the occurrence of a surface wave,” IRE Trans. Antennas Propag., vol. AP-4, pp. 288–293, 1956. [2] L. Cagniard, Reflexion et Refraction des Ondes Seismiques Progressives. Paris, France: Gauthier-Villars, 1939. [3] L. Cagniard, Reflection and Refraction of Progressive Seismic Waves. New York: McGraw-Hill, 1962. [4] A. T. de Hoop, “A modification of Cagniard’s method for solving seismic pulse problems,” Appl. Sci. Res. B, vol. 8, pp. 349–356, 1960. [5] A. T. de Hoop and H. J. Frankena, “Radiation of pulses generated by a vertical electric dipole above a plane, non-conducting, earth,” Appl. Sci. Res. B, vol. 8, pp. 369–377, 1960. [6] H. J. Frankena, “Transient phenomena associated with Sommerfeld’s horizontal dipole problem,” Appl. Sci. Res. B, vol. 8, pp. 357–368, 1960. [7] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Upper Saddle River, NJ: Prentice-Hall, 1973, pp. 523–527.

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[8] K. J. Langenberg, “The transient response of a dielectric layer,” Appl. Phys., vol. 3, no. 3, pp. 179–188, 1974. [9] A. T. de Hoop, “Pulsed electromagnetic radiation from a line source in a two-media configuration,” Radio Sci., vol. 14, no. 2, pp. 253–268, Mar.–Apr. 1979. [10] B. J. Kooij, “The transient electromagnetic field of an electric line source above a plane conducting earth,” IEEE Trans. Electromagn. Compat., vol. 33, no. 1, pp. 19–24, Feb. 1991. [11] H. C. Murrell and A. Ungar, “From Cagniard’s method for solving seismic pulse problems to the method of the differential transform,” Comput. Math. Appl., vol. 8, no. 2, pp. 103–118, 1982. [12] N. Bleistein and J. K. Cohen, “An alternative approach to the Cagniard de Hoop method,” Geophys. Prospect., vol. 40, no. 6, pp. 619–649, 1991. [13] R. Beh-Hador and P. Buchen, “A new approach to Cagniard’s problem,” Appl. Math. Lett., vol. 12, no. 12, pp. 65–72, 1999, no. 8. [14] B. V. Shabat, Introduction to Complex Analysis, American Mathematical Society, p. 371, 1992, new ed. [15] F. G. Friedlander, Sound Pulses. Cambridge, U.K.: Cambridge Univ. Press, 2009, p. 216. [16] G. A. Korn, G. A. , and T. M. Korn, Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, 1961. [17] R. Mittra and S. W. Lee, Analytical Techniques in the Theory of Guided Waves. New York: MacMillan, 1971. [18] G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Computations. Upper Saddle River, NJ: PrenticeHall, 1977, 07632. [19] W. Lihh and S. Nam, “Time-domain electromagnetic fields radiating along the horizontal interface between vertically uniaxial half-space media,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1305–1317, May 2007.

Leonid A. Pazynin was born in Gorki, USSR, in 1946. He received the M.S. degree in radio physics from Kharkov State University, in 1972 and the Ph.D. degree in radio physics from Rostov State University, USSR, in 1987. Since 1976, he has been with the Institute of Radio Physics and Electronics, National Academy of Science of Ukraine, where he is currently a Research Scientist. His current research interests lie in the field of electromagnetic scattering and propagation in complex media.

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Off-Axis Gaussian Beam Scattering by an Anisotropic Coated Sphere Zhen-Sen Wu, Senior Member, IEEE, Zheng-Jun Li, Huan Li, Qiong-Kun Yuan, and Hai-Ying Li

Abstract—An analytical solution to the scattering of an off-axis Gaussian beam incident on an anisotropic coated sphere is proposed. Based on the local approximation of the off-axis beam shape coefficients, the field of the incident Gaussian beam is expanded using first spherical vector wave functions. By introducing the Fourier transform, the electromagnetic fields in the anisotropic layer are expressed as the addition of the first and the second spherical vector wave functions. The expansion coefficients are analytically derived by applying the continuous tangential boundary conditions to each interface among the internal isotropic dielectric or conducting sphere, the anisotropic shell, and the free space. The influence of the beam widths, the beam waist center positioning, and the size parameters of the spherical structure on the field distributions are analyzed. The applications of this theoretical development in the fields of biomedicine, target shielding, and anti-radar coating are numerically discussed. The accuracy of the theory is verified by comparing the numerical results reduced to the special cases of a plane wave incidence and the case of a homogeneous anisotropic sphere with results from a CST simulation and references. Index Terms—Anisotropic layered, electromagnetic scattering, Gaussian beam, off-axis.

I. INTRODUCTION HE scattering of a plane electromagnetic wave by a coated spherical particle has been extensively discussed in many areas such as combustion, biomedicine, chemical engineering, remote sensing, etc. Within a theoretical framework similar to the classic Lorenz-Mie theory [1], [2], rigorous formulations were first developed for scattering using two concentric spheres based on the properties of Bessel functions by Aden and Kerker [3]. This general solution was subsequently specialized for a lossless or lossy dielectric-coated conducting sphere [4]–[8]. Wu further presented numerical results for plane wave scattering by a multilayered isotropic sphere [9]. In recent years, interest in the scattering characteristics of anisotropic media has grown because of their wide applications in optical signal processing, radar cross section (RCS) controlling, microwave device fabrication, etc. Particularly, research on layered anisotropic spheres is of special importance in the target shielding field. In addition to studies on the homogeneous

T

Manuscript received October 15, 2010; revised March 21, 2011; accepted June 02, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. This work was supported National Natural Science Foundation of China under Grant 60771038 and Fundamental Research Funds for the Central Universities. The authors are with the School of Science, Xidian University, Xi’an Shaanxi 710071, China (e-mail: [email protected]; lizhengjuncosabc@yahoo. com.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165489

anisotropic sphere [10]–[14] and other shaped anisotropic particles [15], [16], numerous investigations have focused on the interaction between a plane wave and an anisotropic coated sphere. Using the boundary integral method, Baker studied the electromagnetic scattering by arbitrarily-shaped, two-dimensional, perfectly conducting objects coated with homogeneous anisotropic materials [17]. Subsequently, the plane wave expansion, along with the Fourier transform and the vector wave functions (VWFs), was widely employed in the analysis of uniaxial or plasma coated spheres [18], [19]. The dyadic Green’s functions based on modified spherical VWFs were constructed to investigate multilayered radial anisotropic spheres [20]. However, the excitation source in all these studies is limited to a plane wave. With the advent of lasers and their growing use in the fields of particle sizing, biomedicine, laser fusion, Raman scattering diagnostics, optical levitation, aerosol cloud penetration and near filed scattering and calibration, the scattering problem from shaped beams has drawn considerable attention. Examples of beam applications are the use of radiation pressure to manipulate biological cells and the analysis of the phase Doppler technique. Theories regarding Gaussian beam scattering have been well established on the basis of the decomposition of the incident beam into an infinite series of elementary constituents, with amplitudes and phases given by a set of beam-shape coefficients [1]–[10]. Using the first-order approximation of a Gaussian beam developed by Davis [21] as basis, Barton and Alexander derived the high-order approximate expression for a TEM Gaussian beam, and calculated the internal and scattered fields of a homogeneous isotropic sphere [22]. The generalized Lorenz-Mie theory (GLMT) developed by Gouesbet et al., concerns the expansion of the incident shaped beam as a series of spherical VWFs, which effectively describes the electromagnetic scattering of a beam by a spherical particle; the beam shape coefficients are obtained by applying localized approximation [23], [24]. In the approach presented by Doicu et al., the translational addition theorem for spherical VWFs was used to calculate the beam shape coefficients of an off-axis beam [25]. Khaled et al. used the angular spectrum of plane waves to model a shaped beam and compute the fields inside and outside an isotropic sphere [26]. Although past efforts were primarily spent on the interaction of a Gaussian beam with a homogenous isotropic medium, we have recently made progress in the interactions of an on-axis Gaussian beam (the propagation direction of the Gaussian beam are coincident with the z-axis of particle coordinate system) with a multilayered isotropic sphere [27] and a homogenous uniaxial anisotropic sphere [28].

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distinct regions are thus defined, namely, region 0 for the free space with and , region 1 for the anisotropic medium, and region 2 for the isotropic dielectric or conducting sphere with and . This composite structure is illuminated by an -polarized Gaussian beam that propagates in the direction, and the . center of the beam waist is located at The spatial distribution of the incident electric field (designated by the superscript inc) in the plane is given by (1) Fig. 1. Anisotropic coated isotropic sphere illuminated by an off-axis incident Gaussian Beam.

However, the EM scattering of a Gaussian beam by an anisotropic coated sphere has not yet been studied. Beam propagation and scattering in inhomogeneous anisotropic media are complex, making analysis difficult. Moreover, the expansion terms m are not only equivalent zero for off-axis Gaussian beam (the propagation direction of the Gaussian beam are parallel with the z-axis of particle coordinate system but the center of the particle may not in the propagation direction) scattering, which also enhances the difficulty of the solution and numerical calculations of the anisotropic medium scattered from an off-axis Gaussian beam. Previous theories are no longer applicable to current problems, but provide a referable framework and an effective calibration. This paper aims to develop a precise solution to the general case of off-axis Gaussian beam scattering by an anisotropic coated isotropic dielectric or conducting sphere, study the role of the anisotropy and beam waist center positioning in far-field scattering diagrams, and understand the mechanism of wave-medium interaction. The accuracy of the theory is verified by comparing the numerical results reduced to the special cases of a plane wave incidence and the case of a homogeneous anisotropic sphere with results from a CST simulation and references. CST is a three-dimensional electromagnetic simulation software based on the numerical method of finite integration. This high-performance commercial software was developed by the German CST company in 1992; our laboratory is the Sino-Germany Joint CST Training Centre in Northwestern China. The applications of this theoretical development in the fields of biomedicine, target shielding, and anti-radar coating are numerically discussed. In the subsequent analysis, a time dependence of the form is assumed for all the EM fields, but disregarded throughout the treatment. II. THEORETICAL FORMULATIONS A. Off-Axis Gaussian Beam Expansions Consider an anisotropic coated isotropic dielectric or conducting sphere center located in a Cartesian coordinate system . As Fig. 1 shows, its outer and inner radii are and , respectively. The inner isotropic dielectric or conducting sphere is coated with an anisotropic material characterized by permittivity tensor and permeability tensor with thickness in the primary coordinate system . Three

where is the beam waist radius, and is the amplitude of the electric field at the center of the beam waist, and taken as unity for simplicity. In terms of spherical VWFs, the electromagnetic field of the incident Gaussian beam can be written as [24], [25]

(2)

(3) where (4) in which the expansion coefficients and as the beam shape coefficients. Spherical VWFs are defined as [12]

are known and

(5)

(6) where represents the appropriate kind of spherical Bessel functions , and , for and , respectively. Applying the local approximation of the GLMT [24], [25], the beam shape coefficients can be determined as follows:

(7)

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where

(8)

(12)

The E-field vector wave equation in the source-free anisotropic shell in region 1 can be written as follows:

where the expressions of eigenvalues and cofor the uniaxial efficients anisotropic medium and plasma can be found in [13] and [14], respectively. The EM fields are finite at the origin; thus, using the spherical VWFs, the internal fields of the isotropic sphere in can be expanded as (13)-(14), shown at the region 2 bottom of the page, where is the wave number denotes a conducting sphere. Furtherin region 2, and more, the scattered fields (designated by the superscript ) can be expanded thus [13]:

.

B. Internal and Scattered Fields

(9) where permittivity and permeability tensors and terized in primary system by

are charac-

(10) (15) It is very difficult to solve the characteristic equations to obtain the angular spectral representations of the wave fields if none of the 18 parameters in these two tensors are zero. So uniaxial anisotropic medium and plasma relatively simple and the most common anisotropic medium are considered here. For uniaxial anisotropic medium, the components in (10) have following relations: and . For plasma, the components in (10) have following relations: and . Then using the Fourier transform and spherical VWFs, the fields on coated uniaxial anisotropic medium or , designated by the superscript plasma in region 1 ( 1) can be expanded as [18], [19]

In (11)–(15), the unknown expansion coefficients and can later be determined by working out the matrix equations derived from the boundary conditions. C. Scattering Coefficients On the spherical boundary at and , the tangential components (designated by the subscript t) of the EM fields continue as (16) (17) Substituting (2), (3), (11), and (12) into (16) yields the following relationships at

(11)

(18)

(13)

(14)

WU et al.: OFF-AXIS GAUSSIAN BEAM SCATTERING BY AN ANISOTROPIC COATED SPHERE

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(25)

(19)

(20)

Combining the above mentioned equations, we obtain (26)–(29), shown at the bottom of the following page. From is very hard to ana(26)–(29), expansion coefficient lytically derive, but can be numerically calculated through back into (18) and (20), programs. Then, substituting we obtain the following expressions for scattering coefficients and

(30) (21)

Similarly, the substitution of (11), (12), (13), and (14) into the leads to boundary condition at

(31) as the amplitude of the incident electric Taking field, the radar cross section (RCS) for the far-region scattered field can be calculated as (22) (32)

III. NUMERICAL RESULTS AND DISCUSSION

(23)

(24)

The equations derived in the previous sections for the scattering coefficients are analytically solved. In this section, some numerical solutions to off-axis Gaussian beam scattering by an anisotropic coated sphere are provided. The E-plane corre-plane and the H-plane corresponds to the sponds to the -plane. To verify the accuracy of our theory, we initially make three comparisons. As Figs. 2 and 3 show, the results obtained from our codes reduced to a plane wave incidence for a plasma coated spherical shell and a uniaxial anisotropic coated isotropic sphere are coincident with the results generated in [18] and the CST simulation, respectively. Moreover, the numerical

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Fig. 2. Results reduced to the case of a plasma anisotropic coated dielectric sphere in a plane wave compared with those in [18]. (a = 0:6; a = 0:3; " = "   " = " ;  I;

0

= 2:5" ; " = 1:5" ; " = " = = " ; (x ;y ;z ) = (0; 0; 0); w = 20).

 i" ; 

=

results reduced to a homogenous uniaxial anisotropic sphere agree well with those in [28], as shown in Fig. 4.

Fig. 3. Results reduced to the case of a uniaxial anisotropic coated dielectric sphere in a plane wave compared with those by the CST simulation. (a =

; a

2:4"

= 0:6; " = 1:7"

;

=

;w

"

= 5:3495" = 50).

;"

= 4:9284"

 ;

=

  "  I;

=

The effects of the beam width on the RCS in the E- and H-planes are shown by Fig. 5(a) and (b), respectively. As the

(26)

(27)

(28)

(29)

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Fig. 4. Results reduced to the case of a homogeneous uniaxial anisotropic sphere in a Gaussian beam compared with those in [28]. (a = ; a = = " = 4" 0:00001; " 3:0; (x ;y ;z ) = (0; 0; 0)).

;"

= 2"

;

=

Fig. 5. Effects of the beam width on RCS. (a = ; a

"

1:7"

= 5:3495" ; " = 4:9284" ; (x ;y ;z ) = (0; 0; 0)).

 ;

=

 ;"



 ;w

=

= 0:6; " = 2:4" ; 

= =

=

beam width increases, the intensity of the RCS rises, but the angular distribution exhibits minimal change. The role of the beam waist center positioning in the RCS distributions is shown in Figs. 6 and 7(a), (b), (c), and (d), which are plots for a uniaxial anisotropic coated sphere and plasma coated sphere, respectively. The position offset of the beam waist center weakens the scattering intensity and deflects the scattering angle . Along both and corresponding to the largest RCS from axes, the bias of the beam waist center in the positive and negative directions imposes the same effects on the RCS. This can

Fig. 6. Effects of the beam waist center positioning along the x-axis on   " = 2:4" ;  = 1:7" ; w  =  I; RCS (a = ; a = 0:6;  = 5:3495" ; " = 4:9284" , (b) H-plane, " 2). (a) E-plane, " 5:3495" ; " = 4:9284" , (c) E-plane, " = " = 5:3495" ; " = " = i2" , (d) H-plane, " = " = 5:3495" ; " 7" ; " 7" ; " = " = i2" .

0 0

the = = = =

also be indicated from the derived expression of the beam facinfluences the E-plane tors. Fig. 6 also shows that the value of inRCS more visibly than it does the H-plane. As the value of creases, the scattering angle corresponding to the largest RCS in

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Fig. 8. Results reduced to the case of two concentric isotropic spheres for ; ;  (b) off-axis, a nucleated blood cell. (a) on-axis, x ; y ; z : ; : ; : . a : ; " " x ;y ;z : ; a : " ;   I; " : i : " ; " ;w  . "

(

Fig. 7. Effects of the beam waist center positioning along the y -axis on the : " ;" : " , (b) H-plane, " RCS. (a) E-plane, " : " ;" : " , (c) E-plane, " " : " ;" " i " , (d) H-plane, " " : " ;" " ;" " i " . " ;"

= 5 3495 5 3495 = 4 9284 =0 = 2 7 =0 = 2 7

= 4 9284 = = 5 3495 = = 5 3495

= = =

the E-plane increases as well, whereas the angular distribution in the H-plane exhibits minimal change and remains symmetrical. produces contrasting results (Fig. 7). Conversely, In the field of biology, we may encounter roughly spherical but inhomogeneous cells, essentially composed of nuclei surrounded by a liquid solution and a shell. Shown in Fig. 8 are cal-

( ) = (0 0 0) ) = (1 0 20 1 0) ( = 1 75 = 15 = = = 1 03  = = (1 05 + 0 005) = =2 )

culations made for the reduced case of two concentric isotropic spheres for a nucleated blood cell. Fig. 8(a) is for an on-axis Gaussian beam, which is compared with the GLMT results, whereas Fig. 8(b) is for an off-axis beam. The influence of the beam waist center positioning on the intensity and angular distributions of the scattering fields are further witnessed. The effects of the thickness of the anisotropic coating upon the E-plane RCS are characterized in Fig. 9 for a plasma anisotropic coated conducting sphere when the radius of the inner conducting sphere is constant. Compared with the ho, the presence of the mogeneous conducting sphere plasma anisotropic coating enhances the scattering intensity in the region near the forward and backward directions. Note that the RCS for the coated sphere is highly oscillating and produce blind areas at certain scattering angles, which is of special importance in radar target shielding. The angular distribution is dependent to a great extent on the thickness and the permittivity tensors of the coating; thus, in practical applications, the parameters of the plasma layer should be optimized to generate a better absorption effect. Fig. 10 presents the H-plane scattering of a uniaxial anisotropic coated spherical shell by an off-axis Gaussian beam. The thickness of the anisotropic coating is constant; hence, understanding and characterizing the effects of the inner radius of the shell upon the scattering performance is straightforward. The RCS in the region near the forward direction always increases with the inner radius of the shell. Such

WU et al.: OFF-AXIS GAUSSIAN BEAM SCATTERING BY AN ANISOTROPIC COATED SPHERE

Fig. 9. Effect of the thickness of the anisotropic coating upon the E-plane RCS for a plasma anisotropic coated conducting sphere when the radius = " = of the inner conducting sphere is constant. (a = 0:5; "

= 7" ; " = 5" ; " (0; 0; 0); w = ).

0"

=

 i" ; 

=

  "  I;

! 1; x ; y ; z (

) =

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width considerably influences the intensity of the RCS, but has little effect on the angular distribution. The position offset of the beam waist center weakens the scattering intensity and deflects the scattering angle corresponding to the largest RCS. The bias of the beam waist center along the axes in the positive and negative directions has the same effect on the RCS. The applications of this theoretical development in the fields of biomedicine, target shielding, and anti-radar coating are also numerically discussed. Only the transverse electric (TE) polarization is considered in this paper, while a similar formulation of the transverse magnetic (TM) polarization can be obtained via duality. The results presented in this paper are hopeful to provide an effective calibration for research on the scattering properties of multilayered anisotropic targets. In the paper, we only consider the uniaxial anisotropic medium and plasma coated sphere. Scattering for a general anisotropic-coated sphere in an obliquely incident off-axis Gaussian beam will be discussed in future research endeavors. REFERENCES

Fig. 10. Effects of the inner radius upon the scattering characteristics for a uniaxial anisotropic coated spherical shell when the thickness of the coating is con  " =  =  I; stant. (d = 0:1; " = " = 5:917" ; " = 7:197" ;  " ;  =  ; (x ; y ; z ) = (1:0; 1:0; 1:0); w = 2).

anisotropic coated shells have been widely used as phantom targets in the anti-radar technique. IV. CONCLUSION On the basis of the local approximation of the off-axis beam shape coefficients, the incident Gaussian beam is expanded using first spherical VWFs. By introducing the Fourier transform, the electromagnetic fields in the anisotropic layer are expressed as the addition of the first and the second spherical VWFs. Matching the continuous tangential boundary conditions at each interface among the internal isotropic sphere, the anisotropic shell, and the free space, the expansion coefficients of an off-axis Gaussian beam incident on an anisotropic coated sphere are proposed. The accuracy of the theory is verified by comparing the numerical results reduced to the special cases of a plane wave incidence and the case of a homogeneous anisotropic sphere with results from the CST simulation and references. The influence of the beam widths, the beam waist center positioning, and the size parameters of the spherical structure on the field distributions are analyzed. The beam

[1] G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys., vol. 25, pp. 377–455, 1908. [2] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. New York: Wiley, 1998, pp. 93–97. [3] L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys., vol. 22, pp. 1242–1246, 1951. [4] H. Scharfman, “Scattering from dielectric coated spheres in the region of the first resonance,” J. Appl. Phys., vol. 25, pp. 1352–1356, 1954. [5] W. G. Swarner and L. Peters, “Radar cross sections of dielectric or plasma coated conducting spheres and circular cylinders,” IEEE Trans. Antennas Propag., vol. 11, pp. 558–569, 1963. [6] J. Rheinstein, “Scattering of electromagnetic waves from dielectric coated conducting spheres,” IEEE Trans. Antennas Propag., vol. 12, pp. 334–340, 1964. [7] J. Rheinstein, “Scattering of electromagnetic waves from conducting spheres with thin lossy coatings,” IEEE Trans. Antennas Propag., vol. 13, pp. 983–983, 1965. [8] J. H. Richmond, “Scattering by a ferrite-coated conducting sphere,” IEEE Trans. Antennas Propag., vol. 35, pp. 73–79, 1987. [9] Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci., vol. 26, pp. 1393–1401, 1991. [10] W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E, vol. 47, pp. 664–673, 1993. [11] X. B. Wu and K. Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: An analytical solution,” J. Appl. Phys., vol. 82, pp. 1996–2003, 1997. [12] D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E, vol. 56, pp. 1102–1112, 1997. [13] Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E, vol. 70, pp. 1–8, 2004. [14] Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci., vol. 38, p. 1104, 2003. [15] S. N. Papadakis, N. K. Uzunoglu, and C. N. Capsalis, “Scattering of a plane wave by a general anisotropic dielectric ellipsoid,” J. Opt. Soc. Am. A, vol. 7, pp. 991–997, June 1990. [16] Z. S. Wu, S. C. Mao, and L. Yang, “Two-dimensional scattering by a conduction elliptic cylinder coated with a homogeneous anisotropic shell,” IEEE Trans. Antennas Propag., vol. 57, pp. 1–8, Nov. 2009. [17] B. Beker, K. R. Umashankar, and A. Taflove, “Electromagnetic scattering by arbitrarily shaped two-dimensional perfectly conducting objects coated with homogeneous anisotropic materials,” Electromagnetics, vol. 10, pp. 387–406, 1990. [18] Y. L. Geng, C. W. Q. , and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propag., vol. 57, pp. 572–576, 2009.

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[19] Y. L. Geng, X. B. W. , and L. W. Li, “Characterization of electromagnetic scattering by a plasma anisotropic spherical shell,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 100–103, 2004. [20] C. W. Qiu, S. Zouhdi, and A. Razek, “Modified spherical wave functions with anisotropy ratio: Application to the analysis of scattering by multilayered anisotropic shells,” IEEE Trans. Antennas Propag., vol. 55, pp. 3515–3523, 2007. [21] L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A, vol. 19, pp. 1177–1179, 1979. [22] J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and nearsurface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys., vol. 64, pp. 1632–1639, 1988. [23] J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A, vol. 11, pp. 2503–2515, 1994. [24] G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A, vol. 11, pp. 2516–2525, 1994. [25] Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt., vol. 36, pp. 2971–2978, 1997. [26] E. E. M. Khaled, S. C. Hill, and P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag., vol. 41, pp. 259–303, 1993. [27] Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouestbet, and G. Grehan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt., vol. 36, pp. 5188–5198, 1997. [28] Z. S. Wu, Q. K. Yuan, Y. Peng, and Z. J. Li, “Internal and external electromagnetic fields for on-axis Gaussian beam scattering from a uniaxial anisotropic sphere,” J. Opt. Soc. Am. A, vol. 26, pp. 1779–1788, Aug. 2009.

Zhen-Sen Wu (M’97–SM’04) received the B.Sc. degree in applied physics from Xi’an Jiaotong University, Xi’an, China, in 1969 and the M.Sc. degree in space physics from Wuhan University, Wuhan, China, in 1981. He is currently a Professor at Xidian University, Xi’an, China. From 1995 to 2001, he was invited multiple times as a Visiting Professor to Rouen University, France, for implementing joint study of two projects supported by the Sino-France Program for Advanced Research. His research interests include electromagnetic and optical waves in random media, optical wave propagation and scattering, and ionospheric radio propagation.

Zheng-Jun Li received the B.Sc. degree in applied physics, from Xidian University, in 2007, where he is currently working towards the Ph.D. degree. His current work concerns the electromagnetic and optic scattering by single and multiple anisotropic spheres.

Huan Li, photograph and biography not available at the time of publication.

Qiong-Kun Yuan, photograph and biography not available at the time of publication.

Hai-Ying Li, photograph and biography not available at the time of publication.

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Analytic Propagation Model for Wireless Body-Area Networks Da Ma, Student Member, IEEE, and Wen Xun Zhang, Fellow, IEEE

Abstract—Wave propagation in wireless body area networks (WBAN) is analytically modeled as a polarized point source close to an elliptic lossy dielectric cylinder. Using the Fourier transform along the axes, the expansion in terms of Mathieu functions in cross section, and the impedance boundary condition (IBC) on surface, the field distribution outside the cylinder can be formulated. In particular, the path gain of propagation around the human body is described in detail for 915-MHz and 2.40-GHz bands toward industrial, scientific, and medical (ISM) applications. Index Terms—Fourier transform, impedance boundary condition, Mathieu functions, wave propagation, wireless body area networks.

I. INTRODUCTION

R

ECENTLY, wireless body area networks (WBAN) had drawn increasing attention due to their promising applications in medical sensor systems, home entertainment and military communication [1]. To achieve better performances with adequate reliability, the path gain (or its counterpart, the pass loss) in wave propagation needs to be modeled correctly and studied quantitatively. In the early researches as summarized in [1]–[3] and the successive works in [4]–[11], most reports were based on the data fitting from numerical calculations or computer simulations, and even from statistical measurements. The disadvantages are that they shied away from the mechanisms of wave propagation and lacked clear expatiations in physics. Besides, a few studies have focused on the analytic model of propagating around a circular cylinder [12]–[14]. However, a cylinder with elliptic (rather than circular) cross-section composed of lossy (rather than lossless) dielectrics is in more accordance with the trunk of the human body [15]. In this paper, the human body is treated as a lossy dielectric elliptic cylinder with infinite length; and a small transmitted antenna is simplified to a three-dimensional (3-D) axially polarized point source [12]–[14]. Thus, the propagation model can be treated as a classical electromagnetic boundary-value problem and then be solved using the mature method with the complicated process as described below.

The topic of electromagnetic scattering from an elliptic cylinder illuminated by a plane wave had been studied as a 2-D boundary-value problem [16]–[21]. On the other hand, a 3-D problem of a circular cylinder illuminated by point source had been decomposed into a 2-D problem in the spectral domain by employing the Fourier transform [12]. Correspondingly, a 3-D problem of the elliptic cylinder can also be translated into a 2-D problem. Then, the formal solution of a 2-D problem will be expanded in terms of eigenfunctions in the elliptic cylinder coordinates, i.e., Mathieu functions, using the separation of variables in sequence. Finally, the expansion coefficients will be determined by means of the boundary conditions on the surface of the cylinder. The propagated fields outside the body are focused here, and the surface impedance of human body is easy to be tested. Hence, the impedance boundary condition (IBC) [22]–[24] can be employed to avoid the complex composition inside the human body. Section II introduces the geometry of a simplified model for the WBAN channel; Section III solves the boundary-value problem for field distribution near the lossy dielectric elliptic cylinder; Section IV shows the path gain in the case of on-body source; and Section V provides some numerical results and discussions. is omitted throughout all the A time-dependence factor analyses in the frequency domain.

II. GEOMETRY OF THE PROPAGATION MODEL The geometry of the WBAN channel model is presented in Fig. 1, in which the human body is modeled as a lossy dielectric elliptic cylinder with infinite length along the -axis, and the antenna located near the body is modeled as an axially polarized point source. The top view of the model is shown in Fig. 2, and the Cartesian coordinates can be translated into the elliptic cylinder coordinates by the following equations: (1)

Manuscript received May 17, 2010; revised May 02, 2011; accepted June 20, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. This work was supported in part by the Graduate Research and Innovation Program of Jiangsu Province under Grant CX07B_177z and in part by the National High-tech. Project under the Grant 2007AA01Z264. The authors are with the State Key Lab. of Millimeter Waves, Southeast University, Nanjing 210096, China (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2165473

where is the semifocal length of an elliptical cross section and determined by the semimajor and semiminor . The interface between the body and the free space is described as , and the point source is located at .

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Fig. 1. The geometry of the WBAN propagation model.

Fig. 3. The 2-D model of the linear-source near the body.

the surface of cylinder. Subsections B and C describe the solutions of the scattered and the incident waves, respectively; Subsection D gives total 2-D solutions in the spectral domain; Subsection E describes the IBC in detail; finally, the 3-D solution in the spatial domain is obtained by inverse Fourier transform in Subsection F. B. Potential Vector of Scattered Waves in Spectral Domain The homogeneous equation of the spectral potential vector of the scattered wave in elliptic cylindrical coordinates can be written as

Fig. 2. The top view of the WBAN propagation model.

III. SOLUTIONS FOR THE BOUNDARY-VALUE PROBLEM A. Transformation From a 3-D Problem to a 2-D Problem For a -polarized point source with unit current density lo, the magnetic potential vector is the socated at lution that satisfies the inhomogeneous Helmholtz equation of (2) This 3-D problem can be transformed into a simplified 2-D problem with a linear source (Fig. 3) using the Fourier transform with respect to for both sides of (2). Thus, the 2-D Helmholtz equation in spectral domain appears as

(3) , and . where The solution of (3) consists of two parts: the solution of the corresponding homogeneous equation and a unique solution of the inhomogeneous term. In a physical mechanism, the former means the scattered field from the cylinder, whereas the latter describes the incident field from the line source in free space. By expanding both the unknown scattered field and the given incident field in terms of the Mathieu functions (i.e., eigenfunctions in elliptic cylindrical coordinates), the expansion coefficients of the scattered field can be determined by IBC on

(4) where . Its formal solution is expressed as (5) where and are the radial Mathieu functions and the angular Mathieu functions, respectively; their notations here are the same as in [25]. The potential vector of the scattered wave inside the cylinder is not focused in this topic; but that outside the cylinder can be expressed in terms of the 4th kind of radial Mathieu functions: evenand oddas

(6) where the parameter of Mathieu functions is with , and is the wave number in free space. The expansion coefficients and in (6) will be determined using IBC in the following Subsection E.

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C. Potential Vector of Incident Field in Spectral Domain

For

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, a field point far away from the body

The potential vector of the incident wave as a particular solution of (3) can be determined by the 2nd kind of the Hankel function of zero order as (7) is the distance between where and the field point , both located the source point outside the body. According to the addition theorem of Hankel function in terms of Mathieu functions [20], (7) can be rewritten as follows. , a field point close to the body surface For

(12) Then, the total radiated electric field in spectral domain has component only and can be obtained using the Bromwich method (13) The total magnetic field in the spectral domain has transversal components only and can be obtained from the definition of the magnetic potential vector as (14)

(8) For

(15)

, a field point far away from the body surface E. Impedance Boundary Condition

The surface impedance of the human body approximately equals the characteristic impedance of the body tissue, and the latter is defined as (16) Then, the impedance boundary condition takes the form of (17) (9) D. Resultant Fields in Spectral Domain The resultant potential vector in the spectral domain is the sum of the potential vectors of the incident and the scattered waves. It can be expressed in the form of (10)

where and are the total fields in the spectral domain, and is the outward normal unit vector at the surface of cylinder , where each side of (17) may be simplified as:

(18) Substituting (13), (15), and (18) into (17), it becomes

where the general term can be written as follows. , a field point close to the body For

(19) in which the potential vector can be expressed as (10) . and then expanded in the infinite series of (11) for Let (20)

(11)

which is approximated due to cross section of the human body.

for a practical

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Then, the unknown expansion coefficients and can be determined from (19) directly using the orthogonality of the set of angular Mathieu functions [25]. Hence, the expansion coefficients of the even terms are given in

Substituting (21) into the even part of (24), we obtain

(21) (25)

Similarly, the expansion coefficients of the odd terms are

(22)

where the normalized surface impedance to the wave impedance in free space.

which can be further simplified using the definition of the radial Mathieu functions of the fourth kind and the Wronskian relationship [25]

is relative (26)

F. Inverse Transform for 3-D Solution in Spatial Domain After the expansion coefficients are calculated and then substituted into (13), the field distribution around the dielectric elliptic cylinder due to an infinite linear current source is formally solved in the spectral domain. Then, the field distribution of a point current source parallel in the spatial domain can to the axis of the cylinder be obtained by means of the inverse Fourier transform from the of an infinite linear current source as above solution (23) IV. THE SIMPLIFIED CASE OF ON-BODY SOURCE Usually, the on-body antennas for the WBAN applications are always very close to the body surface, so both the radial coordinates of the source point (transmitter) and of the field point (receiver) can be considered the same as the body surface, . Then, (11) can be rewritten as separate terms i.e., of even and odd Mathieu functions as (24) with

Similarly, to substitute (22) into the odd part of (24), then

(27) Hence, the total 3-D electric field in spatial domain can be expressed in detail by substituting (26) and (27) into (13) and then into (23), which becomes (28), shown at the bottom of the is the height difference between the page, where field point and the source point. V. NUMERICAL RESULTS AND DISCUSSIONS Some numerical results from the aforementioned analytical model involving complicated computations of the Mathieu functions are displayed later. At first, the convergence of the series in integrals of (28) has been investigated, e.g., the series in the even part (29)

(28)

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Fig. 4. Convergence test of infinite series involving the Mathieu function.

Fig. 4 shows the summation in (29) versus the cutoff order in the case of , and . The results show that the real part, the imaginary part, and the absolute magnitude converge to individual proper values very slowly with severe oscillation, due to the higher orders of Mathieu functions being calculated. Hence, the Shanks transform [26], [27] is applied to accelerate the convergence rate of (29), though the stricter convergence condition must be satisfied [28] for getting better precision. Fig. 5 shows the results of summation by Shanks trans. The convergence form under the convergence criterion of of both real and imaginary parts of (29) becomes fast obviously. As both the series under the integration of (28) are even functions with respect to the integral variable , the integration can be simplified as 2 times of the integration on , i.e., (30) shown at the bottom of the page, which on should be calculated numerically by means of contour integration on the complex plane for avoiding the singularity at as shown in Fig. 6. This contour integral is calculated by employing the Gauss-Legendre quadrature algorithm with high numerical precision and stability. The effective surface impedance of lossy human body in previous formulas can be calculated as

Fig. 5. The results of calculation using the Shanks transform. (a) The value of summation. (b) The relative error.

The typical geometric sizes of the elliptic cylinder model for human body are semimajor m, semiminor m, and semifocal length m. The surface of the elliptic m. In this case, the cylinder can be determined as is really satisfied. condition of Once the electric field is calculated through the complicated mathematical procedure including the Shanks transform and contour integration on the complex -plane, the path gain vs. path length from source points can easily be evaluated as

(31)

(32)

, and is the light where and the conductivity of velocity; the relative permittivity human body can be approximated to 2/3 times of that of human muscle [29], [30] as listed in Table I.

(33) (34)

(30)

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Fig. 6. The parabolic contour for integration on the complex plane. TABLE I DIELECTRIC PARAMETERS AND SURFACE IMPEDANCE FROM MUSCLE-EQUIVALENT PHANTOM 2/3

2

where the antenna directivity is for the axial polarized short dipole. Figs. 7 and 8 show the calculated curves of path gain for wave creeping around the human body versus the path length from the or , i.e., path on the body-air source point at interface ( , i.e., 0.85 m in perimeter) for 915 MHz and 2.40 GHz, respectively (there are 64 sampling data within 40 minutes of time consumption for each frequency). Each curve can be divided into three parts, i.e., fast attenuation (region I), slow attenuation (region II), and diffraction region (region III). In region I, the curves decrease the fast versus path length where the direct wave propagates along the line of sight (LOS) from the source to observer. In region II as a transition between LOS and the non-LOS link, the decreasing rates of the curves tend to become slower because of additional contributions of the creeping wave propagation. The curves in region III are oscillated due to the interference of clockwise and anticlockwise creeping waves, and such an oscillation is more serious for higher frequencies. The differences of path gain for different source locations around the body are also disclosed from the calculated results in Figs. 7 and 8 as follows: 1) The region I (LOS) extends slightly farther for the source than that at , but the path gain located at is almost independent of the location of source in region I, except when approaching region II. 2) The region II is slightly wider for the source located at than that at , and the path gain is slightly higher for the source located at than that at , as well as its larger curvature radius. 3) In region III (diffraction), the average path gain for the is slightly higher than that at source located at in the first half, but slightly lower in the second half. is better than at 4) In general, the source located at . 5) The higher frequency leads to the obviously decreased path gain due to the lossy body, thus reducing the ranges of regions I and II.

Fig. 7. Path gain versus path length around the body at 915 MHz.

Fig. 8. Path gain versus path length around the body at 2.40 GHz.

Fig. 9. Comparisons between the curves from analytic calculations and measured fitting at the 2.40-GHz band.

To verify the calculated results from the analytical model with complicated deductions and algorithms, the fitting curves of measured data at 2.40 GHz from [10] and [31] are cited for comparison in Fig. 9. Their trends are consistent with each other. Although the absolute values of analytic path gain in regions I and II are lower than those measured, it may be explained as the assumption that the analytical model sets the antenna directly , which results in more attenuaon the lossy body tion. However, all the path gains are almost in the same level in the diffraction region III, where the creeping waves become the dominant propagation mechanism.

MA AND ZHANG: ANALYTIC PROPAGATION MODEL FOR WIRELESS BODY-AREA NETWORKS

VI. CONCLUSION An analytic model of wave propagation in WBAN is presented. The exact expression of the electric field distribution near the human body is deduced by solving a 3-D boundaryvalue problem in elliptic cylindrical coordinates, where the body area channel is simplified as a model of an axial-polarized point source near an elliptic lossy dielectric cylinder. This model is further applied to calculate the path gain versus path length for different locations of source point at 915 MHz and 2.40 GHz. The calculated curves of path gain around the lossy elliptic cylinder can be divided into three regions with different characteristics of fast attenuation, slow attenuation, and diffraction, which correspond to individual propagation mechanisms. Moreover, the path gain depends on frequency and source location. All the calculated path gain curves are in accordance with the aforementioned measured data. ACKNOWLEDGMENT The authors are grateful to Prof. S. J. Zhang for his valuable suggestions on the calculation and the integration of the Mathieu functions. The authors also would like to thank Prof. Y. Hao, A. Sani, and others of the antennas and electromagnetics group, Queen Mary College, University of London, for their helpful discussions. REFERENCES [1] P. S. Hall and Y. Hao, “Introduction to body-centric wireless communications,” in Antenna and Propagation for Body-Centric Wireless Communications, P. S. Hall and Y. Hao, Eds. Boston: Artech House, 2006, pp. 1–9. [2] A. Alomainy, Y. Hao, X. Hu, C. G. Parini, and P. S. Hall, “UWB onbody radio propagation and system modeling for wireless body-centric networks,” IEE Proc. Commun., vol. 153, no. 1, pp. 107–114, Feb. 2006. [3] P. S. Hall et al., “Antennas and propagation for on-body communication systems,” IEEE Antennas Propag. Mag., vol. 49, no. 3, pp. 41–58, Jun. 2007. [4] A. Sani, Y. Zhao, Y. Hao, A. Alomainy, and C. Parini, “An efficient FDTD algorithm based on the equivalence principle for analyzing on body antenna performance,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1006–1014, Apr. 2009. [5] M. Gallo, P. S. Hall, Y. I. Nechayev, and M. Bozzetti, “Use of animation software in simulation of on-body communications channels at 2.45 GHz,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 321–324, 2008. [6] A. Fort, C. Desset, P. D. Doncker, P. Wambacq, and L. V. Biesen, “An ultra-wideband body area propagation channel model—From statistics to implementation,” IEEE Trans Microw. Theory Tech., vol. 54, no. 4, pp. 1820–1826, Apr. 2006. [7] A. Alomainy, Y. Hao, A. Owadally, C. G. Parini, Y. I. Nechayev, C. C. Constantinou, and P. S. Hall, “Statistical analysis and performance evaluation for on-body radio propagation with microstrip patch antennas,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 245–248, Jan. 2007. [8] Y. P. Zhang and Q. Li, “Performance of UWB impulse radio with planar monopoles over on-human-body propagation channel for wireless body area networks,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2907–2914, Oct. 2007. [9] J. Ryckaert, P. D. Doncker, R. Meys, A. L. Hoye, and S. Donnay, “Channel model for wireless communication around human body,” Electron. Lett., vol. 40, no. 9, pp. 543–544, Apr. 2004.

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[10] A. Fort, C. Desset, P. Wambacq, and L. V. Biesen, “Indoor bodyarea channel model for narrowband communications,” IET Microw. Antennas Propag., vol. 1, no. 6, pp. 1197–1203, Dec. 2007. [11] Y. Hao, A. Alomainy, Y. Zhao, C. G. Parini, Y. I. Nechayev, P. S. Hall, and C. C. Constantinou, “Statistical and deterministic modelling of radio propagation channels in WBAN at 2.45 GHz,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 2006, pp. 2169–2172. [12] G. Roquetal, A. Fort, C. Craeye, and C. Oestges, “Analytical propagation models for body area networkss,” in Proc. IET Seminar on Antennas and Propagation for Body-Centric Wireless Communications, London, U.K., Apr. 2007, pp. 90–96. [13] A. Fort, L. Liu, F. Keshmiri, P. De Doncker, C. Oestges, and C. Craeye, “Analysis of wave propagation including shadow fading correlation for BAN applications,” in Proc. 2nd IET Seminar on Antennas and Propagation for Body-Centric Wireless Communications, London, U.K., Apr. 2009, pp. 1–28. [14] A. Fort, F. Keshmiri, G. R. Crusats, C. Craeye, and C. Oestges, “A body area propagation model derived from fundamental principles: Analytical analysis and comparison with measurements,” IEEE Trans. Antennas Propag., vol. 58, no. 2, pp. 503–513, Feb. 2010. [15] H. Massoudi, C. H. Durney, and C. C. Johnson, “Long-wavelength analysis of planewave irradiation of an ellipsoidal model of man,” IEEE Trans. Microwave Theory Tech., vol. 25, no. 1, pp. 41–46, 1977. [16] P. L. E. Uslenghi and N. R. Zitron, “The elliptic cylinder,” in Electromagnetic and Acoustic Scattering by Simple Shapes, J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Eds. Amsterdam: North-Holland, 1969, pp. 129–180. [17] C. Yeh, “Backscattering cross section of a dielectric elliptical cylinder,” J. Opt. Soc. Amer., vol. 55, pp. 309–314, 1965. [18] S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: Series solution in terms of Mathieu functions,” IEEE Trans. Antennas Propag., vol. 45, no. 6, pp. 926–935, Jun. 1997. [19] S. Caorsi and M. Pastorino, “Scattering by multilayer isorefractive elliptic cylinder,” IEEE Trans. Antennas Propag., vol. 52, no. 1, pp. 189–196, Jan. 2004. [20] P. L. E. Uslenghi, “Exact penetration, radiation, and scattering for a slotted semielliptical channel filled with isorefractive material,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1473–1480, Jun. 2004. [21] D. Erricolo and P. L. E. Uslenghi, “Exact radiation and scattering for an elliptic metal cylinder at the interface between isorefractive halfspace,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2214–2225, Sep. 2004. [22] G. Pelosi and P. Y. Ufimtsev, “The impedance-boundary condition,” IEEE Antennas Propag. Mag., vol. 38, no. 1, pp. 31–35, Feb. 1996. [23] N. G. Alexopoules and G. A. J. Tadler, “Electromagnetic scattering from an elliptic cylinder loaded with continuous and discontinuous surface impedance,” J. Appl. Phys., vol. 42, pp. 1128–1134, 1975. [24] T. B. A. Senior and J. L. Volakis, “Derivation and application of a class of generalized boundary conditions,” IEEE Trans. Antennas Propag., vol. 37, no. 12, pp. 1566–1572, Dec. 1989. [25] S. J. Zhang and J. M. Jin, Computation of Special Functions. New York: Wiley, 1996, pp. 475–535. [26] S. Singh, W. F. Richards, J. R. Zinecker, and D. R. Wilton, “Accelerating the convergence of series representing the free space periodic Green’s function,” IEEE Trans. Antennas Propag., vol. 38, no. 12, pp. 1958–1962, Dec. 1990. [27] D. Erricolo, “Acceleration of the convergence of series containing Mathieu functions using Shanks transformation,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 58–61, 2003. [28] Y. Liu, Z. F. Li, and Y. Song, “Capacitance extraction of periodic structures accelerated by nonlinear Shank’s transformation in high speed circuits,” J. Shanghai Jiaotong Univ., vol. 38, pp. 137–144, Oct. 2004, Sup. [29] IFAC-CNR (Florence, Italy), “Dielectric Properties of Body Tissues” [Online]. Available: http://niremf.ifac.cnr.it/tissprop [30] W. Xia, K. Saito, M. Takahashi, and K. Ito, “Implanted cavity slot antenna for 2.45 GHz applications,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 2008, pp. 1–4. [31] B. Latre, G. Vermeeren, I. Moerman, L. Martens, F. Louagie, S. Donnay, and P. Demeester, “Networking and propagation issues in body area networks,” presented at the Proc. SCVT, 2004.

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Da Ma (S’07) was born in Jiangsu, China, 1983. He received the B.E. degree in electronic science and technology from Nanjing University of Posts and Telecommunications, China, in 2005, and the M.E. and Ph.D. degrees in electromagnetics and microwave technology from Southeast University, China, in 2007 and 2010, respectively. Previously, he worked as a Research Assistant in the State Key Lab. of Millimeter Waves, Southeast University. He was engaged in the research projects on RFID antenna and tunable FSS, and on antenna and propagation for wireless body area networks.

Wen Xun Zhang (F’99) was born in Shanghai, China, in 1937. He graduated from and then joined the faculty of the Radio Engineering Department of Nanjing Institute of Technology, China, in 1958 (renamed as Southeast University in 1988). He has authored three books and one translated book, edited three conferences proceedings, and published over 400 papers. Prof. Zhang is a Fellow of IEEE and IET. He has been involved in the committees of many international conference series, serves in the editorial boards of five international journals and is a Vice Editor of a Chinese journal.

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Experimental Characterization of Microwave Radio Propagation in ICT Equipment for Wireless Harness Communications Masataka Ohira, Member, IEEE, Takayuki Umaba, Member, IEEE, Shoichi Kitazawa, Member, IEEE, Hiroshi Ban, Member, IEEE, and Masazumi Ueba

Abstract—The wireless harness is a new emerging short-range communication to replace wire harness implemented in devices with wireless communications between internal components. In such an in-device wireless harness, the communication distance is up to a couple of meters at most. Through the challenge to apply this wireless communication technology to information and communication technology (ICT) equipment, we found that the radio channel inside the ICT equipment deeply depends on its internal structure more than we expected. In order to understand the radio propagation characteristics inside such equipment, we propose a new modeling technique with using a frequencydependent path loss exponent expressing the near- and far-field propagation, which enables us to successfully extract attenuation factors for the frequency and the propagation distance from the measured data. The results shows the path loss characteristics can be divided into three regions; line-of-sight (LOS), non-line-of-sight (NLOS), and a transition range. The transition range, which appears between the LOS and the NLOS, is caused by a blocking due to the densely-packaged internal components. These findings and equations can be criteria to design a wireless harness communication link. Index Terms—Information and communication technology (ICT) equipment, mechatronics, multipath propagation, radiowave propagation, reflection and scattering.

I. INTRODUCTION N RECENT decades, the wireless technology is growing very rapidly with the commercial success of outdoor and indoor wireless communications, and it is still expanding its application fields to other new areas; for example, wireless communications in aircrafts [1]–[3], body area networks (BANs) [4], [5] with using ultra-wideband (UWB) technologies, and wireless sensor networks [6]. One of new promising areas is a wireless machine-to-machine (M2M) data communication. In conventional communications

I

Manuscript received November 14, 2010; revised April 11, 2011; accepted June 02, 2011. Date of publication October 03, 2011; date of current version December 02, 2011. This work is part of the “Research and development of wireless harness in ICT equipment” supported by the Ministry of Internal Affairs and Communications (MIC), Japan. M. Ohira was with the ATR Wave Engineering Laboratories, Keihanna Science City, Kyoto 619-0288, Japan. He is now with Saitama University, Saitama 338-8570, Japan (e-mail: [email protected]). T. Umaba was with the ATR Wave Engineering Laboratories, Keihanna Science City, Kyoto 619-0288, Japan. He is now with Mitsubishi Electric Corporation, Amagasaki-shi, Hyogo 661-8661, Japan. S. Kitazawa, H. Ban, and M. Ueba are with the ATR Wave Engineering Laboratories, Keihanna Science City, Kyoto 619-0288, Japan. Digital Object Identifier 10.1109/TAP.2011.2165494

between components implemented inside a machine, wire cables are used as transmission channels. The cable assembly is also known as cable harness or wire harness. The use of long cable inside a compact space has been a cause of cable jams and potentially economical loss of cable resources, manufacturing and maintenance processes. Therefore, the replacement of wire harness with wireless harness will lead a great merit from the economical viewpoint as well as the resource saving. To realize the wireless harness in microwave and millimeterwave (mm-wave) bands, some attempts have been already made for automotives [7]–[11] and spacecrafts [12] with using the UWB and Bluetooth technologies, and for wireless intra-connect between internal integrated circuit (IC) boards [13] with using a very fast data mm-wave communication. The radiowave propagation environments of such applications are significantly different from conventional outdoor and indoor ones [14]–[21]. For a wideband characterization of the radiowave propagation, the propagation loss has been measured in some devices and test models of automotives [10], [11] and spacecrafts [12]. In such an in-device wireless harness, the communication distance inside devices is up to a couple of meters at most. Here, it is worth noting that the near-field radiowave propagation [22] as well as the far-filed one needs to be taken into account for the path loss model in a confined volume. The path loss characteristics inside devices may not be approximated with previously reported wideband far-field models [23] so well, since the path loss exponent of the near-field region shows a different frequency dependency from that of the far-field region. Through the challenge to apply the wireless harness technology to information and communication technology (ICT) equipment, this paper proposes a new modeling technique with using a frequency-dependent path loss exponent, which allows one to express the near- and far-field propagation with one path loss equation in a wide frequency range. To verify the effectiveness of the proposed path-loss model, we measure and characterize the microwave radio propagation in mechatronics units of some different types of ICT equipment. The ICT equipment is composed of metal-enclosed, compact, and highly densed mechatronics units of less than 1 m . Interestingly, the radio channel has a large propagation loss owing to a radiowave blocking by the mechatronics components, even in a short range of 1 m. The four examples of the ICT equipment under the test are automated teller machine (ATM), ticket vendor, vending machine, and printer. The rest of the paper is organized as follows. Section II explains briefly the inside structures

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Fig. 1. Images of internal mechatronics structures in ICT equipment. (a) Wire harness and (b) wireless harness.

of the ICT equipment. Section III measures the propagation loss in microwave band. In Section IV, we derive path loss equations for each type of ICT equipment, and successfully extract attenuation factors from the measured data. Finally, Section V concludes our key findings. II. WIRELESS HARNESS OF ICT EQUIPMENT A. Wireless Harness Communication in ICT Equipment The ICT equipment has many sensors inside itself along the internal transfer mechatronics in order to detect papers, coins, and bills. The mechatronics units of less than 1 m are constructed by many mechanically moving components, mortars, electric components in addition to the sensors. As shown in Fig. 1(a), all the sensors are connected to a central control board with cables in the present equipment. The central control unit collects the information from the sensors with point-to-multipoint (P-MP) communication. The role of the sensors is not to detect counterfeit but to check the state of coins and bills along transportation trucks inside the equipment. In the equipment introducing the wireless harness as illustrated in Fig. 1(b), all the sensors are connected wirelessly. One of frequency-band candidates for this communication is a mm-wave band such as 60-GHz band [24]–[26] because a short wavelength is advantageous to narrow spaces and gaps. However, it has drawbacks of a high transmit power required due to a large propagation loss in addition to very high cost for the use of the ICT equipment and for the purpose of the energy saving. In the ICT equipment, severe signal attenuation, reflection and scattering may occur due to many obstacles such as mechatronics components, thus we decided to adopt microwave band for our use. In addition, internal noises such as a clock noise were observed at the frequency lower than 1 GHz when starting up the equipment. Hence, the microwave band higher than 1 GHz is chosen for the wireless harness. B. Internal Structure of ICT Equipment Fig. 2 gives schematic drawings and partial pictures of each equipment, for the purpose of showing measurement points and how confined its internal component is, since the radiowave propagation depends on the internal structure. 1) ATM: The ATM is a typical example of mechatronics system integrated with very long wires. In Fig. 2(a), the shaded box indicates the internal units used for measurements. The position of the control board, which corresponds to the base station in the wireless communication, is also shown in the inset of the

figure. The size of the internal units is 28 cm (width) 90 cm (length) 75 cm (height). About two hundred sensors are implemented in the internal units, which are shielded by the metal case. The internal structure is composed of the free space of 76%, the metal hardware of 13%, and other components such as plastics of 11% in volume. However, the actual structure of the mechatronics is much complicated. The gap between adjacent components is less than a couple of centimeters. The components of the mechatronics are therefore obstacles for the radiowave propagation. 2) Ticket Vendor: The size of the internal units in the ticket vendor, which is used at railroad stations, is 14 cm (width) 80 cm (length) 74 cm (height). Similar to the ATM, mechanically moving components and other hardware composed of plastics and metal plates are densely built up in the ticket vendor. The number of the sensors is about half of the ATM. The ticket vendor is shielded by a metal case as well. Although the width of the mechatronics unit in the ticket vendor is narrower than that of the ATM, there are some spaces between the internal components and the side metal plate. 3) Vending Machine: The vending machine used for selling beverages has a large refrigerator shielded by a metal case, of which the size is 100 cm (width) 53 cm (length) 190 cm (height). The space between adjacent components in the vending machine is much wider than that in the ATM and the ticket vendor. The front door and the refrigerator are separated by a metal internal door. Some sensors are implemented at the back of the front door for detecting coins and bills, and at the refrigerator for controlling the temperature. 4) Printer: The size of a commercial printer is 48 cm (width) 55 cm (length) 48 cm (height). The outsides of the printer are not completely shielded. Some metal plates are inserted at the both sides of the printer, under the space for toner cartridges, at the paper trays and so on. The inside of the printer has some spaces in paper trays. The sensors in the printer are used for detecting papers, though there are not so many sensors implemented. The common features in these four different types of ICT equipment are that the sensors are not visible directly from the control board due to the internal components, except for sensors placed near the control board, and also that the metal-enclosed mechatronics units are scatter-rich environments. III. RADIOWAVE PROPAGATION MEASUREMENTS A. Measurement Setup The measurement setup for the radiowave propagation is shown in Fig. 3. The -parameters of two ports are measured by a vector network analyzer (VNA) (Rohde & Schwarz ZVT8). The frequency range is set from 3 to 8 GHz, since the lower and upper limits are determined by the specifications of antenna probes and the VNA. The measurement conditions are as follows; the dynamic range of 118 dB according to a data sheet, the transmit power of 0 dBm, the IF bandwidth of 1 kHz, the frequency points of 5000, the frequency resolution of 1 MHz, and the resulting noise level of 90 dBm. Both transmit (Tx) and receive (Rx) antennas are commercial chip antennas designed for a UWB system because they

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Fig. 2. Internal structures and measurement points in ICT equipment. (a) ATM, (b) ticket vendor, (c) vending machine, and (d) printer. For simplicity, threedimensional positions of Rx antenna are mapped into two-dimensional plane. The positions and the scale are not exactly shown, and all the measurement points are not shown in the figures.

Fig. 3. Measurement setup for propagation loss in ICT equipment.

are placed in a confined space. They are also suitable to obtain measured data in a wide frequency range. The chip antenna is implemented on a print circuit board of 36 20 mm with an SMA connector. The effective average gain of the antennas in a multipath-rich environment is assumed to be 0 dBi for the same reason as in [1]. Although reflection characteristics of the antenna may be changed when the antenna is located near metal plates of the mechatronics, the effect can be removed from the measured data as described later. The Tx antenna is fixed near the control board, which plays the role of the base station in the wireless communication. The Rx antenna is placed not only at the sensor position but also near the control board. The reference planes at the ports 1 and 2 are set at the ends of the RF cables by the calibration. The probe antennas are connected at the ends of the RF cables, which are 5-m JMCA Junflon MWX311 microwave flexible coaxial cables. The cable losses are 6.0 dB at 3.0 GHz, 8.2 dB at 5.5 GHz, and 9.9 dB at 8.0 GHz. The system margins estimated by a link budget calculation [1] are 18 dB at 3.0 GHz, 14 dB at 5.5 GHz, and 10 dB at 8.0 GHz. Here it is

assumed that the path loss is 60 dB at a propagation distance of 1 m. The measurements were carried out at an experimental room. To avoid external interferences, nonmetal outside surfaces were fully covered with electromagnetic absorbers. The main power switch of the ICT equipment is turned off while measuring the propagation loss. The measurement points or the locations of the Rx antenna are indicated by small circles in Fig. 2. Straight lines from the Tx antenna to the Rx one are schematically drawn to show how the direct paths are blocked by the internal components. We define simply the length of the straight line as the propagation distance , although the line and many components cross each other. The number of the measurement points is 80 in the ATM, 64 in the ticket vendor, 62 in the vending machine, and 70 in the is 1016 mm printer. The maximum propagation distance in the ATM, 770 mm in the ticket vendor, 761 mm in the vending machine, and 629 mm in the printer. B. Calculation of Propagation Loss , and are measured by The -parameters of the above-mentioned method. On the other hand, the received signal power is calculated by the following equation: (1) denotes a transmit signal power in dBm. where and in decibels represent the Tx and Rx cable losses, respectively. As shown in Fig. 3, the reference planes are set at the

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ends of the RF cables with the calibration so that , and can be omitted in (1). and in decibels are the effective gains of the Tx and Rx antennas in a multipath enin decibels represents the propvironment, respectively. agation loss. The gain of the antennas is also expressed by the to show the impedance mismatching equation is defined by explicitly, where the mismatch loss with the reflection coefficient of at the is the effective gain without the misantenna input port, and match loss. Therefore, we get the following equation from (1): (2) where each term is represented by (3) (4) (5) Hence, we define the effective propagation loss bels in the ICT equipment from (2) as

in deci-

(6) Here, the gains and are involved in . This is from the because it is difficult to remove the effective gain measured data at each measurement point. Consequently, the effective propagation loss is obtained from the measured -parameters by (7) The reflection characteristics of the probe antennas are changed due to metal plates of mechatronics units, which operate as large ground planes. By calculating the effective propagation loss with (7), the frequency dependency of the impedance mismatching at the input port of the probe antennas can be de-em. The propagation loss is meabedded from the measured sured as functions of the frequency and the distance.

Fig. 4. Measured frequency characteristics of propagation loss. (a) ATM mm), (b) ticket vendor (770 mm), (c) vending machine (761 (d mm), and (d) printer (629 mm).

= 1016

C. Measured Frequency Characteristics of Propagation Loss We conducted the measurements of the propagation loss in four types of ICT equipment. The frequency characteristics measured at two points in each type of ICT equipment are shown in Fig. 4. The left figures, except for the figure (b), correspond to the case that Rx antenna is located on a line-of-sight (LOS) path from Tx antenna, while the right figures including the left figure of (b) correspond to the case that Rx antenna is located on a non-line-of-sight (NLOS) path. In general, the technical terms LOS and NLOS are used for far-field propagation. This means that in a conventional communication environment, the distance between Tx and Rx antennas is much longer than the wavelength of the frequency used for the wireless communication. In in-device wireless harness, there are some cases where the propagation distance in the ICT equipment is about one wavelength, especially in LOS. Note that the LOS and the NLOS used in this paper includes not only far-field propagation but also near-field

one, as a matter of convenience. To check the near-field effect, we measured the coupling coefficient between the Tx and Rx antennas when they were located about one wavelength apart. The dB in free space. Hence, it is found measured value is about that the antenna coupling effect is relatively low even if the antennas are in the near-field region. The coupling effect is included in the frequency characteristics of the propagation loss shown in Fig. 4, since it is hard to remove the effect from (7) at each measurement point. As can be seen from Fig. 4, the measured propagation losses in the LOS are lower than those in the NLOS, as expected. In addition, many deep nulls are observed in the measured frequency characteristics. The number of the nulls observed in the LOS is smaller than that in the NLOS. Such measured characteristics show that the mechatronics units inside the ICT equipment are typical multipath-rich environments.

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IV. PATH LOSS MODEL FOR ICT EQUIPMENT A. Derivation of Path Loss Equation To analyze the measured frequency characteristics of the propagation loss in the equipment, we derive here a new path loss equation, which considers near-field propagation approximately as well as far-field one. A simple path loss equation obtained from the analogy of Friis’s transmission formula is given by (8) Similar expressions for a wideband model can be found in [23]. The exponents and are attenuation factors ( in free space) to be determined according to propagation environments. The above equation is valid for the far-field propagation, since a path loss exponent in the near field [22] is different from that in the far field. This suggests that and are not independent of the frequency and the propagation distance. In our proposed modeling, we therefore introduce a new term as follows, when the near-field and far-field propagation in a wide frequency range is simply modeled by one equation in such a manner to express the path loss in (8): (9) where and represent the propagation distance in millimeters and the frequency in gigahertz, respectively. More specifically, is expressed by (10) The coefficient shows a degree of dependency of the attenuation factors on both the frequency and the propagation distance. , (9) reduces to (8). From (9) and (10), we In the case of have the following path loss expression in decibels:

(11) where is an attenuation constant, and is a frequencydependent attenuation factor for the propagation distance. On the other hand, the frequency characteristic of the path loss can be readily rewritten from (11) as

(12) is an attenuation factor for the frequency. The where , and are determined by a least-square fitting coefficients approach using a mean path loss of the measured data. As shown later, the radiowave propagation in ICT equipment can be well modeled by the above approximate equations. B. Path Loss Versus Distance at 3 GHz versus As an example, the propagation loss the distance in each type of ICT equipment is shown in

Fig. 5. Path loss characteristics obtained in frequency domain and their approximate lines obtained by least square method in spatial domain at 3 GHz. (a) ATM, (b) ticket vendor, (c) vending machine, and (d) printer.

Fig. 5. The procedure to determine the coefficients , and from the measured frequency data is as follows. 1) In the first step, the mean path loss of the measured frequency data at each measurement point is calculated by a moving average (e.g., [27]) in frequency domain, which is equivalent to a low-pass filter in time domain. The frequency range per one averaging is 400 MHz, which corresponds to the suppression of multiple waves propagating over 75-cm distance, thereby reducing a small-scale fading effect such as deep nulls. 2) In the next step, using the mean path loss from 3 to 8 GHz allows one to determine and in (12) (not , and ) at each measurement point with the least square method. Then, the propagation losses denoted by small circles in Fig. 5 are obtained by substituting [GHz] into (12).

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3) In the final step, the solid lines shown in Fig. 5 are obtained by determining and in (11) at 3 GHz so that they can be fitted to the plotted data (small circles) as a function . The exponent and the attenuation constant of are obtained at each frequency, and therefore they are expressed as a function of the frequency. The coefficients , and in (11) and (12) are determined from the frequency data of and . In Fig. 5, we divide the propagation-loss versus distance characteristics into three regions: LOS, NLOS, and a transition range. In each type of equipment, the transition range separating the LOS and the NLOS depends on the internal structure. 1) ATM: In the ATM, the LOS region is within about 20 cm from Tx antenna. As can be seen from Fig. 5(a), the propagation loss in the LOS region is about 15–35 dB. In the NLOS region, where the propagation distance is longer than 20 cm, the path loss is greater than 50 dB. In the transition range between the LOS and the NLOS, the path loss is drastically increased with a small change of the distance. To compare this path loss characteristic with the structural feature, the internal structure of the ATM is shown in the inset of the figure. The drastic attenuation is caused at the boundary between the internal components and free space at around Tx antenna. In the LOS region near the Tx antenna, the radiowave propagates in a free space shielded by metal plates, while in the NLOS region, the radiowaves from Tx antenna are blocked by the internal components. This radiowave blocking results in the drastic attenuation between LOS and NLOS regions. 2) Ticket Vendor: In the ticket vendor, the path loss characteristic given in Fig. 5(b) has only NLOS region, since the Tx antenna are set at the back of the mechatronics units, and the receiver is invisible from the transmitter. Therefore, no drastic radiowave attenuation can be observed in this figure. 3) Vending Machine: In the vending machine, it can be found from Fig. 5(c) that the radiowave attenuation in the NLOS region is larger than that in the LOS region. The gap between two approximate lines of the LOS and the NLOS are observed in the transition range. The gap which appears in the path loss characteristic is called hereinafter blocking gap. The blocking gap is caused by the internal door composed of metal plates in the vending machine, as shown in Fig. 2(c). 4) Printer: In the printer, the blocking gap is also clearly observed in Fig. 5(d). This is because the printer has toner cartridges in the middle of the structure shown in Fig. 2(d). Therefore, the printer has two different path losses in spite of the same propagation distance at the overlapped region of the LOS and the NLOS, as shown in Fig. 5(d). It can be concluded from the above discussions that the path loss curves and their discontinuities match the structural features of the ICT equipment, and also that they can be characterized by the LOS, the NLOS, and the transition range. C. Path Loss Equations and Attenuation Factors Finally, we derive the wideband path loss equations for four types of ICT equipment, and extract the attenuation factors for the propagation distance and the frequency. The obtained equations are applicable in the measured frequency range of 3 GHz GHz.

1) ATM: The propagation loss in decibels in the ATM can be approximated by the following equations:

(13)

(14)

(15) The first, the second, and the third equations are valid in the mm, mm propagation distance 100 mm mm, and mm mm, respectively. The path losses calculated by (13)–(15) are given in Fig. 6(a). In each region, six lines are plotted with changing the frequency. The frequency range is from 3 to 8 GHz with 1-GHz steps. For comparison, the path losses in free space are also plotted in the same figure. It is found that the path loss of the NLOS is much severer than as expected, in spite of a low frequency of 3 GHz. In the LOS region, the attenuation factor for the distance is 2.5–5.0, and for the frequency is 1.7–3.4. The reason of the large variation is because the received power of the Rx antenna in the near field greatly depends on the frequency and the antenna position. Fig. 6(a) indicates that such a large variation is well expressed with our introduced term in (9). In the transition range, the attenuation factor is 9.0–9.6, while in the NLOS region is 0.4–0.6. The attenuation factors for the frequency do not exceed 2 in the transition range and the NLOS. The radiowave propagation in the shielded narrow space of the ATM may be regarded as an oversized waveguide filled by a lossy material. 2) Ticket Vendor: The propagation loss in decibels in the ticket vendor can be approximated by the following equation:

(16) This equation is valid in the propagation distance mm mm. The path loss calculated by (16) is given in Fig. 6(b). The inside structure in the ticket vendor may be regarded as a parallel-plate waveguide, since there is a narrow free space between the internal metal components and the shield for the frequency is 0.8–1.8, case. The attenuation factor while for the distance is 1.7–2.1. Similar to the NLOS region in the ATM, the attenuation factor for the frequency in the ticket vendor is low, compared with the path loss in free space. 3) Vending Machine: The propagation loss in decibels in the vending machine can be approximated by the following equations:

(17)

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TABLE I ATTENUATION FACTOR  OF PATH LOSS FOR THE PROPGATION DISTANCE IN THE FREQUENCY RANGE OF 3 TO 8 GHZ

TABLE II ATTENUATION FACTOR A OF PATH LOSS FOR THE FREQUENCY

as a free space shielded by a relatively large metal case. It can be seen from Fig. 6(c) that the blocking gap becomes narrower as the wavelength is shorter. The frequency-dependent blocking gap is well modeled by the path-loss exponent introduced in this paper. 4) Printer: The propagation loss in decibels in the printer can be approximated by the following equations:

(19) (20)

Fig. 6. Propagation loss as functions of the frequency and the distance. Frequency step is 1 GHz from 3 to 8 GHz. (a) ATM, (b) ticket vendor, (c) vending machine, and (d) printer.

(18) The first and the second equations are valid in the propagation mm mm and mm distance mm, respectively. The path losses calculated by (17) and (18) are given in Fig. 6(c). In the LOS region, the attenuation factor for the distance is 1.5–4.0, and for the frequency is 0.6–3.5. The reason of the large variations is the same as the ATM. In the NLOS region, the attenuation factor for the distance is 1.9–2.7. On the other hand, the attenuation factor for the frequency is 0.1–1.0, which is less than that in free space. The propagation environment in the vending machine may be regarded

The first and the second equations are valid in the propagation mm mm and mm distance mm, respectively. The path losses calculated by (19) and (20) are given in Fig. 6(d). In the LOS region, the attenuation factor for the distance is 1.0–1.6, and for the frequency is 2.5–3.5. In the NLOS region, for the distance is 2.3–3.3, and for the frequency is 1.1–2.5. The attenuation factor in the NLOS region is the largest value in four types of ICT equipment. The reason may be because the printer is not completely shielded by the metal plates, and is closest to a free-space propagation environment in the four types of ICT equipment. and for the distance and the The attenuation factors frequency in each type of ICT equipment are summarized in Tables I and II, respectively. Using the equations derived here makes it possible to estimate the propagation loss at a specified sensor position in the ICT equipment. V. CONCLUSION For a path loss modeling of the in-device wireless harness communication, this paper has introduced a new path loss equation describing the near- and far-field propagation in a wide frequency range, and has measured and characterized the microwave radio propagation in the ICT equipment. The measured results show that the mechatronics unit in the ICT equipment is a typical multipath-rich environment. Furthermore, the key findings of this paper based on the detailed analysis of the measured data are as follows.

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1) In microwave band of 3–8 GHz measured in this paper, the radio channels in the ICT equipment has a large propagation loss of about 20–60 dB even in a short propagation distance of less than 1 m inside the metal-enclosed confined volume. 2) The attenuation factor for the frequency in the NLOS is lower than 2, except for the printer. The reason may be because the propagation environment inside the metal-enclosed highly densed space of the ICT equipment can be regarded as an oversized waveguide. The attenuation factors extracted from the measured data are summarized in Tables I and II. 3) The path loss characteristic has three regions: the LOS, the transition range and the NLOS. The transition range is caused by the radiowave blocking or the drastic signal attenuation, for example about 20 dB in the ATM, due to the internal components. The path loss models and the modeling technique are very useful for the system design to realize wireless harness in ICT equipment. We think that they are universally applicable for radio channels in a narrow and metal-rich space. ACKNOWLEDGMENT The authors would like to thank S. Ano, ATR Wave Engineering Laboratories, for his help with the measurements, Dr. K. Yano, ATR Wave Engineering Laboratories, for his technical advice, and Dr. S. Shimizu and the project group, OKI Electric Industry Co. Ltd., for their useful discussions and technical support. REFERENCES [1] S. Chiu, J. Chuang, and D. G. Michelson, “Characterization of UWB channel impulse responses within the passenger cabin of a Boeing 737-200 aircraft,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 935–945, Mar. 2010. [2] S. Chiu and D. G. Michelson, “Effect of human presence on UWB radiowave propagation within the passenger cabin of a midsize airliner,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 917–926, Mar. 2010. [3] N. R. Diaz and M. Holzbock, “Aircraft cabin propagation for multimedia communications,” in Proc. EMPS, Sep. 2002, pp. 281–288. [4] S. L. Cotton and W. G. Scanlon, “Characterization and modeling of the indoor radio channel at 868 MHz for a mobile bodyworn wireless personal area network,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 51–55, 2007. [5] K. I. Ziri-Castro, W. G. Scanlon, and N. E. Evans, “Indoor radio channel characterization and modeling for a 5.2-GHz bodyworn receiver,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 219–222, 2004. [6] C. M. Kruesi, R. J. Vyas, and M. M. Tentzeris, “Design and development of a novel 3-D cubic antenna for wireless sensor networks (WSNs) and RFID applications,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3293–3299, Oct. 2009. [7] R. Frank, “Wireless technologies simplify wiring harness,” Auto Electron. pp. 17–22, Jul. 2007 [Online]. Available: http://autoelectronics. com/mag/707AESpecialReport.pdf [8] G. Leen and D. Heffernan, “Vehicles without wires,” Comput. Control Eng. J., vol. 12, no. 5, pp. 205–211, Oct. 2001. [9] K. Akingbehin and N. Patel, “Development of a hybrid automotive wireless harness,” in Proc. 28th Annu. Int. Comput. Software Applicat. Conf., Sep. 2004, vol. 2, pp. 56–57. [10] A. R. Moghimi, H.-M. Tsai, C. U. Saraydar, and O. K. Tonguz, “Characterizing intra-car wireless channels,” IEEE Trans. Veh. Technol., vol. 58, no. 9, pp. 5299–5305, Nov. 2009. [11] T. Kobayashi, “Measurements and characterization of ultrawideband propagation channels in a passenger-car compartment,” IEICE Trans. Fundamentals, vol. E89-A, no. 11, pp. 3089–3094, Nov. 2006.

[12] A. Matsubara, T. Ichikawa, A. Tomiki, T. Toda, and T. Kobayashi, “Measurements and characterization of ultra wideband propagation within spacecrafts,” in Proc. Loughborough Antennas Propag. Conf., Nov. 2009, pp. 565–568. [13] K. Kawasaki, Y. Akiyama, K. Komori, M. Uno, H. Takeuchi, T. Itagaki, Y. Hino, Y. Kawasaki, K. Ito, and A. Hajimiri, “A millimeterwave intraconnect solution,” in Dig. 2010 IEEE Int. Solid-State Circuits Conf., San Francisco, CA, Feb. 2010, pp. 414–415. [14] T. K. Sarkar, Z. Ji, K. Kim, A. Medouri, and M. Salazar-Palma, “A survey of various propagation models for mobile communication,” IEEE Antennas Propag. Mag., vol. 45, no. 3, pp. 1740–1746, Jun. 2003. [15] K. R. Scltaubacli and N. J. Davis, “Microcellular radio-channel propagation prediction,” IEEE Antennas Propag. Mag., vol. 36, no. 4, pp. 1740–1746, Aug. 1994. [16] J. Walfisch and H. L. Bertoni, “A theoretical model of UHF propagation in urban environments,” IEEE Trans. Antennas Propag., vol. AP-36, no. 12, pp. 1788–1796, Dec. 1988. [17] F. Ikegami, S. Yoshida, T. Takeuchi, and M. Umehara, “Propagation factors controlling mean field strength on urban streets,” IEEE Trans. Antennas Propag., vol. AP-32, no. 8, pp. 822–829, Aug. 1984. [18] M. Hata, “Empirical formula for propagation loss in and mobile radio service,” IEEE Trans. Veh. Technol., vol. VT-29, no. 3, pp. 317–325, Jun. 1980. [19] H. H. Xia, H. L. Bertoni, L. R. Maciel, A. Lindsay-Stewart, and R. Rowe, “Radio propagation characteristics for line-of-sight microcellular and personal communications,” IEEE Trans. Antennas Propag., vol. 41, no. 10, pp. 1439–1447, Oct. 1993. [20] C. W. Kim, X. Sun, L. C. Chiam, B. Kannan, F. P. S. Chin, and H. K. Garg, “Characterization of ultra-wideband channels for outdoor office environment,” in Proc. IEEE Wireless Commun. Netw. Conf., Mar. 2005, vol. 2, pp. 950–955. [21] J. W. Wallace and M. A. Jensen, “Modeling the indoor MIMO wireless channel,” IEEE Trans. Antennas Propag., vol. 50, no. 5, pp. 591–599, May 2002. [22] H. G. Schantz, “A near field propagation law & a novel fundamental limit to antenna gain versus size,” in Dig. 2005 IEEE AP-S Int. Symp., Washington, DC, Jul. 2005, vol. 3A, pp. 237–240. [23] A. F. Molisch, D. Cassioli, C.-C. Chong, S. Emami, A. Fort, B. Kannan, J. K. J. Karedal, H. G. Schantz, K. Siwiak, and M. Z. Win, “A comprehensive standardized model for ultrawideband propagation channels,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3151–3166, Nov. 2006. [24] K. Sato, T. Manabe, T. Ihara, H. Saito, S. Ito, T. Tanaka, K. Sugai, N. Ohmi, Y. Murakami, M. Shibayama, Y. Konishi, and T. Kimura, “Measurements of reflection and transmission characteristics of interior structures of office building in the 60-GHz band,” IEEE Trans. Antennas Propag., vol. 45, no. 12, pp. 1783–1792, Dec. 1997. [25] H. Xu, V. Kukshya, and T. S. Rappaport, “Spatial and temporal characteristics of 60-GHz indoor channels,” IEEE J. Sel. Areas Commun., vol. 20, no. 3, pp. 620–630, Apr. 2002. [26] A. Miura, M. Ohira, S. Kitazawa, and M. Ueba, “60-GHz-band switched-beam eight-sector antenna with SP8T switch for 180 azimuth scan,” IEICE Trans. Commun., vol. E93-B, no. 3, pp. 551–559, Mar. 2010. [27] S. W. Smith, The Scientist and Engineer’s Guide to Digital Signal Processing. San Diego, CA: California Technical Publishing, 1999.

Masataka Ohira (S’03–M’06) received the B.E., M.E., and D.E. degrees from Doshisha University, Kyoto, Japan, in 2001, 2003, and 2006, respectively. From 2006 to 2010, he was with ATR Wave Engineering Laboratories, Kyoto, Japan, where he engaged in research and development of millimeter-wave circuits and antennas for 60-GHz band wireless communications and small smart antennas for wireless LAN. In 2010, he joined Saitama University, Saitama, Japan, where he is currently an Assistant Professor. His current research activities are concerned with analysis and design of microwave and millimeter-wave filters and antennas. Dr. Ohira is a member of the Institute of Electronics, Information, and Communication Engineers (IEICE), Japan, and the Institute of Electrical Engineers (IEE), Japan. He received the IEICE Young Engineer Award in 2005. He was a Chair of IEEE MTT-S Kansai Chapter Young Engineers’ Committee for 2008–2009.

OHIRA et al.: EXPERIMENTAL CHARACTERIZATION OF MICROWAVE RADIO PROPAGATION

Takayuki Umaba received the B.E. and M.E. degrees from Kobe University, Kobe, Japan, in 2001 and 2003, respectively. In 2003, he joined Mitsubishi Electric Corporation, and started his career in the field of design and development of communication systems. From 2008 to 2009, he was with ATR Wave Engineering Laboratories and engaged in research and development of ultra-high speed giga-bit rate wireless LAN systems. Since 2010, he has engaged in the development of the wireless communication system for public use with Mitsubishi Electric Corporation. Mr. Umaba is a member of the Institute of Electronics, Information, and Communication Engineers (IEICE), Japan.

Shoichi Kitazawa received the B.E. and M.E. degrees from Kinki University, Higashi-Osaka, Japan, in 1991 and 1993, respectively, and the D.E. degree from Osaka Prefecture University, Sakai, Japan, in 2007. In 1993, he joined Matsushita Nittoh Electric Co., Ltd., Kyoto, Japan, where he has been engaged in development on microwave filters. Since October 2005, he has been with ATR Wave Engineering Laboratories, Kyoto, Japan. His current research interest includes microwave and millimeter-wave devices. Dr. Kitazawa is a member of the Institute of Electronics, Information, and Communication Engineers (IEICE), Japan, and the Institute of Electrical Engineers (IEE), Japan.

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Hiroshi Ban (M’11) received the B.E., M.E., and D.E. degrees in polymer science and technology from the Tokyo Institute of Technology, Tokyo, Japan, in 1982, 1984, and 1991, respectively. In 1984, he joined Ibaraki Electrical Communications Laboratories, Nippon Telegraph and Telephone Corporation, and started his career in research on photosensitive materials and microlithographic technologies. Since 2001, he has expanded his interest to sensor network systems, environmental assessment methodologies, and environmental ICT technologies. He moved to ATR in 2009, and is currently a head of the Department of Environment Communications of ATR Wave Engineering Laboratories Dr. Ban received the Photopolymer Science and Technology Award from the Photopolymer Conference in 1998. . He is a member of the Institute of Electronics, Information, and Communication engineers (IEICE) Japan, the Japan Society of Applied Physics, and the Society of Polymer Science Japan.

Masazumi Ueba received the B.E. and M.S. degrees in aeronautical engineering and the Doctor of Engineering degree from Tokyo University, Tokyo, Japan, in 1982, 1984, and 1996, respectively. In 1984, he joined the Yokosuka Electrical Communication Laboratories, NTT Corporation. He has been engaged in research on a dynamics of antenna pointing control systems of satellites a shape control systems of large antenna reflectors and technologies for next-generation future mobile satellite communication system. Currently he is a director of ATR Wave Engineering Laboratories at Advanced Telecommunications Research Institute International, where he is engaged in research on advanced wireless communications technologies to achieve high-frequency efficiency and to enable new applications. Dr. Ueba is a member of the Institute of Electronics, Information, and Communication Engineers (IEICE), Japan, the Society of Aeronautical and Space Sciences, Japan, and AIAA.

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Sub-Wavelength Focusing at the Multi-Wavelength Range Using Superoscillations: An Experimental Demonstration Alex M. H. Wong and George V. Eleftheriades, Fellow, IEEE

Abstract—We experimentally demonstrate the formation of a superoscillatory sub-wavelength focus at a multi-wavelength working distance. We first discuss and distinguish superlensing, superdirectivity and superoscillation as different methods which, in their respective ways, achieve sub-diffraction resolution. After establishing superoscillation as a potential way towards sub-wavelength focusing at the multi-wavelength range, we proceed to design, synthesize and demonstrate a superoscillatory sub-wavelength focus in a waveguide environment. Our measurements confirm the formation of a focus at 75% the spatial width of the diffraction limited sinc pulse, 4.8 wavelengths away from the source distributions. This working distance is an order of magnitude extended from those of superlenses and related evanescent-wave-based devices, and should pave way to various applications in high-resolution imaging. Index Terms—Diffraction limit, image resolution, sub-wavelength focusing, superdirectivity, superoscillation.

I. INTRODUCTION

T

HE diffractive nature of electromagnetic waves has traditionally been viewed as a fundamental limit on the resolution of imaging systems, blurring image details at length scales smaller than half the imaging wavelength. Notwithstanding, there has been intensive interest towards developing imaging systems with resolution beyond the conventional limit of diffraction. An early example of imaging beyond the diffraction limit was Synge’s proposal [1] which eventually became the near-field scanning optical microscope. More recently, Veselago and Pendry’s proposal of a superlens [2], [3] triggered various clever ideas which used evanescent electromagnetic fields to image beyond the diffraction limit [4]–[11]. While these ideas extend the imaging device’s working distance to about a quarter of the imaging wavelength, this distance remains constrained to the evanescent near-field of the imaging device. On a related front, the superdirective antenna, proposed by Schelkunoff in 1943 [12], could in principle squeeze an antenna main beam beyond that emanating from a uniform aperture of Manuscript received March 22, 2011; revised May 31, 2011; accepted June 16, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. This work was supported in part by the Natural Science and Engineering Research Council of Canada (NSERC). The authors are with the Rogers S. Sr. Dept. 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.2011.2165518

the same size. This constitutes the achievement of directivity beyond the angular diffraction limit. Nonetheless, as antenna pattern formation occurs in the far-field, the superdirective antenna does not really provide imaging capability beyond the spatial diffraction limit. We shall elaborate on these claims later in this paper. The phenomenon of superoscillation holds promise to sub-diffraction imaging beyond the constraining extent of evanescent near-field. While the discovery of superoscillations [13], [14] and their linkage to spatial domain electromagnetic waves [15] predate Pendry’s superlens proposal, superoscillatory wave-based imaging has not seen much research interest, perhaps due to the perception that required sensitivity levels and achievable power efficiencies would render most potential application impractical. Nonetheless, theoretical procedures have been proposed which design superoscillatory waveforms [15]–[17]; experimental observations of pseudorandom superoscillatory waveforms have also been reported [18], [19]. However, it remains to be demonstrated whether custom-designed superoscillatory waveforms can be synthesized in a practical manner conducive to imaging beyond the diffraction limit. This present work aims to achieve two objectives: to elucidate the relationship between the aforementioned methods towards imaging beyond the diffraction limit, and to experimentally demonstrate sub-diffraction focusing by synthesizing a custom-designed superoscillatory waveform at a multi-wavelength image distance—10-fold increased from the image distances of evanescent-field-based imaging devices. In Section II, we begin by reviewing the physics behind the diffraction limit from a Fourier perspective. Upon this platform we then compare the phenomena of superlensing, superdirectivity and superoscillation. Subsequently, in Section III, we focus our attention on superoscillation as our method of choice to demonstrate sub-diffraction focusing at a multi-wavelength image distance. First we review our design formulation reported in previous works, then we describe the motivation towards an in-waveguide experiment and provide fabrication and operational details of our experimental apparatus. We then show results which represent a first experimental demonstration of custom-designed sub-diffraction focusing at a 4.8 wavelength imaging distance. Our ability to achieve sub-wavelength focusing in this extended imaging distance with a designer waveform distinguishes this work from [18], [19], in which randomness prohibits the prediction and design of the location of sub-wavelength foci, and the specification of the field profile

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immediately surrounding the foci. Finally, we offer a brief discussion suggesting future directions to this work and potential areas of application. II. AVENUES TO SUB-DIFFRACTION IMAGING A. Diffraction Limit Before we embark on a comparison of different methods of sub-diffraction imaging, it is helpful to review the physics behind the diffraction limit from a Fourier-transform perspective. Every electromagnetic waveform can be written as a composition of plane waves with varying transverse spatial frequencies (for clarity we consider 2D wave propagation with x and z as the transverse and longitudinal directions respectively)

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The uncertainty principle translates this maximum width into a minimum width to which one can focus a propagating electromagnetic wave; this forms a measure for spatial resolution since such a focus can be viewed as a point spread function , obtained in this confor more intricate imaging systems. text, is called the Abbe diffraction limit or simply the diffraction limit. For most typical metrics of waveform width (3 dB width, peak to null width, etc.), and for numerical apertures apto . proaching unity, the diffraction limit ranges from Angular Diffraction Limit: In a different but related manifestation, diffraction limits the divergence angle of an electromagnetic beam emanating from a source, for example an antenna. The finite extent of the aperture of the source radiation (fixed ) sets a lower bound for through the uncertainty principle. It is well known that in the far-field of an antenna, each propagating component maps into a plane wave whose wave vector makes an angle with the positive x-axis (antenna axis), where (5)

(1)

It is straightforward to show that this relation maps a spectral width into a far-field angular beamwidth through the relation

where we define (2)

(6) as the plane-wave spectrum for , which is its Fourier transform in the domain. The uncertainty principle forbids a function and its Fourier transform pair to simultaneously be localized [20]. In mathematical terms,

where denotes the beamwidth and denotes the beam direction. For small angular beamwidths, (6) simplifies to (7)

(3) where the constant depends on the way in which the widths and are quantified. While this principle was made famous by Heisenberg in a quantum mechanical context, it also causes the diffraction limit for electromagnetic fields, in the following two different contexts. Spatial Diffraction Limit: The evanescent components of an electromagnetic waveform (for which ) decay in the longitudinal direction and thus do not escape the evanescent near-field of an object or a device. As thus, only the propagating spectrum remains beyond the evanescent near-field of an imaging system. Hence the width of such a plane wave spectrum is limited to (4) In many imaging systems, geometrical limitations impose a finite aperture size, and hence prohibit one from accessing the entire spectrum of propagating waves. Imaging systems where is characterized by a propagation is limited to , where is the numerical aperture maximum angle which a plane wave exiting the imaging system can form with the principal axis (z-axis). In such cases (4) becomes (4b)

Combining (7) with (3) yields (8) which shows that a fixed aperture size leads to a lower bound for the emanating antenna beamwidth. This limitation will be referred to as the angular diffraction limit, or the diffraction limit for directivity. In this paper, we use the term “sub-diffraction” to represent length scales or resolutions beyond the spatial or angular diffraction limit. Moreover, we use the term “sub-wavelength” to refer to length scales or resolution beyond the spatial (but not angular) diffraction limit. Also, we focus our discussion on the focusing problem, having already established its equivalency to the corresponding imaging problem. B. Superlensing/Evanescent-Field-Based Focusing The deployment of evanescent waves has been predominant in sub-diffraction imaging devices proposed thus far. The underlying idea is simple: including evanescent waves in the imaging process allows one to access a transverse spatial frequency spectrum much larger than twice the wave number (i.e., ). As a result, the uncertainty relation (3) becomes much , thus allowing one to obtain a much less constraining on finer resolution. Near-field scanning schemes represent a simple

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Fig. 1. Evanescent-field-based sub-diffraction imaging. (a) The field pattern at a plane 0.05 wavelength away from a z-directed small dipole is tightly localized to below 0:01 . (b) The corresponding spatial spectrum, showing large evanescent-wave content. The black dashed circle separates the propagating spectrum (inside) from the evanescent spectrum (outside). (c) Phase progression for a line source that is imaged by a superlens. Solid white lines denote the location of the superlens, while dash white lines denote the locations of the source (to the left of the lens) and image (to the right of the lens). (d) The corresponding spectral evolution, showing large evanescent field components near the output facet of the superlens.

way of employing evanescent waves for measurement purposes. Fig. 1(a) and (b) show the electric field amplitude and transfrom a near-field scanning verse spatial field spectrum at tip, showing that evanescent field components are present and essential for sub-diffraction near-field scanning. Metamaterial superlensing also achieves sub-diffraction resolution imaging due to a resonant enhancement of evanescent fields within a negative-index slab, which compensates for their decay in free space, and thereby increases the working image distance to up to quarter-wavelength in practice (limited mainly due to material losses). Fig. 1(c) and (d) show the phase progression and spectral amplitude distribution as a line source is refocused by a superlens. The abundance of evanescent waves at the output facet of the superlens confirms the participation of evanescent waves in sub-diffraction resolution image formation. Since the proposal of the metamaterial superlens, other schemes have been conceived to employ evanescent waves in clever manners to focus or resolve waveforms beyond the diffraction limit. The metascreen [8] and the near-field focusing plate [9], [21] synthesize the electric field at the output facet of the device which will form a sub-wavelength focus at a predefined image distance. While these devices do not perform point-to-point sub-wavelength imaging in the same sense as the superlens, they can nonetheless form sub-wavelength foci which, if desired, can be used for imaging in a point-scan apparatus. In related developments, near-field gratings [4],

[5], near-field scatterers [22] and the hyperlens [6], [7] are placed very close to the object, to convert evanescent fields into propagating ones and thus resolve them in the far-field. The employment of evanescent waves within the image recording/formation process has indeed been proven effective towards achieving sub-wavelength resolution. However, this comes at a steep price of a constrained image distance. In near-field scanning setups, scan probes are often placed much less than a tenth of a wavelength away from the sample, where the field extending off the tip of the probe remains tightly localized. The metamaterial superlens typically operates with image distances ranging from a twentieth to a quarter of the wavelength. Beyond this distance, such a significant resonant enhancement is needed from the superlens that the achievable resolution quickly degrades due to dispersion [23] and various kinds of losses [23]–[25]. The same is true for all aforementioned imaging methods based on evanescent waves: they must necessarily be limited to the evanescent near-field either in their image distances (metascreens and focusing plates), or their object-to-device distances (near-field scatters, gratings and hyperlens). This fundamental limitation in working distance prevents superlensing from bringing sub-wavelength resolution to many applications. C. Superdirectivity Superdirectivity refers to the achievement of higher directivity than a similar-sized uniform aperture antenna, from which the emanating beam angle can be referred to as the angular diffraction limit. While it was traditionally believed that one can achieve the smallest beam angle with a uniform aperture antenna, Schelkunoff, in his seminal work in 1943, suggested otherwise. He proposed a theory whereby one can design current excitations on an antenna array such that the main beam can be arbitrarily squeezed without increasing overall antenna size. Fig. 2(a) shows current excitations from three arrays of isotropic antennas, all of overall length ; Fig. 2(c) compares their beam patterns with that from a uniform antenna array. The angular diffraction limit is clearly overcome with these superdirective antennas. Underlying Mechanism: The Fourier perspective clearly elucidates how the superdirective antenna overcomes the angular diffraction limit. Fig. 2(b) shows the transverse spatial spectra for array factors of the superdirective antennas displayed in Fig. 2(a). While they contain tight peaks and low sidelobes in , they also contain the region of propagating waves . huge sidebands in the region of evanescent waves Hence the superdirective array-factors have wide waveform widths in the transverse spatial frequency domain, which is in accordance to the uncertainty principle. However, as one maps the near-field spectrum into the antenna’s far-field, only the propagation spectrum is mapped; evanescent waves are invisible to the far-field. In this manner, the narrow peaks in the propagating spectra map into the highly directive beams whose angular widths surpass the diffraction limit. Superdirectivity and the Spatial Diffraction Limit: While it is well known that a superdirective antenna can focus the direction of electromagnetic radiation beyond the angular diffraction limit, one might speculate whether it can also be used for high-

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array used for evanescent-field-based sub-wavelength imaging, similar to [8]. We continue to examine the second observation, which can be cast as the following question: can a superdirective antenna be designed to form a sub-wavelength far-field focal spot? The antenna far-field is nominally described by the relation, (9) where represents the overall antenna size. At this distance the spatial beamwidth is given by (10)

Fig. 2. (a) Array factors for 3 antennas arrays (of 11, 21 and 31 elements respectively) of length 2. (b) The corresponding spatial spectra, showing very high amplitude evanescent components. Here jk =k j  1 represents the propagating spectrum. (c) The corresponding far-field angular distributions, compared to that of a uniform array.

resolution imaging beyond the spatial diffraction limit. Such speculation might stem from two observations: that the superdirective antenna has a large evanescent spectrum, and that the antenna beam angle can, at least conceptually, be arbitrarily squeezed in the antenna pattern. We now examine these two observations and show that they do not give superdirective antennas the ability to image beyond the spatial diffraction limit. We begin with the first observation. It is indeed true that superdirective antennas contain large evanescent spectra, and that sub-wavelength field variations might be observed within its near-field, in similarity with previously discussed evanescent-field-based imaging devices. However, distinction must be made in the evanescent fields’ “purpose of existence” in these two cases of sub-diffraction imaging: in near-field imaging schemes, evanescent-field-dominated near-field profiles perform the imaging, whereas for superdirective antennas, evanescent fields can be viewed as by-products of the antenna pattern design process, which become invisible in the regime where the image (the desired antenna pattern) is formed. Due to this difference in purpose of existence, evanescent waves present at the near-field of a superdirective antenna are not optimized for near-field sub-diffraction imaging purposes. Of course, one can alter the current excitations of a superdirective antenna to achieve near-field sub-wavelength imaging. However in this case the array antenna ceases to become superdirective, hence the device becomes a sub-wavelength spaced antenna

of a superdirective anSince, in principle, the beam angle tenna can be arbitrarily squeezed without changing the antenna size D, one might suppose the spatial beamwidth can also be to zero. This is, however, in arbitrarily squeezed by taking direct contradiction with (8). As noted in [26], the rapid spatial phase variations in a superdirective antenna require better phase agreement between waves reaching an observation point from the central part of the antenna, and waves reaching the same point from the edge of the antenna. This intolerance to phase difference renders the traditional Rayleigh distance—as recorded on the right-hand side of (9)—inadequate; instead one needs to travel even further from the antenna before the far-field antenna pattern is formed. Hence, as one increases the gain of the su, but at the same time perdirective antenna, one decreases does not get arbitrarily squeezed. increases . As a result This fact is illustrated in Fig. 3, which shows field evolutions and , from three superdirective antennas of lengths but nevertheless are designed to have similar far-field beam angles (hence they have different degrees of superdirectivity). If we qualitatively define the onset of the far-field as the place where the first sidelobe decreases to half the field amplitude of the main beam, then we see from Fig. 3 that for these three anaway from the tennas the far-field region begins at around antenna. This clearly shows the inapplicability of (9); furthermore, it shows that increasing an antenna’s superdirective gain also pushes out its far-field regime. Fig. 4 shows the onset of the far-field, as well as the spatial width (the electric field full width at half maximum) of superdirective antennas of varying directivities. We see from this figure that squeezing an antenna’s angular beamwidth does not necessarily squeeze its spatial dimensions in the far-field; it might actually increase it. This analysis is in agreement with the uncertainty principle: one should not be able to obtain sub-wavelength field localization with a waveform limited to propagating waves. We are thus left to conclude that while a superdirective antenna overcomes the angular diffraction limit, it does not also overcome the spatial diffraction limit. D. Superoscillation Whereas the superdirective antenna fails to bring propagating waves into a sub-wavelength focus, superoscillations can do the job. Superoscillation is the phenomenon whereby a waveform locally oscillates faster than its highest constituent frequency component. As a result, combining a range of these waveforms

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oscillations and narrow peaks within a certain stretch of the spatial domain, as long as one tolerates high energy sidebands to occur outside the spatial stretch. This process is exactly analoand spatial fregous to superdirectivity: only now the spatial domains are flipped; moreover, without an equivquency alent of the far-field transformation process which occurs for superdirectivity, the high energy sidebands remain visible [28]. Nonetheless, in this manner one can generate a sub-wavelength focus employing propagation waves, and thus extend spatial sub-diffraction imaging capabilities to working distances beyond the evanescent near-field of the object and the imaging device. In the following we report our efforts towards demonstrating superoscillatory sub-diffraction focusing at five wavelengths away from a source. III. SUPEROSCILLATORY SUB-WAVELENGTH FOCUSING AT THE MULTI-WAVELENGTH RANGE A. Design Formulation

Fig. 3. Field evolutions from three antenna arrays, of electrical sizes 5; 2 and  respectively, which are designed to have similar far-field antenna patterns. The antenna axis is at z = 0.

In the above section we have shown the dual relationship between the phenomena of superdirectivity and superoscillation. In a previous work, we leveraged this duality to design superoscillation signals by adapting established techniques for superdirective antenna design [28]. For completeness we briefly review our method here before proceeding to report experimental results. We refer interested readers to our previous work [28] for finer details. We post our problem in reverse: first we specify the desired superoscillatory waveform in the image plane, then we backpropagate the desired field pattern to determine the source excitation required, and finally we synthesize the desired source excitation. For the first procedure, we expand our target waveform into Tschebyscheff polynomials [8], [9] to determine the set of in the plane of , such that upon a transzeros formation to the x-domain, we would obtain an image waveform with the narrowest central peak width, along with constant sidelobes at 20% the central peak field strength (4% intensity) for a region of half wavelength on both sides of the peak. After is obtained, multiplicative expansion on a product of these roots in a formation akin to the reveals the image waveform array factor in antenna array design. Thereafter a Fourier transform reveals the discrete spatial spectrum

Fig. 4. A plot of the far-field distance, and the corresponding beamwidth at that distance, for Chebyshev superdirective antennas with varying numbers of elements, with fixed electrical lengths of 2.

effectively forms a spectrum which is locally widened, which then allows the formation of foci and other wave patterns with sub-diffraction spatial features. However, along with the desired superoscillatory features come high energy sidebands outside the region of superoscillation, the energy for which varies polynomially with the superoscillatory region’s apparent spectral width, and exponentially with its duration [27]. A simultaneous glance at a superoscillation profile in the spatial and spatial frequency domains proves sufficient for one to understand how superoscillations occur. While the superoscillatory waveform occupies a limited spectral width—namely the region of propagating waves, they can generate arbitrarily fast

(11) where (12) is the weighting of the n’th delta function, is the Here is the number of delta functions in the spectral domain, is spacing between delta functions in the spectral domain, the transverse frequency of the first (most negative) delta function and is a phase constant which simplifies the notation in (12). We choose the latter three parameters such that all harmonics (or all delta functions in the spatial spectrum) lie within the propagation region. This allows us to back propagate these

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harmonics to the source location at a multi-wavelength distance away, and obtain the source spatial spectrum , inverse Fourier transforming which gives the source field profile .

(13) (14) When one chooses to synthesize this source field profile with an array of sources, the source spatial spectrum can be viewed as the spatial spectrum of the individual element, multiplied by that of the array. As such the spatial spectrum of the array would be given by

(15) Adequately sampling the inverse Fourier transform of will give the desired source excitation coefficients. B. Waveguide Implementation Motivation: In achieving sub-wavelength resolution it is essential to employ an imaging device with a high-numerical aperture (NA), such that the working spatial spectrum covers as much of the propagating region as possible. However, to obtain a high NA with an image distance in the multi-wavelength range, one would require an imaging device with a lateral aperture on the order of tens of wavelengths. Such a large electrical size, along with the corresponding large number of array elements, makes it inconvenient to demonstrate microwave superoscillation in free-space. On the other hand, a rectangular waveguide is an ideal platform for demonstrating microwave superoscillamode of a rectangular waveguide has a transtion. The verse spatial frequency given by

Fig. 5. (a) Superoscillatory image waveform across a 3 (half-period) crosssection. (b) A close up of (a) in the design region of x 2 [0=2; =2]. (c) A plot of the corresponding spatial spectral amplitudes.

spectral lines in our design, which gives us freedom to zeros in our design procedure. We place 4 place zeros to generate the focus as explained in the previous section, to align an electric and place the remaining zero at field null point with the waveguide sidewalls. Substituting these parameters into (11) and (12) gives the superoscillatory image waveform in Fig. 5(a). A close up on the design region is shown in Fig. 5(b), and the corresponding spatial spectrum is shown in Fig. 5(c). We proceed to design an imaging device that produces this away, as depicted in Fig. 6(a). waveform from a distance We find using (13) and (14), then synthesize it with an array of current line sources, placed at half wavelength intervals and embedded within a rectangular waveguide. The TE10, TE30 and TE50 waveguide modes form the desired harmonics depicted in Fig. 5(c). We use a waveguide with a width of in the x-direction, which corresponds to choosing (17)

(16) denotes the width of the waveguide in x-direction. where Since these waveguide modes space uniformly in the domain, they form a good candidate for synthesizing the source spectrum (13): one only needs to appropriately excite them to generate the desired source spectrum, which will then propagate into the desired superoscillation profile. In an equivalent view, the walls of a rectangular waveguide act as mirrors which image the source to mimic an infinite array. Hence with this platform, one can access the entire propagating spectrum and perform sub-wavelength focusing with several source elements. This makes for a relatively compact device. With these considerations in mind, we proceed to design and demonstrate a sub-wavelength superoscillatory focus at the multi-wavelength range in a waveguide environment. Design: Here we outline the design parameters for designing our sub-wavelength superoscillatory waveform. We will employ

This ensures the existence of the TE50 mode, but cuts off higherorder modes. The field emanating from a line source is described by a zero order Hankel function of the second kind, for which the corresponding spatial spectrum is [29] (18) Substituting (18) into (15) gives

(19) (20)

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Fig. 7. A photograph showing the experimental apparatus.

Fig. 6. (a) A schematic of the experimental setup. (b) A diagram of the feed network.

TABLE I CURRENT EXCITATION COEFFICIENTS NEEDED TO GENERATE A SUB-WAVELENGTH FOCUS AT AN IMAGE DISTANCE OF 5.

Finally, we sample at half wavelength intervals to obtain the required line source excitation coefficients h[n], which we have tabulated in Table I. (3.2 mm) Fabrication and Experimentation: We bent a sheet of stainless steel into a waveguide section of width 297 mm, height 12 mm and length 1 m. Due to a fabrication error, the waveguide width is slightly narrowed from the intended 300 for the experimental frequency mm, which would represent of 3 GHz. To reflect this change, we modify our calculations above and obtain adjusted values for h[n], which are also tabulated in Table I. We embedded five metallic posts (1.2 mm diameter) spaced at 50 mm each along the x-direction to form an array of line end of the waveguide. Each post is sources 200 mm into the formed by connecting two inner conductors which extrude from their respective SMA receptacles which are welded above the top and beneath the bottom waveguide walls. Holes of sufficient sizes are drilled through the top and bottom waveguide walls at the location of the posts, such that the inner conductors, when fitted through from both sides, connect without forming electrical contact with the waveguide. The SMA receptacles provide access ports. In our experiment, we use them to feed current into the metallic posts and to monitor their current levels. We feed the aforementioned array of line sources using the feed network schematized in Fig. 6(b). The microwave source originates from port 1 of a programmable network analyzer (PNA),

and splits into three signal paths through a 1-to-3 power splitter. One output path from this power splitter directly feeds into port 3 of the waveguide (see Fig. 6(a) for port enumeration); outputs from the other two paths each go through a variable attenuator and a variable phase shifter, then get power divided and fed into ports 1 and 5, and 2 and 4. In this manner, we obtain freedom to tune the complex current inputs into ports 1 to 5. Since we operate with closely spaced antenna elements embedded in a waveguide environment, we need to account for mutual coupling effects amongst our antenna elements. To accomplish this, we monitor fields coupled into ports 6 to 10 by connecting these ports sequentially to port 2 of our PNA. Since the coupled currents are, by and large, proportional to the currents on the posts, we tune our feed network such that the set of for to match those for h[n] (as listed values in Table I) for the experimental frequency 3 GHz. This tuning procedure allows us to account for mutual coupling, and thus synthesize the desired current excitation on the source array. Fig. 7 shows a photograph of the experimental apparatus. With the aforementioned feeding network we drive predesigned currents into the source array, and thus generate the TE10, TE30 and TE50 waveguide modes in appropriate proportions. (The TE20 and TE40 modes are rejected due to excitation symmetry.) We place microwave absorbers at the ends of the waveguide to quench reflected components, but leave a small gap at the end in direction, through which we insert a coaxial cable probe the to measure the y-directed electric field at the image region 500 away from the source. mm C. Simulation and Experimental Results mm, we scan the probe With the source located at to measure the electric field at a series of cross-sections from mm to mm, with spacing of 10 mm. We mm scan the probe across each cross-section from mm at step sizes of 2.5 mm. We leave a space of to 18.5 mm on either side of the waveguide cross-section to avoid colliding the probe with the waveguide walls. The measured electric field profile across this region is shown in Fig. 8(c). We compare our measurement with full-wave simulation results obtained using Ansoft HFSS. In our simulation, we provide wave port excitation to SMA connectors, which in turn couple currents into the metallic posts, thus forming the source array. Here, we account for mutual coupling effects by setting the wave as port excitation weights

WONG AND ELEFTHERIADES: SUB-WAVELENGTH FOCUSING AT THE MULTI-WAVELENGTH RANGE USING SUPEROSCILLATIONS

= 500

Fig. 8. Electric field profiles near the image plane z mm (denoted by white dash lines). (a) Simulation with absorbers covering the entire waveguide end of end facets. (b) Simulation with a partial gap in the absorber at the the waveguide (see Fig. 6 for pictorial depiction. (c) Measured electric field magnitude. The color bar to the left applies to (a) and (b); the one to the right applies to (c).

+z

(21) describes a selected portion of the simulated 10-port where S-parameter from which we distill information on mutual coupling effects (again, see Fig. 6(a) for port enumeration). The source array excites relevant waveguide modes in appropriate proportions, which are guided through the stainless steel waveguide, and terminated by perfect matching layers (PMLs). These PML blocks span the entire cross-section of the waveguide, and are placed at the same longitudinal locations as the absorbers in the experiment. The resulting electric-field magnitude is plotted in Fig. 8(a). It displays near-perfect agreement with an analytical field magnitude calculation of the interference of prescribed portions of the TE10, TE30 and TE50 modes. The sub-diffraction central peak, as well as the high energy sidebands, are visible across this entire region; the fixed ripple sidelobe region mm. also appears around the designed image plane at To reconcile the apparent difference between the simulated and experimental results (Fig. 8(a) and (c)), we run a modified simulation where we introduce a 6 mm gap in the y-direction for the PML situated at the end facet of the waveguide, from whence our coaxial cable probe is inserted in the experiment. Fig. 8(b) shows the electric-field magnitude of this modified simulation. Here, the presence of a reflected wave generates observable standing-wave patterns. As a result, the sideband ammm—and the cenplitude is decreased—particularly at tral peak is widened. In this spirit, we see that Fig. 8(c) resembles Fig. 8(b) in that all major peaks in Fig. 8(b) are observed

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Fig. 9. A comparison on the measured and simulated superoscillatory foci at the image plane. The simulated field profiles are taken at the design image plane mm. z mm, while the measured field profile is taken at

= 500

z = 480

in Fig. 8(c). However, the seemingly more significant reflected wave components cause a sideband cancellation at mm, producing a central peak which becomes much more pronounced in amplitude, but no longer sub-wavelength. Nevertheless, a sub-wavelength pattern is formed in the immediate mm; we will examine the experimental vicinity at focal quality at this distance in the following figures. Some experimental factors which lead to reflected waves, or otherwise disturb the superoscillatory wave interference pattern, include the intrusion of the coaxial cable probe, the receiving characteristic of the small, but finite-sized, probe tip, rounded waveguide corners, and slight asymmetry in feed network distribution and source array construction. Notwithstanding, the field measured at cross-sections slightly before the designed imaging distance remains superoscillatory despite interference with reflected waves. Figs. 9 and 10 plot and compare the measured and simulated mm and a nearby waveform cross-sections at mm . Simulation with a full (PML) abplane end of the waveguide leads to a superoscillation sorber at the focus with an electric field full-width-half-maximum (FWHM) . This electric field profile is in exact agreeof 37 mm ment with the target waveform, apart from slight deviations surrounding the outer pair of nulls. However, when we employ end of the waveguide, the image the partial absorber at the profile widens to a FWHM of 45 mm . Whereas rediflected wave components and slight asymmetry in the rection obscure our superoscillatory focusing measurement at mm, the measured electric field cross-section at mm clearly demonstrates superoscillatory behavior. This profile is plotted in Figs. 9 and 10 for comparison purposes. Here the electric field FWHM of the central peak is squeezed to —comparable to the partial absorber simulation 45 mm and at 75% of the diffraction limit of 61 mm , formed from a uniform, in-phase superposition of all symmetric modes within the waveguide. Even though experimental imperfections prevented us from obtaining a region of low sidelobe ripples, we have nevertheless successfully demonstrated superoscillatory sub-diffraction focusing at a multi-wavelength image distance.

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Fig. 10. A close up of Fig. 9, comparing the measured and simulated superoscillatory foci across the design interval. The diffraction-limited focus in this waveguide environment is also included for comparison. As in Fig. 9, the simmm, while the measured field profile ulated field profiles are taken at z mm. is taken at

z = 480

= 500

Fig. 11. Sensitivity of the superoscillatory focus. The red curve shows the design superoscillatory focus, while dotted curves show typical electric field amplitude distributions when a 2.5% error is introduced in the waveguide mode weightings.

D. Discussion We begin our discussion with a comment on sensitivity. It has been well known that sensitivity issues prevent superdirective antennas from arbitrarily squeezing the width of a beam emanating from an antenna. We have therefore tested the sensitivity of the superoscillatory waveform we generated, to confirm that reasonable levels of spot size reduction can be achieved with tolerances which are practical for implementation. To characterize the sensitivity of our proposed method, we vary the excitation coefficients by adding to each waveguide mode a randomly phased white Gaussian component with mean amplitude 2.5% of the strongest excitation coefficient. Typical resulting waveforms are plotted in Fig. 11. Despite the appearance of an uneven and increased sidelobe level, our achieved subwavelength focal width is unaffected by this level of perturbation. A similar analysis on the waveguide excitation currents has been conducted in [28], where it was found that a precision level of 1% is sufficient to obtain the desired sub-diffraction focus. As this required level of precision was met by our device, we believe degradations observed in Figs. 8–10 are not caused by the sensitivity of the superoscillatory waveform, but rather due to the reasons outlined in the previous section. In practice, the precise excitation of current elements (and hence the relevant waveguide modes) comes easier for superoscillatory waveforms than it does for superdirective antennas, since no radiative energy is stored, and reduce the sigsince relatively wide antenna spacings nificance of mutual coupling. Hence, the aforementioned level of precision is by no means stringent in current imaging systems. Thus we believe superoscillatory sub-diffraction focusing can be implemented in a wide range of practical platforms. We proceed to comment on the regime in which we performed sub-diffraction focusing. Unlike evanescent-field-based imaging devices, we have formed a sub-diffraction focus far beyond the region of existence and dominance of evanescent waves. However, the focus formed does not exist in the far-field either—in the sense that our sub-diffraction focal distribution is not a scaled copy of its spatial spectrum, hence the Fraunhofer approximation does not apply. Rather our sub-diffraction focus lies in the radiating near-field—where only radiating

waves exist, but the dynamics of their interference determines the overall spatial waveform. Notwithstanding, that the domain of focus formation is not in the far-field should not be seen as a limitation. While we have experimentally demonstrated focusing at 4.8 wavelengths (480 mm), the radiating near-field can extend many-fold beyond this distance to cover practical ranges conducive to many applications. We now discuss the waveguide environment in which we have performed superoscillatory sub-diffraction focusing. Indeed, we have demonstrated, for ease of experimentation, superoscillatory sub-wavelength focusing in a waveguide environment. However, our design formulation can be readily used for sub-wavelength focusing in a free-space environment. Waveguide walls in our experiment serve as mirrors for the source electric distribution. Hence they can be extended to any integral multiples of the waveform’s spatial period (odd multiples for anti-mirrors), as long as the source distribution is also extended. In the limit of infinite extension, the waveguide solution converges to a free-space solution. Moreover, even with a finite extension, the waveguide cross-section can be enlarged at will to contain the object which is to be imaged. This finite extension renders the waveguide environment practical for imaging implementations at microwave frequencies. Finally, we discuss the possibility of multi-dimensional superoscillatory sub-diffraction focusing. Although in this work we demonstrate sub-wavelength focusing in only one direction, an extension to multi-dimensional focusing is very conceivable—because a superoscillation is but a special superposition of propagating waves. Extension to 2D focusing in the waveguide environment is conceptually straightforward: one only needs to excite the waveguide with a 2D array, and tune the waveguide height to also allow the propagation of modes with y-directed oscillations. Excitation weightings can be readily determined by our proposed method, treating a 2D array as a y-directed array where each element is in itself an x-directed array. Thus with the incorporation of a more elaborate feeding network one can perform 2D sub-wavelength focusing with the waveguide environment proposed in this work.

WONG AND ELEFTHERIADES: SUB-WAVELENGTH FOCUSING AT THE MULTI-WAVELENGTH RANGE USING SUPEROSCILLATIONS

It has been shown that evanescent-field-based focusing devices cannot form a 3D sub-diffraction focus, on grounds of an inevitable violation of the consistency relation [30]. This limitation, however, does not restrict the formation of superoscillatory 3D sub-diffraction foci, since superoscillatory waveforms can be formed by the sole interference of propagating waves. The unique ability to form a 3D sub-wavelength focus, combined with the ability to form such a focus at multi-wavelength distances, make superoscillatory sub-wavelength focusing attractive to a wide range of imaging and sensing applications. IV. CONCLUSION In this work, we have elucidated the difference among evanescent-field-based focusing, superdirectivity and superoscillation. In particular, we have shown that superdirectivity and superoscillation are dual phenomena in the spatial and spatial frequency domains. This property gives superoscillation the potential to overcome the spatial diffraction limit—to form a sub-wavelength focus with propagating waves. Realizing this as the avenue to push sub-wavelength focusing capabilities to working distances far beyond the evanescent near-field, we have formulated a proof-of-principle design to perform superoscillation-based 1D sub-wavelength focusing in a waveguide environment, and presented corresponding results from full-wave simulations and experimental measurements. We have experimentally achieved superoscillatory sub-wavelength electric field FWHM (75% of the diffraction focusing to from the source—an order of maglimit) at a distance nitude increase from sub-wavelength focusing schemes with evanescent-field-based devices. While we have demonstrated 1D sub-wavelength focusing in a waveguide environment, our formulation can be extended to multi-dimensional sub-wavelength focusing in waveguide or free-space environments in conceptually straightforward manners. We believe the ability to form a sub-wavelength focus at a working distance much longer than the evanescent near-field regime could become very attractive for high-resolution imaging and sensing applications. REFERENCES [1] E. H. Synge, “A suggested method for extending microscopic resolution into the ultra-microscopic region,” Philos. Mag., vol. 6, pp. 356–362, Aug. 1928. [2] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of eps and mu,” Sov. Phys. Usp., vol. 10, pp. 509–514, Jan. 1968. [3] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett., vol. 85, no. 18, pp. 3966–3969, Apr. 2000. [4] V. Eckhouse, Z. Zalevsky, N. Konforti, and D. Mendlovic, “Subwavelength structure imaging,” Opt. Eng., vol. 43, no. 10, pp. 2462–2468, Oct. 2004. [5] Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett., vol. 7, no. 2, pp. 403–408, Feb. 2007. [6] A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B, vol. 74, p. 075103, Aug. 2006. [7] Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Farfield imaging beyond the diffraction limit,” Opt. Express, vol. 14, no. 18, pp. 8247–8256, Sep. 2006. [8] L. Markley, A. M. H. Wong, Y. Wang, and G. V. Eleftheriades, “Spatially shifted beam approach to subwavelength focusing,” Phys. Rev. Lett., vol. 101, pp. 113–901, Sep. 2008.

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[9] A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: Subdiffraction focusing with patterned surfaces,” Science, vol. 320, pp. 511–513, Apr. 2008. [10] R. Gordon, “Proposal for superfocusing at visible wavelengths using radiationless interference of a plasmonic array,” Phys. Rev. Lett., vol. 102, pp. 207–402, May 2009. [11] G. V. Eleftheriades and A. M. H. Wong, “Holography-inspired screens for sub-wavelength focusing in the near field,” IEEE Microwave Wireless Compon. Lett., vol. 18, pp. 236–238, Apr. 2008. [12] S. A. Schelkunoff, “A mathematical theory of linear arrays,” Bell Syst. Tech. J., vol. 22, pp. 80–107, Jan. 1943. [13] Y. Aharonov, J. Anandan, S. Popescu, and L. Vaidman, “Superpositions of time evolutions of a quantum system and a quantum time-translation machine,” Phys. Rev. Lett., vol. 64, pp. 2965–2968, Jun. 1990. [14] M. V. Berry, “Faster than Fourier,” in Quantum Coherence and Reality: In Celebration of the 60th Birthday of Yakir Aharonov, J. S. Anandan and J. L. Safko, Eds. Singapore: World Scientific, 1994, pp. 55–65. [15] M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A: Math. Gen., vol. 27, pp. L391–L398, Jun. 1994. [16] P. J. S. G. Ferreira, A. Kempf, and M. J. C. S. Reis, “Construction of Aharonov-Berry’s superoscillations,” J. Phys. A: Math. Gen., vol. 40, pp. 5141–5147, May 2007. [17] F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett., vol. 9, pp. 1249–1254, Jan. 2009. [18] F. M. Huang, N. Zheludev, Y. Chen, and F. J. G. De Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett., vol. 90, p. 091119, Feb. 2007. [19] M. R. Dennis, A. C. Hamilton, and J. Courtial, “Superoscillation in speckle patterns,” Opt. Lett., vol. 33, pp. 2976–2978, Dec. 2008. [20] G. B. Folland and A. Sitaram, “The uncertainty principle: A mathematical survey,” J. Fourier Analy. Applicat., vol. 3, pp. 207–238, May 1997. [21] R. Merlin, “Radiationless electromagnetic interference: Evanescentfield lenses and perfect focusing,” Science, vol. 317, pp. 927–929, Aug. 2007. [22] G. Lerosey, J. Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science, vol. 23, pp. 1120–1122, Feb. 2007. [23] R. Marqués and J. Baena, “Effect of losses and dispersion on the focusing properties of left-handed media,” Microw. Opt. Technol. Lett., vol. 41, pp. 290–294, May 2004. [24] D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett., vol. 82, pp. 1506–1508, Mar. 2003. [25] A. Grbic and G. V. Eleftheriades, “Practical limitations of subwavelength resolution using negative-refractive-index transmission-line lenses,” IEEE Trans. Antennas Propag., vol. 53, pp. 3201–3209, Oct. 2005. [26] A. C. Marvin, A. P. Anderson, and J. C. Bennett, “Inadequacy of the Rayleigh range criterion for superdirective arrays,” Electron. Lett., vol. 12, pp. 415–416, Aug. 1976. [27] P. J. S. G. Ferreira and A. Kempf, “Superoscillations: Faster than the Nyquist rate,” IEEE Trans. Signal Process., vol. 54, pp. 3732–3740, Oct. 2006. [28] A. M. H. Wong and G. V. Eleftheriades, “Adaptation of Schelkunoff’s superdirective antenna theory for the realization of superoscillatory antenna arrays,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 315–318, Apr. 2010. [29] F. Oberhettinger, Tables of Fourier Transforms and Fourier Transforms of Distributions. Berlin, Germany: Springer-Verlag, 1990, p. 21. [30] R. Marqués, M. J. Freire, and J. D. Baena, “Theory of three-dimensional subdiffraction imaging,” Appl. Phys. Lett., vol. 89, p. 211113, Nov. 2006. Alex M. H. Wong received the B.A.Sc. degree in engineering science and the M.A.Sc. degree in electrical engineering from the University of Toronto, Toronto, ON, Canada, in 2006 and 2009, respectively, where he is working toward the Ph.D. degree. His research interests include high resolution imaging using electromagnetic waves, sub-wavelength focusing devices, metascreens and superoscillations.

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George V. Eleftheriades (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, Lausanne. Currently, he is a Professor in the Department of Electrical and Computer Engineering, University of Toronto, ON, Canada, where he holds the Canada Research Chair/Velma M. Rogers Graham Chair in Engineering. Prof. Eleftheriades was elected a Fellow of the Royal Society of Canada, in 2009. He was the recipient of the 2008 IEEE Kiyo Tomiyasu Technical

Field Award, the 2001 Ontario Premiers’ Research Excellence Award, and the 2001 Gordon Slemon Award presented by the University of Toronto, and the 2004 E.W.R. Steacie Fellowship presented by the Natural Sciences and Engineering Research Council of Canada. He is the co-recipient of the inaugural 2009 IEEE Microwave and Wireless Components Letters Best Paper Award. One of the papers he coauthored received the RWP King Best Paper Award in 2008. He has served as an IEEE Antennas and Propagation-Society (AP-S) Distinguished Lecturer (2004–2009) and as a member of the IEEE APS Administrative Committee (AdCom, 2008–2010). He is an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and a member of Technical Coordination Committee MTT-15 (Microwave Field Theory). He has been the General Chair of the 2010 IEEE Int. Symposium on Antennas and Propagation and CNC/USNC/URSI Radio Science Meeting held in Toronto.

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Three-Dimensional Near-Field Microwave Holography Using Reflected and Transmitted Signals Reza K. Amineh, Member, IEEE, Maryam Ravan, Member, IEEE, Ali Khalatpour, Student Member, IEEE, and Natalia K. Nikolova, Fellow, IEEE

Abstract—A new 3-D holographic microwave imaging technique is proposed to reconstruct targets in the near-field range. It is based on the Fourier analysis of the wideband transmission and reflection signals recorded by two antennas scanning together along two rectangular parallel apertures on both sides of the inspected region. The complex scattering parameters of the two antennas are collected at several frequencies and then processed to obtain a representation of the 3-D target in terms of 2-D slice images at all desired range locations. No assumptions are made about the incident field and Green’s function, which are derived either by simulation or by measurement. Furthermore, an approach is proposed to reduce the image artifacts along range. To validate the proposed technique, predetermined simulated targets are reconstructed. The effects of random noise, number of sampling frequencies, and dielectric contrast of the targets are also discussed. Index Terms—Image reconstruction, microwave holography, near-field microwave imaging.

I. INTRODUCTION HE relatively long wavelength of microwave and millimeter waves allows for penetration inside dielectric bodies where visible light cannot reach. Various techniques have been proposed to harness the ability of microwaves for 2-D and 3-D imaging in a wide variety of applications such as biomedical imaging [1], concealed weapon detection [2], through-the-wall imaging [3], nondestructive testing and evaluation [4], etc. All these methods operate on the transmitted and/or received signals at the antenna terminals but they differ significantly in the employed algorithms. Microwave imaging techniques in general could be categorized into: optimization-based microwave imaging (e.g., see [5]), confocal radar-based imaging (e.g., see [6]), microwave tomography (e.g., see [7]), and microwave holography [8]. In optimization-based imaging, the exact nonlinear diffraction relation between the measured scattered field and the complex permittivity is found iteratively. The distribution of dielectric properties is sought within the imaged region employing a forward model. These techniques are in principle capable of achieving super-resolution (e.g., 0.1 in [9]). However, their

T

Manuscript received November 23, 2010; revised March 08, 2011; accepted June 02, 2011. Date of publication August 30, 2011; date of current version December 02, 2011. The authors are with the Department of Electrical and Computer Engineering, McMaster University, Hamilton ON, L8S 4K1, Canada (e-mail: [email protected]; [email protected]; [email protected]; khalata@mcmaster. 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.2011.2165496

high computational cost (e.g., see [10]) precludes fast performance. Also, they suffer from ill-posedness (e.g., see [11]) and their performance depends critically on the fidelity of the forward model. It is often the case that microwave imaging setups are difficult to align well with electromagnetic numerical simulation. Confocal microwave imaging aims to detect strong scatterers inside the inspected region [6]. A wide-band pulse is transmitted and the reflections from the targets are focused synthetically. The focusing algorithm constructs a map of the scatterers (if present) inside the interrogated region. While very fast, these techniques are sensitive to clutter [12] and the heterogeneity of the inspected region [13]. Microwave tomography includes computed tomography (CT) [14]–[17] and diffraction tomography (DT) [18]–[22]. Both approaches acquire the transmitted signals in multiple planes intersecting the imaged object. The signals acquired in each plane are used to reconstruct a 2-D slice image in this plane only. Thus, the reconstruction is carried out slice by slice, each slice being processed independently. Both approaches exploit linearization approximations to achieve direct (non-iterative) image reconstruction. Microwave CT [14] employs a straight-path assumption for the rays between the transmitter and receiver and applies filtered backprojection (FBP), similar to X-ray CT. As the straight-ray assumption is often inadequate, microwave CT often employs chirp-pulse radar [15]–[17]. Microwave DT employs the linear Born approximation [23] to the solution of the scattering problem which is valid for weak scatterers. This approximation allows for image reconstruction through a Fourier transform (FT). The processing is in quasi-real time but artifacts are often present if the scatterers are not weak. Also, the spatial resolution is limited by the wavelength. Note that similar Fourier-based reconstruction is employed in synthetic aperture radar (SAR) (e.g., see [24], [25]). The conventional Gabor holography (optical holography) [26], [27] has been extended to microwave frequencies [28], [29] and to acoustic problems [30], [31]. In these approaches, the interference pattern of a reference wave with the wave scattered from the target is recorded on an aperture called hologram. The hologram is illuminated with the reference wave to reconstruct an image of the object. Modern microwave holography [32]–[35], however, is quite distinct from these conventional approaches. Coherent (magnitude and phase) back-scattered signals are acquired on a surface similarly to the conventional holography. However, the reconstruction of the target is based on a sequence of direct and inverse FTs. This reconstruction is similar to DT in the sense that it employs the linear Born approximation and FTs. However, unlike DT, where reconstruction is done slice by slice,

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the data acquired on the surface is used simultaneously in a single reconstruction process to obtain the 3-D reflectivity distribution of the target. Thus, microwave holography provides a framework where, with wideband frequency information, a 3-D image of the target is obtained. In the case of a single-frequency measurement, microwave holography can provide a 2-D image of the target’s cross-section in a plane parallel to the acquisition plane (e.g., see [32] and [35]). Microwave holographic imaging based on rectangular and cylindrical aperture scanning has proven reliable and is now widely employed in concealed weapon detection [32]–[34]. Recently, we extended the single-frequency 2-D holographic image reconstruction to near-field microwave imaging [35]. This method employs not only the back-scattered (as in [32], [33]) but also the forward-scattered signals. The additional information from the forward-scattered signals improves the image quality and enables localization of the target in the range direction. In contrast to previous work, this method does not make any assumptions about the incident field such as plane, spherical or cylindrical wave representations. The incident field is derived in a numeric form either through simulation or measurement. This is important in near-field imaging where the target is close to the antenna and the planar or spherical approximations of the illuminating wave are not valid. Here, we extend the 2-D near-field holographic imaging technique [35] to full 3-D imaging when wideband information is available. The proposed method has a number of distinct features and advantages compared to the existing 3-D holographic techniques. First, the method allows for incorporating forward-scattered signals in addition to the back-scattered signals. This additional information leads to more accurate reconstruction results and also allows for the significant suppression of image artifacts in the range direction. Second, the method allows for an incidentfield distribution represented in numeric form. This distribution can be obtained either through simulation or measurement with the particular antenna setup and medium. Third, it also allows for numeric input of the Green’s function, i.e., the set of signals due to point scatterers in the given medium and received by the given antennas. These can be efficiently obtained through simulation as explained later. The accurate representations of the incident field and Green’s function for the particular problem at hand are crucial in near-field imaging where analytical approximations such as plane or spherical waves are not adequate. Fourth, the numerical formats of the incident field and the Green function necessitate a new inversion procedure. Previous 3-D holography methods [32], [33] rely on the analytical (exponential) form of the incident field and the Green function in order to cast the inversion expression in the form of a 3-D inverse FT. Resampling of the data in -space is also necessary, which may lead to errors. This procedure is inapplicable with numeric representations of the incident field and the Green’s function. Instead, we solve a and system of equations in each spatial frequency pair apply 2-D inverse FT to the least-square solution at each desired range location. The systems of equations have much smaller dimensions compared to the systems of equations in regular optimization-based microwave imaging techniques. This reduces the ill-posedness of the proposed imaging technique significantly. Thus, the 3-D target is reconstructed as a set of 2-D slice images

Fig. 1. 3-D Microwave holography setup.

in parallel planes along the range. Fifth, the algorithm is fast, accurate, and robust to high levels of noise. We examine the performance of the 3-D image reconstruction technique by a number of simulation examples. II. THEORY The microwave holography set-up in this work consists of two -polarized antennas and a 3-D target in between as shown in Fig. 1. The formulation of the vectorial scattering problem is presented in the Appendix. Here, the radiation field of the -polarized antennas (dipoles) can be reasonably approximated by polarization. Thus, we consider the -components of the incident and scattered fields only. This leads to a scalar Green’s element of the full dyadic. The anfunction which is the tennas perform a 2-D scan while moving together on two sepaand . rate parallel planes positioned at Assume that at any measurement frequency , we know the incident field at any point in the inspected volume when the transmitting antenna is at (0,0,0). In is known addition, the Green’s function and measured for an -polarized point source at at . This information is obtained via simulations as explained later. For brevity, we set (1) (2) Let the signal be the scattered wave received at , i.e., where the scattered field is calculated as in (A.8). Note that this implies that the since it moves together with transmitting antenna is at the receiving antenna. The incident field and the Green functions can be obtained for the case where the antenna pair is at from those in (1) and (2) by a simple translation: (3) (4) According to (A.8),

is expressed as

(5)

AMINEH et al.: THREE-DIMENSIONAL NEAR-FIELD MICROWAVE HOLOGRAPHY USING REFLECTED AND TRANSMITTED SIGNALS

Equation (5) can then be rewritten as

(6) where (7) (8) as the contrast function. Here, the inWe refer to tegral over and can be interpreted as a 2-D convolution inis written as tegral. Thus, the 2-D FT of

respectively. Second, we assume that the variation of the is known a priori. This contrast function with frequency function can be approximated for example by fitting Cole-Cole or Debye models (e.g., see [36]) to the dielectric properties of would approximate the the background medium. Then, frequency behavior of both the background medium and the targets. In many applications where the operating frequency band . is narrow (or in nondispersive mediums), we assume This is the case for the examples presented in this paper. Taking the 2-D FT of both sides of (11) with respect to and leads where to is the 2-D FT of . Thus, replacing with in (11) leads to the following system of equations: .. .

(9) where

and

are the 2-D FTs of and , respectively; and are the Fourier variables with respect to and , respectively. To reconstruct the contrast function, we first approximate the reconintegral in (9) by a discrete sum with respect to for struction planes

(10) is the distance between two neighboring reconstrucwhere tion planes. frequencies, writing Since we perform measurements at equations at each spatial (10) for all frequencies leads to , see (11). frequency pair

.. .

(11) decoupled equations from which finding diIn (11), we have rectly the frequency dependant function and , is not feasible. However, under certain assumptions, a system of equations can be constructed from which the contrast function is found. First, we assume that the contrast function can be expressed as (12) where the variation of the contrast function with space and and , frequency is separated into the functions

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(13)

where (14) (15) (16) This system of equations is solved in the least-square sense , at each spatial freto find . Then, inverse 2-D FT is applied to quency pair , to reconstruct a 2-D slice at each plane. Then, the of the function , where normalized modulus of is the maximum of for all , is plotted versus the spatial coordinates and to obtain a 2-D image of the target plane. By putting together all 2-D slice images, at each a 3-D image of the target is obtained. For the setup shown in Fig. 1, the acquired scattered waves are obtained by measuring the four complex -parameters at the two antenna terminals at each scanning position on the apertures. These four -parameters constitute four separate scatexpressed in (5), from each one a comtered signals plete set of 2-D slice images can be obtained. The -paramis the scattered signal at the terminals of eter the th antenna when the th antenna transmits. The parameters and are referred to as the reflection -parameters while and are the transmission -parameters. , , and . In the proposed reconstruction process, we assumed that , we at any measurement frequency know the incident field and the Green function at any target point when the transmitting and , respecand the receiving antennas are at tively. In prior work exploiting microwave holography, these are approximated as which is suitable for far-field scattering problems. Here, these functions are obtained via simulation of the case , we place when the targets are absent. To obtain the transmitting antenna at the origin of the corresponding aperture plane and record the -component of the electric field in all reconstruction planes and at all frequencies of interest, i.e., and .

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One approach to obtain is to move an -polarized point source on all reconstruction planes along all pixels and to record the field at the receiving antenna terminals. This approach, however, is extremely inefficient because it requires as many simulations as the number of imaged pixels. Another more efficient approach is using the reciprocity principle [37]. Only one simulation is performed for each transmitting antenna where the infinitesimal -polarized source is at the origin of the respective aperture plane while the field is sampled simultaneously at all pixels of all reconstruction planes. The collected data from each -parameter, i.e., , can be processed separately as described above to create an image of the target. In this case, we can obtain four separate images at each plane. Another approach is to reconstruct a single image from the data sets of all -parameters. To implement this, we write equations using the data collected from each (12) to get -parameter at each . Considering that (or in (12)), (unknowns) are the same in all four systems of equations, these four systems of equations are combined to construct a single system of equations with rows, each rows correspond to one of the -parameters. This system of equations is usually over-determined since at equations need to be solved simultaneeach unknowns . ously to find Again, a least-square solution is sought to find these unknowns. III. RANGE AND CROSS-RANGE RESOLUTIONS We approximate the resolution at the center frequency of the . Following the approach system bandwidth presented in [35] for a single-frequency holographic imaging, the cross-range resolution is calculated as [35] (17) where is the wavelength corresponding to the frequency , and and are the angles subtended by the transmitting and receiving apertures or the full beamwidths of the antennas, whichever is less. for small targets is approximately The range resolution [32] (18) where is the wave velocity in the medium and quency bandwidth.

is the fre-

is the wavelength corresponding to the maximum where . frequency The required frequency sampling is determined in a similar way. The phase shift resulting from a change in the wavenumber is , where is the maximum target range. Requiring that this phase shift is less than yields [32] (20) where

is the frequency sampling interval. V. EXAMPLES AND ARTIFACTS SUPPRESSION

To validate the accuracy of the proposed 3-D microwave (at the center frequency of holography technique, two operation) -polarized dipole antennas with targets in between are simulated in FEKO Suite 5.4 [38] as shown in Fig. 2. The antennas perform a 2-D scan by moving together on two parallel aperture planes and collecting wideband data at each position. We refer to the apertures as aperture 1 and aperture 2 corresponding to antenna 1 and antenna 2, respectively. The apertures have a size of 60 mm 60 mm with their centers being on the axis. for the two antennas are The -parameters acquired in the presence of the target and recorded for every position of the antenna pair. The numerical noise in the acquired data is estimated through the numerical convergence error of the -parameters which is 0.02. The proposed holography technique is applied to the calibrated -parameters. To perform calibration, the same setup is simulated without the target to obtain the background -param. Then, the calibrated -parameters are eters calculated as (21) Here, due to the uniform background, the background simulapotions need to be performed only once in a sample sition since in all other positions they are the same. Since the dipoles are -polarized, the -component of the simulated inciand the Green function are recorded dent electric field at the reconstruction planes as described in Section II. These planes are of size 80 mm 80 mm. To evaluate the quality of the target shape reconstruction process, we define a reconstruction error (RE) paremeter as (22)

IV. SPATIAL AND FREQUENCY SAMPLING In order to make use of all frequency components of the measured signals, the sampling rate of the scan has to satisfy the Nyquist criterion, i.e., the phase shift from one sample point to the next must be less than . For a spatial sampling interval of along a cross-range direction, the worst case is a phase shift . Therefore, the sampling criterion can be expressed of as [32] (19)

for each slice of the reconstructed image is the for all . The maximum of function represents the exact shape of the target. It is 1 inside the target and 0 elsewhere. Also, to have an overall estimate of the reconstruction error, we define the parameter as (23)

AMINEH et al.: THREE-DIMENSIONAL NEAR-FIELD MICROWAVE HOLOGRAPHY USING REFLECTED AND TRANSMITTED SIGNALS

Fig. 2. Dielectric targets in free space scanned by two =2 (at 35 GHz) horizontally-polarized (x-polarized) dipoles; dipole 1 is moving on the z = 50 mm plane while dipole 2 is moving on the z = 0 mm plane. The simulated S -parameters are recorded in the frequency band of 25 GHz to 45 GHz for: (a) two similar cuboids with sides of 3 mm, centered at (0; 4:5; 25) mm and (0,4.5,25) mm with dielectric properties of " = 2 and  = 0 S/m; (b) four similar cuboids with sides of 3 mm, centered at ( 4:5; 0; 11) mm, (4.5,0,11) mm, (0; 4:5; 27) mm, and (0,4.5,27) mm and all having dielectric properties of " = 2 and  = 0 S/m; (c) two similar X-shape objects with square cross-sections 2 mm on a side and length of each arm 16 mm, parallel to the x - y plane, one centered at (0,0,27) mm with its arms along the x and y axes, the other one centered at (0,0,35) mm, with the arms rotated by 45 degrees with respect to the x and y axes, both targets having dielectric properties of " = 2 and  = 0 S/m; (d) two similar cuboids with sides of d, centered at ( 4:5; 0; 35) mm and (4.5,0,35) mm, with dielectric properties " = 5 and  = 0 S/m.

0

0

0

0

A. Reconstruction Examples for a Background Medium With and S/m In this section, we present a number of reconstruction examples for dielectric targets in air. Fig. 2 shows the simulation setups. In all examples, the aperture planes are located at mm and . The center frequency is 35 GHz. The wideband data is in the range of 25 GHz to 45 GHz. This gives a range mm as per (18). Since must resolution of be fulfilled, in all examples, we set the distance between the reconstruction planes as 8 mm. Since the maximum frequency in the simulations is 45 GHz, as per (19), the spatial sampling rate for the 2-D scan has to be smaller than 1.6 mm. Therefore, a sampling rate of 1.5 mm along and is adopted. Also, we approximate the minimum number of frequencies from (20) which

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. For example, for being 40 mm, the depends on is 1.8 GHz. Here, we use GHz which maximum leads to 11 frequencies in the range of 25 GHz to 45 GHz. In the first example, two dielectric cuboids with a side of 3 mm and wall-to-wall distance of 6 mm are placed midway bemm as shown in Fig. 2(a). The tween the two antennas at and S/m constitutive parametres of the cuboids are while the background is free space. The separation distance beplanes, tween the reconstruction planes along the range ( ) is 8 mm. The system of equations in (13) is when the -pasolved at each spatial frequency pair rameters are used separately to reconstruct the 2-D images on 5 planes along the axis. Fig. 3 shows these images. It is observed and reconstruct the target that the images obtained from mm properly but also show some artifacts at the plane 8 mm away. The images obtained from are not conclusive mm plane is good. at all although the image obtained at leads to similar results as . Using Next, we solve the reconstruction problem when all the -parameters are used simultaneously. In this case, at each , 44 equations are available to find the 5 unknowns . Fig. 4(a) show the reconstructed images. It is observed that the targets are reconstructed mm. At the same time, the properly at the plane images at the other range locations do not contain artifacts. The computed values for RE (shown above slice images) and (shown in the figure captions) also confirm the improvement in the image quality. For this example, the cross-range resolution estimated from (17) is 3.8 mm. Thus, the two targets, which are 6 mm apart, are resolved well. From now on, we only present the results when the data collected from all -parameters contribute simultaneously to the image reconstruction. In the next example, a more challenging recosntruction problem is solved. As shown in Fig. 2(b), a pair of cuboids mm and similar to those in Fig. 2(a) are placed at another similar pair of cuboids, rotated by 90 about the axis, mm. The constitutive parameters of the are placed at and S/m while the background is free cuboids are space. Fig. 4(b) shows the reconstructed images. It is observed mm that the targets are recovered well in the planes at mm. Also, the images at the other range locations and do not show artifacts. A third example is shown in Fig. 2(c) where an “X” shape mm. The object parallel to the - plane is placed at “X” shape has two orthogonal arms of length 16 mm and square cross-sections 2 mm on a side. Another similar “X” shape object is rotated by 45 about the axis and is placed at mm. The constitutive parameters of the cuboids are and S/m while the background is free space. The distance between the two objects is 8 mm, which is very close to the mm. Fig. 4(c) shows the reconrange resolution limit mm structed images. The targets are recovered well at mm. The images formed at other range locations and show weak artifacts. Later, we present an approach to reduce mm, the arms along the the artifacts. For the target at axis have been reconstructed better than the arms along the axis. This is due to the fact that the dipoles are oriented along

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Fig. 3. Reconstructed images of the dielectric targets in Fig. 2(a) when the system of equations in (13) is solved using only: (a) s (RE = 8176).

378), (c) s

(RE = 380), (b) s

(RE =

Fig. 4. Reconstructed images when using all S -parameters simultaneously for the dielectric targets in: (a) Fig. 2(a) (RE = 143), (b) Fig. 2(b) (RE = 302), and (c) Fig. 2(c) (RE = 393).

and thus the incident field is -polarized. Such field interacts better with the arms oriented along . It is worthwhile to compare the proposed technique to the holography technique in [32]. Fig. 5 shows the reconstructed

images for the targets in Fig. 2(c) when applying the 3-D holography technique presented in [32]. Note that this technique makes use of the reflection (back-scattered) signals only. The same number of frequencies is adopted and the reconstructed

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Fig. 5. Reconstructed images obtained for the dielectric targets in Fig. 2(c) when applying the 3-D holography technique in [32] to: (a) s (b) s (RE = 1076).

(RE = 804) and

Fig. 6. Reconstructed images obtained for the dielectric targets in Fig. 2(c) when applying our proposed 3-D holography technique to: (a) s (b) s (RE = 728).

(RE = 781) and

images are shown on the same planes. As observed, artifacts are present and the target shapes are not recovered well. For the sake of comparison, Fig. 6 shows the reconstructed images obtained from our proposed technique where only the reflection -parameters are processed. The comparison of the RE values for the images in Fig. 5 with those in Fig. 6 and confirms the advantages of using adequate representation of the incident field and the Green’s function in our formulation instead of using the plane-wave assumptions as in [32], which is inadequate in near-field imaging. In addition, the possibility of using transmission -parameters in conjunction with the reflection ones in our technique leads to even better results as confirm this shown in Fig. 4(c). The values of RE and observation. For the three examples presented above, we have used the data recorded at 11 frequencies in the range of 25 GHz to 45 GHz). We observe that when using the data at GHz ( GHz, the more frequencies, e.g., 21 frequencies or quality of the reconstructed images remains almost the same. Also, we observe that very similar results with slight degrada-

tion of quality are obtained when using the simulated data only GHz). However, at 5 frequencies in the same band GHz) leads reducing the number of frequencies to 3 to serious degradation of the reconstructed images. As an example, Fig. 7 shows the results of reconstructing the targets in Fig. 2(b) with the data recorded at 5 and 3 frequencies. As shown values increase with decreasing in the figure, the RE and the number of sampling frequencies. B. Reconstruction Example for a Background Medium with and S/m Since the late 1970’s, when Larsen and Jacobi pioneered the use of scattering parameter imagery for biomedical applications [39], such applications have attracted a lot of attention. Here, we present an example in which the background medium is lossy dielectric. We chose the dielectric properties and the operating frequency band to be close to those considered in the microwave imaging of biological tissues. In this mm and example, the aperture planes are located at . As shown in Fig. 8, the targets are X-shape objects cre-

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N = 5; 1f = 5 GHz) (RE = 448) and (b) 3

Fig. 7. Reconstructed images obtained for the dielectric target in Fig. 2(b) when using: (a) 5 frequencies ( GHz) . frequencies (

N = 3;1f = 10

(RE = 973)

being 70 mm, the maximum is 0.26 GHz. Here, for GHz which leads to 29 frequencies in the we use range of 3 GHz to 10 GHz. Fig. 9 shows the reconstructed mm and images. The targets are recovered well at mm. The images formed at other range locations show some artifacts, which are weaker than the images at the slices containing the targets. These artifacts can be reduced further by a method described later. C. Reconstruction at Higher Contrasts and Larger Sizes

" = 32 " = 16  = 0:5 z=0

=1 z = 80 S

Fig. 8. Dielectric targets with and S/m in a background medium with and S/m scanned by two (at 6.5 GHz) -polarized dipoles; dipole 1 is moving on the mm plane while dipole 2 is moving on the mm plane. The simulated -parameters are recorded in the frequency band from 3 GHz to 10 GHz for two similar X-shape objects. Each arm of the X-shape has a square cross-section 2 mm on a side and a length of 20 mm. The arms are parallel to the - plane. One centered at (0,0,30) mm with its arms along the and axes. The other one is centered at (0,0,54) mm, with the arms rotated by 45 degrees with respect to the and axes.

x

x

y

=2

xy

x

y

ated from two arms of length 20 mm and square cross-sections mm and the 2 mm on a side. One target is placed at other one, rotated by 45 , is placed at the range position mm. The constitutive parameters of the targets are and S/m while the background is with and S/m. The -parameters are collected in the range of 3 GHz to 10 GHz (the center frequency is 6.5 GHz) and this leads to .4 mm, according to (18). a range resolution limit of must be fulfilled, in all examples, we set the Since distance between the reconstruction planes as 6 mm. Since the maximum frequency in the simulations is 10 GHz, from (19), the spatial sampling rate for the 2-D scan has to be smaller than 1.9 mm. Therefore, a sampling rate of 1.5 mm along both and is adopted. Also, we approximate the minimum number of . For example, frequencies from (20) which depends on

In all examples considered so far, the ratio of the permittivity of the target to that of the background medium was 2. To investigate the quality of the imaging for targets with higher contrasts, a pair of cuboids are placed at mm as shown in Fig. 2(d). The constitutive parameters of the cuboids in this example are and S/m while the background is free space. Three cases are investigated where the cuboids have a side of mm, 2.6 mm, and 3 mm. Fig. 10(a) shows the reconstructed images for the case of mm. It is clear that the objects are mm reconstructed well. However, for larger objects ( and 3 mm), the quality degrades, i.e., artifacts appear and the values increase. RE and The degradation of the images when increasing the size of the cuboids is due to the fact that the criterion for the first-order Born approximation is violated for larger cuboids [40]. This approximation is indeed used in deriving (5) when approximating the field inside the targets with the incident field. The condition for applying first-order Born approximation is that the radius of a sphere enclosing the target and its refractive index satisfy [40] (24) For . Thus, for GHz ( mm), the radius of the object needs to be smaller than 1.4 mm for the linear Born approximation to hold. For the targets with mm and 3 mm, this condition does not hold and thus the quality of the images degrades. We note, however, that the detection capability (or the sensitivity) of the algorithm remains good even at high contrasts.

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Fig. 9. Reconstructed images for the example shown in Fig. 8 (RE = 966).

Fig. 10. Reconstructed images obtained from the simulation of dielectric targets in Fig. 2(d) with: (a) d = 2 mm (RE = 324), (b) d = 2:6 mm (RE = 378), and (c) d = 3 mm (RE = 753).

D. Effect of Random Noise on the Reconstruction Results In order to investigate the effect of random noise on the proposed 3-D holography technique, we consider two types of noise: 1) white Gaussian noise added to the magnitude of the -parameters (such noise could be produced by electronic

devices or the environment), 2) white Gaussian noise added only to the phase of the -parameters (such noise could be due to the mechanical vibrations of the antennas). For both types, we apply high levels of noise. Fig. 11 shows the reconstructed images for the targets in Fig. 2(c) when adding noise of type 1 and 2, respectively. For

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Fig. 11. Reconstructed images for the example of Fig. 2(c) when: (a) adding noise to the complex S -parameters with SNR = adding noise to only to the phase of the S -parameters with SNR = 10 dB (RE = 896).

010 dB (RE

= 1101), and (b)

SNR value as low as dB for the first type of noise and 10 dB for the second type, the quality of the reconstructed images is still comparable to that of the images obtained from noiseless degrade. This indidata [see Fig. 4(c)] although RE and cates that the proposed reconstruction technique is very robust dB and 10 dB for noise to noise. SNR values lower than of type 1 and 2, respectively, lead to visible degradation of the reconstructed images. E. Suppressing Artifacts Along Range To improve the quality of the image reconstruction, we make use of a minimization procedure which was first proposed in [35] for the purpose of locating a target along the range in single-frequency 2-D holography imaging. Here, the method is used in the 3-D reconstruction algorithm to suppress artifacts along the range. The process involves the data, the images following steps. First, using only the , are obtained. Second, using only the data, the images , and imare obtained. Third, the ages are subtracted at each reconstruction plane and the 2-norm of the difference is taken as

Fig. 12. Variation of cost function in (25) for the example of Fig. 4(c).

As an example, Fig. 12 shows the cost function (25) computed for the example of Fig. 2(c). It has the lowest values at mm and mm, where the actual targets are. From the reconstructed images shown in Fig. 4(c), an object is also observed mm. However, the large value of the cost function at at this range location indicates that this object is an artifact. Fig. 13 with the original compares the enhanced images reconstructed images. Note that the RE and values reduce mm in Fig. 4(c) disappears. and the artifact seen at

(25) Assuming that the artifacts are not similar in and unlike the targets, which should appear consistently in both images, we expect that the function in (25) has minima in the true positions of the targets. This can be employed to suppress the artifacts as explained next. can be used to weigh properly the slice The function is images along the range. Images at range positions where is large. small are assigned larger weight than those where This is implemented using (26) where

is the enhanced image.

VI. CONCLUSION A general 3-D holography technique for near-field microwave imaging of dielectric targets is proposed, which allows for incorporating both transmission and reflection data. The data are acquired by two antennas moving on two parallel apertures on the opposite sides of the inspected region and recording wideband data at each position. This technique, compared to previously proposed 3-D holography techniques, has a number of distinct features and advantages. First, the forward-scattered signals contribute to the image reconstruction in addition to the back-scattered signals. This additional information leads to reconstructed images of higher fidelity. It also allows for the efficient suppression of artifacts along range. Second, the incident field is supplied by measurements or simulations and need not be in the ana-

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Fig. 13. (a) Original reconstructed images when using all S -parameters simultaneously for the dielectric targets (RE = 393) and (b) enhanced images using (26) (RE = 277).

lytical form of plane, cylindrical, or spherical waves. Third, the Green function of the scattering problem is supplied by simulation and need not be approximated by the free-space Green’s function. The acquisition of the Green function exploits the reciprocity principle and requires only one simulation. The above two features make this technique suitable for near-field imaging. This advantage is especially important when the target is in the near zone of the antenna because, in this case, the analytical incident-wave representations are inadequate [41]. Fourth, the numerical formats of the incident field and Green’s function necessitate a new approach to the solution of the inverse problem, which exploits a least-square solution at each spatial frequency . We should emphasize that this approach provides pair systems of equations with much smaller number of unknowns compared to the systems of equations constructed in regular optimization-based techniques (where the number of unknowns is equal to the number of reconstructed voxels). This reduces the ill-posedness of the proposed technique significantly. Fifth, the algorithm is fast, accurate, and robust to high levels of noise. The execution time is in the order of few minutes on a Pentium 4 with CPU speed of 3 GHz and 4 GB of RAM. As long as the criterion for the first-order Born approximation is fulfilled, high-quality artifact-free images of the targets are obtained. The technique is capable of providing high-quality 3-D images with the minimum required number of sampling frequencies. It facilitates robust, reliable, and fast near-field microwave imaging of dielectric bodies with possible application in biomedical imaging where the acquisition of both reflected and transmitted signals is feasible. It is worth noting that using cross-polarized data leads to more accurate reconstructed images. This involves using dual-polarization antennas. APPENDIX FORMULATION OF INVERSE SCATTERING PROBLEM IN A DIELECTRIC MEDIUM

(A.2) where and are the electric and magnetic field vectors, respectively, and and are the permeability of the medium and the angular frequency, respectively. From (A.1) and (A.2), the vector Helmholtz equation for the -field is obtained: (A.3) , and and are the permittivity where and the conductivity of the (isotropic) background medium. Consider two scenarios: 1) the target is not present and 2) the target is present. For these two cases, (A.3) gives (A.4) (A.5) and are the incident and total field vectors, where and are the complex wavenumbers in respectively, and the investigated region for the cases without and with the target, respectively. From (A.4) and (A.5), we obtain (A.6) where

. The solution for (A.6) is [23]. (A.7)

where is the target volume and is the dyadic Green’s in (A.7) can be approxfunction. Under certain conditions, imated by the incident field, i.e., . This is known as the linear Born approximation for weak or small scatterers [23]. In this case, (A.7) is approximated as

In a time-harmonic regime and assuming only electric current excitation J in a nonmagnetic medium, the field is governed by Maxwell’s equations (A.1)

(A.8)

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REFERENCES [1] E. C. Fear, S. C. Hagness, P. M. Meaney, M. Okoniewski, and M. A. Stuchly, “Enhancing breast tumor detection with near-field imaging,” IEEE Microw. Mag., vol. 3, no. 1, pp. 48–56, 2002. [2] N. H. Farhat and W. R. Guard, “Millimeter wave holographic imaging of concealed weapons,” Proc. IEEE, vol. 59, pp. 1383–1384, 1971. [3] E. J. Baranoski, “Through-wall imaging: Historical perspective and future directions,” J. of the Franklin Institute, vol. 345, no. 6, pp. 556–569, 2008. [4] R. Zoughi, Microwave Non-Destructive Testing and Evaluation. : Kluwer Academic Publishers, 2000. [5] T. Rubek, P. M. Meaney, P. Meincke, and K. D. Paulsen, “Nonlinear microwave imaging for breast-cancer screening using Gauss-Newton’s method and the CGLS inversion algorithm,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2320–2331, 2007. [6] X. Li, E. J. Bond, B. D. Van Veen, and S. C. Hagness, “An overview of ultra-wideband microwave imaging via space-time beamforming for early-stage breast-cancer detection,” IEEE Antennas Propag. Mag., vol. 47, no. 1, pp. 19–34, 2005. [7] W. Tabbara, B. Duchene, C. Pichot, D. Lesselier, and L. Chommeloux, “Diffrcation tomography: Contribution to the analysis of some applications in microwave and ultrasonics,” Inverse Problems, vol. 4, pp. 305–331, 1988. [8] D. M. Sheen, D. L. McMakin, and T. E. Hall, “Near field imaging at microwave and millimeter wave frequencies,” in IEEE/MTT-S Int. Microwave Symp., 2007, pp. 1693–96. [9] M. A. Ali and M. Moghaddam, “3-D nonlinear super-resolution microwave inversion technique using time-domain data,” IEEE Trans. Antennas Propag., vol. 58, no. 7, pp. 2327–2336, 2010. [10] Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation,” IEEE Trans. Med. Imag., vol. 23, no. 4, pp. 475–584, 2004. [11] P. Lobel, L. Blanc-Feraud, C. Pichot, and M. Barlaud, “A new regularization scheme for inverse scattering,” Inverse Problems, vol. 13, pp. 403–410, 1997. [12] D. J. Kurrant and E. C. Fear, “An improved technique to predict the time-of-arrival of a tumor response in radar-based breast imaging,” IEEE Trans. Biomed. Eng., vol. 56, no. 4, pp. 1200–1208, 2009. [13] M. O’Halloran, M. Glavin, and E. Jones, “Effects of fibroglandular tissue distribution on data-independent beamforming algorithms,” Progress In Electromagnetics Research, PIER, vol. 97, pp. 141–158, 2009. [14] I. Yamaura, Non-Invasive Thermometry by Means of Microwave Active Imaging Tech. Rep. IEICE, MW85-81, 1985, pp. 81–100. [15] M. Miyakawa, “Tomographic change of temperature change in phantoms of the human body by chirp radar-type microwave computed tomography,” Med. Biol. Eng. Comput., vol. 31, pp. S31–S36, 1993. [16] M. Bertero, M. Miyakawa, P. Boccacci, F. Conte, K. Orikasa, and M. Furutani, “Image restoration in chirp-pulse microwave CT (CP-MCT),” IEEE Trans. Biomed. Eng., vol. 47, no. 5, pp. 690–699, 2000. [17] A. M. Massone, M. Miyakawa, M. Piana, F. Conte, and Bertero, “A linear model for chirp-pulse microwave computed tomography application conditions,” Inverse Problems, vol. 22, pp. 2209–2222, 2006. [18] C. Pichot, L. Jofre, G. Peronnet, A. Izadnegahdar, and J. C. Bolomey, “An angular spectrum method for inhomogeneous bodies reconstruction in microwave domain,” in IEEE/AP-S and URSI Symp., 1982. [19] M. F. Adams and A. P. Anderson, “Synthetic aperture tomographic (SAT) imaging for microwave diagnostics,” in IEEE Proc., 1982, vol. 129, pp. 83–88. [20] J. C. Bolomey, A. Izadnegahdar, L. Jofre, C. Pichot, G. Peronnet, and M. Solaimani, “Microwave diffraction tomography for biomedical applications,” IEEE Trans. Microwave Theory Tech., vol. 30, no. 11, pp. 1998–2000, 1982. [21] J. C. Bolomey, L. Jofre, and G. Peronnet, “On the possible use of microwave-active imaging for remote thermal sensing,” IEEE Trans. Microwave Theory Tech., vol. 31, no. 9, pp. 777–781, 1983. [22] J. M. Rius, C. Pichot, L. Jofre, J. C. Bolomey, N. Joachimowicz, A. Broquetas, and M. Ferrando, “Planar and cylindrical active microwave temperature imaging: Numerical simulations,” IEEE Trans. Med. Imag., vol. 11, no. 4, pp. 457–469, 1992. [23] W. Chew, Waves and Fields in Inhomogeneous. Media Piscataway, NJ: IEEE Press, 1995.

[24] M. Soumekh, “Bistatic synthetic aperture radar inversion with application in dynamic object imaging,” IEEE Trans. Signal Processing, vol. 39, pp. 2044–2055, 1991. [25] D. C. Munson, J. D. O’Brien, and W. K. Jenkins, “A tomographic formulation of spotlight-mode synthetic aperture radar,” Proc. IEEE, vol. 71, no. 8, pp. 917–925, 1983. [26] D. Gabor, “A new microscopic principle,” Nature, vol. 161, pp. 777–779, 1948. [27] E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Amer., vol. 52, pp. 1123–1130, 1962. [28] N. H. Farhat, “Microwave holography and its applications in modern aviation,” in Proc. SPIE Eng. Applicat. Holography Symp., 1972, pp. 295–314. [29] G. Tricoles and N. H. Farhat, “Microwave holography: Applications and techniques,” Proc. IEEE, vol. 65, no. 1, pp. 108–121, 1977. [30] B. P. Hildebrand and K. A. Haines, “Holography by scanning,” J. Opt. Soc. Amer., vol. 59, pp. 1–6, 1969. [31] B. P. Hildebrand and B. B. Brenden, An Introduction to Acoustical Holography. New York: Plenum, 1972. [32] D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microwave Theory Tech., vol. 49, no. 9, pp. 1581–1592, 2001. [33] D. M. Sheen, D. L. McMakin, and T. E. Hall, “Near-field three-dimensional radar imaging techniques and applications,” Applied Optics, vol. 49, no. 19, pp. E83–E93, 2010. [34] J. Detlefsen, A. Dallinger, and S. Schelkshorn, “Effective reconstruction approaches to millimeter-wave imaging of humans,” in XXVIIIth General Assembly of Int. Union of Radio Science (URSI), 2005, pp. 23–29. [35] M. Ravan, R. K. Amineh, and N. K. Nikolova, “Two-dimensional nearfield microwave holography,” Inverse Problems, vol. 26, no. 5, 2010. [36] M. Lazebnik, M. Okoniewski, J. H. Booske, and S. C. Hagness, “Highly accurate Debye models for normal and malignant breast tissue dielectric properties at microwave frequencies,” IEEE Trans. Microwave Theory Tech., vol. 17, no. 12, pp. 822–824, 2007. [37] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [38] EM Softwares & Systems-S.A. (Pty) Ltd. [Online]. Available: http:// www.feko.info [39] L. E. Larsen and J. H. Jacobi, “Microwave scattering parameter imagery of an isolated canine kidney,” Med. Phys., vol. 6, no. 5, pp. 394–403, 1979. [40] M. Slaney, A. C. Kak, and L. E. Larsen, “Limitation of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech., vol. 32, no. 8, pp. 860–874, 1984. [41] C. A. Balanis, Antenna Theory, 3rd ed. Hoboken, New Jersey: Wiley, 2005.

Reza K. Amineh (S’08-M’11) received the B.Sc., M.Sc., and Ph.D. degrees, all in electrical engineering from Sharif University of Technology, Amirkabir University of Technology, and McMaster University in 2001, 2004, and 2010, respectively. From 2004 to 2006 he was a Researcher with the telecommunications industry in Iran. From 2006 to 2010, he was a Research Assistant and Teacher Assistant at the Department of Electrical and Computer Engineering, McMaster University, Canada. During the winter of 2009 he was a Ph.D. intern with the Advanced Technology Group, Research In Motion (RIM), Waterloo, Canada. He is the recipient of the McMaster Internal Prestige Scholarship “Clifton W. Sherman” for two consecutive years in 2008 and 2009. He is also the recipient of the Ontario Ministry of Research and Innovation (MRI) post-doctoral fellowship for 2010 and 2011. He is the co-author of an honorable mention paper presented at the IEEE APS/URSI, 2008. His research interests include forward and inverse solutions in electromagnetics with applications in biomedical imaging and non-destructive testing, antenna design and measurements, and high-frequency computer-aided analysis and design. He has authored and co-authored over 40 journal and conference papers and a book chapter.

AMINEH et al.: THREE-DIMENSIONAL NEAR-FIELD MICROWAVE HOLOGRAPHY USING REFLECTED AND TRANSMITTED SIGNALS

Maryam Ravan received the Ph.D. degree from Amirkabir University of Technology, Tehran, Iran, in 2007. From May 2007 to December 2007 she was a Postdoctoral Fellow with the Department of Electrical and Computer Engineering, University of Toronto, Canada. From January 2008 to April 2011 she was a Postdoctoral Fellow and Lecturer with the Department of Electrical and Computer Engineering and the School for Computational Engineering and Science, McMaster University, Canada. Since August, 2009, she has been a Postdoctoral Fellow with the Department of Electrical Engineering, University of Toronto. Her research interests include biomedical signal and image processing, neural and wavelet networks, machine learning, optimization techniques, MIMO radar systems and space-time adaptive processing, power spectrum estimation, and non-destructive testing. She has authored and co-authored over 30 journal and conference papers and a book chapter.

Ali Khalatpour received the B.Sc. and M.Sc. degrees in electrical engineering from Amirkabir University of Technology (Tehran Polytechnic) and McMaster University in 2009 and 2011, respectively. In July 2011, he joined the Department of Electrical and Computer Engineering, McMaster University where he is currently working as a research associate in the Computational Electromagnetics Research Laboratory. His research interests include forward and inverse solutions in electromagnetics with applications in biomedical imaging, antenna design and measurements, and high-frequency computer-aided analysis and design.

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Natalia K. Nikolova (S’93–M’97–SM’05–F’11) received the Dipl. Eng. degree from the Technical University of Varna, Bulgaria, in 1989, and the Ph.D. degree from the University of Electro-Communications, Tokyo, Japan, in 1997. From 1998 to 1999, she held a Postdoctoral Fellowship of the Natural Sciences and Engineering Research Council of Canada (NSERC), during which time she was initially with the Microwave and Electromagnetics Laboratory, DalTech, Dalhousie University, Halifax, Canada, and, later, for a year, with the Simulation Optimization Systems Research Laboratory, McMaster University, Hamilton, ON, Canada. In July 1999, she joined the Department of Electrical and Computer Engineering, McMaster University, where she is currently a Professor. Her research interests include theoretical and computational electromagnetism, inverse scattering and microwave imaging, as well as methods for the computer-aided analysis and design of high-frequency structures and antennas. Prof. Nikolova was the recipient of a University Faculty Award of NSERC from 2000 to 2005. Since 2008, she is a Canada Research Chair in High-frequency Electromagnetics. She is a Fellow of the IEEE and is currently serving as a Distinguished Microwave Lecturer. She is also a member of the Applied Computational Electromagnetics Society (ACES) and a correspondent of the International Union of Radio Science (URSI).

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A Multiplicative Regularized Gauss–Newton Inversion for Shape and Location Reconstruction Puyan Mojabi, Member, IEEE, Joe LoVetri, Senior Member, IEEE, and Lotfollah Shafai, Life Fellow, IEEE

Abstract—A multiplicative regularized Gauss–Newton inversion algorithm is proposed for shape and location reconstruction of homogeneous targets with known permittivities. The data misfit cost functional is regularized with two different multiplicative regularizers. The first regularizer is the weighted -norm total variation which provides an edge-preserving regularization. The second one imposes a priori information about the permittivities of the objects being imaged. Using both synthetically and experimentally collected data sets, we show that the proposed algorithm is robust in reconstructing the shape and location of homogeneous targets. Index Terms—Gauss–Newton inversion, microwave tomography, regularization.

I. INTRODUCTION

I

N some applications of microwave tomography, there may exist a priori information about the objects being imaged. Proper incorporation of such information into the utilized inversion algorithm can improve reconstruction results as compared to the results obtained from blind inversion algorithms. There are different ways of incorporating such information into the inversion algorithm; e.g., by introducing a dummy variable over which to perform minimization [1], [2] or by utilizing an appropriate regularization term [3], [4]. In this paper, we consider one type of a priori information and attempt to incorporate it within the Gauss–Newton inversion algorithm via an appropriate regularization term. The a priori information considered herein is that the object of interest (OI) consists of some homogeneous scatterers with known permittivity values. The goal is then to find the shape and location of these scatterers. This problem is sometimes referred to as shape and location reconstruction. This approach can be very useful for nondestructive testing applications such as detection of voids in concrete [5], [6]. For the so-called binary shape and location reconstruction, where all the homogeneous scatterers have the same known permittivity value, Crocco and Isernia [4] introduced an additive regularizer for the contrast source inversion (CSI) algorithm which pushes each pixel in the discretized imaging domain to have either a contrast corresponding to the known permittivity value of the scatterers or a contrast of zero. Allowing the inversion algorithm to converge to a zero contrast

is important as part of the imaging domain which is not occupied by the OI has the contrast of zero. Thus, the binary inversion algorithm attempts to find the spatial distribution of these two different contrasts within the imaging domain. The weight of this additive regularizer was chosen using an ad hoc algorithm [4]. Based on this algorithm, Abubakar and van den Berg [3] introduced a multiplicative regularizer (MR) which can provide an adaptive regularization [7] in the framework of the CSI algorithm. They also extended their algorithm for the case when there are several homogeneous targets inside the imaging domain. That is, it is more than a binary inversion algorithm which is only capable of reconstructing the shape and location of some homogeneous targets with the same known permittivity value. Inspired by the work of Abubakar and van den Berg [3], we introduce a Gauss–Newton inversion (GNI) algorithm for shape and location reconstruction. As will be seen, the proposed algorithm is capable of incorporating a priori information about several homogeneous targets inside the imaging domain. The proposed inversion algorithm utilizes two different MRs. The first one is the weighted -norm total variation which provides an edge-preserving regularization. The second regularizer, which is similar to the one used in the CSI algorithm for shape and location reconstruction, attempts to push the GNI algorithm to select one of the known permittivity values in any particular region of the imaging domain. Using both synthetically and experimentally collected data, we show that the proposed algorithm is robust in reconstructing the shape and location of homogeneous targets. Within the framework of this paper, we consider the 2-D transverse magnetic (TM) formulation and assume a time factor of . The paper is organized as follows. The mathematical formulation of the microwave tomography problem within the framework of the GNI algorithm is presented in Section II. The multiplicative regularized Gauss–Newton inversion algorithm, a blind inversion algorithm, is briefly explained in Section III. In Section IV, we present the proposed Gauss–Newton inversion algorithm for shape and location reconstruction. Sections V and VI provide the reconstruction results. Conclusions are provided in Section VII. II. PROBLEM FORMULATION

Manuscript received October 05, 2010; revised May 09, 2011; accepted June 16, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. This work was supported by the Natural Sciences and Engineering Research Council of Canada. The authors are with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB R3T5V6 Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165487

Consider a bounded imaging domain which contains a nonmagnetic OI. Denoting the relative complex permittivity of the homogeneous background medium by , the complex contrast function is defined as (1) where

is the position vector.

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MOJABI et al.: A MULTIPLICATIVE REGULARIZED GAUSS–NEWTON INVERSION FOR SHAPE AND LOCATION RECONSTRUCTION

The goal is to find the unknown contrast function from the measured scattered field on the measurement domain which is located outside the OI. To collect the scattered field data, the OI is successively interrogated by some known incident fields where denotes the number of the active transmitter. Interaction of the incident field with the OI results in the total field . Measuring incident and total electric fields on the measurement domain , the scattered electric field on is found as . The microwave tomography problem may then be formulated as the minimization over of the least squares data misfit cost functional (2) is the simulated scattered field on the measurewhere ment domain corresponding to the contrast and the th transmitter, is the total number of transmitters, and denotes the -norm on . The weighting is chosen to be (3) is nonlinear and ill-posed. The data misfit cost functional The ill-posedness of the cost functional can be treated by different regularization techniques [7]. The nonlinearity of the problem is handled by utilizing iterative techniques such as the Gauss–Newton inversion (GNI) method. In the GNI algorithm, which is based on the Newton optimization but ignores the second derivative of the scattered electric field with respect to , the contrast at the th iteration is updated as where is the predicted contrast at the th iteration, is an appropriate step-length, and is the correction. In the discrete setup, we discretize the imaging domain into cells using 2-D pulse basis functions. Thus, the contrast function is represented by the complex vector . Assuming the number of measured data to be , the measured scattered data on the discrete measurement domain are denoted by the complex vector which is the stacked version of the measured scattered fields for each transmitter. The vector is formed by stacking the discrete forms of . The data misfit cost functional maps spaces of complex functions defined on into a real number . The discrete form of this cost functional, which maps the complex vector into a real number , is denoted by the same symbol used in the continuous domain. That is, represents the discrete form of . III. MULTIPLICATIVE REGULARIZED GAUSS–NEWTON INVERSION (MR-GNI) We may regularize the data misfit cost functional by the weighted -norm total variation MR. This regularizer, which was first developed for the CSI algorithm [8], was recently adapted to the GNI algorithm [1], [7], [9], [10]. Utilizing this regularizer with the GNI method, we construct the following cost functional at the th iteration of the algorithm: (4)

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The weighted -norm total variation MR , which changes at each iteration of the algorithm, is given as [1], [7], [9] (5) is the area of the imaging domain and the gradient where is taken with respect to the position vector . The steering parameter is chosen to be [1], [9] (6) where is the area of a single cell in the uniformly discretized domain . In the discrete domain, we minimize over the complex vector . The complex correction vector may be found from [1], [7], [9] (7) where is the Jacobian matrix. This matrix is formed by stacking matrices where represents the discrete form of the derivative of the scattered field with respect to the contrast and evaluated at . That is, the matrix represents the discrete form of . The discrepancy vector is given as (8) and . The regularization operator represents the discrete form of the operator “ ” where “ ” is the divergence operator and (9) ” We note that the weighted Laplacian operator “ provides edge-preserving characteristics for the inversion algorithm [9], [11]. Also, it should be noted that the null space of the operator does not intersect with that of [see (7)], thus ensuring a unique solution at each iteration of the algorithm. (For more explanation, see [12, pp. 53–54].) Having found the correction , the contrast is updated in the form of where is the step length determined via an appropriate line search algorithm. This completes the brief explanation of the multiplicative regularized Gauss–Newton inversion which we refer to as the MR-GNI method in this paper. IV. GAUSS–NEWTON INVERSION FOR SHAPE AND LOCATION RECONSTRUCTION (SL-GNI) Assume that the imaging domain consists of homogeneous targets, each of which has a known contrast of where . The goal is to reconstruct the shape and location of these objects using the measured scattered field on . In the framework of the CSI algorithm, Abubakar and van den Berg proposed a multiplicative regularization term which pushes each pixel of the imaging domain to be one of these known contrast values [3]. This MR can be written as (10)

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In their work, the steering parameter was chosen so that the regularization factor is more dominant as the number of CSI iterations increases [3]. in the In our work, we use the regularization term framework of the GNI algorithm. To make the inversion algorithm more stable, we add one more level of regularization -norm total variation multiplicative using the weighted . That is, we construct the regularized regularization term at the th iteration of the inversion cost functional algorithm as (11) Thus, the data misfit cost functional is regularized with two and . different multiplicative regularization terms: The latter imposes the a priori information about the OI. The former is mainly used to impose the weighted Laplacian operat different iterations of the GNI algorithm so as to make ator the inversion algorithm more stable. To the best of our knowledge, it is the first time that two different MRs have been used to regularize the microwave tomography problem. to be In our work, we choose the steering parameter (12) is similar to the choice of , both of which This choice for decrease as the predicted contrast converges to the true solution. The only difference between and is that the former is dewhereas the latter is not. This can be justified pendent on by noting that is added to [see (5)], which depends on the area of each pixel within the imaging domain. Thus, this choice of minimizes the discretization dependency in the performance of . However, it is not the case for which is added to ; see (10). over the complex vector , the comMinimizing may be found from (see Appendix I plex correction vector for the derivation)

(13) is a vector of all ones. The matrix where is a diagonal matrix given as (14) where

and location reconstruction, we use the line search algorithm explained in [13] to find an appropriate step length. It should be noted that the computational complexity of the proposed GNI algorithm for shape and location reconstruction is similar to that matrices are all diagonal. (The of the MR-GNI method as computational complexity analysis of the MR-GNI method can be found in [7, App. B].) This completes the brief explanation of the Gauss–Newton inversion for shape and location reconstruction. In this paper, we refer to this algorithm as the SL-GNI algorithm. It is worth noting that this algorithm uses two different regularizers to improve the reconstruction results. In comparison, the CSI algorithm for shape and location reconstruction, herein regreferred to as the SL-CSI method, utilizes only the ularizer [3]. From our experience with the GNI method, utiand results in a more robust reconstruclizing both tion than utilizing only the regularizer. This will be shown in Sections V-B and VI-B. Moreover, although the shape and used in the SL-GNI method is simlocation regularizer ilar to the one used in the SL-CSI algorithm, its incorporation into the GNI algorithm requires that the steering parameter be chosen in a different way. This is due to the fact that the steering parameter used in the SL-CSI algorithm is chosen to be the normalized error in the so-called domain equation [3, eq. 16]. However, in the SL-GNI algorithm, there is no explicit domain equation in the cost functional to be minimized. Thus, we to be so that this pachoose the steering parameter rameter decreases as the algorithm gets closer to the solution. We would also like to add that the main disadvantage of the SL-CSI method is that there is no closed-form formula to find the step length [3, p. 6] as opposed to the multiplicative regularized contrast source inversion (MR-CSI) method where the step length is found using a closed-form formula (see [8, eq. (33) and (41)]), thus, in the SL-CSI algorithm, a numerical line search algorithm needs to be used. This can increase the computational complexity of the SL-CSI significantly as this algorithm usually requires a few hundred iterations (the number of iterations is usually set to 1024 in the contrast source inversion algorithm). On the other hand, although both the SL-GNI and MR-GNI algorithms require numerical line searches to calculate the step length, the number of iterations for these two algorithms is much lower than that required for the SL-CSI and MR-CSI methods (the number of iterations is usually lower than 20 in both SL-GNI and MR-GNI methods). Thus, utilizing a numerical line search algorithm does not add a large computational burden to the SL-GNI and MR-GNI algorithms. Specially, if adaptive regularization, like the one used in our paper, is incorporated with an appropriate line search algorithm, the number of calls to the numerical line search algorithm is minimal in the GNI algorithm (for more discussion, see [14, Sec. V.D]).

is the discretized form of (15)

As in the standard MR-GNI method, the contrast is updated as where is an appropriate step length. For both the MR-GNI method and the GNI method for shape

V. SYNTHETIC DATA RESULTS In this section, we test the performance of the proposed algorithm against two different synthetically collected data sets. To avoid any inverse crime, the synthetic data sets are generated on a different grid than the ones used in the inversion algorithm.

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We also add 3% root mean square (RMS) additive white noise to the synthetic data set using [15]

(16) where is the scattered field on the measurement domain due to the th transmitter obtained by the forward solver, and are two real vectors whose elements are uniformly distributed zero-mean random numbers between and , and . The vector , constructed by stacking the vectors , is then used to test the inversion algorithm against synthetic data sets. Finally, it should be mentioned that the starting guess for the contrast to be found is set to be zero in both MR-GNI and SL-GNI algorithms for these synthetic data sets. In all examples, including the experimental data results, we ensure that we have at least ten cells per wavelength. We note that the proposed algorithm is robust with respect to oversampling of the unknown contrast. This is discussed in Appendix II. A. Concentric Squares We consider a similar scatterer which has been used in [14] and [16]–[18]. The scatterer consists of two concentric squares located in free space, the inner square having dimension of 0.3 m 0.3 m with a relative complex permittivity of , which corresponds to a contrast of . The inner square is surrounded by an exterior square having dimension of 0.6 m 0.6 m with a relative complex permittivity of , which corresponds to a contrast of . The exact relative complex permittivity profile is shown in Fig. 1. We consider this target in two different scenarios distinguished by their frequency of operation. In both scenarios, the synthetic data, which include 16 transmitters (line sources) and 16 receivers per transmitter evenly placed on the measurement circle of radius 0.7 m, are generated using a grid of 80 80 square pulses in a 0.9 m 0.9 m square. The imaging domain is chosen to be a 0.94 m 0.94 m and is discretized into 61 61 square pulses. It should also be noted that the contrast of the object being imaged is assumed to be the same in both frequencies of operation. In the first scenario, the frequency of operation is chosen to be 100 MHz. The inversion of this data set using the MR-GNI method is shown in Fig. 2(a) and (b). As can be seen, the MR-GNI method is not capable of resolving the concentric squares. We also utilize the SL-GNI algorithm with three different values for which correspond to the true contrast values within the imaging domain; i.e., , and . Utilizing these three values for , the shape and location reconstruction of this target are shown in Fig. 2(c) and (d). Although the SL-GNI algorithm is not capable of reconstructing the square shape of the two scatterers, it does resolve two regions and provides a good location reconstruction. It is worth noting that the number of MR-GNI and SL-GNI iterations required for the convergence is 8 and 11, respectively. (The inversion algorithms are terminated when the difference between two successive data misfit values becomes less than .) The inversion algorithms were implemented in object-oriented MATLAB® and were

Fig. 1. Exact relative complex permittivity profile for the first synthetic test and (b) . case (concentric squares): (a)

running on a PC workstation with two Intel® Xeon® quad-core 2.8-GHz processors. With this machine, the first iterations of the MR-GNI and SL-GNI algorithms took about 7 and 3 s, respectively. In the second scenario, we choose the frequency of operation to be 1 GHz. The MR-GNI reconstruction of this target as well as its SL-GNI reconstruction using the three true values for are shown in Fig. 3. As can be seen, both of these algorithms can reconstruct the scatterer very well at this frequency of operation. To test the robustness of the SL-GNI algorithm to the chosen values for , we run this algorithm with 20% error in the utilized values for . Specifically, we assume 20% error in and 20% error in . That is, we consider and to be and , respectively. Of course, the value of , which corresponds to the contrast of the background medium, is kept to be 0. Utilizing these values for , the inversion results for the two frequencies of operation are shown in Fig. 4. As can be seen, the inversion results (shape and location) are very similar to the case where the true values of are incorporated to the SL-GNI algorithm. In order to test the performance of the SL-GNI algorithm when the a priori information about the number of contrast values is wrong, we consider two different cases. In the first case, an extra contrast value is given to the SL-GNI algorithm: in addition to , and , one extra contrast value is also given to the algorithm. That is, we utilize a quaternary inversion algorithm instead of a trinary inversion algorithm. In Fig. 5(a)–(d), we have shown the performance of the SL-GNI algorithm for this situation for two different values of at 100 MHz. For ,

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Fig. 2. Reconstruction of and of the concentric squares when 100 MHz: (a)–(b) the MR-GNI reconstructhe frequency of operation is tion, and (c)–(d) the SL-GNI reconstruction.

the algorithm resolves the two scatterers but converges to instead of . Also, the overall dimension of the reconstructed inner scatterer is wrong. For , the inversion re-

Fig. 3. Reconstruction of and of the concentric squares when 1 GHz: (a)–(b) the MR-GNI reconstruction, the frequency of operation is and (c)–(d) the SL-GNI reconstruction.

sult is very similar to the one obtained using only the three true contrast values [see Fig. 2(c)–(d)]. In the second case, we give only two contrast values to the SL-GNI algorithm, as opposed to the three contrast values cor-

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Fig. 5. The SL-GNI reconstruction of the concentric squares at 100 MHz when the a priori information about the number of contrast values is wrong and right: ]. Case I: In addition to the three true contrast [left: , and , one extra contrast values is also given to the SL-GNI algorithm: (a)–(b) , value . Case II: Only two contrast values are given to and (c)–(d) and , and (g)-(h) and . the SL-GNI algorithm: (e)–(f)

Fig. 4. The SL-GNI reconstruction of and of the concentric squares with 20% error in the utilized values for : (a)–(b) when the frequency of operation is 100 MHz, and (c)–(d) when the frequency of operation is 1 GHz.

responding to the target; i.e., utilizing a binary inversion algorithm instead of a trinary inversion algorithm. In Fig. 5(e)–(h), we have shown the performance of the SL-GNI algorithm for this situation for two sets of binary values at 100 MHz.

Giving and to the SL-GNI algorithm, the reconstruction result does not resolve the two scatterers. This inversion result is, in fact, similar to the blind inversion of this data set [see Fig. 2(a)–(b)]. Giving and to the SL-GNI algorithm, the algorithm does resolve the two scatterers and provides a reconstruction result which is very similar to the one obtained using the three true contrast values [see Fig. 2(c)–(d)]. These two different cases show that the SL-GNI algorithm can be very sensitive to the utilized values for if is chosen to be a wrong number. As the focus of this paper is for the case where the correct value of is known, we will not consider this case anymore. The usefulness of this algorithm when is not known requires further study and is not within the scope of this paper.

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TABLE I NUMBER OF SL-GNI AND MR-GNI ITERATIONS REQUIRED FOR THE CONVERGENCE OF THE FIRST SYNTHETIC DATA SET IN SOME REPRESENTATIVE SCENARIOS

(as opposed to 16 16 data points considered earlier). The reconstruction results using this new data set are shown in Fig. 6 for MR-GNI and SL-GNI. Comparing Fig. 6 with Fig. 2, it can be seen that only slight changes can be observed in the reconstruction results of this target by increasing the number of data points on the measurement circle. Thus, we speculate that we have reached the maximum amount of information that these two algorithms can extract from the given target at this specific frequency when the measured data, contaminated by 3% noise, is collected on the given measurement circle. That is, we speculate that increasing the number of transceivers from 16 to 32 has added redundant scattering information about this target for the given configurations. It should also be noted that although it is, in general, advantageous to increase the number of data points by having more coresident transmitters and receivers, it is in direct conflict with another design criteria which is the minimization of the mutual coupling between the coresident antenna elements [19, Ch. 7], [20]. That is also one of the reasons why the state-of-the-art microwave breast cancer imaging system at Dartmouth College utilizes only 16 monopole antennas [21]. Our microwave tomography system utilizes 24 Vivaldi antennas. Even with 24 Vivaldi antennas, we were not able to image at some frequencies due to the high mutual coupling between the coresident antenna elements at those frequencies [20]. Finally, the number of iterations for this synthetic example in some representative scenarios is given in Table I. The main criterion governing the number of iterations is the difference between two successive data misfit values: if the difference between two successive data misfit values becomes less than , the inversion algorithm is terminated. B. 6-Ary Target

Fig. 6. Reconstruction of and of the concentric squares when 100 MHz: (a)–(b) MR-GNI reconstrucilluminated by 32 transceivers at tion, and (c)–(d) SL-GNI reconstruction.

To see if any changes can be observed in the reconstruction result of the first scenario ( 100 MHz) using denser near-field sampling, the concentric squares are illuminated with 32 transmitters and the resulting scattered field is collected at 32 receivers per transmitter, thus, providing 32 32 data points

We consider the target shown in Fig. 7(a) and (b). The scatterer consists of a square having dimension of 0.03 m 0.03 m with a relative complex permittivity of which is located inside a cylinder of diameter 0.07 m with a relative complex permittivity of . Three smaller cylinders of diameter 0.031 m with three different relative complex permittivities, namely, , and , are located external to the larger cylinder. The background medium has a relative permittivity of 3 at the frequency of operation which is chosen to be 3 GHz. The target is interrogated using 32 transmitters (line sources) and 32 receivers which are evenly placed on the measurement circle of radius 0.12 m. The synthetic data are then generated using a grid of 100 100 square pulses

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(as opposed to minimizing over the contrast), the reconstruction result, shown in Fig. 7(g) and (h), is not satisfactory. VI. EXPERIMENTAL DATA RESULTS We consider two different experimental data sets. The first one is the FoamDielIntTM data set from the second Fresnel experimental data set [22] collected by the Institut Fresnel, France. The second data set is collected from the University of Manitoba air-filled microwave tomography system [20]. In both cases, the measured data are calibrated for the TM polarization so that the antennas can be represented by 2-D line sources. The calibration procedure adopted to calibrate the Fresnel data set is that explained in [23]. The calibration method utilized to calibrate the University of Manitoba data set is outlined in [20]. Similar to the inversion of synthetic data sets, the starting guess for the contrast to be found is set be zero in both MR-GNI and SL-GNI algorithms. A. The Second Fresnel Data Set: FoamDielIntTM

Fig. 7. The 6-ary target [left: and right: ]: (a)–(b) the true relative complex permittivity profile, (c)–(d) the MR-GNI reconstruction, (e)–(f) the SL-GNI reconstruction, and (g)–(h) the shape and location reconstruction . method without the use of

in a 0.15 m 0.15 m square. The imaging domain is chosen to be a 0.154 m 0.154 m and is discretized into 71 71 square pulses. The MR-GNI reconstruction of this target is shown in Fig. 7(c) and (d). As can be seen, the square has not been resolved in the MR-GNI real-part reconstruction. The MR-GNI imaginary-part reconstruction shows the presence of the square scatterer within the large cylinder; however, its dimension is very different than the true size of the square scatterer (i.e., 0.03 m 0.03 m). The SL-GNI reconstruction of this target utilizing the contrast of zero as well as the five true contrast values of the scatterer is shown in Fig. 7(e) and (f). As can be seen, both realand imaginary-part reconstructions resolve the square scatterer. Although the shape of the square scatterer is not reconstructed, its location and its approximate size have been reconstructed well. It is worth noting that if we remove the regularization term from (11) and minimize over the contrast

For this data set, the transmitting and receiving antennas are both wide-band ridged horn antennas and are located on a circle with radius 1.67 m. The target [see Fig. 8(a)] consists of a lossless cylinder of diameter 0.031 m with the relative permittivity of which is located inside another lossless cylinder of diameter 0.08 m with the relative permittivity of . This target is illuminated from eight different transmitter locations and the scattered data are collected at 241 locations per transmitter. The background medium is free space and the frequency of operation is chosen to be 3 GHz. The imaging domain is a 0.15 m 0.15 m square and is discretized into 60 60 square pulses. The MR-GNI reconstruction of this single-frequency data set is shown in Fig. 8(b) and (c). Although the MR-GNI algorithm resolves the two different cylinders, the periphery of the outer cylinder is blurred, thus, it is difficult to deduce its radius from the reconstructed image. The SL-GNI reconstruction of this target using three values for , namely, , and , is shown in Fig. 8(d) and (e). As can be seen, the periphery of both circles is very clear. Also, the reconstructed radii for both cylinders are very accurate: the radius of the reconstructed outer cylinder is 0.078 m and that of the reconstructed inner cylinder is 0.03 m. We also run the SL-GNI algorithm with 20% error in the utilized values for . That is, the corresponding values of for the two cylinders utilized in the SL-GNI algorithm are chosen to be 2.4 and 0.54. The contrast of zero, which is the contrast of the background medium, is kept to be zero. The SL-GNI reconstruction results using these values for are shown in Fig. 8(f) and (g). As can be seen, the two cylinders are resolved. However, the radii of the inner cylinder and outer cylinders are underestimated: 0.024 m and 0.073 m. Finally, we note that all of these reconstructions show artifacts in the reconstructed imaginary parts. These artifacts are emphasized in the SL-GNI reconstructions at the boundary of the two cylinders. However, in the MR-GNI reconstruction, this artifact contains the whole inner cylinder but with a smaller magnitude compared to the magnitude of the artifact in the SL-GNI reconstruction. Finally,

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Fig. 9. University of Manitoba’s microwave tomography system with two nylon cylinders with a separation of 8 mm (frequency of operation is 5 GHz).

Fig. 8. (a) The FoamDielIntTM target from the second Fresnel experimental 3 GHz). Reconstruction [left: data set (frequency of operation is and right: ] using (b)–(c) the MR-GNI method, (d)–(e) the SL-GNI method, and (f)–(g) the SL-GNI method with 20% error in the utilized values . for

TABLE II NUMBER OF SL-GNI AND MR-GNI ITERATIONS REQUIRED FOR CONVERGENCE OF THE FIRST EXPERIMENTAL DATA SET

THE

the number of iterations for this experimental example is given in Table II. B. Two Nylon Rods This data set is collected using 24 co-resident Vivaldi antennas. The measured data are calibrated assuming that the ra-

dius of the measurement domain is 13.5 cm. Similar to [24], the two canonical targets are nylon-66 cylinders, 0.038 m in diameter and 0.44 m in height. The two cylinders were placed in the imaging system with the separation of the two targets being 8 mm as shown in Fig. 9. The background medium is free space and the frequency of operation is 5 GHz. Thus, this separation corresponds to where is the wavelength in the background medium. For each active transmitter, the scattered data are collected using the remaining antennas. That is, the measured data consist of 24 23 data points. At 5 GHz, the nylon has a measured relative complex permittivity of . The imaging domain is chosen to be 0.104 m 0.104 m and is discretized into 60 60 square pulses. The MR-GNI reconstruction of this target is shown in Fig. 10(a) and (b). As can be seen, these two cylinders are resolved and their reconstructed real-part permittivity value is very close to its expected value. However, the imaginary part of the permittivity profile has not been reconstructed due to its very small value and the limited signal-to-noise ratio of the measured data. The SL-GNI reconstruction of this target with two values for , namely, 0 and 2, is shown in Fig. 10(c) and (d). Similar to the MR-GNI reconstruction, the nylon rods are resolved. It is worth noting that the number of MR-GNI and SL-GNI iterations required for the convergence is 10 and 21; the first iterations of the MR-GNI and SL-GNI algorithms took about 17 and 13 s, respectively. To check whether the SL-GNI reconstruction is capable of reconstructing the very small imaginary part of the permittivity profile, we have run the SL-GNI algorithm with and . However, it was still not able to reconstruct the imaginary part (not shown here). It should be noted that the SL-GNI algorithm used for this target may be referred to as a binary inversion algorithm as it deals with two different contrast values: the contrast of the background medium (which is zero) and that of the nylon rods. It is worth noting that if we remove the regularization term from (11) and optimize over , the reconstruction result, which is shown in Fig. 10(e) and (f), is not satisfactory. To show the performance of the SL-GNI algorithm when the utilized for nylon rods contains a high error, we run the SL-GNI algorithm with 50% error in the utilized for the two nylon rods. That is, we assume that the contrast of the nylon

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values of the objects being imaged. Using synthetically and experimentally collected data, we have shown that this algorithm is robust and can outperform the standard multiplicative regularized Gauss–Newton inversion algorithm in reconstruction the shape and location of the object of interest. APPENDIX I DERIVATIVE OPERATORS Herein, we derive the derivatives for given in (10). The derivatives for and can be found in [12, App. D]. The derivatives for the regularized cost functional can then be obtained using the product rule. We denote the spaces of complex functions defined on by with the norms and inner products defined as and

(17)

where the superscript denotes the complex conjugate operator. At the th iteration of the GNI algorithm, we start with finding the limit (18) The above limit can be written as

(19) The above limit can be simplified to (the argument dropped for simplicity)

Fig. 10. The reconstructed relative complex permittivity of the two nylon and right: ] using (a)–(b) the MR-GNI method, cylinders [left: (c)–(d) the SL-GNI method, (e)–(f) the shape and location reconstruction , (g)–(h) the SL-GNI method with 50% error in method without the use of . the utilized value for

rods is 1 (instead of 2). Using this wrong value, the SL-GNI reconstruction of the target is shown in Fig. 10(g) and (h). Although the SL-GNI algorithm was not capable of reconstructing the target with this utilized for the nylon rods, it shows that this value of is wrong. This can be deduced by noting that the reconstructed image shows three totally different contrast values: 1) contrast of zero (background medium), 2) contrast of 1, and 3) contrast of 2.5. This is in contradiction with the a priori information which assumes only one contrast value for the scatterers. VII. CONCLUSION We have presented a multiplicative regularized Gauss– Newton inversion algorithm for shape and location reconstruction which utilizes a priori information about the permittivity

has been

(20) After mathematical simplifications, this can be written as

(21) Noting that (22) expression (21) may be written as (23)

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it can be concluded that

Writing the above limit as

(31) (24)

and (32)

the derivative operators may then be written as (25)

Using a similar procedure, we can derive

and

(26)

(33) and

. We note that is an auxwhere iliary cost functional which treats and as two independent functions. The cost functional is complex differentiable with respect to for a fixed and vice versa. This method, sometimes referred to as Wirtinger calculus [25]–[28], [12, App. C], is one way of handling the fact that is not complex differentiable with respect to the complex function . To find the derivatives

(34) Having found the derivative operators in the continuous domain, the discretized forms of these operators can easily be found. For example, the discretized form of (25) can be written as (35)

and

we start with finding the limit

(27) Utilizing (25), the above limit may be written as

where the superscript “ ” denotes the transposition operator. The SL-GNI method also requires the derivatives of and with respect to and . Using the same procedure as explained above, these derivatives can be derived. The closedform expressions of these derivative operators are given in [12, App. D]. Using the first-order derivative operators in the discrete domain, we can form the gradient of the cost functional which is the negative of the vector given in the right-hand side of (13). To form the Hessian matrix, i.e., the matrix in the left-hand side of (13), we use the product rule. However, only the second derivative operators which make the Hessian matrix nonnegative definite are kept. APPENDIX II OVERSAMPLING OF THE CONTRAST

(28) which can be simplified as (29) Writing the above expression as (30)

The proposed algorithm is robust with respect to oversampling of the unknown contrast. This can be explained as follows. Assume that we have a discrete ill-posed problem as where , and the unknown vector is in . (In our problem, is the number of measured data and is the number of discretized elements in the imaging domain.) To solve this ill-posed problem, we use multiplicative regularization. It can be shown that multiplicative regularization when applied to the discrete ill-posed problem is equivalent to the following minimization [7, Sec. V]: (36)

MOJABI et al.: A MULTIPLICATIVE REGULARIZED GAUSS–NEWTON INVERSION FOR SHAPE AND LOCATION RECONSTRUCTION

where is the regularization operator, is the regularization weight, and is some form of the initial guess (which is equal to the reconstructed contrast at the previous iteration of the GNI algorithm [7]). The above minimization is then equivalent to solving the following damped least squares problem: (37)

As can be seen in (37), the unknown vector whereas the matrix

belongs to

belongs to . Thus,

whatever we choose the number of discretized elements in the imaging domain (i.e., ), the number of rows of the matrix will be more than the number of elements in the vector . This makes the algorithm robust to oversampling of the unknown contrast. ACKNOWLEDGMENT The authors would like to thank the Institut Fresnel, France, for providing the FoamDielIntTM experimental data set. REFERENCES [1] A. Abubakar, T. Habashy, V. Druskin, L. Knizhnerman, and D. Alumbaugh, “2.5D forward and inverse modeling for interpreting low-frequency electromagnetic measurements,” Geophysics, vol. 73, no. 4, pp. F165–F177, Jul.–Aug. 2008. [2] P. Meaney, N. Yagnamurthy, and K. D. Paulsen, “Pre-scaled twoparameter Gauss-Newton image reconstruction to reduce property recovery imbalance,” Phys. Med. Biol., vol. 47, pp. 1101–1119, 2002. [3] A. Abubakar and P. M. van den Berg, “The contrast source inversion method for location and shape reconstructions,” Inverse Probl., vol. 18, pp. 495–510, 2002. [4] L. Crocco and T. Isernia, “Inverse scattering with real data: Detecting and imaging homogeneous dielectric objects,” Inverse Probl., vol. 17, pp. 1573–1583, 2001. [5] K. Belkebir, R. Kleinman, and C. Pichot, “Microwave imaging-location and shape reconstruction from multifrequency scattering data,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 4, pp. 469–476, Apr. 1997. [6] K. Belkebir, C. Pichot, J. C. Bolomey, P. Berthaud, G. Cottard, X. Derobert, and G. Fauchoux, “Microwave tomography system for reinforced concrete structures,” in Proc. 24th Eur. Conf. , 1994, vol. 2, pp. 1209–1211. [7] 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. [8] A. Abubakar, P. M. van den Berg, and J. J. Mallorqui, “Imaging of biomedical data using a multiplicative regularized contrast source inversion method,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 7, pp. 1761–1777, Jul. 2002. [9] P. Mojabi and J. LoVetri, “Microwave biomedical imaging using the multiplicative regularized Gauss-Newton inversion,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 645–648, 2009. [10] P. Mojabi and J. LoVetri, “A novel microwave tomography system using a rotatable conductive enclosure,” IEEE Trans. Antennas Propag., vol. 59, no. 5, pp. 1597–1605, May 2011. [11] P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process., vol. 6, no. 2, pp. 298–311, Feb. 1997.

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[12] P. Mojabi, “Investigation and development of algorithms and techniques for microwave tomography” Ph.D. dissertation, Dept. Electr. Comput. Eng., Univ. Manitoba, Winnipeg, MB, Canada, 2010 [Online]. Available: http://mspace.lib.umanitoba.ca/handle/1993/3946 [13] T. M. Habashy and A. Abubakar, “A general framework for constraint minimization for the inversion of electromagnetic measurements,” Progr. Electromagn. Res., vol. 46, pp. 265–312, 2004. [14] P. Mojabi and J. LoVetri, “Comparison of TE and TM inversions in the framework of the Gauss-Newton method,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1336–1348, Apr. 2010. [15] A. Abubakar, P. M. van den Berg, and S. Y. Semenov, “A robust iterative method for Born inversion,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 2, pp. 342–354, Feb. 2004. [16] P. M. van den Berg and R. E. Kleinman, “A contrast source inversion method,” Inverse Probl., vol. 13, pp. 1607–1620, 1997. [17] P. M. van den Berg, A. L. van Broekhoven, and A. Abubakar, “Extended contrast source inversion,” Inverse Probl., vol. 15, pp. 1325–1344, 1999. [18] B. J. Kooij and P. M. van den Berg, “Nonlinear inversion in TE scattering,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 11, pp. 1704–1712, Nov. 1998. [19] J. Stang, “A 3D active microwave imaging system for breast cancer screening,” Ph.D. dissertation, Dept. Electr. Comput. Eng., Duke Univ., Durham, NC, 2008. [20] C. Gilmore, P. Mojabi, A. Zakaria, M. Ostadrahimi, C. Kaye, S. Noghanian, L. Shafai, S. Pistorius, and J. LoVetri, “A wideband microwave tomography system with a novel frequency selection procedure,” IEEE Trans. Biomed. Eng., vol. 57, no. 4, pp. 894–904, Apr. 2010. [21] T. Rubæk, P. M. Meaney, P. Meincke, and K. D. Paulsen, “Nonlinear microwave imaging for breast-cancer screening using Gauss-Newton’s method and the CGLS inversion algorithm,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2320–2331, Aug. 2007. [22] 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. [23] 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. [24] C. Gilmore, P. Mojabi, A. Zakaria, S. Pistorius, and J. LoVetri, “On super-resolution with an experimental microwave tomography system,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 393–396, 2010. [25] W. Wirtinger, “Zur formalen theorie der funktionen von mehr komplexen veränderlichen,” Mathematische Annalen, vol. 97, no. 1, pp. 357–375, 1927. [26] H. Li and T. Adali, “Complex-valued adaptive signal processing using nonlinear functions,” EURASIP J. Adv. Signal Process., 2008, DOI: 10.1155/2008/765615. [27] D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” Inst. Electr. Eng. Proc. F and H, vol. 130, no. 1, pp. 11–16, 1983. [28] A. van den Bos, “Complex gradient and Hessian,” Inst. Electr. Eng. Proc.—Vision Image Signal Process., vol. 141, no. 6, pp. 380–383, 1994.

Puyan Mojabi (S’09–M’10) received the B.Sc. degree in electrical engineering from the University of Tehran, Tehran, Iran, in 2002, the M.Sc. degree in electrical engineering from Iran University of Science and Technology, Tehran, Iran, in 2004, and the Ph.D. degree in electrical engineering from the University of Manitoba, Winnipeg, MB, Canada, in 2010. He is currently an Assistant Professor with the Electrical and Computer Engineering Department, University of Manitoba. His current research interests are computational electromagnetics, antenna design, and inverse problems.

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Joe LoVetri (S’84–M’84–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 EMI/EMC Engineer at Sperry Defence Division, Winnipeg, MB, Canada. From 1986 to 1988, he held the position of TEMPEST Engineer at the Communications Security Establishment, Ottawa, ON, Canada. 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, London, ON, Canada. In 1997–1998 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.

Lotfollah Shafai (S’67–M’69–SM’75–F’88–LF’07) received the B.Sc. degree from the University of Tehran, Tehran, Iran, in 1963 and the M.Sc. and Ph.D. degrees from the Faculty of Applied Sciences and Engineering, University of Toronto, Toronto, ON, Canada, in 1966 and 1969, respectively, all in electrical engineering. In November 1969, he joined the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, Canada, as a Sessional Lecturer, Assistant Professor (1970), Associate Professor (1973), and Professor (1979). Since 1975, he has made special effort to link the University research to the industrial development, by assisting

industries in the development of new products or establishing new technologies. To enhance the University of Manitoba contact with industry, in 1985 he assisted in establishing “The Institute for Technology Development,” and was its Director until 1987, when he became the Head of the Electrical Engineering Department. His assistance to industry was instrumental in establishing an Industrial Research Chair in Applied Electromagnetics at the University of Manitoba in 1989, which he held until July 1994. Dr. Shafai has been a participant in nearly all antennas and propagation symposia and participates in the review committees. He is a member of Commission B of the International Union of Radio Science (URSI) and was its Chairman during 1985–1988. In 1986, he established the Symposium on Antenna Technology and Applied Electromagnetics (ANTEM) at the University of Manitoba that is currently held every two years. He has been the recipient of numerous awards. In 1978, his contribution to the design of a small ground station for the Hermus satellite was selected as the Third Meritorious Industrial Design. In 1984, he received the Professional Engineers Merit Award, and in 1985, “The Thinker” Award from Canadian Patents and Development Corporation. From the University of Manitoba, he received the “Research Awards” in 1983, 1987, and 1989, the Outreach Award in 1987 and the Sigma Xi, Senior Scientist Award in 1989. In 1990, he received the Maxwell Premium Award from the Institution of Electrical Engineers (IEE; London, U.K.), and in 1993 and 1994, the Distinguished Achievement Awards from Corporate Higher Education Forum. In 1998, he received the Winnipeg RH Institute Foundation Medal for Excellence in Research. In 1999 and 2000, he received the University of Manitoba, Faculty Association Research Award. He was elected a Fellow of The Royal Society of Canada in 1998. He was a recipient of the IEEE Third Millennium Medal in 2000, and in 2002, was elected a Fellow of The Canadian Academy of Engineering and Distinguished Professor at The University of Manitoba. In 2003, he received an IEEE Canada “Reginald A. Fessenden Medal” for “outstanding contributions to telecommunications and satellite communications,” and a Natural Sciences and Engineering Research Council (NSERC) Synergy Award for “development of advanced satellite and wireless antennas.” He holds a Canada Research Chair in Applied Electromagnetics and was the International Chair of Commission B of URSI for 2005–2008. In 2009, he was elected a Fellow of the Engineering Institute of Canada, and was the recipient of an IEEE Chen-To-Tai Distinguished Educator Award. In 2011, he received a Killam Prize in Engineering from The Canada Council for the Arts, for his “outstanding Canadian career achievements in engineering, and his work in antenna research.”

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Application of DCIM on Marine Controlled-Source Electromagnetic Survey Hanji Ju, Guangyou Fang, Zhiwei Lin, Student Member, IEEE, Feng Zhang, Ling Huang, and Yicai Ji

Abstract—The discrete complex image method (DCIM) is used to calculate the mixed-potential electric field integral equation. The DCIM has been used broadly in high-frequency domain, and now is successfully introduced to the extremely low-frequency situation of marine controlled-source electromagnetic (MCSEM) survey. The integral path in high-frequency situation does not hold valid for extremely low-frequency situation. The reason can be ascribed to the change of wave number formulas. A new integral path suitable for the extremely low-frequency situation is proposed. The appropriate ranges of the parameters used in DCIM are given. The computation is stable at these considerably large ranges. The results of a five-layer model calculated with DCIM are compared with those with the filter method and the direct integral method. The computation time with DCIM is comparable to the time with filter method. The effects of water depth, upper-sediment thickness, and oil-layer thickness are studied with DCIM. The results show that electric field varies with the change of thickness. The effects accord to the physical meanings of MCSEM survey. It is safe to conclude that the DCIM with the new path is valid in MCSEM survey. Index Terms—Discrete complex image method (DCIM), extremely low frequency, integral path, marine controlled-source electromagnetic (MCSEM), mixed-potential electric field integral equation (MPIE).

I. INTRODUCTION

D

URING recent years, marine controlled-source electromagnetic (MCSEM) survey has been extensively used for offshore petroleum/hydrocarbon exploration [1]–[5]. The electromagnetic (EM) field is emitted by an electric dipole and recorded by the EM receivers deployed at the seafloor along the survey profile line. The EM field is guided by the high resistive petroleum/hydrocarbon layer between the sediments [6]. It can transmit to a long offset with little attenuation due to the guiding-wave effect. The seawater is so conductive that the EM field will attenuate greatly in high-frequency situation. Extremely low-frequency (ELF) EM field, typically within range of 0.1–10 Hz, is used [1], [2]. Manuscript received September 22, 2010; revised April 25, 2011; accepted June 13, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. This work was supported by the National Natural Science Foundation of China under Grant 60871052. H. Ju was with the Key Laboratory of Electromagnetic Radiation and Detection Techniques, Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China. He is now with the Center of Metrology, North China Grid Company Limited, Beijing 100045, China (e-mail: [email protected]). G. Fang, Z. Lin, F. Zhang, L. Huang, and Y. Ji are with the Key Laboratory of Electromagnetic Radiation and Detection Techniques, Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165485

The planar-layered model is used in MCSEM. The EM field in multilayer model is presented by Sommerfeld-type integrals [1], [7]. The Hankel transform filter technique is an important method to compute Sommerfeld-type integrals [8]–[10]. In this paper, the electric field of a horizontal electric dipole (HED) source is expressed based on the mixed-potential electric field integral equation (MPIE) [11]. The calculation of vector and scalar potential Green’s functions is such a time-consuming task that the discrete complex image method (DCIM) is employed to obtain the spatial domain closed-form Green’s functions. The generalized pencil-of-function (GPOF) method is used to present the integral kernel with exponential summation [12]. The DCIM has been used broadly in high-frequency analysis, such as microwave integrated circuits and microstrip antennas [13]–[17]. In this paper, it is introduced into the ELF situation of MCSEM. The validity of ELF DCIM is studied. The integral path in ELF situation is totally different from that in high-frequency situation because of the change of wave number formulas. A new integral path suitable for the ELF two-level DCIM is proposed and examined. The effects of water depth, upper-sediment thickness, and oil-layer thickness are discussed with DCIM to validate this method. II. MIXED-POTENTIAL ELECTRIC FIELD INTEGRAL EQUATION The advantages of the MPIE over the other electric field integral equation (EFIE) are pronounced in multilayer medium [16]. The MPIE only involves the potential forms of Green’s function, which are less singular than their derivatives needed in the other forms of the EFIE. The MPIE of the electric field [11] is as follows:

(1) where is the current density, is vector potential Green’s function, is scalar potential Green’s function, and is the inner product. The traditional form of vector potential Green’s function is used [16] (2) where represents the field in direction- created by a -directed source. The current density of the HED is . The three components of the electric field can be expressed as [11]

0018-926X/$26.00 © 2011 IEEE

(3)

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Fig. 1. The new path in extremely low-frequency DCIM. (a) Paths on plane. (b) Paths and on plane.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 12, DECEMBER 2011

and

(4)

(5) The term interrelated with scalar potential in (3)–(5) can be denoted as

(6) , and stand for scalar potential Green’s where functions of -, -, and -directions, respectively. In this paper, there is only the component, because the dipole is placed in -direction. The formulas of Green’s functions are given in the Appendix. III. A NEW INTEGRAL PATH SUITABLE FOR EXTREMELY LOW-FREQUENCY DCIM After spatial domain Green’s functions are attained, the DCIM and the GPOF [12] can be used to approximate the integral in terms of complex exponentials. Then, the analytical evaluation of the integral becomes possible via the Sommerfeld identity. In this paper, two-level DCIM [13] is adopted. The GPOF method requires uniform samples along a real variable. Paths and on plane are defined as a mapping of a real variable onto the complex plane by two parametric equations. Fig. 1 shows the path on plane and the corresponding path on plane, where and are the truncation points of paths and , respectively. The difference of DCIM between high-frequency and ELF situation is examined. The path and parametric equations of high-frequency DCIM adopted by most of DCIM researchers

can be found in [13]–[16]. In this paper, the path should be shifted from the first and third quadrants to the second and fourth because the different forms of Green’s functions are used here. There is a multivalue function in the integrands of (A-1)–(A-3). There are two Riemann sheets on plane [7], [17]. The branch points and branch cuts on plane are shown in Fig. 1. The square roots of are chosen for mathematical convenience. There are two requirements for the integral path. First, paths on plane must avoid the first and third quadrants to avoid the branch points. Second, in order to satisfy the radiation condition (decay in amplitude and delay in phase at a distant region) [7], must satisfy and along the integral paths. Neither of these requirements is satisfied if the same parametric equations as high-frequency situation are used. It is well known that the wave number formula is suitable for any frequency. In high-frequency situation, , the displacement current is larger than the conduction current. The second term in wave number formula is negligible and the wave number can be simplified to . The wave number is a real number [7], [10]–[14]. In ELF situation, . The conduction current is larger than the displacement current. The first term is negligible and the wave number can be simplified to . The wave number is a complex number [1], [15]. However, it should be noted that . In high-frequency situation, is a real number. When , is a real number; when is a pure imaginary number. But these relations do not hold true in ELF situation. At least one of and is a complex number because is a complex number. The path in ELF situation is different from that in high frequency. The new path suitable for ELF application is shown in Fig. 1. The parametric equations describing new paths and are For (7) For (8) where is the running variable sampled uniformly on the corresponding path, and are the truncation points of paths and , respectively, and is the wave number in the source layer. In order to simplify the calculation in GPOF, is chosen as a linear function of . IV. TWO-LEVEL DISCRETE COMPLEX IMAGE METHOD Spatial domain Green’s functions in formulas (A-1)–(A-3) have the same form as follows: (9) and are spatial and spectral domain where Green’s functions, respectively, and SIP is Sommerfeld integral path.

JU et al.: APPLICATION OF DCIM ON MARINE CONTROLLED-SOURCE ELECTROMAGNETIC SURVEY

In order to use Sommerfeld identity,

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is transformed to

(10) For two-level DCIM, there are several parameters that should be selected correctly, including and , the number of samples and on path and respectively, and the number of exponential terms and to be used in GPOF. Ideally, all the results should be independent of these parameters. However, due to the limitation of numerical algorithm, this is rarely the case. After numerous tests, it was found that are safe to keep the computation stable. The tests also show that the parameters on path have a stronger influence than those on path . The choice of is more critical than . The steps of two-level DCIM are outlined as follows. 1) Choose and and and . 2) Sample the along path and approximate it by the GPOF method

(11)

(12) where and are the coefficients and exponentials obtained from the GPOF method, and and are the coefficients and exponentials of . 3) Subtract function from the original function . Sample the remaining function along path and approximate it by GPOF method (13)

(14) 4) Rewrite (9) to

(15) in an exponential summation form 5) Approximate by utilizing Sommerfeld identity (16)

Fig. 2. A planar multilayer medium model.

Equation (15) can be expressed as follows: (17) (18) where is the radial distance between the electric dipole source and the receiver. V. NUMERICAL RESULTS WITH THE NEW INTEGRAL PATH In order to verify the new integral path, vector and scalar potentials Green’s functions for a five-layer model in Fig. 2 are calculated. The results of DCIM are compared with those of the filter method and the direct integral method. The model parameters are: 1 S/m, 0.01 S/m, 1 S/m, 4 S/m, 1000 m, 1200 m, 1700 m, 2700 m, 1750 m, 1705 m, where are conductivity and permeability, is the height of the th layer from the bottom of is the height of the source, and is the height of the receiver. The reflections from the first and last layers are assumed zero. The frequency of source signal is 1 Hz. For any numerical scheme, there are two factors of importance. One is the evaluation accuracy, or the smallest fields which can be evaluated. The other is the computation speed. For DCIM, parameters are used. For the filter method, the filter with lengths 241 provided by Kong [8] is used. The direct integral method is carried out along the real axis of plane except the origin to avoid the pole at the origin. The integral range is and the integral step is . With a smaller step, the process would be tedious. The computation time of the DCIM, the filter method, and the direct integral method is about 4 s, 3 s, and more than 30 min, respectively, on a 1.6-GHz Genuine Intel(R) CPU with 1-GB memory. It is well known that the biggest advantage of the filter method is its computation speed. The time of the DCIM and the filter method is almost the same. It is safe to say that the DCIM is also efficient in ELF situation. Vector potentials and are shown in Figs. 3 and 4, respectively. Scalar potential is shown in Fig. 5. In Fig. 3, the difference of the three methods appears in the zone

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Fig. 3. Vector potential component

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 12, DECEMBER 2011

. Fig. 5. Scalar potential component

Fig. 4. Vector potential component

.

.

farther away than 5000 m. After 5000 m, although all the three fields began to decay slower, the field of the DCIM decays faster than the other two methods. One reason for the slower decay rate at the distant offset is the existence of the oil layer. It accords to the physical meaning of the MCSEM. Another reason is the drawback of the numerical scheme itself. It has been examined by Kong [8] that the result with the Sommerfeld identity is more accurate than that with the filter method at the distant offset. The DCIM is just a method utilizing the Sommerfeld identity. That is a good reason to think that the field decay rate of the DCIM at the distant offset can be more accurate. That means that with the DCIM a weaker field can be evaluated. The same trend can be found in Figs. 4 and 5. In Fig. 4, the difference appears in the zone farther away than 1000 m. In Fig. 5, the difference is not as obvious as that in Figs. 3 and 4. Electric fields Ex and Ez with three methods are compared in Fig. 6. The amplitude of the field with the DCIM is larger than those with the other methods between the offset of 500 and 2000 m. Further efforts should be taken upon to eliminate the difference. After 2000 m, the decay rate with the DCIM is a little faster. With the filter method, an apparent valley of Ez at 1000 m can be seen in Fig. 6(b), which may be a contribution from . In Fig. 4, an abrupt change of occurs at 1000 m, which results in the abrupt change in Ez. It is noticed that the distance is just twice the thickness of upper sediment. A phase inversion may just occur at this position. The electric fields through different paths counteract with each other, and then the smallest field occurs.

Fig. 6. Comparison of the electric field magnitude versus offset. (a) Comparison of Ex field. (b) Comparison of Ez field.

In general, the agreement of the results with three methods is good. It is safe to conclude that the two-level DCIM approach in this paper is effective and can be used in the MCSEM. The DCIM can evaluate a weaker field than the filter method with nearly the same calculation time. Some further efforts should be taken upon to eliminate the difference between the offset of 500 and 2000 m.

JU et al.: APPLICATION OF DCIM ON MARINE CONTROLLED-SOURCE ELECTROMAGNETIC SURVEY

Fig. 7. Effect of the water depth on the electric field magnitude. (a) Effect on Ex field. (b) Effect on Ez field.

VI. EFFECT OF DIFFERENT MODELS MARINE CESM SURVEY

ON

In this section, the DCIM is used to simulate the electric field distribution over seafloor with different models. A basic five-layer model in Fig. 2 is used. The effects of water depth [18], [19], upper-sediment thickness, and oil-layer thickness are discussed. The results show that the intensity of the electric field varies with the change of these parameters. A. Effect of Water Depth The water depth changes from 1000 to 50 m when the other layer thicknesses keep constant. It can be seen from Fig. 7 that both electric fields Ex and Ez become stronger with the decrease of the water depth. This means that the wave from air layer dominates the received signal when the water is shallow. In the deep-water environment, the response from the air is smaller due to the heavy damping of the signal in the water column. The curves of the electric field overlap when the water depths are 500 and 1000 m. It means that the effect of the air layer is negligible when the water depth is larger than 500 m. The recorded signal is dominated by the oil layer. The results show that the deep-water environment is favored to detect the signal from the oil layer. It must be stressed that this conclusion is only right for the frequency of 1 Hz used here. If the frequency changes, the depth at which the air effect can be negligible will change.

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Fig. 8. Effect of the upper sediment thickness on the electric field magnitude. (a) Effect on the Ex field. (b) Effect on the Ez field.

It also can be seen that the difference becomes more obvious with the increase of the source–receiver distance. This means that the direct wave from the water becomes weaker and the wave from the oil layer dominates the signal at a distant offset. In the near-field region, the direct wave dominates the signal and the usable signal is concealed. In order to record the signal from the oil layer, an offset larger than 2000 m is preferred. B. Effect of Upper-Sediment Thickness The effect of the upper-sediment thickness on the electric field is discussed. The thickness changes from 100 to 4000 m when the other layer thicknesses keep constant. It can be seen from Fig. 8 that both electric fields Ex and Ez become weaker with the increase of the thickness. The upper sediment decreases the detecting ability for the oil layer. When the thickness is larger than 3000 m, the curves of the electric field overlap with each other. It means that the signal from the oil layer is completely concealed and the oil layer cannot be detected. C. Effect of the Oil-Layer Thickness The effect of the oil-layer thickness on the electric field is discussed. The thickness changes from 50 to 200 m when the other layer thicknesses keep constant. From Fig. 9, it can be seen that both electric fields Ex and Ez become stronger with the increase of the thickness. It means that a thick oil layer can be detected more easily than a thin one. From the comparison

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show that the intensity of the electric field varies with the changes of these parameters. The effects accord to the physical meaning of the MCSEM survey. The results with the DCIM reveal the difference of different geological formation. It is safe to conclude that the two-level DCIM in this paper is valid.

APPENDIX SPATIAL-DOMAIN GREEN’S FUNCTIONS MULTILAYER MODEL

FOR

The planar multilayer model is shown in Fig. 2. The derivation of Green’s functions follows the similar procedure as [7]. The time harmonic factor is , and the corresponding Green function is in uniform space. Green’s function of this form is widely used in geophysics, but in electromagnetic domain, is preferred. Green’s functions in a layered medium due to a horizontal electric dipole (HED) are as follows:

(A-1)

Fig. 9. Effect of the oil layer thickness on the electric field magnitude. (a) Effect on the Ex field. (b) Effect on the Ez field.

of Figs. 8 and 9, it can be found that the effect of the oil layer thickness is not as strong as the effect of the sediment thickness. It means that even for an oil layer as thin as 50 m, the response can be detected easily. VII. CONCLUSION Vector and scalar potentials Green’s functions of the HED in a planar multilayer conductive medium are derived in this work. The electric field is expressed based on the mixed-potential electric field integral equation. The DCIM, which has been used broadly in high-frequency analysis, is successfully introduced into MCSEM survey. In ELF situation, if the same parametric equations as in high-frequency situation are used, the computation is not stable. A new integral path suitable for the ELF two-level DCIM is proposed. The selection of parameters is important for DCIM computation. The appropriate ranges of these parameters are given after numerous tests. The computation is stable at these considerably large ranges. In order to validate the DCIM, the results are compared with those of the filter method and the direct integral method. In general, the agreements are good. The results with the DCIM exhibits a faster field decay rate at a distant offset than the results with the filter method. The computation time with the DCIM is comparable to that with the filter method. The difference of the electric field between the offset of 500 and 2000 m shows that some further improvement should be made. The effects of the water depth, upper-sediment thickness, and oil-layer thickness are discussed with the DCIM. The results

(A-2)

(A-3) where is the Hankel function of the first kind. The coefficients are functions of the generalized reflection coefficient

(A-4)

(A-5)

(A-6)

(A-7) (A-8)

JU et al.: APPLICATION OF DCIM ON MARINE CONTROLLED-SOURCE ELECTROMAGNETIC SURVEY

(A-9) (A-10) (A-11) where

is the generalized reflection coefficient, and represents the electric/magnetic reflection coefficient . For the at the interface of domainand domaincorrespond to the reflection coefficient of TE mode, HED, and correspond to the TM mode. ACKNOWLEDGMENT

The authors would like to thank Dr. F. N. Kong of Norwegian Geotechnical Institute for providing the filter coefficients and Prof. F. Li for her valuable discussion. The authors would also like to thank the three anonymous reviewers for useful suggestions that improved this study. Dr. Y. Liu made great effort to improve the English language of this paper. REFERENCES [1] M. S. Zhdanov, Geophysical Electromagnetic Theory and Method. Oxford, U.K.: Elsevier, 2009, ch. 3, 5, and 15. [2] S. Constable and L. J. Srnka, “An introduction to marine controlledsource electromagnetic methods for hydrocarbon exploration,” Geophysics, vol. 72, no. 2, pp. WA3–WA12, 2007. [3] F. N. Kong, S. E. Johnstad, and T. Roesten, “Characteristics of scattered fields from hydrocarbon layers in seabed logging,” Progr. Electromagn. Res., vol. 2, no. 6, pp. 585–588, 2006. [4] E. S. Um and D. L. Alumbaugh, “On the physics of the marine controlled-source electromagnetic methods,” Geophysics, vol. 72, no. 2, pp. WA13–WA26, 2007. [5] F. N. Kong, H. Westerdahl, S. Ellinsgrud, T. Eidesmo, and S. Johansen, “Seabed logging: A possible direct hydrocarbon indicator for deep-sea prospects using EM energy,” Oil Gas J., vol. 100, no. 19, pp. 30–38, 2002. [6] F. N. Kong, S. E. Johnstad, and J. Park, “Wavenumber of the guided wave supported by a thin resistive layer in marine controlled-source electromagnetics,” Geophys. Prospect., vol. 58, no. 4, pp. 711–723, 2010. [7] W.-C. Chew, Wave and Fields in Inhomogeneous Media. New York: IEEE Press, 1995, ch. 2 and 7. [8] F. N. Kong, “Hankel transform filters for dipole antenna radiation in a conductive medium,” Geophys. Prospect., vol. 55, no. 1, pp. 83–89, 2007. [9] W. L. Anderson, “A hybrid fast Hankel transform algorithm for electromagnetic modeling,” Geophysics, vol. 54, no. 2, pp. 263–266, 1989. [10] D. Guptasarma and B. Singh, “New digital linear filters for Hankel and transforms,” Geophys. Prospect., vol. 45, no. 5, pp. 745–762, 1997. [11] X. L. He, S. X. Gong, and Q. Z. Liu, “Fast calculation of closed form Green’s functions of planar-layered media,” Chin. J. Radio Sci., vol. 19, no. 6, pp. 761–766, 2004. [12] Y. B. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag., vol. 37, no. 2, pp. 229–234, Feb. 1989. [13] M. I. Aksun, “A robust approach for the derivation of closed-form Green’s functions,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 5, pp. 651–658, May 1996.

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[14] G. Dural and M. I. Aksun, “Closed-form Green’s functions for general sources and stratified media,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 7, pp. 1545–1552, Jul. 1995. [15] K. A. Michalski and M. I. Aksun, “Discrete complex image method for planar multilayers with uniaxial anisotropy,” in Proc. 2nd Eur. Conf. Antennas Propag., Edinburgh, U.K., 2007, pp. 331–337. [16] K. A. Michalski and D. L. Zheng, “Electromagnetic scattering and radiation by surface of arbitrary shape in layered media—Part I: Theory,” IEEE Trans. Antennas Propag., vol. AP-38, no. 3, pp. 335–344, Mar. 1990. [17] J. J. Yang, Y. L. Chow, and D. G. Fang, “Discrete complex images of a three-dimensional dipole above and within a lossy ground,” Inst. Electr. Eng. Proc. H, vol. 138, no. 4, pp. 319–326, 1991. [18] C. S. Liu, M. E. Everett, J. Lin, and F. D. Zhou, “Modeling of seafloor exploration using electric-source frequency-domain CSEM and the analysis of water depth effect,” Chin. J. Geophys., vol. 53, no. 8, pp. 1940–1952, 2010. [19] M. Deng, W. Wei, W. Zhang, Y. Sheng, Y. Li, and M. Wang, “Electric field responses of different gas hydrate models excited by ahorizontal electric dipole source with changing arrangements,” Petroleum Explorat. Develop., vol. 37, no. 4, pp. 438–442, 2010. Hanji Ju was born in Qingdao, China, in 1982. He received the B.S. degree in electronic information engineering from Qingdao University, Qingdao, Shandong, China, in 2005, the M.S. degree in physical electronics from University of Electronic Science and Technology of China (UESTC), Chengdu, Sichuan, China, in 2008, and the Ph.D. degree in electromagnetic field and microwave technology from the Institute of Electronics, Chinese Academy of Sciences, Beijing, China. Currently, he is with the Center of Metrology, North China Grid Company Limited, Beijing, China. His research interests include electromagnetic wave propagation in seawater, marine controlled-source electromagnetic survey, microwave heating, and EMC/EMI.

Guangyou Fang received the B.S. degree in electrical engineering from Hunan University, Changsha, China, in 1984 and the M.S. and Ph.D. degrees in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 1990 and 1996, respectively. From 1990 to 1999, he worked as an Engineer, an Associate Professor, and a Professor in the China Research Institute of Radiowave Propagation. From 2000 to 2001, he was a Visiting Scholar at the University of Trieste, Trieste, Italy, and the International Center for Science and High Technology—United Nations Industrial Development Organization, Trieste. From 2001 to 2003, he was a Special Foreign Research Fellow of Japan Society for Promotion of Science (JSPS), working with Prof. M. Sato at Tohoku University, Sendai, Japan. Since 2004, he has been a Professor with the Institute of Electronics, Chinese Academy of Sciences (CAS), Beijing, China, and the Director of the Key Lab of Electromagnetic Radiation and Sensing Technique, CAS. He is the author of more than 100 publications. His research interests include ultrawideband radar, ground-penetrating radar signal processing and identification methods, and computational electromagnetics.

Zhiwei Lin (S’09) was born in Beijing, China, in 1984. He received the B.S. degree in electrical engineering from Peking University, Beijing, China, in 2006. Currently, he is working towards the Ph.D. degree at the Key Laboratory of Electromagnetic Radiation and Detection Techniques, Chinese Academy of Sciences, Beijing, China. His research interests are electromagnetic (EM) scattering from rough surfaces, EM scattering from inhomogeneous media, and EM inverse problem.

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Feng Zhang was born in Shaanxi, China, in 1981. He received the B.S. and M.S. degrees in electrical engineering from Northwestern Polytechnical University, Xi’an, Shaanxi, in 2004, and 2007, respectively. Currently, he is working towards the Ph.D. degree at the Key Laboratory of Electromagnetic Radiation and Detection Techniques, Institute of Electronics, Chinese Academy of Science (CAS), Beijing, China. His main interest is theory, optimization, design, and application research of wideband antenna.

Ling Huang was born in Nanning, China, in 1981. He received the B.S., M.S., and Ph.D. degrees in applied geophysics from Jinlin University, ChangChun, China, in 2005, 2007, and 2010, respectively. Currently, his research work focuses on GPR, electromagnetic, and other geophysical method in engineering and environment application.

Yicai Ji was born in Shandong, China, in 1974. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from Xidian University, Shaanxi, China, in 1998, 2001, and 2004, respectively. Currently, he is an Associate Researcher in the Key Laboratory of Electromagnetic Radiation and Detection Techniques, Institute of Electronics, Chinese Academy of Science (CAS), Beijing, China. His research interests including numerical calculation of antennas and electromagnetic compatibility.

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Communications RF MEMS Switchable Slot Patch Antenna Integrated With Bias Network Ilkyu Kim and Yahya Rahmat-Samii

Abstract—The functionality of the patch antenna with switchable slot (PASS) is realized by integrating a commercially packaged MEMS (microelectromechanical system) switch and a novel bias network. The proposed antenna includes the fully integrated dc bias network capable of actuating the MEMS switch on the slot. The bias network can minimize the interference between dc bias and RF radiation mechanism with a negligible loss. By using a MEMS switch and the bias network, the patch antenna operates at 4.57 GHz and 4.88 GHz. The fabricated design shows acceptable return losses and directivities at the broadside direction when the switch is on and off. The influence of RF MEMS switch on the antenna impedance and radiation pattern is addressed. Index Terms—Patch antenna, PCB substrate, radio frequency microelectromechanical system (RF MEMS), reconfigurable antenna, slot.

I. INTRODUCTION In recent years, researchers have incorporated RF switches in antenna configurations to attain more diverse functionalities [1]–[6]. Multiple functionalities have been achieved by using patch antennas loaded with switchable slots and a two-terminal diode switch [7]. Even though the two-terminal structure is preferable in the slot-loaded structure, diode switches exhibit relatively high insertion loss and low linearity [8]. In this aspect, RF MEMS switch can be an attractive candidate for designing reconfigurable devices [9]–[11]. However, inclusion of a three-terminal commercial packaged MEMS switch into the antenna can be a critical issue in designing reconfigurable antennas. In particular, when MEMS switches are placed in the slot-loaded structure, the bias line has the limited access to the switch in the slot. To overcome this problem, the floating dc ground [12], [13] has been used to increase the accessibility to the slot-structure. This ground, however, may not be desirable for building a fully-integrated structure due to the lack of the necessary bias line. In this communication, an innovative dc bias line is integrated into slot-loaded structure to actuate the MEMS switch in the slot. By including dc bias line in the design, the design accommodates easy supply of dc potential to the RF switch and full integration on one layer. Additionally, the bias network consists of the spiral and interdigital gap filters that are utilized to minimize the interference. The proposed bias network is applicable to any other slot loaded structure for commercial three terminal MEMS switches. The prototype of the proposed antenna is fabricated and measured. The effect of adding wire-bonded a MEMS switch on the radiating structure is also discussed. Manuscript received February 23, 2011; revised April 01, 2011; accepted May 09, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. The authors are with the Department of Electrical Engineering, University of California, Los Angeles, CA 90095-1594 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165512

II. MEMS IMPLEMENTATION AND BIAS CIRCUIT The patch antenna with the switchable slot (PASS) incorporated with RF switch is a compact and simple structure that can achieve tunable dual-band and dual-polarization performance [14]. The slot-loaded patch antennas exhibit two resonant frequencies shifting from 4.57 GHz to 4.88 GHz, according to the direct or meandering current paths of the slot. The topology of the proposed PASS antenna co-designed with a packaged RF MEMS switch is shown in Fig. 1. The patch antenna with a slot is printed on the substrate, and SMA probe is connected to the patch antenna for the excitation. The position of the probe feed is placed at the position, (x; y) = (4:8mm; 0) for the best match to 50 . The size of the patch antenna (L 2 W) is 17.7 mm 2 20.5 mm and the slot has the dimension (Ls 2 Ws) of 19.6 mm 2 2.2 mm with Wg = 1 mm. The substrate used in this antenna is Rodgers Duroid 5880, which has thickness of 3.18 mm, size of 55 mm 2 55 mm, and relative dielectric constant "r = 2:2 with a loss tan  = 0:0009 at 5 GHz. The RF MEMS switch placed in the slot alters effective electrical lengths of the patch antenna. When the switch is at the off-state, the current meanders around the slot which result in the resonance at 4.57 GHz. The switch at the on-state creates the direct current path across the slot for operating at 4.88 GHz. The RF MEMS switch and bias circuit for the actuation of the switch will be discussed next. A. MEMS Switch and Integrated Components Radant MEMS SPST-RMSW 100 packaged switch is placed in the center of the slot. The copper pad (1.5 mm 2 1.5 mm) is used to attach the gold base of the MEMS switch. The actuation voltage 90 VDC between the gate and source is applied to the pad in the bias line. When the switch is at the on-state, the RF continuity is generated with approximately 0.23 dB insertion loss. In the off-state, 20 dB isolation between source and drain creates RF discontinuity. To guarantee proper operation, source and drain pads are connected to the microstrip antenna with two wire-bondings. This method can reduce the relatively high-impedance of the wire-bonding line and increase the current flow. The width of the bias line is 0.2 mm which results in high characteristic impedance Zline > 200 at the operating frequencies of the antenna from 4.6 GHz to 5 GHz. To actuate the switch, the source and drain of the MEMS switch is required to connect to the dc ground plane. The RF ground plane is utilized for the dc ground plane for the simplicity of the circuitry. The dc continuity between patch antenna and RF ground plane is realized by utilizing the =4 length stub line short to the RF ground plane through a via structure. The stub is placed at 4.4 mm above from the center of the patch antenna. The shorted line appears as open-state for the antenna at the junction of the line and antenna due to its electrical length. The overall structure including the bias network is incorporated in a layer and only single via is used for dc ground. These features are amenable for the fully-integrated structure with minimum cost of fabrication. B. Operation of the Bias Network The slot-loaded structure complicates implementing a proper bias network integrated in the antenna. This is because the microstrip patch around the slot limits the accessibility of the dc bias line. To increase the accessibility, the area that RF and dc signals coexist is placed around the slot as shown in Fig. 2. The design makes it feasible to maintain

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Fig. 2. The dimension and configuration of the bias network: bias network consists of the dc bias line, spiral filter and inter-digital gap filter.

TABLE I DIMENSIONS OF ANTENNA AND BIAS NETWORK

Fig. 1. Patch antenna with switchable slot (PASS), bias network and RF MEMS switch. (a) Top and side view of the PASS antenna configuration. (b) RF MEMS PASS Antenna, Bias network and MEMS switch.

the high dc potential from the bias line to the gate of the switch and to continue the radiating current flow around the slot. The bias network is capable of strictly discerning the radiating and dc bias mechanism with the help of the high impedance filters in dc and microwave regions. The bias network includes the dc bias line, interdigital capacitor and spiral filter as shown in Fig. 2. The interdigital capacitor [15] is widely used for many applications due to its high Q-response as the low band stop filter. It also exhibits the low insertion loss in the pass band of the RF operation. In this design, the capacitor is used to isolate high dc potential from the microstrip patch and to flow the radiating current with small loss. Another important component in the bias network is the spiral filer [16]. The filter exhibits useful features like steep band rejection characteristics and compact size compared with widely used quarter-wave open stub. High impedance characteristics at the operating frequencies of the antenna prevent the leakage of the radiation current into the bias line. In Fig. 3, the operating mechanism of the dc and radiating current is presented. The dc potential is supplied along bias line without intruding other RF radiating regions. In the case of the radiating current, the interference with the bias line is minimized and its continuity on the patch is maintained. The S21 characteristics of the single filter component is shown in Fig. 4. The insertion loss of the interdigital capacitor is from 0.3 dB to 0.6 dB at the operating frequencies of the antenna from 4.6 GHz to 5.1 GHz. It exhibits the band rejection characteristics at the low frequency bands. The filter offers the sufficient band rejection characteristics against radiation current. The S21 of the inductor is lower than 012 dB from 4.57 GHz to 5.1 GHz, the range of dual-band operations of the antenna. Spiral filter should be placed in a position close to the patch to prevent standing-waves.

Fig. 3. The configuration of the dc potential distributions and radiating current distributions of the bias network. (a) DC potential distribution, (b) radiating current distribution.

III. EFFECTS OF SWITCH AND BIAS ON RF OPERATION The effects of the MEMS switch and its connection to the antenna are critical for the RF operation of the antenna. The original impedance and radiation pattern of the antenna without any switches can be altered when thin wire-bonding and switch is loaded with the antenna. To evaluate the changes caused by the wire-bondings and switch, the operation of the antenna loaded with RF MEMS switch should be compared with the antennas with an ideal connection. In Fig. 5, two simulated cases when the switch is on or off are presented. For the case of the ideal connection, thin copper microstrip is placed across the middle of the slot to create ideal shortest path of the radiating current as shown in Fig. 5(a). The ideal switch-off state is represented by placing the isolated copper pad as depicted in Fig. 5(c). In Fig. 5(b), the wire bonded RF MEMS switch model, which is closer to the real MEMS switch,

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Fig. 4. The operation of two filter components (spiral inductive filter and interdigital gap capacitive filter) of the bias network.

is shown. The membrane of the MEMS switch, designed with simplified 0.075 mm width microstrip line, is placed on the silicon substrate with a height of 0.25 mm. Two gold wires with a 0.01 mm radius are used to connect the membrane and the patch antenna. On or off-state of the RF MEMS switch is represented by attaching or detaching the middle of membrane, respectively. The S-parameters of the antenna for the each case are simulated and compared in HFSS. All simulated configurations are identical except for using different kinds of connections across the slot: Ideal connection and the wire-bonded RF MEMS switch. In Fig. 6, the S-parameters with ideal and MEMS loaded connection are compared. At the off-state, the maximum resonance of the ideal and MEMS loaded connection occurs at 4.59 GHz and 4.62 GHz, respectively. At the on-state, the patch antenna operates at 4.89 GHz and 4.92 GHz with MEMS load and ideal connection, respectively. It is seen that a design loaded with a RF MEMS switch does not exhibit the significant shift of the resonant frequency. Next, the change of the radiation pattern should be carefully considered to examine the loss of the radiation when the MEMS switch is loaded in the design. In Fig. 7, the simulated radiation patterns at frequencies of maximum resonances for each case are shown. The comparison was made with co-polarization and cross-polarization pattern at the off-state and on-state of the switch, respectively. The radiation patterns of MEMS switch are normalized to that of the ideal connection case. It is observed that the radiation patterns of three cases agree well with each other. The distortion caused by RF MEMS switch is still tolerable when the switch is in both off and on-state. At the broad-side direction, the loss of the co-polarized radiation pattern at switch off-state and on-state is less than 0.3 dB after loading the MEMS switch. Since all the cuts of the radiation pattern is not investigated, it is hard to generalize the change of the cross-polarization; however, from the Fig. 7, roughly 1–3 dB rise in the cross-polarization is observed with a RF MEMS switch. IV. EXPERIMENTAL RESULTS The proposed antenna was fabricated and measured to demonstrate its operational characteristics. The photograph of the prototype antenna is shown in Fig. 8. The return loss of the antenna loaded with the Radant MEMS switch was also measured. Using an Agilent 8720ES vector network analyzer, the return loss of the antenna was measured. To observe the reconfigurability, the voltage supplier was also connected to the bias network while maintaining the connection of the RF cable connection to the network analyzer. The voltage 3 VDC from the power

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Fig. 5. The simulated configurations for each case: (a) ideal switch-off connection, (b) wire-bonded MEMS switch-off, (c) ideal switch-on connection, (d) wire-bonded MEMS switch-on.

Fig. 6. The comparison of the return loss characteristics: (1) ideal connection, (2) wire-bonded MEMS switch. (a) switch-off state, (b) switch-on state.

supplier Agilent E3631A was amplified to 90 VDC through an amplifier circuit. The actuation voltage 90 VDC from the circuit was then applied to pad A through a soldered electrical wire. Similarly, a pad B was connected to the ground of dc power supply to make sure that the patch was grounded. When the voltage was applied, the MEMS switch is actuated and created the short current path across the slot at 4.88 GHz. When no voltage was applied, the current around slot created the long current path at 4.57 GHz. The simulated and measured return loss of the antenna is shown in Fig. 9. For more accuracy in simulation, the design was loaded with wire-bonded MEMS switch model as shown in Fig. 5(b) and (d). The antenna resonated at 4.57 GHz at the switch-off state and 4.88 GHz at the switch-on state. There was a 30 MHz discrepancy between simulated and measured S-parameters. The over-etching of the interdigital capacitor is assumed to be the main reason for the mismatch. The interdigital gap was originally designed as 80 m, however, it is over-etched to 100 m gap in the prototype. Using HFSS simulation, it is found that additional 0.3–0.5 dB loss is generated from the over-etching. The measured frequency ratio of two

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Fig. 9. Simulated and measured return loss of the proposed antenna.

Fig. 7. Simulated normalized co-polarized and cross-polarized radiation patterns at switch off-state and on-state for each case: (1) ideal connection, (2) wire-bonded MEMS switch.

Fig. 10. Measured normalized radiation pattern of the ideal off-state at 4.57 GHz and the ideal on-state at 4.89 GHz and of the MEMS off-state at 4. The radiation pattern with linear polarization was measured at 8 = 0 degree and 8 = 90 degree. Fig. 8. The photograph of the fabricated design loaded with RF MEMS switch, dc bias line and via structure.

operation states is 4:88=4:57 = 1:07 and the separation between two states is 300 MHz. The radiation pattern was measured with the fabricated patch which includes ideal off-state and on-state connection. As discussed in Section III, this model was verified to have almost similar return loss and radiation pattern to the case of actual MEMS switch. The normalized measured directivity pattern of the proposed antenna is shown in Fig. 10. The directivity pattern was measured with ideal off-state and MEMS off-state design at 4.57 GHz and with ideal on-state design at 4.88 GHz. At the off-state, the measured patterns with ideal connection and MEMS switch are very similar. The measured directivities of the antenna at the off-state and on-state are 7.1 dBi and

7.6 dBi at the broadside direction, respectively. At the switch on-state, the cross-polarization is 20 dB lower than the co-polarization. However, at the switch of off-state, the cross-polarization becomes higher because the current meandering around the slot is relatively more sensitive to the asymmetrical structure of the bias network. The radiation pattern with cross-polarization can be enhanced with positioning bias network near the center of the antenna. V. CONCLUSION The design of the patch antenna with switchable slot (PASS) is implemented with commercially an available RF MEMS switch and a novel bias network. A unique dc bias lines is integrated into the slotloaded structure to actuate switch placed in the slot. The inclusion of

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the dc bias lines brings out the necessity of the additional circuit components for the proper operation of the bias network. The filter components were added to generate the dc voltage difference for actuation and to maintain the continuity of the current on the patch with minimized leakage of those signals. The proposed design was fabricated, and measured at off-state and at on-state of the switch. The antenna operates at dual-frequency bands, 4.57 GHz and 4.88 GHz when the switch is off and on, respectively. The measured directivities at broadside direction are 7.1 dBi and 7.6 dBi at switch off and on-state, respectively. Next, the influence of the integration of the MEMS switch and bias network was addressed. It is verified that adding a MEMS switch and its connection to the patch antenna has minor effects on the radiation mechanism of the patch. The result also provides the validity of the measurement of the radiation pattern with the simple design of the ideal connection. The proposed design can be applicable to any other structure with slot and can also be expanded to the phased arrays applications.

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[13] H. Rajagopalan, Y. Rahmat-Samii, and W. A. Imbriale, “RF MEMS actuated reconfigurable reflectarray patch-slot element,” IEEE Trans. Antennas Propag., vol. 56, no. 12, pp. 3689–3699, Dec. 2008. [14] F. Yang and Y. Rahmat-Samii, “Patch antennas with switchable slots (PASS) in wireless communications: Concepts, designs, and applications,” IEEE Antennas Propag. Mag., vol. 47, no. 2, pp. 13–29, Apr. 2005. [15] G. D. Alley, “Interdigital capacitors and their application to lumpedelement microwave integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. 18, no. 12, pp. 1028–1033, Feb. 1979. [16] D. Ahn, J.-S. Park, C.-S. Kim, J. Kim, Y. Qian, and T. Itoh, “A design of the low-pass filter using the novel microstrip defected ground structure,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 1, pp. 86–93, Jan. 2001.

Reconfigurable Small-Aperture Evanescent Waveguide Antenna Peter Ludlow and Vincent Fusco

ACKNOWLEDGMENT The authors would like to thank Rogers Corporation for providing sample substrates and H. Rajagopalan for his assistance in measuring the antenna.

REFERENCES [1] M. Nishio, T. Itoh, S. Sekine, H. Shoki, M. Nishigaki, T. Nagano, and T. Kawakubo, “A study of wideband built-in antenna using RF MEMS variable capacitor for digital terrestrial broadcasting,” in Proc. IEEE Antennas Propag. Society Int. Symp., Jul. 2006, pp. 3943–3946. [2] J. H. Park, Y. D. Kim, Y. H. Park, H. C. Lee, H. Kwon, H. J. Nam, and J. U. Bu, “A tunable planar inverted-F antenna with an RF MEMS switch for the correction of impedance mismatch due to human hand effects,” in Proc. IEEE 20th Int. Conf. on Micro Electro Mechanical Systems, 2007, pp. 163–166. [3] P. Panaïa, C. Luxey, G. Jacquemod, R. Staraj, L. Petit, and L. Dussopt, “Multistandard reconfigurable PIFA antenna,” Microw. Opt. Technol. Lett., vol. 48, no. 10, pp. 1975–1977, Oct. 2006. [4] K. R. Boyle and P. G. Steeneken, “A five-band MEMS switched PIFA for mobile phones,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3300–3309, Nov. 2007. [5] D. Peroulis, S. Pacheco, K. Sarabandi, and L. P. B. Katehi, “Tunable lumped components with applications to reconfigurable MEMS filters,” in IEEE MTT-S Int. Microwave Symp. Digest, May 2001, vol. 1, pp. 341–344. [6] G. H. Huff and J. T. Bernhard, “Integration of packaged RF MEMS switches with radiation pattern reconfigurable square spiral microstrip antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 464–469, Feb. 2006. [7] N. Jin, F. Yang, and Y. Rahmat-Samii, “A novel patch antenna with switchable slot (PASS): Dual-frequency operation with reversed circular polarizations,” IEEE Trans. Antennas Propag., vol. 54, no. 3, pp. 1031–1034, Mar. 2006. [8] P. D. Grant, M. W. Denhoff, and R. R. Mansour, “A comparison between RF MEMS switches and semiconductor switches,” in Proc. MEMS, NANO and Smart Systems ICMENS Int. Conf., 2004, pp. 515–521. [9] G. M. Rebeiz, RF MEMS Theory, Design, and Technology. New York: Wiley, 2003. [10] J. Papapolymerou, K. L. Lange, C. L. Goldsmith, A. Malczewski, and J. Kleber, “Reconfigurable double-stub tuners using MEMS switches for intelligent RF front-ends,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 271–278, Jan. 2003. [11] J. P. Gianvittorio and Y. Rahmat-Samii, “Reconfigurable patch antennas for steerable reflectarray applications,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1388–1392, May 2006. [12] S. Nikolaou, A. Amadjikpe, J. Papapolymerou, and M. M. Tentzeris, “UWB elliptical monopoles with a reconfigurable band notch using MEMS switches actuated without bias lines,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2242–225, Aug. 2009.

Abstract—We present a reconfigurable small-aperture evanescent waveguide antenna. Tuning and waveguide matching to free space are simultaneously achieved by placing a printed iris with a shunt varactor diode connected across the aperture of a below-cutoff waveguide. A design approach using the Imaginary Smith Chart is used to synthesize the antenna. The maximum dimension of the antenna is 0 36 at 2.49 GHz and it operates over an operating frequency range of 2.05–2.49 GHz with maximum realized gain of 5 dBi. The antenna—due to its small size and compact feed arrangement—can be used in phased array/cognitive radio applications. Also its tuneable return loss characteristics enable it to act as an electromagnetic switch (shutter), or variable RCS aperture. Index Terms—Electrically small, evanescent waveguide, imaginary Smith chart, reconfigurable.

I. INTRODUCTION There is presently much interest in the design of reconfigurable antennas, i.e., antennas that have the ability to have their electrical and/or radiation characteristics manipulated in order to accommodate changes in the frequency spectrum environment in which they operate. It is evident that such antennas would have numerous applications, such as in software defined/cognitive radio systems, in situations where interfering bands require rejection, or where adaptive RF front ends are required. There are several ways in which a reconfigurable antenna may be realized: (a) using RF switches that may selectively switch in, or out, parts of the antenna structure [1], (b) by adjusting the loading or matching of the antenna externally (e.g., using varactor diodes [2], or, by deploying MEMS-tuneable capacitors [3]), and (c) by changing the antenna’s geometry through mechanical movement [4]. Evanescent waveguides operate below the cutoff frequency defined by the waveguide’s physical cross-sectional dimensions. The wave impedance of an evanescent waveguide is reactive in nature and the propagation constant is real; therefore a section of unmodified Manuscript received January 17, 2011; revised May 16, 2011; accepted June 02, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. This work was supported in part by the DEL Strengthening the All-Island Research Base Programme, Mobile Wireless Futures, and in part by the U.K. Engineering and Physical Science Research Council under Grant EP/E01707X/1. The work of P. Ludlow was supported by Powerwave Technologies and by the Department of Learning (DEL) for Northern Ireland. The authors are with the Institute of Electronics, Communications and Information Technology, Queen’s University Belfast, Queen’s Island, Belfast BT3 9DT, Northern Ireland, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2165505

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Fig. 2. Variation in aperture impedance of WR159 waveguide radiating into free space with frequency.

Fig. 1. Imaginary Smith Chart (impedance coordinates) [10].

evanescent waveguide will not permit effective wave propagation. Craven and Mok in 1971 [5] showed that an evanescent waveguide can be made to propagate if the appropriate terminating conditions (i.e., a conjugate reactance or susceptance) for the evanescent mode are synthesized. Since then evanescent waveguide has found application as filters [5], [6], metamaterial guided-wave structures [7] and as phased array antenna elements [8], [9]. In phased array applications they are used since their compact size leads to wider scan angles and reduction of grating lobe possibilities. A method was proposed in [10] by which evanescent-mode structures may be analyzed using an Imaginary Smith Chart. This involves normalizing the impedances being plotted to the characteristic impedance of the evanescent waveguide, jX0 . In this communication we present a reconfigurable evanescent openended waveguide antenna that has been designed using the Imaginary Smith Chart, thereby simplifying the design process greatly. The proposed design bears some similarities to cavity-backed slot antennas (CBSAs) since much of the aperture is metallized; however there are notable differences, with an iris rather than a slot forming the radiating aperture in our design. Furthermore, the cavity width of CBSAs must generally be chosen to be close to c =2 at the operating frequency to enable efficient radiation [11]. The waveguide used in our design operates below its dominant-mode cutoff frequency and therefore has a width 1, if capacitive impedances are to be represented. As shown in Fig. 1, the impedance is purely real and positive for all points lying on the lower half of the j0j = 1 circle of the Imaginary Smith Chart. Furthermore, while a length of above cutoff waveguide rotates an impedance point around the Smith Chart, a length of evanescent waveguide moves an impedance point radially closer to the centre of the Imaginary Smith Chart by the factor e02 s , due to the propagation constant being purely real. If the aperture impedance of the evanescent waveguide can be transformed such that for a given frequency it intersects with the lower half of the j0j = 1 circle of the Imaginary Smith Chart at a plane in the waveguide then it is possible to match to the real impedance at this

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Fig. 4. Geometry of the reconfigurable evanescent open-ended waveguide antenna (a) side view, (b) printed iris. Fig. 3. Matching the aperture impedance using a series resonant obstacle with the Imaginary Admittance Smith Chart.

plane using a coaxial probe. By using a shunt obstacle that is series resonant in nature e.g., an inductive iris with a series lumped capacitance, it is possible to obtain a match at a single frequency. The insertion of a varactor diode rather than a lumped capacitance means that the frequency of the match may be electrically tuned. The equivalent circuit of an obstacle in cutoff waveguide is not necessarily the same as the one that applies in the propagating region, with the effect of the imaginary guide wavelength (which occurs below cutoff) altering the sign of the obstacle susceptance [5]. Note that an iris consisting of horizontal metal strips placed orthogonal to the electric field of the TE10 mode is a resonant iris with its resonance centred at cutoff; it is capacitive above cutoff but inductive below cutoff [5], [14]. This iris configuration is used in the design of the antenna. Fig. 3 shows the transformations, along the Imaginary Admittance Smith Chart, of the aperture admittance of the WR159 waveguide at 2.1 GHz and 2.3 GHz during matching–initially a series resonant obstacle is added at the aperture of the waveguide and transforms the aperture admittance along a circle of constant conductance; then a length of evanescent waveguide is added which transforms the admittance back towards the centre of the chart such that it becomes purely real at 2.3 GHz and may therefore be easily matched into 50 ohms at this frequency. By increasing the inductance of the series resonant obstacle i.e., by making the inductive iris extend across all of the broad dimension of the waveguide or else by increasing the size of the aperture of the inductive iris, it is possible to decrease the resonant frequency at which the optimal match may be obtained. Likewise, the capacitance of the series resonant obstacle may be increased—and the resonant frequency decreased - by decreasing the reverse voltage of the varactor diode or using a varactor diode with a higher maximum capacitance value. Decreasing the length of evanescent waveguide between the obstacle and the feed probe also causes the operating resonant frequency to decrease. This may be explained with reference to the locus formed on the Imaginary Smith Chart by plotting the aperture admittance of WR159 waveguide across the operating frequency range of the antenna (as shown for two frequencies in Fig. 3). For higher frequencies, the susceptance of points on this locus becomes more positive. The addition of the shunt LC obstacle at the aperture therefore transforms aperture admittance points at higher frequencies further from the centre of the Imaginary Smith Chart. A shorter length of evanescent waveguide between obstacle and feed probe is therefore required to allow transformed admittance points at lower frequencies to intersect with the j0j = 1 circle and thereby be matched. CST was used to optimize the parameters of the design. The primary considerations in the design were: (i) to obtain matching within the 1.8–3 GHz band, (ii) to allow the resonant frequency to be tuned over as much of this band as possible, (iii) to allow ease of fabrication of the design, and (iv) to make the design as compact as possible. It was

TABLE I DIMENSIONS OF THE RECONFIGURABLE EVANESCENT OPEN-ENDED WAVEGUIDE ANTENNA [mm]

decided that an SMA connector would be used as the feeding probe, with its inner conductor extending into the waveguide such that the TE10 mode is excited; this did however place a limitation on the design as the flange of the connector is 12.7 mm2 and so to allow it to be electrically connected to the waveguide wall the distance from aperture to probe has to be at least 6.35 mm. It was desired that the cavity be as shallow as possible such that the design remained compact, so therefore the distance between probe and obstacle was set at a value close to 6.35 mm. The varactor diode used in the design was chosen after preliminary investigations to determine the range of capacitance values required, and with the requirement that its parasitic resistance be as low as possible, to ensure that the antenna’s radiation efficiency wasn’t adversely affected. A further consideration was that the tuning range of capacitance values be as great as possible to enable a wider operating frequency range. The varactor diode chosen was the Skyworks SMV 1408, which has a capacitance that varies from 0.95–4.08 pF as the reverse voltage is varied from 0–30 V and a low value of series resistance (0.6

at 4 V, 500 MHz). The other main parameter to be adjusted in the design to achieve the desired matching was the dimension of the inductive iris, which is printed onto RT Duroid 5880 dielectric with "r = 2:2 and 0.508 mm thickness. To ensure that the anode and cathode of the varactor diode are isolated from the walls of the waveguide (which are electrically connected to the outer conductor of the SMA), a 1 mm gap is introduced between the inductive iris and the side wall of the waveguide. The dimension of the aperture of the inductive iris was then adjusted to obtain a lower frequency for the operating frequency range of close to 1.8 GHz. The final dimensions of the design are as shown in Fig. 4 and Table I. The simulated variation in matching of the antenna as the reverse voltage on the varactor diode is tuned from 0 V to 30 V is as shown in Fig. 5, with the antenna matched at 1.86 GHz for Vr = 0 V and 2.31 GHz for Vr = 30 V i.e., the match may be varied across an operating frequency range of 22%. The varactor diode has a specified value of series resistance of 0.6 at 500 MHz/4 V, estimated at 2 GHz to be 1.5

at 0 V, 1.0 at 4 V and 0.5 at 30 V. The electric field distribution at the obstacle plane is modified by the addition of gap, i, used to isolate the varactor diode from the waveguide walls. The cross-polarized component of the electric field is in anti-phase on either side of the broad dimension of the iris—these components interfere destructively at boresight, giving a null. The converse is true at the sides of the antenna. Cross-polarization level increases as i

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Fig. 5. Variation in simulated reflection coefficient of the antenna with frequency as the varactor diode’s reverse voltage is varied.

Fig. 6. Fabricated reconfigurable evanescent open-ended waveguide antenna (a) front view, (b) printed iris with shunt varactor diode.

Fig. 7. Variation in measured reflection coefficient of the antenna with frequency as the varactor diode’s reverse voltage is varied.

is increased, while decreasing as the width of the aperture of the inductive iris, wa , is increased. It was also found that simulated directivity is increased slightly when i is decreased or wa increased. III. MEASURED RESULTS The antenna design outlined in Section II was fabricated as shown in Fig. 6. The varactor diode was excited using 0.255 mm diameter copper wire (not shown in Fig. 6) placed orthogonal to the TE10 mode, to avoid coupling to the feeding probe. Fig. 7 shows the measured variation in the matching of the antenna as the reverse voltage on the varactor diode is tuned from 0 V to 30 V. Here the antenna is matched at 2.05 GHz for Vr = 0 V and 2.49 GHz for Vr = 30 V i.e., over an operating frequency range of 19.4%. The operating frequency range over which matching is possible is similar to that simulated, albeit across a slightly higher range of frequencies due to varactor capacitance tolerance. These results indicate that suitable choice of varactor bias can be used to select or reject specific frequencies within the overall operating

Fig. 8. Variation in measured peak realized gain of the antenna with frequency as the varactor diode’s reverse voltage is varied.

passband of the antenna. Also they show that within its operating band the antenna can be made to reflect or accept energy (i.e., behave as an agile RCS or EM shutter device), or indeed, act as an AM modulator by appropriate selection of the dc bias applied to the varactor diode. The radiation patterns of the antenna were measured in the x-y ( = 90 ) plane and the x-z (' = 0 ) plane. The variation in the measured peak realized gain of the antenna with frequency as the reverse voltage applied to the varactor diode is tuned is shown in Fig. 8. When configured with Vr = 0 V the antenna is matched at 2.05 GHz, with a realized gain of 03:03 dBi; Vr = 4 V match is at 2.23 GHz, with a realized gain of 3.4 dBi; with Vr = 30 V the antenna is matched at 2.49 GHz, with a realized gain of 5.1 dBi. The low realized gain observed when the antenna is matched at 2.05 GHz is indicative of low radiation efficiency. This is caused by the parasitic resistance of the varactor diode (which is one of the main sources of loss in the design) being greater with lower values of reverse voltage applied. This is reflected in the simulated farfield results, where radiation efficiency is equal to 27.9%, 59% and 89.3% with reverse voltages of 0 V, 4 V and 30 V applied respectively. The feeding probe is also electrically smaller at the lower end of the operating frequency range so will be a less efficient radiator, while directivity will be lower as the aperture size becomes electrically smaller, contributing to the lower realized gain observed at 2.05 GHz. The radiation patterns of the antenna for the above range of Vr are shown in Fig. 9 for both the x-y ( = 90 ) plane and the x-z (' = 0 ) plane. It may be observed that cross-polarization is fairly low at boresight, with a cross-polar ratio >15 dB observed at boresight across the operating frequency range for the patterns shown; however, cross-polarization is considerably higher at the sides of the antenna. This is most likely caused by the gap, i, introduced between the waveguide walls and inductive iris to isolate the varactor diode and SMA outer conductor, as was discussed in Section II. The level of cross-polarization could be reduced by using a smaller value of i in the design. This value was dependent on fabrication tolerances and in hindsight could have been made smaller. Using a wider aperture in the inductive iris may also improve the cross-polarization characteristics of the antenna but this would also change the matching frequency. The high level of back radiation observed is similar to that seen in the designs of [11], [12] and is most likely due to the size of the antenna, which is less than 0:370 in dimension at the highest frequency of operation. IV. CONCLUSION This communication has demonstrated a simple reconfigurable evanescent open-ended waveguide antenna that has a tuneable resonant frequency, achieved through placing a printed iris with a varactor diode connected across it at the aperture of a below-cutoff waveguide. The Imaginary Smith Chart was used in the design of the antenna. It was shown that the resonant frequency may be electronically tuned from 2.05–2.49 GHz as the voltage supplied to the varactor diode

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Fig. 9. Radiation patterns for reconfigurable evanescent open-ended waveguide antenna measured at (a) 2.05 GHz, V = 0 V,  = V = 0 V, ' = 0 plane, (c) 2.23 GHz, V = 4 V,  = 90 plane, (d) 2.23 GHz, V = 4 V, ' = 0 plane, (e) 2.49 GHz, V (f) 2.49 GHz, V = 30 V, ' = 0 plane.

is varied. Radiation pattern measurements show that a realized gain of up to 5.03 dBi may be attained (when matched at 2.49 GHz).

ACKNOWLEDGMENT The authors would like to thank G. Rafferty, M. Major and J. Megarry for the construction of the antenna elements used in this work.

REFERENCES [1] M. Ali, T. M. Sayem, and V. K. Kunda, “A reconfigurable stacked microstrip patch antenna for satellite and terrestrial links,” IEEE Trans. Vehicular Tech., vol. 56, no. 2, pp. 426–435, Mar. 2007. [2] Z. Zhou and K. L. Melde, “Frequency agility of broadband antennas integrated with a reconfigurable RF impedance tuner,” IEEE Antennas Wireless Propag. Lett., vol. 6, no. 11, pp. 56–59, 2007. [3] K. R. Boyle and P. G. Steeneken, “A five-band reconfigurable PIFA for mobile phones,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3300–3309, Nov. 2007. [4] G. Ruvio, M. J. Ammann, and Z. N. Chen, “Wideband reconfigurable rolled planar monopole antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1760–1767, June 2007.

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90 plane, (b) 2.05 GHz, = 30 V,  = 90 plane,

[5] G. F. Craven and C. K. Mok, “The design of evanescent mode waveguide bandpass filters for a prescribed insertion loss characteristic,” IEEE Trans. Microwave Theory Tech., vol. 19, no. 3, pp. 295–308, Mar. 1971. [6] S. Park, I. Reines, C. Patel, and G. Rebeiz, “High-Q RF-MEMS 4–6 GHz tunable evanescent-mode cavity filter,” IEEE Trans. Microwave Theory Tech., vol. 58, no. 2, pp. 381–389, Feb. 2010. [7] I. A. Eshrah, A. A. Kishk, A. B. Yakovlev, and A. W. Glisson, “Rectangular waveguide with dielectric filled corrugations supporting backward waves,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 11, pp. 3298–3304, Nov. 2005. [8] A. R. Lopez, “Wideband dual-polarised element for a phased array antenna,” Wheeler Laboratory Hazeltine Corp, Tech. Rep. AFAL-TR-74000, Jun. 1974. [9] S. J. Foti and M. W. Shelley, “An experimental wideband polarisation diverse phased array,” Proc. Military Microw,, no. 7, pp. 263–271, 1990. [10] E. DeJersey, “Imaginary Smith Chart for evanescent-mode structures,” Electron. Lett., vol. 16, no. 3, pp. 93–94, Jan. 1980. [11] W. Hong, N. Behdad, and K. Sarabandi, “Size reduction of cavitybacked slot antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1461–1466, May 2006. [12] C. White and G. M. Rebeiz, “A shallow varactor-tuned cavity-backed slot antenna with a 1.9:1 tuning range,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 633–639, Mar. 2010. [13] C. R. Boyd, “Impedance matching of open-ended waveguide radiating elements,” presented at the SBMO Int. Microwave Symp., Rio de Janeiro, Brazil, Jul. 1987. [14] G. F. Craven and R. F. Skedd, Evanescent Mode Microwave Components. Boston, MA: Artech House, 1987, pp. 13–16.

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Plastic-Based Supershaped Dielectric Resonator Antennas for Wide-Band Applications Massimiliano Simeoni, Renato Cicchetti, Alexander Yarovoy, and Diego Caratelli

Abstract—Novel cylindrical dielectric resonator antennas (DRAs) having supershaped base contour and adopting plastic materials for the resonator are studied. The specialization of the supershaped DRAs to the generation of linearly and circularly polarized waves is discussed, analyzed and experimentally verified. The resulting antennas exhibit wide-band (WB) performance in terms of input impedance matching, radiation patterns, realized gain and polarization properties. The proposed class of antennas shows broadside radiation with broad and smooth patterns stable over frequency, efficient and stable radiation and wide matching bandwidths. These antennas can potentially find application as access points for indoor multimedia radio systems. Index Terms—Circular polarization, dielectric resonator antennas (DRAs), plastics, ultra wideband antennas.

I. INTRODUCTION The ubiquitous nature of current and future wireless systems continuously rises the demand for better antenna designs. In particular, improvements are stimulated for enhancing the electrical performance and for reducing the cost of these components. Currently, antennas for wireless applications are mainly based on planar and monopole radiators, it is however the authors’ conviction that dielectric resonator antennas (DRAs) can play an increasingly important role in this field. This belief is based on the many intrinsic advantages introduced by DRAs over more traditional implementations. An interesting account supporting this point of view can be found in [1]. Since their first appearance, DRAs have been found attractive in reason of their reduced losses (virtually no metal loss), ease of fabrication, broadside radiation patterns, wide frequency band of operation, ease of integration with planar circuitry and the possibility that they offer of achieving high radiation efficiency. A good summary of DRAs different implementations and related features is provided in [2]. In the recent years the most standard dielectric resonators (DRs) configurations (hemispherical, cylindrical and rectangular) have been modified mostly in the quest for enhanced operational bandwidths. Multi-segment DRs [3], stacked DRs [4], as well as more elaborated structures (see for instance [5] and [6]) led to enhanced (relative) bandwidths of about 35–40%. In [7] stacked and embedded DRs were proposed showing the possibility to achieve remarkable operational bandwidths (up to more than 60%) at the expense of an increased complexity in the antenna manufacturing process. The use of multiple resonating modes was adopted in [8]. A complete account on the different designs is obviously beyond the scope of this discussion. More recently a new class of DRAs, the so-called supershaped DRA (S-DRAs) [9] was introduced. These antennas feature an increased Manuscript received July 02, 2010; revised May 18, 2011; accepted June 02, 2011. Date of publication August 18, 2011; date of current version December 02, 2011. M. Simeoni, A. Yarovoy, and D. Caratelli are with the Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, 2628 CD Delft, The Netherlands (e-mail: [email protected]). R. Cicchetti is with the Department of Electronic Engineering, “Sapienza” University of Rome, 00184 Rome, Italy. Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165477

Fig. 1. Supershaped DRA geometry (a = b = 1; m = 4; n = n = 1=2) and relevant reference system. The DR is shown in light gray.

n

=

number of degrees of freedom for the design of the DR geometry, paving the way towards a wider variety of DRAs. In [9] the basic idea of S-DRAs was presented together with a numerical investigation of a small sample of such radiators. S-DRAs were shown to introduce good flexibility in terms of radiation patterns and matching bandwidths while preserving broadside radiation and ease of manufacturing. In this communication the basic idea of S-DRAs is associated to the use of plastic, easily available and machinable materials. This choice leads towards the realization of inexpensive DRAs with superior experimentally verified performance in terms of matching bandwidths, radiation efficiency and stability of the radiation patterns. A set of S-DRAs, specifically designed to operate as circularly polarized radiators, featuring wide-band behavior in terms of impedance matching (complying to the WiMEDIA standard [10]), radiation patterns and polarization purity is analyzed and experimentally validated. II. SUPERSHAPED DRAS A. Geometry Description The proposed supershaped DRA, as shown in Fig. 1, consists of a DR placed on top of a circular metal plate (having radius g and thickness tg ) acting as ground plane. The antenna is fed from the bottom side by means of a coaxial connector turning, at the ground plane level, into an electric probe of length hp and diameter dp . The probe is located at (xp ; yp ; 0). The DR is a prism with its axis aligned along the z -direction and its base profile defined by the polar function

d (') =

1

a

cos

m 4

'

n +

1

b

sin

m 4

'

n

01=n (1)

where d (') is a curve located in the xy -plane and ' 2 [0; 2 ) is the angular coordinate (see Fig. 1). Equation (1) is a generalization of the ellipse’s polar equation and is known in literature with the name of superformula [11]. The shape of the DR profile can be modified by acting on a sextet of real and positive numerical parameters1 6 (a; b; m; n1 ; n2 ; n3 ) 2 + , with a; b 6= 0. For a sample of the large variety of shapes that can be described by (1) see [11]. 1This translates into the possibility of automatically reshaping the DR profile. Since the antenna behavior is dependent on the DR profile, an automated optimization procedure could be devised in order to identify the sextet (a; b; m; n ; n ; n ) yielding optimum antenna performance.

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B. Design Guidelines The design of a S-DRA is carried out by assimilating it to an ordinary cylindrical DRA, its base effective radius is defined as

1 e = 2

2 0

 2d

(')d'

(2)

where d (') is given by (1). The resonator height (hd ) is chosen to be about a wavelength in the dielectric material at the central operating frequency of the antenna. Based on the study reported in [8], in order to achieve wide frequency bands of operation, the cross-sectional dimensions of the DR are set so that the following aspect ratio is obtained

 = e =hd

= 0:4:

(3)

The location and the length of the probe are heuristically determined for maximum operational bandwidth. III. PLASTIC-BASED S-DRAS Usually, low-loss microwave ceramics are chosen for manufacturing DRs. This follows naturally from the exigence of having DRs exhibiting low dielectric losses and compact dimensions, two of their most advantageous features. However, the ceramic materials commonly used have a rather high permittivity ("r exceeding 10), this causing an increase in the DR’s quality factor (Q) that eventually translates in a narrow operational bandwidth (see [2, Ch. 2]). Moreover, the use of ceramic materials seriously limits the possibility of shaping the DRs such to exploit the design flexibility offered by non canonical (cylindrical and rectangular) shapes. Being ceramic materials prone to chip and fracture, the machining of ceramic blocks is a delicate, time consuming and expensive task. An alternative approach would be firing the ceramic compounds directly in the desired shapes, this, however, would result in higher production costs. In view of what stated above, we hereby advocate the use of plastic materials for the manufacturing of DRAs, in particular, polyvinyl chloride (PVC) has been chosen. Polyvinyl chloride is a thermoplastic polymer produced in enormous quantities in reason of its low cost and ease of processing. Moreover, PVC is extremely durable and resistant to chemical corrosion thus very well adapted to outdoor applications. Additionally, PVC can be blended with different materials providing a wide range of physical properties. The use of PVC can then result in an inexpensive possibility of manufacturing intricately shaped DRAs. It is worth noting that in [9] the material of choice for the DR was Teflon, ensuring very good electrical performance but being considerably more expensive than PVC. Not only PVC enables the implementation of complicated DR geometries but, in reason of its low permittivity ("r = 2:8 6 20% [12]), it also favors a lowering of the DR Q factor and consequently a widening of the antenna working frequency band. Obviously, the use of low permittivity materials also causes an increase in the geometrical dimensions of the DR (see Section II.B). Despite the DR low permittivity it is important to stress that the proposed antennas truly perform as DRAs. In fact, the broadside radiation patterns shown in Section VI clearly witness the effect of the resonator in the radiation process. The DR results in a wave-guiding of the electromagnetic energy along the z -direction, conversely a radiation null would be observed due to the monopole-like behavior of the excitation probe. It is worth noting that since PVC has not been so far used for applications where its permittivity value was of primary importance, it is difficult to find this material with well controlled values of "r . This, in principle, could affect the performance of PVC-based DRAs. However, as it will become evident in the forthcoming sections, in our case taking "r = 2:8 and tan  (@10 GHz) ' 0:003 according to the ex-

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periments reported in [12] leads to measured antenna performance in good agreement with the numerical predictions. IV. A FAMILY OF S-DRAS As stated in Section II.A, by changing the numerical parameters in (1), a virtually infinite variety of shapes can be generated. The exhaustive investigation of the electrical properties of S-DRAs generated in this way is clearly out of our reach and outside the scope of this work. We limit therefore our investigation to a family of antennas obtained by setting (a; b; n1 ; n2 ; n3 ) = (a; a; 1=2; 1=2; 1=2) and m = 0; 1; . . . ; 5. This set of shapes represents a family of pseudo-polygons with m representing the number of pseudo-vertexes, m = 0 corresponding to the case of the circle. The study that follows, carried out by means of the full-wave commercial software CST Microwave Studio, enables us to estimate the impact of the DR shape (in this case the number of pseudo-vertexes) on the electrical performance of the S-DRAs. In particular, the different antennas are compared in terms of their fractional bandwidth (FBW) evaluated for a return-loss level at the connector input larger than 10 dB. The six different antennas are designed following the guidelines reported in Section II.B to operate in the frequency band allocated in Europe and China for the upcoming WiMedia standard for wireless multimedia applications [10]. The probe length hp = 6:5 mm has been set such to maximize the operational bandwidth of the cylindrical DRA (m = 0) and has been kept unchanged for all the configurations. The DR height is set to be hd = 27 mm, slightly exceeding 0;d , the wavelength in the dielectric material at the central frequency (f0 = 7:65 GHz). The effective radius is the same for all the six configurations and amounts to e = 1:08 cm, this obviously results in the fact that all the different DRs occupy the same volume. A parametric study on the location of the excitation probe has been conducted for the considered six antennas. The probe was allowed to explore the DR footprint, provided that the connector’s insulator was not exposed on the ground plane. The probe length has been kept constant in order to limit the number of optimization parameters even though different DR geometries could require different values of hp for yielding maximum operational bandwidth. Fig. 2 shows the results of the parametric study. The fractional bandwidth is represented as function of the probe location. The highlighted areas represent the probe locations yielding FBW exceeding 60%. As it is clearly visible from the different plots, the cylindrical DRA never reached the condition FBW 60%. This result clearly demonstrates how S-DRAs outperform the classical cylindrical DRAs in terms of FBW. It is to be noticed that, thanks to the adoption of suitable plastic materials, such remarkable performance is achievable with no impact on the DR machining process and, hence, no additional manufacturing costs. This feature is particularly desirable in a mass-production context typical of new-generation wireless multimedia applications with demanding requirements in terms of operational bandwidth and data transmission rate. For the different DR geometries the maximum value of the antenna realized gain (G0 ) has been numerically estimated within the frequency range of operation with a resolution of 1 GHz. For every antenna configuration the probe location yielding maximum impedance bandwidth has been considered. The numerical results listed in Table I show that S-DRAs combine wide impedance matching operation (FBW > 60%) and values of G0 in the range of 10 dBi, making them very efficient wide-band radiators. It is worth noting that for different antenna geometries the maximum gain is obtained at different frequencies, this is due to the fact that the radiation patterns undergo slight modifications as the frequency changes. A closer look at how the DR shape influences the field radiation, especially towards the broadside direction, has been taken by analyzing

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 12, DECEMBER 2011

Fig. 3. The spatial distribution of the real part of the vertical component of the ) in the plane = for = 11 GHz. Different DR Poynting vector ( configurations: (a) = 4, (b) = 0.

(1=2)(a=c) for 

 1 and (t=a) < 0:57.

(14) (15) (16)

(20)

The equivalent internal inductance is then (t=a) in parallel with the unit external inductance yielding

QTE =e Q  1 + 1=((t=a)opt ):

(21)

The Q values are not too sensitive to the layer thickness so that a single thickness can be adjusted to provide low Q performance for couTE modes. The table illustrates that the Q’s of both the pled TM individual and coupled modes decrease towards the external (Chu) limit as  increases, coming within a few percent at ka = 0:5 and  = 1000. If one of the antenna configurations in [7]–[12] were placed over an appropriate magnetic shell and retuned, it would have a correspondingly lower Q. As an ancillary benefit, these antennas would have space inside for electronic circuitry that would not affect the Q or pattern. An example of scattering of coupled TM TE modes from a  covered conducting sphere is given in ([2], Section V).

$

$

B. Solid Material Core The effect of material uniformly filling the sphere can be analyzed by multiplying all inductances and capacitances in the entire internal circuit ([2], Fig. 1) by  and ", respectively, but for small values of ; ", and ka only the leading elements are needed. The series TM and parallel TE resonances occur at the same frequencies, and vice versa, regardless of  and " so that there is no option to optimize both modes with the same filling. The lowest resonance occurs at

p

kc a = 2:744= ":

(22)

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 12, DECEMBER 2011

An expression for QTE based only on the first internal element was previously reported [3], [14] and is quite accurate to just over half of the first resonance. That is

QTE

=e Q = 1 + 2=;

for (") < 2:5=(ka)2 :

TM11 doublets which

VII. CONCLUSION (23)

For example at ka = 0:5;  = 10, and " = 1, (23) gives 1.200 compared to the exact value of 1.205; for,  = 5 and " = 2 the corresponding results are 1.400 and 1.409. As the " product increases beyond this level, the Q does not decrease as rapidly as predicted by (23), and when it reaches the resonance condition from (22), the Q starts to rise rapidly [15, Fig. 3(c)]. At this point replacing the solid magnetic core with a shell over a conducting or hollow core allows the Q to continue to decrease roughly in accordance with (23). Although increasing  of a solid core lowers QTE , it raises QTM until it becomes infinite at (series) resonance (22) as illustrated by the circuit in Fig. 1. The values of  in the table lower QTE at the expense of a slight increase in QTM in order to equalize them and would be appropriate, e.g., for a circularly polarized antenna with a (lossless) gain of 3.0 and a Q lower than the single-mode external e Q (Chu). For ka = 0:5, a circularly polarized antenna could approach 1:398e QD = (1:398)(6:00) = 8:4 (5) compared to e Q = 10 (4). Much higher values of  with a conductive lining could do even better. A multiarm spherical helix antenna wound on a solid magnetic core is described in [15]. C. Material Shell With Hollow Core A hollow core acts as a small inductive reactance to a TE mode, approximating the effect of a conducting core. As a result, the thickness and Q are essentially the same for both configurations. On the other hand, it acts as a large capacitive reactance for a TM mode, resulting in a thicker layer, more internal energy, and a higher Q as shown in the table. Specific cylindrical and spherical geometries with thin, high  shells are reported in [16]. VI. NON-SPHERICAL SHAPES For a spherical antenna the three n = 1 TM modes have a common Q, as do the three TE modes. As a result, for example, a TM mode can be combined with any TE mode(s) to modify the pattern without impacting the Q as explained in the previous analyses and tables. When the shape deviates from a sphere, these degeneracies are upset; the Q of some modes may increase to the point that they can not contribute to the pattern. If the antenna is symmetrical about the z axis, the two m = 1 modes remain degenerate, but the m = 0 does not. Hansen and Adams provide equations and curves that show how the e QTM bound varies for the m = 0 mode as the shape transitions from an oblate spheroid to a sphere to a prolate spheroid [17, Eqs. (1), (2)], ([18], Fig. 2). Table VI gives the bounds normalized to the e Q of a circumscribed sphere as a function of the height-to-diameter ratio (b/a) 1, based on external energy only. As the shape deviates from for ka spherical, the volume between the spheroid and the sphere increases and the (tighter) spheroidal bound rises above the sphere value. As the prolate spheroid becomes more slender, the bound for the TM10 mode rises more slowly than for the other five modes (although exact values are not provided in the references) so that eventually it dominates, and there is no freedom to control the pattern or to reduce the net Q by coupling to a TE mode. On the other hand, as b/a decreases the oblate



spheroid TM10 Q rises more rapidly than the might be used for circular polarization.

The bounds on the Q of a small antenna depend on the polarization and gain. The bounds derived including the internal energy can be approached quite closely as evidenced by the referenced results. Moderate  lowers the TE Q, but raises the TM. A thin layer of high  material excludes the internal energy and could allow the Chu limit to be realized for both TE and TM. Non-spherical envelopes limit the number of modes and thus the options for synthesizing low Q patterns. The analyses can be extended to n = 2 which would bound virtually all antennas up to ka = 1.

REFERENCES [1] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl Phys., vol. 19, pp. 1163–1175, Dec. 1948. [2] H. L. Thal, “Exact circuit analysis of spherical waves,” IEEE Trans. Antennas Propag., vol. 26, no. 2, pp. 282–287, Mar. 1978. [3] H. L. Thal, “New radiation Q limits for spherical wire antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2757–2763, Oct. 2006. [4] R. L. Fante, “Quality factor of general ideal antennas,” IEEE Trans. Antennas Propag., vol. 17, no. 2, pp. 151–155, Mar. 1969. [5] D. Pozar, “New results for minimum Q, maximum gain, and polarization properties of electrically small arbitrary antennas,” in Proc. 3rd Eur. Conf. on Antennas and Propag., 2009, pp. 1993–1996, EuCap. [6] H. L. Thal, “Gain and Q bounds for coupled TM-TE modes,” IEEE Trans. Antenna Propag., vol. 57, no. 7, pp. 1879–1885, Jul. 2009. [7] J. J. Adams and J. T. Bernhard, “A low Q electrically small spherical antenna,” presented at the IEEE Int. Symposium Antennas Propag., Jul. 5–11, 2008, paper 209.1. [8] S. R. Best, “Low Q electrically small linear and elliptical polarized spherical dipole antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1047–1053, Mar. 2005. [9] S. R. Best, “A low Q electrically small magnetic (TE mode) dipole,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 572–575, 2009. [10] O. S. Kim, “Low-Q electrically small spherical magnetic dipole antennas,” IEEE Trans. Antennas Propag., vol. 58, no. 7, pp. 2210–2217, Jul. 2010. [11] H. L. Thal, “A reevaluation of the radiation Q bounds for loop antennas,” IEEE Antennas Propag. Mag., vol. 51, no. 3, pp. 47–52, Jun. 2009. [12] A. Erentok and O. Sigmund, “Topology optimization of sub-wavelength antennas,” IEEE Trans. Antennas Propag., vol. 59, no. 1, pp. 58–69, Jan. 2011. [13] H. L. Thal, “Comments on ‘topology optimization of sub-wavelength antennas’,” IEEE Trans. Antennas Propag., vol. 59, no. 7, pp. 2752–2752, Jul. 2011. [14] H. A. Wheeler, “The spherical coil as an inductor, shield or antenna,” Proc. IRE, vol. 46, pp. 1595–1602, Sep. 1958, (correction Mar. 1960). [15] O. S. Kim, O. Breinbjerg, and A. D. Yaghjian, “Electrically small magnetic dipole antennas with quality factors approaching the Chu lower bound,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 1898–1906, Jun. 2010. [16] H. Stuart and A. Yaghjian, “Approaching the lower bounds on Q for electrically small electric-dipole antennas using high permeability shells,” IEEE Trans. Antennas Propag., vol. 58, no. 12, pp. 3865–3872, Dec. 2010. [17] P. M. Hansen and R. Adams, “Minimum radiation Q for spheroids—Extension to cylinder, comparison to spherical formulas and practical antennas,” presented at the IEEE Int. Symposium Antennas Propag., Jul. 11–17, 2010, paper 522.7. [18] P. M. Hansen and R. Adams, “The minimum value for the quality factor of an electrically small antenna in spheroidal coordinates—TM case,” presented at the IEEE Int. Symp. Antennas Propag., Jul. 11–17, 2010, paper 522.6.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 12, DECEMBER 2011

On the Robustness to Element Failures of Linear ADS-Thinned Arrays

on ADS when practical architectures, possibly comprising faulty elements, are at hand.

Matteo Carlin, Giacomo Oliveri, and Andrea Massa

Abstract—The robustness of the analytical thinning based on almost difference sets (ADSs) is analyzed when faulty elements are present in linear arrays. Peak sidelobe level (PSL) statistics of the damaged layouts are numerically inferred from representative simulations dealing with different aperture sizes and element failure rates.

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II. BACKGROUND ON ADS THINNING AND FAULTS MODELING Let us consider a broadside-steered linear thinned array of active elements located on a regular lattice of positions axis according to the binary sequence of weights along the

K

N z W = [w(0); . . . ; w(n); . . . ; w(N 0 1)] (w(n) 2 f0; 1g; n = 0; . . . ; N 0 1). The power pattern of such an arrangement

is equal to [1]

PP(u) = jW 1 B(u)j2

Index Terms—Almost difference sets, element failure, thinned arrays.

where B( ) [1 . . . 2indu . . . 2i(N 01)du ] being the lattice spacing in wavelength and = sin( ). An ADS array over such a element lattice is unequivocally defined by the following weight vector [12]

; e

u

I. INTRODUCTION AND MOTIVATION Antenna arrays defined over wide apertures are of great interest in several applications including radar, biomedical imaging, remote sensing, radio-astronomy, and satellite communications [1]. These systems usually impose severe constraints on beamwidth, sidelobe level, and shape of the radiation pattern [1]. To comply with these requirements, several design techniques have been developed [1] starting from the assumption that no failed elements are present [1], [2]. However, it is not an exceptional situation in real working conditions to have some failures due to unforeseen reasons that cause sharp variations of the field intensity across the array aperture [2] yielding distorted radiation patterns with higher sidelobes [3], [4]. Such a problem is even more significant when thinned architectures [1] are at hand. Thinning turns on and off elements in uniformly-spaced lattices in order to radiate a desired low-sidelobe pattern [5]. As a consequence, they are inherently less robust to failures because of the lower “element redundancy” compared to their fully-populated counterparts [5], [6]. Although the above considerations hold true whatever the design technique (e.g., random [6], stochastic [7], [8], or hybrid [9], [10]), thinning approaches exploiting the “regularity” of the associated layouts, such as analytical methods [11]–[13], could be even weaker to element faults. Indeed, analytically-thinned arrays, such as those derived from almost difference sets (ADSs) [14], guarantee low and predictable peak sidelobe levels (PSLs) thanks to the autocorrelation properties of the associated binary sequences [12], [13]. Such properties can be lost whether one or more elements exhibit the so-called “on-off” fault (i.e., the faulty element does not radiate at all) and no a-priori information can be deducted on the performances of the damaged versions of the original ADS thinned arrays. Nevertheless, although of great relevance for its practical implications and the widespread exploitation of ADSs to thin arrays [12], [13] as well as for other array designs [15], [16], no studies have been performed so far on the features of ADS-based layouts when failures occur. In this communication, the robustness of linear ADS-thinned arrays to failures is analyzed. The paper is not aimed at presenting a failure correction approach for thinned layouts, but rather at providing a-priori guidelines for the application of analytical design approaches based Manuscript received February 08, 2011; revised April 23, 2011; accepted June 02, 2011. Date of publication August 22, 2011; date of current version December 02, 2011. The authors are with the ELEDIA Research Group @DISI, University of Trento, Povo 38050 Trento, Italy (e-mail: [email protected]; giacomo. [email protected]; [email protected]) Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2165510

(1)

u

; ;e 

;d

N

W = [(0 2 A); . . . ; (n 2 A); . . . ; (N 0 1 2 A)]

(2)

n 2 A) is equalNto 1 if n belongs to A and 0 otherwise. Morefak 2 ; al 6= ah ; k; l;h = 0; . . . ; K 0 1g is an (N;K; 3;t)-ADS, that is a K subset of Z N whose autocorrelation is equal to ( ) = W 1 W( ) K  =0 = 3 t values of  2 [0; . . . ; N 0 1] (3) where ( over, A

3 + 1 elsewhere

W



; ;n 

; ;N



being ( ) [( cN 2 A) . . . ( + cN 2 A) . . . ( 01+ cN 2 A)] the -positions cyclically shifted version of , and 1cN the “reminders by ” operator. Thanks to (3), the ADS sidelobes are predictable just knowing the ADS descriptors, 3, and [12]. More( ) 2 [0 . . . 0 1] is still an ADS, each reference over, since sequence A can be used to define up to thinned layouts. The optimal ( ) , is related to the shift value [12] one, opt

W

W

N

W

;

W

W

N; K;

; ;N

t

N

W

opt = arg min PSL 

( )

(4)

u=

u

where PSL( () ) = (maxjuj